MODERN NUCLEAR
CHEMISTRY
MODERN NUCLEAR
CHEMISTRY
WALTER D. LOVELAND
Oregon State University
DAVID J. MORRISSEY
Michigan State University
GLENN T. SEABORG
University of California, Berkeley
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright # 2006 by John Wiley & Sons, Inc. All rights reserved
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Library of Congress Cataloging-in-Publication Data.
Loveland, Walter D.
Modern nuclear chemistry / Walter D. Loveland, David J. Morrissey, Glenn T.
Seaborg.
p. cm.
Includes bibliographical references and index.
ISBN-13 978-0-471-11532-8 (cloth: alk. paper)
ISBN-10 0-471-11532-0 (cloth: alk. paper)
1. Nuclear chemistry--Textbooks. I. Morrissey, David J. II. Seaborg, Glenn
Theodore, 1912– III. Title.
QD601.3.L68 2005
5410
.38--dc22
2005022036
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
at www.wiley.com.
CONTENTS
PREFACE xv
CHAPTER 1 INTRODUCTORY CONCEPTS 1
1.1 Introduction / 1
1.2 The Atom / 2
1.3 Atomic Processes / 3
1.3.1 Ionization / 3
1.3.2 X-ray Emission / 4
1.4 The Nucleus Nomenclature / 6
1.5 Survey of Nuclear Decay Types / 8
1.6 Modern Physical Concepts Needed in Nuclear Chemistry / 11
1.6.1 Types of Forces in Nature / 11
1.6.2 Elementary Mechanics / 12
1.6.3 Relativistic Mechanics / 13
1.6.4 De Broglie Wavelength, Wave–Particle Duality / 17
1.6.5 Heisenberg Uncertainty Principle / 19
1.6.6 Units and Conversion Factors / 19
1.7 Particle Physics / 20
1.8 Exchange Particles and Force Carriers / 24
Problems / 24
Bibliography / 26
v
CHAPTER 2 NUCLEAR PROPERTIES 29
2.1 Introduction / 30
2.2 Nuclear Masses / 30
2.3 Terminology / 32
2.4 Binding Energy Per Nucleon / 33
2.5 Separation Energy Systematics / 35
2.6 Abundance Systematics / 36
2.7 Semiempirical Mass Equation / 36
2.8 Nuclear Sizes and Shapes / 42
2.9 Quantum Mechanical Properties / 44
2.9.1 Nuclear Angular Momenta / 44
2.10 Electric and Magnetic Moments / 47
2.10.1 Magnetic Dipole Moment / 47
2.10.2 Electric Quadrupole Moment / 50
Problems / 53
References / 56
Bibliography / 56
CHAPTER 3 RADIOACTIVE DECAY KINETICS 57
3.1 Basic Decay Equations / 58
3.2 Mixture of Two Independently Decaying Radionuclides / 65
3.3 Radioactive Decay Equilibrium / 67
3.4 Branching Decay / 75
3.5 Natural Radioactivity / 77
3.6 Radionuclide Dating / 81
Problems / 87
References / 89
Bibliography / 89
CHAPTER 4 RADIOTRACERS 91
4.1 Introduction / 91
4.2 Design of a Radiotracer Experiment / 92
4.2.1 Basic Design Criteria / 92
4.2.2 Practical Considerations / 95
4.3 Preparation of Radiotracers and Their Compounds / 97
4.3.1 Chemical Synthesis / 99
4.3.2 Biosynthesis / 100
4.3.3 Tritium Labeling / 100
4.3.4 Radiolysis of Labeled Compounds / 101
vi CONTENTS
4.4 Tracing of Physical Process / 101
4.5 Chemical Applications of Tracers / 102
4.6 Isotope Effects / 104
4.7 Biological Applications / 107
4.8 Environmental Applications / 109
4.9 Industrial Use of Radiotracers / 113
4.10 Nuclear Medicine / 113
4.11 Isotope Dilution Analysis / 122
4.11.1 Direct IDA / 122
4.11.2 Inverse IDA / 123
4.11.3 General Comments / 124
4.11.4 Special IDA Techniques / 124
4.12 Radiometric Techniques / 125
Problems / 127
References / 128
Bibliography / 128
CHAPTER 5 NUCLEAR FORCES 129
5.1 Introduction / 129
5.2 Characteristics of the Strong Force / 130
5.3 Charge Independence of Nuclear Forces / 132
Problems / 134
Reference / 135
CHAPTER 6 NUCLEAR STRUCTURE 137
6.1 Nuclear Potentials / 139
6.2 Schematic Shell Model / 140
6.3 Independent Particle Model / 152
6.4 Collective Model / 154
6.5 Nilsson Model / 160
6.6 Nucleus as a Fermi Gas / 163
Problems / 171
References / 174
Bibliography / 174
CHAPTER 7 a DECAY 177
7.1 Energetics of a Decay / 179
7.2 Theory of a Decay / 183
7.3 Hindrance Factors / 192
CONTENTS vii
7.4 Heavy Particle Radioactivity / 193
7.5 Proton Radioactivity / 195
Problems / 197
References / 198
Bibliography / 198
CHAPTER 8 b DECAY 199
8.1 Introduction / 199
8.2 Neutrino Hypothesis / 200
8.3 Derivation of Spectral Shape / 203
8.4 Kurie Plots / 207
8.5 b-Decay Rate Constant / 208
8.6 Electron Capture Decay / 213
8.7 Parity Nonconservation / 214
8.8 Neutrinos / 215
8.9 b-Delayed Radioactivities / 216
8.10 Double-b Decay / 217
Problems / 219
References / 220
Bibliography / 220
CHAPTER 9 g-RAY DECAY 221
9.1 Introduction / 221
9.2 Energetics of g Decay / 222
9.3 Classification of Decay Types / 223
9.4 Electromagnetic Transition Rates / 226
9.5 Internal Conversion / 232
9.6 Angular Correlations / 235
9.7 Mo¨ssbauer Effect / 241
Problems / 247
References / 248
Bibliography / 248
CHAPTER 10 NUCLEAR REACTIONS 249
10.1 Introduction / 249
10.2 Energetics of Nuclear Reactions / 250
10.3 Reaction Types and Mechanisms / 254
10.4 Nuclear Reaction Cross Sections / 255
viii CONTENTS
10.5 Reaction Observables / 264
10.6 Rutherford Scattering / 265
10.7 Elastic (Diffractive) Scattering / 268
10.8 Direct Reactions / 270
10.9 Compound Nucleus Reactions / 272
10.10 Photonuclear Reactions / 278
10.11 Heavy Ion Reactions / 279
10.11.1 Coulomb Excitation / 280
10.11.2 Elastic Scattering / 281
10.11.3 Fusion Reactions / 282
10.11.4 Deep Inelastic Scattering / 286
10.11.5 Incomplete Fusion / 286
10.11.6 Reactions Induced by Radioactive Projectiles / 287
10.12 High-Energy Nuclear Reactions / 288
10.12.1 Spallation/Fragmentation / 288
10.12.2 Multifragmentation / 291
10.12.3 Quark–Gluon Plasma / 292
Problems / 293
References / 296
Bibliography / 297
CHAPTER 11 FISSION 299
11.1 Introduction / 299
11.2 Probability of Fission / 302
11.2.1 Liquid Drop Model / 302
11.2.2 Shell Corrections / 304
11.2.3 Spontaneous Fission / 306
11.2.4 Spontaneously Fissioning Isomers / 308
11.2.5 Transition Nucleus / 310
11.3 Fission Product Distributions / 316
11.3.1 Total Kinetic Energy (TKE) Release in Fission / 316
11.3.2 Fission Product Mass Distributions / 316
11.3.3 Fission Product Charge Distributions / 318
11.4 Excitation Energy of the Fission Fragments / 322
11.5 Dynamical Properties of the Fission Fragments / 325
Problems / 329
References / 329
CONTENTS ix
CHAPTER 12 NUCLEAR REACTIONS IN NATURE:
NUCLEAR ASTROPHYSICS 331
12.1 Introduction / 331
12.2 Elemental and Isotopic Abundances / 332
12.3 Primordial Nucleosynthesis / 336
12.4 Stellar Evolution / 338
12.5 Thermonuclear Reaction Rates / 342
12.6 Stellar Nucleosynthesis / 344
12.6.1 Introduction / 344
12.6.2 Hydrogen Burning / 345
12.6.3 Helium Burning / 348
12.6.4 Synthesis of Nuclei with A , 60 / 349
12.6.5 Synthesis of Nuclei with A . 60 / 351
12.7 Solar Neutrino Problem / 354
12.7.1 Introduction / 354
12.7.2 Expected Solar Neutrino Sources, Energies,
and Fluxes / 355
12.7.3 Detection of Neutrinos / 357
12.7.4 Solar Neutrino Problem / 359
12.7.5 Solution of the Problem—Neutrino Oscillations / 359
12.8 Synthesis of Li, Be, and B / 361
Problems / 362
References / 363
Bibliography / 363
CHAPTER 13 ANALYTICAL APPLICATIONS OF
NUCLEAR REACTIONS 365
13.1 Activation Analysis / 366
13.1.1 Basic Description of Method / 366
13.1.2 Advantages and Disadvantages of Activation
Analysis / 367
13.1.3 Practical Considerations in Activation Analysis / 368
13.1.4 Applications of Activation Analysis / 372
13.2 Particle-Induced X-ray Emission / 373
13.3 Rutherford Backscattering (RBS) / 376
Problems / 379
References / 380
Bibliography / 380
x CONTENTS
CHAPTER 14 REACTORS AND ACCELERATORS 383
14.1 Nuclear Reactors / 384
14.1.1 Neutron-Induced Reactions / 384
14.1.2 Neutron-Induced Fission / 387
14.1.3 Neutron Inventory / 388
14.1.4 Light Water Reactors / 390
14.1.5 The Oklo Phenomenon / 395
14.2 Neutron Sources / 395
14.3 Neutron Generators / 396
14.4 Accelerators / 397
14.4.1 Ion Sources / 397
14.4.2 Electrostatic Machines / 399
14.4.3 Linear Accelerators / 403
14.4.4 Cyclotrons, Synchrotrons, and Rings / 406
14.5 Charged Particle Beam Transport and Analysis / 412
14.6 Radioactive Ion Beams / 417
14.7 Nuclear Weapons / 421
Problems / 426
References / 427
Bibliography / 427
CHAPTER 15 THE TRANSURANIUM ELEMENTS 429
15.1 Introduction / 429
15.2 Limits of Stability / 429
15.3 Element Synthesis / 431
15.4 History of Transuranium Element Discovery / 438
15.5 Superheavy Elements / 447
15.6 Chemistry of the Transuranium Elements / 449
15.7 Environmental Chemistry of the Transuranium Elements / 457
Problems / 462
References / 463
Bibliography / 464
CHAPTER 16 NUCLEAR REACTOR CHEMISTRY 465
16.1 Introduction / 465
16.2 Fission Product Chemistry / 466
CONTENTS xi
16.3 Radiochemistry of Uranium / 470
16.3.1 Uranium Isotopes / 470
16.3.2 Metallic Uranium / 470
16.3.3 Uranium Compounds / 470
16.3.4 Uranium Solution Chemistry / 471
16.4 Nuclear Fuel Cycle—The Front End / 472
16.4.1 Mining and Milling / 472
16.4.2 Refining and Chemical Conversion / 475
16.4.3 Enrichment / 475
16.4.4 Fuel Fabrication / 478
16.5 Nuclear Fuel Cycle—The Back End / 479
16.5.1 Properties of Spent Fuel / 479
16.5.2 Fuel Reprocessing / 481
16.6 Radioactive Waste Disposal / 483
16.6.1 Classification of Radioactive Waste / 483
16.6.2 Amounts and Associated Hazards / 484
16.6.3 Storage and Disposal of Nuclear Waste / 485
16.7 Chemistry of Operating Reactors / 492
16.7.1 Radiation Chemistry of Coolants / 493
16.7.2 Corrosion / 493
16.7.3 Coolant Activities / 494
Problems / 494
References / 495
Bibliography / 496
CHAPTER 17 INTERACTION OF RADIATION
WITH MATTER 497
17.1 Introduction / 497
17.2 Heavy Charged Particles (A  1) / 499
17.3 Electrons / 514
17.4 Electromagnetic Radiation / 518
17.4.1 Photoelectric Effect / 520
17.4.2 Compton Scattering / 522
17.4.3 Pair Production / 524
17.5 Neutrons / 526
17.6 Radiation Exposure and Dosimetry / 530
Problems / 533
References / 535
Bibliography / 535
xii CONTENTS
CHAPTER 18 RADIATION DETECTORS 537
18.1 Detectors Based on Ionization / 540
18.1.1 Gas Ionization Detectors / 540
18.1.2 Semiconductor Detectors (Solid-State
Ionization Chambers) / 548
18.2 Scintillation Detectors / 558
18.3 Nuclear Track Detectors / 564
18.4 Nuclear Electronics and Data Collection / 565
18.5 Nuclear Statistics / 567
18.5.1 Rejection of Abnormal Data / 574
18.5.2 Setting Upper Limits When No
Counts are Observed / 576
Problems / 576
References / 577
Bibliography / 577
CHAPTER 19 RADIOCHEMICAL TECHNIQUES 579
19.1 Unique Aspects of Radiochemistry / 580
19.2 Availability of Radioactive Material / 584
19.3 Targetry / 584
19.4 Measuring Beam Intensity and Fluxes / 589
19.5 Recoils, Evaporation Residues (EVRs),
and Heavy Residues / 591
19.6 Radiochemical Separation Techniques / 595
19.6.1 Precipitation / 595
19.6.2 Solvent Extraction / 596
19.6.3 Ion Exchange / 599
19.6.4 Extraction Chromatography / 602
19.6.5 Rapid Radiochemical Separations / 602
19.7 Low-Level Measurement Techniques / 603
19.7.1 Introduction / 603
19.7.2 Blanks / 604
19.7.3 Low-Level Counting—General Principles / 605
19.7.4 Low-Level Counting—Details / 605
19.7.5 Limits of Detection / 608
Problems / 609
References / 610
Bibliography / 611
CONTENTS xiii
APPENDIX A FUNDAMENTAL CONSTANTS AND
CONVERSION FACTORS 613
APPENDIX B NUCLEAR WALLET CARDS 617
APPENDIX C PERIODIC TABLE OF ELEMENTS 641
APPENDIX D LIST OF ELEMENTS 643
APPENDIX E ELEMENTS OF QUANTUM MECHANICS 645
INDEX 667
xiv CONTENTS
PREFACE
There are many fine textbooks of nuclear physics and chemistry in print at this time.
So the question can be raised as to why we would write another textbook, especially
one focusing on the smaller discipline of nuclear chemistry. When we began this
project over 5 years ago, we felt that we were at a unique juncture in nuclear chemistry
and technology and that, immodestly, we had a unique perspective to offer to
students.
Much of the mainstream of nuclear chemistry is now deeply tied to nuclear
physics, in a cooperative endeavor called nuclear science. At the same time, there
is a large, growing, and vital community of people who use the applications of
nuclear chemistry to tackle a wide-ranging set of problems in the physical, biological
and environmental sciences, medicine, and engineering. We thought it was
important to bring together, in a single volume, a rigorous, detailed perspective
on both the “pure” and “applied” aspects of nuclear chemistry. As such, one
might find more detail about any particular subject than one might like. We hope
this encourages instructors to summarize the textbook material and present it in a
manner most suitable to a particular audience. The amount of material contained
in this book is too much for a one-quarter or one-semester course and a bit too
little for a year-long course. Instructors can pick and choose which material
seems most suitable for their course.
We have attempted to present nuclear chemistry and the associated applications
at a level suitable for an advanced undergraduate or beginning graduate student. We
have assumed the student has prior or, concurrent instruction in physical chemistry
or modern physics and has some skills in handling differential equations. We
have attempted to sprinkle solved problems throughout the text, as we believe
that one learns by working problems. The end-of-the-chapter homework problems
xv
are largely examination questions used at Oregon State University. They should be
considered an integral part of the textbook as they are intended to illustrate or
amplify the main points of each chapter. We have taken some pains to use
quantum mechanics in a schematic way, that is, to use the conclusions of such
considerations without using or demanding a rigorous, complete approach. The
use of hand-waving quantum mechanics, we believe, is appropriate for our
general audience. We summarize, in the appendices, some salient features of
quantum mechanics that may be useful for those students with limited backgrounds.
Our aim is to convey the essence of the ideas and the blend of theory and experiment
that characterizes nuclear and radiochemistry. We have included some more
advanced material for those who would like a deeper immersion in the subject. Our
hope is that the reader can use this book for an introductory treatment of the subject
of interest and can use the end-of-chapter references as a guide to more advanced
and detailed presentations. We also hope the practicing scientist might see this
volume as a quick refresher course for the rudiments of relatively unfamiliar
aspects of nuclear and radiochemistry and as an information booth for directions
for more detailed inquiries.
It is with the deep sense of loss and sadness that the junior authors (WDL, DJM)
note the passing of our dear friend, colleague, and co-author, Prof. Glenn T. Seaborg
before the completion of this work. Glenn participated in planning and development
of the textbook, wrote some of the text, and reviewed much of the rest. We deeply
miss his guidance and his perspective as we have brought this project to conclusion.
We regret not paying closer attention to his urging that we work harder and faster as
he would remark to us “You know I’m not going to live forever.” We hope that the
thoughts and ideas that he taught us are reflected in these pages.
We gratefully acknowledge the many colleagues and students who have taught us
about nuclear chemistry and other things. Special thanks are due to Darrah Thomas
and the late Tom Sugihara for pointing out better ways to discuss some material. We
acknowledge the efforts of Einar Hagebø who used an early version of this book in
his classes and gave us important feedback. We gratefully acknowledge the helpful
comments of D. Peterson, P. Mantica, A. Paulenova, and R. A. Schmitt on various
portions of the book. One of us (WDL) wishes to acknowledge the hospitality of the
National Superconducting Cyclotron Laboratory at Michigan State University for its
hospitality in the fall of 1999 during which time a portion of this book was written.
WALTER LOVELAND
Corvallis, Oregon
DAVID MORRISSEY
East Lansing, Michigan
xvi PREFACE
CHAPTER 1
INTRODUCTORY CONCEPTS
1.1 INTRODUCTION
Nuclear chemistry consists of a four-pronged endeavor made up of (a) studies of the
chemical and physical properties of the heaviest elements where detection of radioactive
decay is an essential part of the work, (b) studies of nuclear properties such
as structure, reactions, and radioactive decay by people trained as chemists, (c)
studies of macroscopic phenomena (such as geochronology or astrophysics) where
nuclear processes are intimately involved, and (d) the application of measurement techniques
based upon nuclear phenomena (such as nuclear medicine, activation analysis
or radiotracers) to study scientific problems in a variety of fields. The principal activity
or “mainstream” of nuclear chemistry involves those activities listed under part (b).
As a branch of chemistry, the activities of nuclear chemists frequently span
several traditional areas of chemistry such as organic, analytical, inorganic, and
physical chemistry. Nuclear chemistry has ties to all branches of chemistry. For
example, nuclear chemists are frequently involved with the synthesis and preparation
of radiolabeled molecules for use in research or medicine. Nuclear analytical
techniques are an important part of the arsenal of the modern analytical chemist. The
study of the actinide and transactinide elements has involved the joint efforts of
nuclear and inorganic chemists in extending knowledge of the periodic table.
Certainly, the physical concepts and reasoning at the heart of modern nuclear
chemistry are familiar to physical chemists. In this book we will touch on many
of these interdisciplinary topics and attempt to bring in familiar chemical concepts.
1
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
A frequently asked question is “What are the differences between nuclear physics
and nuclear chemistry?” Clearly, the two endeavors overlap to a large extent, and in
recognition of this overlap, they are collectively referred to by the catchall phrase
“nuclear science.” But we believe that there are fundamental, important distinctions
between these two fields. Besides the continuing close ties to traditional chemistry
cited above, nuclear chemists tend to study nuclear problems in different ways than
nuclear physicists. Much of nuclear physics is focused on detailed studies of the
fundamental interactions operating between subatomic particles and the basic
symmetries governing their behavior. Nuclear chemists, by contrast, have tended
to focus on studies of more complex phenomena where “statistical behavior” is
important. Nuclear chemists are more likely to be involved in applications of
nuclear phenomena than nuclear physicists, although there is clearly a considerable
overlap in their efforts. Some problems, such as the study of the nuclear fuel cycle in
reactors or the migration of nuclides in the environment, are so inherently chemical
that they involve chemists almost exclusively.
One term that is frequently associated with nuclear chemistry is that of radiochemistry.
The term radiochemistry refers to the chemical manipulation of radioactivity
and associated phenomena. All radiochemists are, by definition, nuclear
chemists, but not all nuclear chemists are radiochemists. Many nuclear chemists
use purely nonchemical, that is, physical techniques, to study nuclear phenomena,
and thus their work is not radiochemistry.
1.2 THE ATOM
Before beginning a discussion of nuclei and their properties, we need to understand
the environment in which most nuclei exist, that is, in the center of atoms. In elementary
chemistry, we learn that the atom is the smallest unit a chemical element can be
divided into that retains its chemical properties. As we know from our study of chemistry,
the radii of atoms are approximately 1–5 Â 10210
m, or 1–5 A˚ . At the center of
each atom we find the nucleus, a small object (r  1–10 Â10215
m) that contains
almost all the mass of the atom (Fig. 1.1). The atomic nucleus contains Z protons,
where Z is the atomic number of the element under study, Z being number of
protons and is thus the number of positive charges in the nucleus. The chemistry of
the element is controlled by Z in that all nuclei with the same Z will have similar
chemical behavior. The nucleus also contains N neutrons, where N is the neutron
number. Neutrons are uncharged particles with masses approximately equal to the
mass of a proton (%1 u). Each proton has a positive charge equal to that of an electron.
The overall charge of a nucleus is þZ electronic charge units.
Most of the atom is empty space in which the electrons surround the nucleus.
(Electrons are small, negatively charged particles with a charge of 21 electronic
charge units and a mass of about 1/1840 of the proton mass.) The negatively
charged electrons are bound by an electrostatic (Coulombic) attraction to the positively
charged nucleus. In a neutral atom, the number of electrons in the atom equals
the number of protons in the nucleus.
2 INTRODUCTORY CONCEPTS
Quantum mechanics tells us that only certain discrete values of E, the total
electron energy, and J, the angular momentum of the electrons are allowed. These
discrete states have been depicted in the familiar semiclassical picture of the atom
(Fig. 1.1) as a tiny nucleus with electrons rotating about it in discrete orbits. In
this book, we will examine nuclear structure and will develop a similar semiclassical
picture of the nucleus that will allow us to understand and predict a large range of
nuclear phenomena.
1.3 ATOMIC PROCESSES
The sizes and energy scales of atomic and nuclear processes are very different.
These differences allow us to consider them separately.
1.3.1 Ionization
Suppose one atom collides with another atom. If the collision is inelastic (the kinetic
energies of the colliding nuclei are not conserved), one of two things may happen.
They are (a) excitation of one or both atoms to an excited state involving a change in
electron configuration or (b) ionization of one or both atoms, that is, removal of one
or more of the atom’s electrons to form a positively charged ion. For ionization to
occur, an atomic electron must receive an energy that is at least equivalent to its
binding energy, which for the innermost or K electrons is (Zeff/137)2
(255.5) keV,
where Zeffective is the effective nuclear charge felt by the electron (and includes
the effects of screening of the nuclear charge by other electrons). This effective
nuclear charge for K electrons can be approximated by the expression (Z 2 0.3).
As one can see from these expressions, the energy necessary to cause ionization
Figure 1.1 Schematic representation of the relative sizes of the atom and the nucleus.
1.3 ATOMIC PROCESSES 3
far exceeds the kinetic energies of gaseous atoms at room temperature. Thus, atoms
must be moving with high speeds (as the result of nuclear decay processes or acceleration)
to eject tightly bound electrons from other atoms through collisions.
1.3.2 X-ray Emission
The term X-ray refers to the electromagnetic radiation produced when an electron in
an outer atomic electron shell drops down to fill a vacancy in an inner atomic electron
shell (Fig. 1.2), such as going from the M shell to fill a vacancy in the L shell.
The electron loses potential energy in this transition (in going to a more tightly
bound shell) and radiates this energy in the form of X-rays. (X-rays are not to be
confused with generally more energetic g rays, which result from transitions
made by the neutrons and protons in the nucleus of the atom, not in the atomic electron
shells.) The energy of the X-ray is given by the difference in the binding energies
of the electrons in the two shells, which, in turn, depends on the atomic number
of the element. Thus X-ray energies can be used to determine the atomic number of
the elemental constituents of a material and are also regarded as conclusive proof of
the identification of a new chemical element.
In X-ray terminology, X-rays due to transitions from the L to K shell are called
Ka X-rays; X-rays due to transitions from the M to K shells are called Kb X-rays.
[In a further refinement, the terms Ka1
, Ka2
refer to X-rays originating in different
subshells (2p3/2, 2p1/2) of the L shell.] X-rays from M to L transitions are La
X-rays, and so forth. For each transition, the changes in orbital angular momentum,
Dl, and total angular momentum, Dj, are required to be
Dl ¼ +1
D j ¼ 0, +1
(1:1)
Figure 1.2 Schematic diagram to show X-ray emission to fill vacancy caused by nuclear
decay. An L-shell electron (A) is shown filling a K-shell vacancy (B). In doing so, it emits
a characteristic K X-ray.
4 INTRODUCTORY CONCEPTS
The simple Bohr model of the hydrogen-like atom (one electron only) predicts
that the X-ray energy or the transition energy, DE, is given as
DE ¼ Einitial À Efinal ¼ R1hcZ2 1
n2
initial
À
1
n2
final
 
(1:2)
where R1, h, c, and n denote the Rydberg constant, Planck constant, the speed of
light, and the principal quantum number for the orbital electron, respectively.
Since the X-ray energy, Ex , is actually 2DE, we can write (after substituting
values for the physical constants)
Ex ¼ 13:6Z2 1
n2
final
À
1
n2
initial
 
eV (1:3)
where Ex is given in units of electron volts (eV).
For Ka X-rays from ions with only one electron
EK
x ¼ 13:6
1
12
À
1
22
 
Z2
eV (1:4)
while for La X-rays, we have
EL
x ¼ 13:6
1
22
À
1
32
 
Z2
eV (1:5)
In reality, many electrons will surround the nucleus, and we must replace Z by
Zeffective to reflect the screening of the nuclear charge by these other electrons.
This correction was done by Moseley who showed that the frequencies, n, of the
Ka series X-rays could be expressed as
n1=2
¼ const(Z À 1) (1:6)
while for La series X-rays
n1=2
¼ const(Z À 7:4) (1:7)
Moseley thus demonstrated the X-ray energies (¼hn) depend on the square of some
altered form (due to screening) of the atomic number. Also, the relative intensities of
the Ka1
, Ka2
, and so on, X-rays will be proportional to the number of possible ways
to make the transition. Thus, we expect the Ka1
=Ka2
intensity ratio to be 2 as the
maximum number of electrons in the 2p3/2 level is 4 while the maximum number
of electrons in the 2p1/2 level is 2. The relative intensities of different X-rays
depend on the chemical state of the atom, its oxidation state, bonding with
1.3 ATOMIC PROCESSES 5
ligands, and other factors that affect the local electron density. These relative
intensities are, thus, useful in chemical speciation studies. We should also note, as
discussed extensively in Chapters 7–9, that X-ray production can accompany
radioactive decay. Radioactive decay modes, such as electron capture or internal
conversion, directly result in vacancies in the atomic electron shells. The resulting
X-rays are signatures that can be used to characterize the decay modes and/or the
decaying species.
1.4 THE NUCLEUS NOMENCLATURE
A nucleus is said to be composed of nucleons. There are two “kinds” of nucleons, the
neutrons and the protons. A nucleus with a given number of protons and neutrons is
called a nuclide. The atomic number Z is the number of protons in the nucleus, while
N, the neutron number, is used to designate the number of neutrons in the nucleus.
The total number of nucleons in the nucleus is A, the mass number. Obviously
A ¼ N þ Z. Note that A, the number of nucleons in the nucleus, is an integer
while the actual mass of that nucleus, m, is not an integer.
Nuclides with the same number of protons in the nucleus but with differing
numbers of neutrons are called isotopes. (This word comes from the Greek
iso þ topos, meaning “same place” and referring to the position in the periodic
table.) Isotopes have very similar chemical behavior because they have the same
electron configurations. Nuclides with the same number of neutrons in the
nucleus, N, but differing numbers of protons, Z, are referred to as isotones. Isotones
have some nuclear properties that are similar in analogy to the similar chemical
properties of isotopes. Nuclides with the same mass number, A, but differing
numbers of neutrons and protons are referred to as isobars. Isobars are important
in radioactive decay processes. Finally, the term isomer refers to a nuclide in an
excited nuclear state that has a measurable lifetime (.1029
s). These labels are
straightforward, but the term isotope is frequently misused. For example, radioactive
nuclei (radionuclides) are often incorrectly referred to as radioisotopes, even though
the nuclides being referenced do not have the same atomic numbers.
The convention for designating a given nuclide (with Z protons, N neutrons) is to
write
A
Z
Chemical symbol
N
with the relative positions indicating a specific feature of the nuclide. Thus, the
nucleus with six protons and eight neutrons is 6
14
C8 , or completely equivalently,
14
C. (The older literature used the form ZChemical symbolA
, so 14
C was designated
as C14
. This nomenclature is generally extinct.) Note that sometimes the atomic
charge of the entity containing the nuclide is denoted as an upper-right-hand
superscript. Thus, a doubly ionized atom containing a Li nucleus with three
protons and four neutrons and only one electron is designated sometimes as 7
Li2þ
.
6 INTRODUCTORY CONCEPTS
Sample Problem 1.1 Consider the following nuclei:
60
Cm
; 14
C; 14
N; 12
C; 13
N
Which are isotopes? Isotones? Isobars? Isomers?
Answer 60
Cm
is the isomer, 14
C and 12
C are isotopes of C, 13
N and 14
N are isotopes
of N, 14
C and 14
N are isobars (A ¼ 14), while 12
C and 13
N are isotones
(N ¼ 6).
We can now make an estimate of two important quantities, the size and the
density of a typical nucleus. We can write
r ; density ¼
mass
volume
%
A amu
4
3 pR3
(1:8)
if we assume that the mass of each nucleon is about 1 u and the nucleus can be
represented as a sphere. It turns out (Chapter 2) that an empirical rule to describe
the radii of stable nuclei is that radius R is
R ¼ 1:2 Â 10À13
A1=3
cm (1:9)
Thus, we have
r ¼
(A u)(1:66 Â 10À24
g=uÞ
4
3 p (1:2 Â 10À13 A1=3cm)3
where we have used the value of 1.66 Â 10224
g for 1 u (Appendix A). Before evaluating
the density r numerically, we note that the A factor cancels in the expression,
leading us to conclude that all nuclei have approximately the same density. This is
similar to the situation with different sized drops of a pure liquid. All of the molecules
in a drop interact with each other with the same short-range forces, and the
overall drop size grows with the number of molecules. Evaluating this expression
and converting to convenient units, we have
r % 200, 000 metric tons=mm3
A cube of nuclear matter that is 1 mm on a side contains a mass of 200,000 tonnes.
WOW! Now we can realize what all the excitement about the nuclear phenomena is
about. Think of the tremendous forces that are needed to hold matter together with
this density. Relatively small changes in nuclei (via decay or reactions) can release
large amounts of energy. (From the point of view of the student doing calculations
with nuclear problems, a more useful expression of the nuclear density is
0.14 nucleons/fm3
.)
1.4 THE NUCLEUS NOMENCLATURE 7
1.5 SURVEY OF NUCLEAR DECAY TYPES
Nuclei can emit radiation spontaneously. The general process is called radioactive
decay. While this subject will be discussed in detail in Chapters 3, 7, 8, and 9, we
need to know a few general ideas about these processes right away (which we
can summarize below).
Radioactive decay usually involves one of three basic types of decay, a decay, b
decay, or g decay in which an unstable nuclide spontaneously changes into a more
stable form and emits some radiation. In Table 1.1, we summarize the basic features
of these decay types.
The fact that there were three basic decay processes (and their names) was discovered
by Rutherford. He showed that all three processes occur in a sample of
decaying natural uranium (and its daughters). The emitted radiations were designated
a, b, and g to denote the penetrating power of the different radiation types.
Further research has shown that in a decay, a heavy nucleus spontaneously emits
a 4
He nucleus (an a particle). The emitted a particles are monoenergetic, and, as
a result of the decay, the parent nucleus loses two protons and two neutrons and
is transformed into a new nuclide. All nuclei with Z . 83 are unstable with
respect to this decay mode.
Nuclear b decay occurs in three ways, b2
, bþ
, and electron capture (EC). In these
decays, a nuclear neutron (or proton) changes into a nuclear proton (or neutron) with
the ejection of a neutrino (or antineutrino) and an electron (or positron). In electron
capture, an orbital electron is captured by the nucleus, changing a proton into a
neutron with the emission of a neutrino. The total number of nucleons, A, in the
nucleus does not change in these decays, only the relative number of neutrons
and protons. In a sense, this process can “correct” or “adjust” an inbalance
between the number of neutrons and protons in a nucleus. In bþ
and b2
decays,
the decay energy is shared between the emitted electron, the neutrino, and the recoiling
daughter nucleus. Thus, the energy spectrum of the emitted electrons and
neutrinos is continuous, ranging from zero to the decay energy. In EC decay, essentially
all the decay energy is carried away by the emitted neutrino. Neutron-rich
nuclei decay by b2
decay, whereas proton-rich nuclei decay by bþ
or EC decay.
bþ
decay is favored in the light nuclei and requires the decay energy to be
greater than 1.02 MeV (for reasons to be discussed later), whereas EC decay is
found mostly in the heavier nuclei.
Nuclear electromagnetic decay occurs in two ways, g decay and internal conversion
(IC). In g-ray decay a nucleus in an excited state decays by the emission of a
photon. In internal conversion the same excited nucleus transfers its energy radiationlessly
to an orbital electron that is ejected from the atom. In both types of
decay, only the excitation energy of the nucleus is reduced with no change in the
number of any of the nucleons.
Sample Problem 1.2 Because of the conservation of the number of nucleons
in the nucleus and conservation of charge during radioactive decay (Table 1.1),
8 INTRODUCTORY CONCEPTS
TABLE1.1CharacteristicsofRadioactiveDecay
Decay
Type
Emitted
ParticleDZDNDA
Typical
Energyof
EmittedParticleExampleOccurrence
a4
He2þþ
2222244Ea10MeV238
U!234
ThþaZ.83
bÀ
EnergeticeÀ
;neþ12100EbÀ2MeV14
C!14
NþbÀ
þneN=Z.(N=Z)stable
bþ
Energeticeþ
,ne21þ100Ebþ2MeV22
Na!22
Neþbþ
þne(N=Z),(N=Z)stable;lightnuclei
ECne21þ100En2MeVeÀ
þ207
Bi!207
Pbþne(N=Z),(N=Z)stable;heavynuclei
gPhoton0000:1Eg2MeV60
NiÃ
!60
NiþgAnyexcitednucleus
ICElectron0000:1,Ee,2MeV125
Sb
m
!125
SbþeÀ
Caseswhereg-rayemissionisinhibited
9
it is relatively easy to write and balance nuclear decay equations. For example,
consider the
b2
decay of 90
Sr
a decay of 232
Th
bþ
decay of 62
Cu
EC decay of 256
Md
These decay equations can be written, using Table 1.1, as
90
38 Sr À! 90
39 Yþ
þ bÀ
þ ne
232
90 Th À! 228
88 Ra þ
4
2He
62
29 Cu À! 62
28 NiÀ
þ bþ
þ ne
eÀ
þ 256
101 Mdþ
À! 256
100 Fm þ ne
Besides its qualitative description, radioactive decay has an important quantitative
description. Radioactive decay can be described as a first-order reaction, that is,
the number of decays is proportional to the number of decaying nuclei present. It is
described by the integrated rate law
N ¼ N0eÀlt
(1:10)
where N is the number of nuclei present at time t while N0 is the number of nuclei
present at time t ¼ 0. The decay constant l, a characteristic of each nucleus, is
related to the half-life, t1/2, by
l ¼ ln 2=t1=2 (1:11)
The half-life is the time required for the number of nuclei present to decrease by a
factor of 2. The number of decays that occur in a radioactive sample in a given
amount of time is called the activity A of the sample. The activity is equal to the
number of nuclei present, N, multiplied by the probability of decay per nucleus,
l, that is, A ¼ lN. Therefore, the activity will also decrease exponentially with time,
A ¼ A0eÀlt
(1:12)
where A is the number of disintegrations per unit time at time t, and A0 is the activity
at time t ¼ 0. The half-lives of nuclei with respect to each decay mode are often used
to identify the nuclei.
10 INTRODUCTORY CONCEPTS
Sample Problem 1.3 14
C decays to 14
N by b2
decay with a half-life of 5730 y.
If a 1-g sample of carbon contains 15.0 disintegrations per minute, what will be
its activity after 10,000 y?
Solution
A ¼ A0eÀlt
l ¼
ln 2
5730 y
¼ 1:210 Â 10À4
yÀ1
A ¼ (15 dis=min) eÀ(1:210 Â 10À4
)(10, 000)
¼ 4:5 dis=min
All living things maintain a constant level of 14
C per gram of carbon through
exchange with their surroundings. When they die, this exchange stops and the
amount of 14
C present decreases exponentially with time. A measurement of the
14
C content of a dead object can be used to determine the age of the object. This
process and other geologically important decay processes are discussed in
Chapter 3.
1.6 MODERN PHYSICAL CONCEPTS NEEDED IN
NUCLEAR CHEMISTRY
While we shall strive to describe nuclear chemistry without using extensive mathematics
and physics, there are several important concepts from modern physics
that we need to review because we will use these concepts in our discussions.
1.6.1 Types of Forces in Nature
Let us review briefly some physical concepts that we shall use in our study of nuclear
chemistry. First, we should discuss the types of forces found in nature. There are
four fundamental forces in nature (Table 1.2). As far as we know, all the interactions
in the universe are the result of these forces. The weakest force is gravity, which is
most significant when the interacting objects are massive, such as planets, stars, and
the like. The next strongest force is the weak interaction, which is important in
nuclear b decay. The familiar electromagnetic force, which governs most behavior
in our sensory world, is next in strength while the nuclear or strong interaction is the
TABLE 1.2 Types of Force Encountered in Nature
Force Range (m) Relative Strength Force Carrier
Gravitational 1 10238
Graviton
Weak 10218
1025
W+
, Z0
Electromagnetic 1 a ¼ 1/137 Photon
Strong 10215
1 Gluon
1.6 MODERN PHYSICAL CONCEPTS NEEDED IN NUCLEAR CHEMISTRY 11
strongest force. Please note, as indicated earlier in our discussion of nuclear
densities, that the strong or nuclear force is more than 100 times stronger than the
electromagnetic force holding atoms together.
In the 19th century, electricity and magnetism were linked together. The 20th
century has seen the demonstration that the electromagnetic and weak forces are
just two different aspects of the same force, called the electroweak force. Current
efforts are directed at unifying the strong and electroweak forces in a so-called
grand unified theory, or GUT. The final step in this direction would be to include
gravity in a theory of everything. Discussion of these unified theories is beyond
the scope of this book; however, the relative strength and character of the forces
will form an important part of our discussion of nuclear phenomena.
1.6.2 Elementary Mechanics
Let us recall a few elementary relationships from classical physics that we shall use.
Force can be represented as a vector, F, that describes the rate of change of the
momentum with time:
F ¼
dp
dt
(1:13)
where the momentum p ¼ mv and where m is the mass and v is the velocity of the
particle. Neglecting relativistic effects (Section 1.6.3) that are important for particles
whose velocity approaches the speed of light, we can say that the kinetic energy of a
moving body T is given as
T ¼
1
2
mv2
(1:14)
For the situation depicted in Figure 1.3 for the motion of a particle past a fixed point,
we can say that the orbital angular momentum of the particle, l, with mass m with
respect to the point Q is
l ¼ r  p (1:15)
The quantity l is a vector whose magnitude is mvr for circular motion. For motion
past a stationary point, the magnitude is mvb where b is the distance of closest
approach called the impact parameter.
Let us also recall the relationship between the magnitude of a force F(r) that
depends on the distance between two objects, r, and the potential energy, V(r),
F ¼
À@V
@r
(1:16)
Thus, if the Coulomb potential energy between two charged objects is given as
V ¼
þkq1q2
r12
(1:17)
12 INTRODUCTORY CONCEPTS
where r12 is the distance separating charges q1 and q2 (and where k is a constant), we
can say the magnitude of the Coulomb force, FC, is
FC ¼
À@V
@r
¼
kq1q2
r2
12
(1:18)
Since forces are usually represented as vectors, it is more convenient when discussing
nuclear interactions to refer to the scalar, potential energy. From the above discussion,
we should always remember that a discussion of potential energy V(r) is
also a discussion of force F(r).
1.6.3 Relativistic Mechanics
As Einstein demonstrated, when a particle moves with a velocity approaching that of
light, the classical relations (Section 1.6.2) describing its motion in a stationary
system are no longer valid. Nuclear processes frequently involve particles with
such high velocities. Thus, we need to understand the basic elements of relativistic
mechanics. According to the special theory of relativity, the mass of a moving
particle changes with speed according to the equation
mÃ
¼ g m0 (1:19)
O
I = r × p
z
x
y
r
P
m
p
θ
Figure 1.3 Particle of mass m, moving with velocity v has a linear momentum p ¼ mv.
Relative to the point O, the particle has an angular momentum of l ¼ r  p, where r is a
vector connecting the point O and the particle. At the point of closest approach, r is equal
to the impact parameter, b.
1.6 MODERN PHYSICAL CONCEPTS NEEDED IN NUCLEAR CHEMISTRY 13
where mÃ
and m0 are the mass of a particle in motion and at rest, respectively. The
Lorentz factor g is given as
g ¼ (1 À b2
)À1=2
(1:20)
where b is the speed of the particle, v, relative to the speed of light, c, so that
b ¼ v/c. Thus, as the speed of the particle increases, the mass also increases;
making further increases in speed more difficult. Since the mass mÃ
cannot be imaginary,
no particle can go faster than the speed of light. The total energy of a particle,
ETOT, is given as
ETOT ¼ mÃ
c2
(1:21)
Since the total energy equals the kinetic energy plus the rest mass energy, we can
write
ETOT ¼ T þ m0c2
(1:22)
where T is the particle’s kinetic energy. Thus,
T ¼ (g À 1) m0c2
(1:23)
The space–time coordinates (x, y, z, t) of a point in a stationary system are,
according to the special theory of relativity, related to the space–time coordinates
in a system moving along the x axis (x0
, y0
, z0
, t0
) by the relations
x0
¼ g(x À bct)
y0
¼ y
z0
¼ z
t0
¼ g ½t À (b=c)xŠ
(1:24)
These transformations from the stationary to the moving frame are called the
Lorentz transformations. The inverse Lorentz transformation is obtained by reversing
the sign of v, so that
x ¼ g (x0
þ bct0
)
y ¼ y0
z ¼ z0
t ¼ g ½t þ (b=c)x0
Š
Dt ¼ t1 À t2 ¼ g ½Dt0
þ (b=c)DxŠ
Dx ¼ Dx0
=g
(1:25)
14 INTRODUCTORY CONCEPTS
Since g . 1, time is slowed down for the stationary observer, and distance in the x
direction is contracted.
One application of these equations in nuclear chemistry involves the decay of
rapidly moving particles. The muon, a heavy electron, has a lifetime, t, at rest, of
2.2 ms. When the particle has a kinetic energy of 100 GeV (as found in cosmic
rays), we observe a lifetime of gt or about 103
t. (This phenomenon is called time
dilation and explains why such muons can reach the surface of Earth.)
A series of relationships have been derived between the stationary coordinate
system (the scientist in his or her laboratory) and a moving (intrinsic, invariant)
coordinate system that can be compared to classical calculations of dynamic
variables (Table 1.3).
Note that for a particle at rest
ETOT ¼ m0c2
(1:26)
where m0 is the rest mass and c is the speed of light. For a massless particle, such as a
photon, we have
ETOT ¼ pc (1:27)
where p is the momentum of the photon. These equations make it clear why the
units of MeV/c2
for mass and MeV/c for momentum can be useful in nuclear
calculations.
An important question is when do we use classical expressions and when do we
use relativistic expressions? A convenient, but arbitrary, criterion for making this
decision is to use the relativistic expression when g ! 1.1. This corresponds
roughly to the point at which a 13% error occurs in the classical expression. What
does this criterion mean, in practice? In Table 1.4, we indicate the values of the
kinetic energy at which g ¼ 1.1 for different particles. Thus, one should always
TABLE 1.3 Comparison of Relativistic and Classical Expressions for a
Free Particle Moving in x Direction
Classical Expression Relativistic Expression
x x ¼ g (x0
þ bct0
)
y y ¼ y0
z z ¼ z0
t t ¼ g (t0
þ b=cx0
)
Dt ¼ t2 À t1 Dt0
¼ g Dt
Mass m m ¼ gm0 (m0 ; rest mass)
Momentum p ¼ mv p ¼ gmv
T ; kinetic energy ¼ 1
2 mv2
T ¼ (g À 1)m0c2
Total energy ETOT ¼ Ek (free particle) ETOT ¼ gm0c2
Energy momentum relationship E ¼ p2
/2m E2
TOT ¼ p2
c2
þ m2
0c4
1.6 MODERN PHYSICAL CONCEPTS NEEDED IN NUCLEAR CHEMISTRY 15
use the relativistic expressions for photons, neutrinos, and electrons (when
Te . 50 keV) or for nucleons when the kinetic energy/nucleon exceeds 100 MeV.
Sample Problem 1.4 Relativistic Mechanics Consider a 20
Ne ion with a kinetic
energy of 1 GeV/nucleon. Calculate its velocity, momentum, and total energy.
Solution Total kinetic energy ¼ 20 Â 1 GeV/nucleon ¼ 20 GeV ¼ 20,000 MeV.
But we know: T ¼ (g 2 1)m0c2
The rest mass is approximately 20 u or (20)(931.5) MeV/c or 18,630 MeV. So we
can say
g ¼
T
m0c2
þ 1 ¼ 1 þ
20, 000
18, 630
¼ 2:07
But we know
g ¼ (1 À b2
)À1=2
So we can say
b ¼ 1 À
1
g2
 1=2
¼ 0:88
So the velocity v is 0.88c or (0.88)(3.00 Â 108
m/s) ¼ 2.6 Â 108
m/s. The momentum
is given as
p ¼
mv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À b2
p ¼ gmv
¼ (2:07)(20)(1:67 Â 10À27
g)(2:6 Â 108
)
¼ 1:8 Â 10À17
kg Á m=s
TABLE 1.4 When Does One Use
Relativistic Expressions?
Particle
T (MeV)
when g ¼ 1.1
g, n 0
e 0.051
m 11
p 14
p, n 94
d 188
a 373
16 INTRODUCTORY CONCEPTS
or in other units
pc ¼
mcv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À b2
p ¼ (931:5)(20)(0:88)(2:07)
¼ 33:9 GeV
p ¼ 33:9 GeV=c
The total energy
ETOT ¼ Ek þ m0c2
¼ gm0c2
¼ (2:07)(20)(931:5) ¼ 38:6 GeV
1.6.4 De Broglie Wavelength, Wave–Particle Duality
There is no distinction between wave and particle descriptions of matter. It is simply
a matter of convenience, which we choose to use in a given situation. For example, it
is quite natural to describe matter in terms of particles with values of momenta,
kinetic energies, and so forth. It is also natural to use a wave description for light.
However, associated with each material particle, there is a wave description in
which the particle is assigned a wavelength (the de Broglie wavelength l) whose
magnitude is given as
l ¼
h
p
(1:28)
where p is the momentum of the particle and h is Planck’s constant. (Note that
Planck’s constant is extremely small, 6.6 Â 10234
J s. Thus, the wavelength of a particle
is only important when the momentum is extremely small, such as with electrons
whose mass is 9 Â 10231
kg.) The expression for the de Broglie wavelength
may be written in rationalized units
l ¼
hÀ
p
(1:29)
where hÀ
is h/2p. The above expressions are classical and should be replaced by their
relativistic equivalents where appropriate, that is,
l ¼
hÀ
c
½Ek(Ek þ 2 m0c2)Š1=2
(1:30)
We can calculate typical magnitudes of these wavelengths of particles encountered
in nuclear chemistry (Table 1.5). Given typical nuclear dimensions of 10213
cm,
the data of Table 1.5 indicate the energies at which such particles might have a
1.6 MODERN PHYSICAL CONCEPTS NEEDED IN NUCLEAR CHEMISTRY 17
wavelength similar or smaller than nuclear dimensions. These particles can be used
as probes of nuclear sizes and shapes.
In a similar manner, it is quite natural to associate a wave description to photons
(Table 1.4). Here we recall that
l ¼
c
n
¼
hc
Eg
(1:31)
where n is the frequency associated with the wave of length l. A convenient form of
this equation is
l (cm) ¼
1:2397 Â 10À10
Eg (MeV)
(1:32)
which was used to calculate the values in Table 1.5. But it is often useful to speak of
photons as particles particularly when they are emitted or absorbed by a nucleus,
when we write
Eg ¼ hn ¼ pc (1:33)
Sample Problem 1.5 de Broglie Wavelength Consider the case of a beam of
1 eV neutrons incident on a crystal. First-order Bragg reflections are observed at
11.88. What is the spacing between crystal planes?
Solution Low-energy neutrons are diffracted like X-rays. The Bragg condition
is that nl ¼ 2d sin Q where the index n ¼ 1 for first-order diffraction.
l ¼ 2d sin Q
d ¼
l
2 sin Q
¼
h
p
2 sin Q
¼
h
(2 mEk)1=2
2 sin Q
d ¼
6:63 Â 1034
J s
(2 1:67 Â 10À27 kg 1:60 Â 10À19 J)1=2
2 sin (11:88)
¼ 7:0 Â 10À11
m
TABLE 1.5 Typical Magnitudes of de Broglie Wavelengths
Energy (MeV)
Wavelength (cm)
Photon Electron Proton
0.1 1.2 Â 1029
3.7 Â 10210
9.0 Â 10212
1 1.2 Â 10210
8.7 Â 10211
2.9 Â 10212
10 1.2 Â 10211
1.2 Â 10211
0.9 Â 10212
100 1.2 Â 10212
1.2 Â 10212
2.8 Â 10213
1000 1.2 Â 10213
1.2 Â 10213
0.7 Â 10213
18 INTRODUCTORY CONCEPTS
1.6.5 Heisenberg Uncertainty Principle
Simply put, the Heisenberg uncertainty principle states that there are limits on
knowing both where something is and how fast it is moving. Formally, we can write
Dpx † Dx ! hÀ
Dpy † Dy ! hÀ
Dpz † Dz ! hÀ
DE † Dt ! hÀ
(1:34)
where Dpx, Dx are the uncertainties in the x component of the momentum and the x
coordinate, respectively, whereas Dt is the lifetime of a particle and DE is the uncertainty
in its total energy. These limits on our knowledge are not due to the limitations
of our measuring instruments. They represent fundamental limits even with ideal or
perfect instruments.
It is instructive to consider a practical example to see the effect of these limits.
Consider an electron with a kinetic energy of 5 eV. Its speed can be calculated
(nonrelativistically):
v ¼
2Ek
m
 1=2
¼
(2)(5)(1:602 Â 10À19
J=eV
9:11 Â 10À31 kg
 1=2
¼ 1:33 Â 106
m=s
Its momentum is then
p ¼ mv ¼ 1:21 Â 10À24
kg m=s
Assume the uncertainty in its measured momentum is 1%. The uncertainty principle
then tells us
Dx ¼
hÀ
Dp
¼
1:06 Â 10À34
J=s
1:21 Â 10À26 kg m=s
¼ 8:8 Â 10À9
m
which is about 40 atomic diameters. In short, if you know the momentum relatively
well, you do not know where the electron is in space.
1.6.6 Units and Conversion Factors
Every field has its own special units of measure and nuclear chemistry is no different.
The unit of length is the femtometer (10215
m), which is called a fermi. The unit
of mass is the atomic mass unit (amu or u), which has a numerical value of 1:66 Â
10À24
g or expressed in units of MeV/c2
, it is 931.5 MeV/c2
. The unit of energy
is MeV (106
eV), which is 1:602 Â 10À13
J, the energy gained when a proton is
1.6 MODERN PHYSICAL CONCEPTS NEEDED IN NUCLEAR CHEMISTRY 19
accelerated through a potential of 106
V. Appendix A contains a list of the exact
numerical values of these and other convenient units. Special attention is called to
five very useful quantities:
e2
4p10
¼ 1:43998 MeV fm
hÀ
¼ 6:58212 Â 10À22
MeV s
c ¼ 2:9979 Â 1023
fm sÀ1
¼ 29:979 cm=ns
hÀ
c ¼ 197:3 MeV fm
1 year (sidereal) ¼ 3:1558 Â 107
s % p  107
s
1.7 PARTICLE PHYSICS
Elementary particle physicists (“high-energy physicists”) study the fundamental
particles of nature and the symmetries found in their interactions. The study of
elementary particle physics is an important endeavor in its own right and beyond
the scope of this book. But we need to use some of the concepts of this area of
physics in our discussion of nuclei.
Particles can be classified as fermions or bosons. Fermions obey the Pauli
principle and have antisymmetric wave functions and half-integer spins. (Neutrons,
protons, and electrons are fermions.) Bosons do not obey the Pauli principle and
have symmetric wave functions and integer spins. (Photons are bosons.)
Particle groups, like fermions, can also be divided into the leptons (such as
the electron) and the hadrons (such as the neutron and proton). The hadrons
can interact via the nuclear or strong interaction while the leptons do not.
(Both particle types can, however, interact via other forces, such as the electromagnetic
force.) Figure 1.4 contains artistic conceptions of the standard model,
a theory that describes these fundamental particles and their interactions.
Examples of bosons, leptons, hadrons, their charges, and masses are given in
Table 1.6.
There are six different kinds of leptons (light particles) (Table 1.6), and they can
be arranged in three pairs. The electron (e), the muon (m), and the tau lepton (t) each
carry a charge of 2e and have associated with them the electron (ne), muon (nm), and
tau neutrinos (nt). These neutrinos are electrically neutral and have small or zero rest
mass. The actual mass of the neutrinos is a subject of current research (see Chapter
12). The electron neutrino is seen in nuclear phenomena such as b decay, whereas
the other neutrinos are involved in higher energy processes.
One important aspect of leptons is that their number is conserved in nuclear
processes. Consider, for example, the decay of the free neutron
n À! pþ
þ eÀ
þne
20 INTRODUCTORY CONCEPTS
NUCLEUS
2 TYPES
The protons and
neutrons of an
atom’s nucleus are
themselves complex
structures, made up
of groups of three
basic particles
called quarks.
Quarks can also
bind with antiquarks
to make other
particles called
mesons.
Leptons are not
made of quarks,
and include
the electrons
that orbit the atomic
nucleus, and their
more esoteric
relatives, like
muons, taus and
neutrinos.
LEPTONS
QUARKS
STRUCTURE OF
THE ATOM
There appear to be four basic forces at work:
A family of particles
called gauge bosons
transmit the forces
between the fermions.
There is a different kind
of particle for each
force:
Photons (the particles
of light) carry the
electromagnetic force;
Gluons carry the strong
force;
W and Z bosons carry
the weak force;
Gravitonsænot yet
observedæare believed
to be responsible for
gravity, which is not a
part of the Standard
Model.
Strong force is responsible for holding together protons and neutrons.
Weak force causes certain forms of radioactivity.
Electromagnetic force holds atoms and molecules together.
Gravity is responsible for the large-scale structure of the Universe,
binding stars and galaxies together.
ELECTRON
QUARK
NEUTRON
PROTON
The Standard Model(a)
Fermions Bosons
All the matter in the Universe,
including atoms, stars, rocks,
plants and animals is made of...
The particles carrying
the forces between
the matter are...
PARTICLES
FORCES
Figure 1.4 Two artists’ conceptions of the standard model. (a) From New York Times,
22 September, 1998. Reprinted by permission of the New York Times. (b) From “Nuclear
Science,” Contemporary Physics Education Project (CPEP), LBNL. (Figure 1.4(b) also
appears in color figure section.)
1.7 PARTICLE PHYSICS 21
(The symbol ne indicates the antiparticle of the electron neutrino.) In this equation,
the number of leptons on the left is zero, so the number of leptons on the right must
also be zero. This equivalence can only be true if we assign a lepton number L of 1 to
the electron (by convention) and L ¼ 21 to the ne (being an antiparticle). Consider
the reaction
ne þ pþ
À! eþ
þ n
Here L ¼ 21 on both sides of the equation where we assign lepton numbers of þ1
for every lepton and 21 for every antilepton (eþ
is an antilepton). By contrast, the
reaction
ne þ pþ
= eþ
þ n
is forbidden by lepton conservation. The law of lepton conservation applies separately
to electrons, muons, and tau muons.
Sample Problem 1.6 Is the reaction mÀ
À! eÀ
þ ne þ nm possible?
Solution
Left-hand side Lm ¼ 1 Le ¼ 0
Right-hand side Le ¼ 1 þ (À1) ¼ 0
Lm ¼ þ1
Yes, the reaction is possible.
If we focus our attention on the neutrons and protons (the nucleons), we note they
have similar masses (1 u). We also note the neutron is slightly (0.14%) more
massive than the proton with the mass difference being 1.29 MeV/c2
(Appendix
A). (This energy difference causes a free neutron to decay to a proton with a half-life
of approximately 10 min.) As remarked earlier, the neutron has no net electric
charge, whereas the proton has a positive charge equal in magnitude to the charge
on the electron. The electric charge on the proton is uniformly and symmetrically
distributed about the center of the proton with a charge radius of about 0.8 fm.
The neutron, although electrically neutral, also has an extended charge distribution
with a positive charge near the center being canceled out by a negative charge at
TABLE 1.6 Table of Leptons and Their Properties
Flavor Mass (GeV/c2
) Electric Charge
ne electron neutrino ,1 Â 10211
0
e electron 0.000511 21
nm muon neutrino ,0.0002 0
m muon 0.106 21
nt tau neutrino ,0.02 0
t tau 1.7771 21
22 INTRODUCTORY CONCEPTS
larger values of the radius. The values of the magnetic dipole moment of the neutron
and proton are also indications of their complex structure (Chapter 2). As far as their
interaction via the nuclear or strong force, the neutrons and protons behave alike (the
“charge independence” of the nuclear force). They can be regarded collectively as
“nucleons.” The nucleon can be treated as a physical entity with a mass of
938 MeV/c2
. One can speak of excited states of the nucleon such as the one with
a mass of 1232 MeV/c2
(which is called the D state).
The fermionic hadrons (called baryons) are thought to be made up of three fundamental
particles called quarks. There are six different kinds (or flavors) of quarks:
u (up), d (down), s (strange), c (charm), t (top), and b (bottom). The masses and
charges of the quarks are given in Table 1.7. The size of each quark is thought to
be ,10218
m. The lightest two quarks, the u and d quarks, are thought to make
up the nucleons. The proton is a uud combination with a charge of ( 2
3 þ 2
3 À 1
3 )e,
whereas the neutron is a udd combination with a charge of ( 2
3 À 1
3 À 1
3 )e. The up
and down quarks are light (m  5–10 MeV/c2
) and pointlike. The quarks
account for 2% of the mass of the proton. The rest of the mass is in gluons,
which “connect” the quarks. The most massive of the quarks is the top quark with
a mass approximately equivalent to that of a 197
Au nucleus and a short lifetime
(10224
s).
Like the leptons, there is a number conservation law for baryons. To each baryon,
such as the neutron or proton, we assign a baryon number B ¼ þ1 while we assign
B ¼ 21 to each antibaryon, such as the antiproton. Our rule is that the total baryon
number must be conserved in any process. Consider the reaction
pþ
þ pþ
À! pþ
þ n þ pþ
On the left, B ¼ 2 as it does on the right (the pþ
is a meson and has B ¼ 0).
As well as binding three quarks (antiquarks) together to make baryons (antibaryons),
the nuclear or strong interaction can bind a quark and an antiquark to
form unstable particles called mesons (q, q). The pþ
and p2
mesons (ud, du) are
of special importance in nuclear science. The quark/antiquark pairs in the p
mesons couple to have zero spin, and thus these mesons are bosons. In fact, all
mesons have integer spins and are thus bosons.
TABLE 1.7 Table of Quarks and Their Properties
Flavor Approx. Mass (GeV/c2
) Electric Charge
u up 0.003 2/3
d down 0.006 21/3
c charm 1.3 2/3
s strange 0.1 21/3
t top 175 2/3
b bottom 4.3 21/3
1.7 PARTICLE PHYSICS 23
1.8 EXCHANGE PARTICLES AND FORCE CARRIERS
The force carrier (or “exchange”) particles are all bosons. These particles are responsible
for carrying the four fundamental forces. This family includes the strong interaction
carrier, the gluon; the weak interaction carriers, the W+
and Z0
; the carrier of
the electromagnetic force, the photon; and the postulated but unobserved carrier of
the gravitational force, the graviton.
To understand how these force carriers work, let us consider the electromagnetic
force acting between two positively charged particles. Quantum electrodynamics
tells us that the force between these two particles is caused by photons passing
between them. At first one may find that idea nonsensical because the emission of
a photon should change the energy of the emitter/source (but exchange of a
force-carrier does not). The trick is that the uncertainty principle allows the emission
of virtual particles (which violate energy conservation) if such emission and absorption
occur within a time Dt that is less than that allowed by the uncertainty principle
Dt ¼
hÀ
DE
(1:35)
where DE is the extent to which energy conservation is violated. We will consider
the range of forces in Chapter 5.
PROBLEMS
1. Define or describe the following terms or phenomena: radiochemistry, isotone,
internal conversion, gluon, lepton.
2. Define or describe the following phenomena: electron capture, exchange forces,
time dilation.
3. Define or describe the following terms: quark, hadron, baryon, lepton, meson.
4. In an experiment one observes the characteristic Ka X-rays of two elements at
energies of 6.930 and 7.478 eV. The higher energy line is due to Ni. What
element is responsible for the lower energy line?
5. Using the Bohr theory, calculate the ratio of the energies of the Ka X-rays of I
and Xe.
6. Given the following energies of the Ka X-rays for the following elements, make
a Moseley plot of the data
V 4:952 eV
Cr 5:415
Mn 5:899
Fe 6:404
7. Predict the mode of decay of the following nuclei: 14
C, 3
H, 11
C, 233
U, 138
La.
24 INTRODUCTORY CONCEPTS
8. Write complete, balanced equations for the following decays:
a. a decay of 230
Th
b. b2
decays of 95
Zr
c. bþ
decay of 17
F
d. EC decay of 192
Au
9. Consider the decay of 238
U to 206
Pb. How many a particles and b2
particles are
emitted in this decay?
10. If a rock has a ratio of 206
Pb to 238
U of 0.6, what is the age of the rock?
11. How long will it take for a sample of 239
Pu (t1/2 ¼ 24, 119 y) to decay to 1
10 its
original amount?
12. If a radioactive sample of 59
Fe (t1=2 ¼ 44:496 d) has an activity of 1000 disintegrations
per minute, what weight of 59
Fe is present?
13. The environmental concentration of 239
Pu (t1=2 ¼ 24, 119 y) in a lake is
3.7 Â 1026
disintegrations/s/liter. What is the molarity of the solution?
14. 32
P (t1=2 ¼ 14:262 d) is a popular tracer in biochemistry. If I need to have
0.1 Â 106
disintegrations/s 60 days from now, how much 32
P tracer must I
purchase today?
15. Calculate the speed of a particle whose kinetic energy is three times its rest
energy.
16. Calculate the speed parameter b and the Lorenz factor g for the following
particles: an electron with EK ¼ 1 MeV; a proton with EK ¼ 1 MeV; a 12
C
nucleus with EK ¼ 12 MeV.
17. Consider the following free particles: a 1-eV photon, a 1-MeV electron, and a
10-MeV proton. Which is moving the fastest? Slowest? Has the most momentum?
The least momentum?
18. How much energy is necessary to increase the speed of a proton from 0.2c to
0.3c? From 0.98c to 0.99c?
19. A nonrelativistic particle is moving five times as fast as a proton. The ratio of
their de Broglie wave lengths is 10. Calculate the mass of the particle.
20. What are the wavelengths of a 500-MeV photon, a 500-MeV electron, and a
500-MeV proton?
21. What is the wavelength of a “thermal” neutron? (The kinetic energy of
the neutron can be taken to be 3
2 kT where T is the absolute room temp-
erature.)
22. Consider a nuclear excited state with a lifetime of 10 ps that decays by the
emission of a 2-MeV g ray. What is the uncertainty in the g-ray energy?
PROBLEMS 25
23. Which of the following decays are allowed by conservation laws?
a. p À! eþ
þ g d. p þ n À! p þ p þ pÀ
b. p À! pþ
þ g e. p þ p À! p þ p þ p þ p
c. n À! p þ g
24. What is the quark composition of the antiproton and the antineutron?
BIBLIOGRAPHY
There are many fine textbooks for nuclear and radiochemistry that cover the material
covered in this book. A limited selection of some of the authors’ favorites appears
below.
Simple Introductions to Nuclear Chemistry
Ehmann, W. D. and D. E. Vance. Radiochemistry and Nuclear Methods of Analysis, Wiley,
New York, 1991. An up-to-date survey of nuclear chemistry that emphasizes its
applications in analytical chemistry.
Harvey, B. G. Nuclear Chemistry, Prentice-Hall, Englewood Cliffs, NJ, 1965. A dated but
elegant summary of the essential features of nuclear science.
Loveland, W. Nuclear Chemistry, in Encyclopedia of Physical Science and Technology,
Vol. 11, Academic, Orlando, FL, 1992. A microversion of this text.
Wang, C. H., D. L. Willis, and W. D. Loveland. Radiotracer Methodology in the Biological,
Environmental and Physical Sciences, Prentice-Hall, Englewood Cliffs, NJ, 1975. A
somewhat out-of-date survey of radiotracer methods that includes an introduction to
nuclear science for life scientists.
History
Romer, A. Radiochemistry and the Discovery of Isotopes, Dover, New York, 1970. An
intriguing view of the beginning of nuclear chemistry.
Romer, A. The Discovery of Radioactivity and Transmutation, Dover, New York, 1964.
A presentation of the earliest explorations of radioactivity.
Seaborg, G. T. and W. Loveland. Nuclear Chemistry, Hutchinson-Ross, Stroudsberg, 1982.
Reprints of the most significant papers in nuclear chemistry from the earliest work to
present with annotations and English translations.
Intermediate-Level Textbooks—Similar to This Book
Choppin, G. R., J. O. Liljenzin, and J. Rydberg. Radiochemistry and Nuclear Chemistry, 3rd
ed., Butterworth-Heineman, Oxford, 2001. A very good, broad discussion of nuclear
chemistry that is oriented toward nuclear power and nuclear power applications.
26 INTRODUCTORY CONCEPTS
Cohen, B. L. Concepts of Nuclear Physics, McGraw-Hill, New York, 1971. This book is
especially noted for its discussion of the shell model and direct reactions.
Evans, R. The Atomic Nucleus, McGraw-Hill, New York, 1955. A dated, but encyclopedic,
treatment of nuclear science that has set the standard for its successors.
Friedlander, G., J. Kennedy, J. M. Miller, and E. S. Macias. Nuclear and Radiochemistry,
Wiley, New York, 1981. The bible of nuclear chemistry.
Harvey, B. G. Introduction to Nuclear Physics and Chemistry, 2nd ed., Prentice-Hall,
Englewood Cliffs, NJ, 1969. A wonderful, clear description of the physics of nuclei and
their interaction that is somewhat dated.
Keller, C. Radiochemistry, Harwood, 1981. A very condensed presentation of radioactivity
and its applications.
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1987. A clear, relatively up-todate
discussion from the point of view of a practicing experimental nuclear physicist.
Meyerhof, W. Elements of Nuclear Physics, McGraw-Hill, New York, 1967. A very concise
summary of the essential ideas of nuclear science.
Mukhin, K. N. Experimental Nuclear Physics, Vol. I, Mir, Moscow, 1987. Nuclear physics
described from a more formal Russian viewpoint.
Valentin, L. Subatomic Physics: Nuclei and Particles, North-Holland, Amsterdam, 1981.
An eclectic treatment at a slightly more advanced level.
More Advanced Textbooks
Burcham, W. E. and M. Jobes. Nuclear and Particle Physics, Longman, Brnt Mill, 1995.
A more comprehensive treatment with extensive discussion of particle physics.
de Shalit, A. and H. Feshbach. Theoretical Nuclear Physics, Vol. I: Nuclear Structure, Vol. II:
Nuclear Reactions, Wiley, New York, 1974. A comprehensive treatment of the theory of
nuclear structure and reactions.
Frauenfelder, H. and E. M. Henley. Subatomic Physics, 2nd ed., Prentice-Hall, Englewood
Cliffs, NJ, 1991. A treatment of both nuclear and elementary particle physics.
Heyde, K. Basic Ideas and Concepts in Nuclear Physics, 2nd ed., IOP, Bristol, 1999.
An excellent treatment of many newer aspects of nuclear physics.
Hodgson, P. E., E. Gadioli, and E. Gadiolo Erba. Introductory Nuclear Physics, Clarendon,
Oxford, 1997. Emphasis on nuclear reactions.
Marmier, P. and E. Sheldon. Physics of Nuclei and Particles, Vol. I and Vol. II, Academic,
New York, 1969. A dated, but accessible, treatment aimed at experimentalists.
Segre, E. Nuclei and Particles, 2nd ed., Benjamin, Reading, 1977. Remarkable for its breadth
and insight in nuclear physics.
Wong, S. S. M. Introductory Nuclear Physics, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ,
1998. A very up-to-date, readable treatment of nuclear physics.
General Physics Textbooks
Halliday, D., R. Resnick, and K. S. Krane. Physics, 4th ed., Vol. I and Vol. II, Extended,
Wiley, New York, 1992. A remarkable encyclopedic treatment of introductory physics.
BIBLIOGRAPHY 27
General References
Anderson, H. L. Ed. A Physicist’s Desk Reference, AIP, New York, 1989. Helpful summaries
of many types of data and directions as to their proper use.
Browne, E. and R. B. Firestone. Table of Radioactive Isotopes, Wiley, New York, 1986.
An authoritative compilation of radioactive decay properties. Do note that the spontaneous
fission half-lives are missing for several heavy nuclei.
Firestone, R. B. and V. S. Shirley. Table of Isotopes, 8th ed., Wiley, New York, 1996.
Although available on the Web, this reference is still useful because it contains simplified
energy level schemes not easily found in other places.
Web References
Living Textbook for Nuclear Chemistry (http://livingtextbook.orst.edu) A compilation of
supplemental materials related to nuclear and radiochemistry.
28 INTRODUCTORY CONCEPTS
CHAPTER 2
NUCLEAR PROPERTIES
Chapter 7. THE STRUCTURE OF THE NUCLEUS OF THE ATOM
“What?” exclaimed Roger, as Karen rolled over on the bed and rested her warm body against
his. “I know some nuclei are spherical and some are ellipsoidal, but where did you find out that
some fluctuate in between?”
Karen pursed her lips. “They’ve been observed with a short-wavelength probe . . .” From
S. Harris, Chalk Up Another one. Copyright # 1992 by AAAS Press. Reprinted by
permission of AAAS Press.
29
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
2.1 INTRODUCTION
In this chapter we will turn to a systematic look at the general properties of nuclei,
including their masses and matter distributions. A very large number of nuclei have
been studied over the years, and the general size, shape, mass, and relative stability
of these nuclei follow patterns that can be understood and interpreted with two complimentary
models of nuclear structure. The average size and stability of a nucleus
can be described by the average binding of the nucleons to each other in a macroscopic
model while the detailed energy levels and decay properties can be understood
with a quantum mechanical or microscopic model. We will consider the
average behavior in this chapter, and a detailed description of nuclear structure is
given later in Chapter 6.
2.2 NUCLEAR MASSES
One of the most important nuclear properties that can be measured is the mass.
Nuclear or atomic masses are usually given in atomic mass units (amu or u) or
their energy equivalent. The mass unit u is defined so that the mass of one atom
of 12
C is equal to 12.0000 . . . u. Note we said atom. For convenience, the masses
of atoms rather than nuclei are used in all calculations. When needed, the nuclear
mass mnucl
can be calculated from the relationship
mnucl
c2
¼ Matomic
c2
À ½Zm0c2
þ Be(Z)Š
where m0 is the rest mass of the electron, and Be(Z) is the total binding energy of all
the electrons in the atom. Be(Z) can be estimated on the basis of the Thomas–Fermi
model of the atom as
Be(Z) ¼ 15:73Z7=3
eV
Because the values of Be(Z) are generally small relative to the masses of the nuclei and
electrons, we shall neglect this factor in most calculations. Let us make a few calculations
to illustrate the use of masses in describing nuclear phenomena. Consider the
b2
decay of 14
C, that is, 14
C ! 14
Nþ
þ bÀ
þ ne þ energy. Neglecting the mass of
the electron antineutrino, thought to be a few electron volts or less, we have
Energy ¼ ½(m(14
C) þ 6m0) À (m(14
N) þ 6m0) À m(bÀ
)Šc2
where m0 is the electron rest mass and m(X) is the mass of the nucleus X. Substituting
in atomic masses as appropriate, recognizing that the b2
is an electron, we have
Energy ¼ ½M(14
C) À M(14
N)Šc2
where M(X) is the atomic mass of X.
30 NUCLEAR PROPERTIES
Let us now consider the case of the bþ
decay of 64
Cu, 64
Cu ! 64
NiÀ
þ bþ
þ
ne þ energy. Writing the equation for the energy release in the decay, we have
Energy ¼ ½(m(64
Cu) þ 29m0) À (m(64
Ni) þ 28m0) À (m0) À m(bþ
)Šc2
Substituting in atomic masses, and noting that the mass of a position equals the mass
of an electron
Energy ¼ ½M(64
Cu) À M(64
Ni) À 2m0Šc2
Our straightforward bookkeeping has shown us that for bþ
decay, the difference
between the initial and final nuclear masses, must be at least 2m0c2
(1.02 MeV)
for the decay to be energetically possible. This energy represents the cost of creating
the positron.
To complete our survey of the energy release in b decay, let us consider the case
of electron capture, that is, the electron capture decay of 207
Bi : e2
þ 207
Biþ
!
207
Pb þ ne þ energy. For the energy release in the decay, we have
Energy ¼ ½(m(207
Bi) þ 83m0) À (m(207
Pb) þ 82m0)Šc2
where we have recognized that 207
Bi captures one of its orbital electrons. Substituting
in atomic masses, we have
Energy ¼ ½M(207
Bi) À M(207
Pb)Šc2
The energy release in nuclear reactions is called the Q value of the reaction, and its
calculation is strictly a matter of bookkeeping. If we consider the reaction
56
Fe þ 4
He À! 59
Co þ 1
H þ energy
we can use the atomic masses to get:
Q ¼ ½M(56
Fe) þ M(4
He) À M(59
Co) À M(1
H)Šc2
Note that the sign convention used in nuclear chemistry and physics that assigns a
positive Q value for exoergic reactions is opposite to that used in chemistry where
exoergic reactions have negative values of DH and DE.
2.2 NUCLEAR MASSES 31
Calculate the energy release in the bÀ
and bþ
decay of 64
Cu:
(Energy release in 64
Cu bÀ
decay) ¼ M(64
Cu) À M(64
Zn)
 Ã
c2
¼ ½À65:421 À (À65:999)Š
¼ 0:578 MeV
(Energy release in 64
Cu bþ
decay) ¼ M(64
Cu) À M(64
Ni) À 2m0
 Ã
c2
¼ ½À65:421 À (À67:096) À 1:022Š
¼ 0:653 MeV
By convention QbÀ ¼ energy release and Qbþ ¼ energy release þ 1:022 MeV. So
QbÀ ¼ 0:578 MeV and Qbþ ¼ 1:675 MeV
2.3 TERMINOLOGY
The difference between the actual nuclear mass and the mass of all the individual
nucleons, which must be assembled to make the nucleus, is called the total binding
energy, Btot(A, Z). It represents the work necessary to dissociate the nucleus into separate
nucleons or the energy that would be released if all the nucleons came together
to form the nucleus. We write
Btot(A, Z) ¼ ½ZM(1
H) þ (A À Z)M(n) À M(A, Z)Šc2
where M(A, Z) is the atomic mass of A
Z, M(n) and M(1
H) are the mass of a neutron
and a hydrogen atom, respectively. The average binding energy per nucleon, Bave
(A, Z) is given by
Bave(A, Z) ¼ Btot(A, Z)=A
In many tabulations of nuclear properties, such as that in Appendix B, the quantity
that is tabulated is the mass excess or mass defect rather than the mass. The mass
excess, D, is defined as M(A, Z) 2 A, usually given in units of the energy equivalent
of mass. Since in most, if not all calculations, the number of nucleons will remain
constant, the use of mass excesses in the calculations will introduce an arithmetic
simplification. Another term that is sometimes used is the mass excess per
nucleon or the packing fraction [;(M 2 A)/A].
The work necessary to separate a neutron, proton, or a particle from a nucleus is
called the (neutron, proton, or a particle) separation energy S. For a neutron
Sn ¼ ½M(A À 1, Z) þ M(n) À M(A, Z)Šc2
32 NUCLEAR PROPERTIES
Such separation energies can be expressed in terms of the total binding energy by
Sn ¼ Btot(A, Z) À Btot(A À 1, Z)
Calculate the neutron separation energy of 236
U and 239
U.
Sn ¼ M(A À 1, Z) þ M(n) À M(A, Z)½ Šc2
For 236
U
Sn ¼ M235U
þ Mn À M236U
 Ã
c2
¼ 40:914 þ 8:071 À 42:441
¼ 6:544 MeV
For 239
U
Sn ¼ M238U
þ Mn À M239U
 Ã
c2
¼ 47:304 þ 8:071 À 50:596
¼ 4:779 MeV
Notice that the neutron separation energy of A
Z is the excitation energy of the
nucleus A
Z produced when A21
Z is irradiated with “zero energy” neutrons. Thus,
when even–odd 235
U is irradiated with neutrons, the 236
U is produced at excitation
energy of 6.5 MeV while the same process with 238
U gives an excitation energy of
4.8 MeV. If it takes 5–6 MeV to cause these nuclei to fission, 235
U is “fissionable”
with zero energy neutrons while 238
U is not.
2.4 BINDING ENERGY PER NUCLEON
The binding energy per nucleon is a measure of the relative stability of a nucleus.
The more tightly bound a nucleus is, the greater the binding energy per nucleon
is. A plot of the average binding energy per nucleon as function of the mass
number is shown in Figure 2.1. Several features of this plot are worth noting. The
greatest stability is associated with medium mass nuclei, with the most stable
nucleus being 62
Ni. The heaviest nuclei could increase their stability by fissioning
while the lightest nuclei could increase their stability by fusing to make nuclei in
the Fe–Ni region.
The most striking feature of Figure 2.1 is the approximate independence of A as
the average binding energy per nucleon for most nuclei (ranging from 7.4 to
8.8 MeV). This is a direct consequence of the short range, saturation character of
the nuclear force. Suppose that the nuclear force was long range and not saturated.
Suppose further that the binding energy of one nucleon to every other nucleon was
some constant K. In a nucleus with A nucleons, there would be A(A À 1)=2 “bonds”
2.4 BINDING ENERGY PER NUCLEON 33
and thus the total binding energy would be KA(A À 1)=2 with the binding energy per
nucleon being K(A À 1)=2. In other words, one would predict that the average
binding energy per nucleon would increase linearly with A. That does not happen,
as shown in Figure 2.1. Thus, one concludes that the nuclear force saturates and
is short range.
In Figure 2.1, one also observes definite peaks in the average binding energy per
nucleon for certain values of A. This is quite reminiscent of the plots of ionization
potential vs. Z for atoms and suggests that there are certain special stable nucleonic
configurations similar to the inert gas structures of atoms. The general decrease of
Bave at higher values of A is due to the increasing influence of the Coulomb force.
0
7.5
8.0
B/A(MeV)
B/A(MeV)
9
8
7
6
5
4
3
2
1
0
0 10 20 30
8.5
9.0 = 20
|| || || ||||
||| ||| ||| |||
28N
Z 28
50
50
82
82
126
= 20
50 100 150 200 250
A
A
Figure 2.1 Average binding energy per nucleon vs. mass number A for the stable nuclei
from Valentin, Subatomic physics: nuclei and particles. Copyright # 1981 by NorthHolland
Publishing Company. Reprinted by permission of North-Holland Publishing
Company.
34 NUCLEAR PROPERTIES
Example of Binding Energies Contrast the binding energy per nucleon in a 16
O
nucleus with the binding energy per molecule in liquid water.
B(16, 8)=16 ¼ ½8 Ã M(1
H) þ 8 à M(n) À M(16, 8)Š à 931:5=16
¼ 7:97 MeV
For water:
DHvaporization=NA ¼ 40700 J=mol=6:02 Â 1023
=mol=1:602 Â 10À19
J=eV
¼ 0:42 eV
Note that these are both constants per particle, only different by approximately 7
orders of magnitude.
2.5 SEPARATION ENERGY SYSTEMATICS
Figure 2.2 shows a plot of the neutron separation energy for several isotopes of lead.
For a given Z, Sn is larger for even N compared to that for odd N. Similarly for a
given N, Sp is larger for even Z compared to that for odd Z. This effect is caused
by that part of the nuclear force that likes to have neutrons paired with neutrons
(with antiparallel spin) and to have protons paired with protons but have no n–p
pairing. This pairing of like nucleons causes even–even nuclei (Z even, N even)
to be more stable than even–odd or odd–even nuclides which, in turn, are more
stable than odd–odd nuclei.
Figure 2.2 Neutron separation energy for the lead isotopes.
2.5 SEPARATION ENERGY SYSTEMATICS 35
2.6 ABUNDANCE SYSTEMATICS
In Figure 2.3, we compare the positions of the known stable nuclides of odd A with
those of even A in the chart of the nuclides. Note that as Z increases, the line of
stability moves from N ¼ Z to N=Z  1:5 due to the influence of the Coulomb
force. For odd A nuclei, only one stable isobar is found while for even A nuclei
there are, in general, no stable odd–odd nuclei. This is further demonstrated by
the data of Table 2.1 showing the distribution of stable isotopes.
2.7 SEMIEMPIRICAL MASS EQUATION
C. F. von Weizsa¨cker developed a crude theory of nuclear masses in 1935. The
theory takes as its basis the idea that nuclei behave like incompressible uniformly
charged liquid drops. How can we account for the variation of nuclear masses?
We begin by stating that
M(Z, A)c2
¼ ½Z Ã M(1
H) þ (A À Z) à M(n)Šc2
À Btot(Z, A)
Figure 2.3 Positions of the stable odd A and even A nuclei in a Segre chart from
W. E. Meyerhof, Elements of Nuclear Physics. Copyright # 1967 by McGraw-Hill Book
Company, Inc. Reprinted by permission of McGraw-Hill Book Company, Inc.
TABLE 2.1 Distribution of Stable Nuclides
N Even Odd Even Odd
Z Even Even Odd Odd
Number 160 53 49 4
36 NUCLEAR PROPERTIES
Weizsa¨cker’s mass equation has evolved into what is called the semiempirical mass
equation, which begins by parameterizing the total binding energy of species Z, A as
Btot(A, Z) ¼ avA À asA2=3
À ac
Z2
A1=3
À aa
(A À 2Z)2
A
+ d
The justification of this representation of the total binding energy of the nucleus is as
follows:
1. Since there are A nucleons in the nucleus and the nuclear force saturates, we
expect each nucleon to contribute to the total binding energy. This term is
known as the volume term. The coefficient av is the energy by which a
nucleon in the interior of the nucleus is bound to its nearest neighbors and
is a parameter to be determined experimentally.
2. However, not all nucleons are in the interior. Those nucleons on the surface
are less tightly bound because they do not have a full complement of neighbors.
We need a correction term to the binding energy proportional to the
surface area of the nucleus. The surface area of the nucleus can be taken
to be 4pR2
. If, as asserted earlier, R / A1=3
, then 4pR2
/ A2=3
. (Notice that
the volume is 4
3 pR3
, which is proportional to A. Hence, the form of the
first term.) The A2/3
factor is multiplied by another coefficient, as, that is to
be determined experimentally.
3. The third term reflects the decrease in binding due to the Coulomb repulsion
between the protons. The Coulomb energy of a uniform sphere can be written as
ECoul ¼
3
5
Z2
e2
R
(2:1)
If we denote R as r0A1/3
, we can substitute for R in the above equation, arriving
at the point that ECoul ¼ 0.72 Z2
/A1/3
MeV. (The usual fitted or adjusted value
of the coefficient ac is 0.7 rather than 0.72.)
4. The fourth term along with the fifth term represent effects on the binding energy
that are quantum mechanical in origin. The fourth term, the asymmetry energy,
is the difference in energy of a nucleus with N neutrons and Z protons (N = Z)
and one where Z ¼ N ¼ A/2. To evaluate this term, we remember that neutrons
and protons occupy orbitals in the nucleus at well-defined energies and that the
neutrons and protons obey the Pauli principle for fermions. A simple model
(Fig. 2.4) should suffice to calculate this energy. Assume the neutron and
proton levels of a nucleus are equidistant with spacing D and that we can
have only one nucleon per level. To make the nucleus A
Z (where Z = N)
from the nucleus with N ¼ Z ¼ A/2, we must take q protons and transform
them into neutrons. Thus, we have N ¼ q þ A/2, Z ¼ A/22q and therefore
q ¼ (N 2 Z)/2. Each of the q protons must be raised in energy an amount
qD. The work needed to transform the N ¼ Z nucleus into the nucleus A
Z is
q2
D ¼ (N 2 Z)2
D/4. Note that we could have made exactly the same argument
by replacing neutrons with protons. We finish the argument by noting that the
2.7 SEMIEMPIRICAL MASS EQUATION 37
energy levels get closer together as the total number of nucleons increases and
thus D / 1=A. As a final matter of notation, we replace (N À Z) by (A À 2Z).
5. The last term represents the special stability associated with completely paired
spins in a nucleus. The pairing energy term is chosen to be zero for odd A
nuclides; for even–even nuclides, use the positive form, for odd–odd
nuclides, use the negative form.
The constants of the semiempirical binding energy equation can be determined
by fitting the data on the masses of nuclei. A recent set of values of the coefficients
are av ¼ 15.56 MeV, as ¼ 17.23 MeV, ac ¼ 0.7 MeV, aa ¼ 23.285 MeV, and
d ¼ 11/A1/2
MeV. The relative contribution of each term to the binding energy per
nucleon is shown in Figure 2.5. Note the large constant contribution of the volume
energy to the average binding energy per nucleon. The surface energy correction is
most important for the lighter nuclei where the fraction of nucleons in the surface
is greatest. Similarly, the Coulomb energy correction is most important for the
heaviest nuclei since it depends on Z2
. The asymmetry energy is a smaller effect
that is most important in the heaviest nuclei where the N/Z ratio is the greatest.
Example Problem Calculate the average binding energy per nucleon of 58
Fe
using the semiempirical mass equation:
Solution
Btot(A, Z) ¼ avA À asA2=3
À ac
Z2
A1=3
À aa
A À 2Zð Þ2
A
+ d
Btot(58, 26) ¼ 15:56(58) À 17:23 582=3
À Á
À 0:7
262
581=3
 
À 23:285
(58 À 52)2
58
 
þ
11
581=2
Btot(58, 26) ¼ 902:48 À 258:17 À 122:25 À 14:45 þ 1:44 ¼ 509:05 MeV
Btot=A ¼ 509:05=58 ¼ 8:78 MeV
Notice the relative contribution of the various terms of the binding energy.
Figure 2.4 Schematic model of how the nucleus A
Z is assembled from an N ¼ Z nucleus
from W. E. Meyerhof, Elements of Nuclear Physics. Copyright # 1967 by McGraw-Hill
Book Company, Inc. Reprinted by permission of McGraw-Hill Book Company, Inc.
38 NUCLEAR PROPERTIES
Myers and Swiatecki (1966) have proposed a modification of the semiempirical
mass equation that gives a better description of the experimental masses. This
modification can be summarized in the following equation:
BEtot(Z, A) ¼ c1A 1 À k
N À Z
A
 2
" #
À c2A2=3
1 À k
N À Z
A
 2
" #
À
c3Z2
A1=3
þ
c4Z2
A þ d
(2:2)
where c1 ¼ 15.677 MeV, c2 ¼ 18.56 MeV, c3 ¼ 0.717 MeV, c4 ¼ 1.211 MeV,
k ¼ 1.79, and d ¼ 11/A1/2
. What Myers and Swiatecki (1966) have done is to
add an asymmetry energy correction term to the surface energy and to add a term
(the c4 term) that is a correction to the Coulomb energy term due to the diffuseness
of the nuclear surface.
We will now look at some of the predictions of the semiempirical mass equation.
The first question we pose is what happens if we hold A constant and vary Z (neglecting
for a moment the pairing term). We can write
M(Z, A) ¼ Z Ã M(1
H) þ (A À Z)M(n) À Btot(Z, A) (2:3)
Btot(Z, A) ¼ avA À asA2=3
À acZ2
=A1=3
À aa(A À 2Z)2
=A (2:4)
Figure 2.5 Relative contributions of the various terms in the semiempirical mass equation
to the average binding energy per nucleon from W. E. Meyerhof, Elements of Nuclear
Physics. Copyright # 1967 by McGraw-Hill Book Company, Inc. Reprinted by permission
of McGraw-Hill Book Company, Inc.
2.7 SEMIEMPIRICAL MASS EQUATION 39
Let us expand the asymmetry energy term as follows:
aa(A À 2Z)2
A
¼ aa
A2
À 4AZ þ 4Z2
A
¼ aa A À 4Z þ
4Z2
A
 
(2:5)
Substituting back into the equation for the mass and collecting terms, we have
M ¼ A M(n) À av þ
as
A1=3
þ aa
h i
þ Z½(M(1
H) À M(n) À 4Zaa)Š þ Z2 ac
A1=3
þ
4aa
A
 
(2:6)
Thus, the mass equation at constant A takes on the form of a parabola (a þ bZ þ gZ2
)
in Z. The third term, g, is positive, and so the parabola goes through a minimum
for some value of Z, which is termed ZA. Note that ZA is not necessarily an integer.
We can now ask ourselves what is the value of ZA? We can evaluate this by minimizing
M with respect to Z at constant A, that is, we get a simple partial differential
equation:
@M
@Z
 
Z=A
¼ 0 ¼ b þ 2gZA (2:7)
ZA ¼
Àb
2g
¼ À
M(1
H) À M(n) À 4aa
2
ac
A1=3
þ
4aa
A
  (2:8)
Substituting numerical values for the coefficients in this expression, we can show that
ZA
A

1
2
81
80 þ 0:6A2=3
(2:9)
Thus, as A goes to 0, ZA/A becomes equal to 1
2, or Z ¼ N ¼ A/2. As A gets large, ZA/A
is less than 1
2, typically, about 0.4. The underlying physics behind this trend is that, in
the absence of the Coulomb repulsion between the protons, we would expect equal
numbers of neutrons and protons due to the asymmetry energy term. When Z gets
large, the Coulomb energy becomes large. The nucleus can gain stability by converting
protons into neutrons. Stability results when we have a balance between the excess
Coulomb energy and the asymmetry energy.
Let us now consider the case where A ¼ 111. From the above relations, we can
calculate ZA ¼ 47.90. In Figure 2.6, we show the actual masses of the nuclei with
A ¼ 111. The expected parabolic dependence of mass upon Z is observed. The
most stable nucleus has Z ¼ 48 (Cd). All the A ¼ 111 nuclei that have more
neutrons than 111
Cd release energy when they decay by b2
decay while the nuclei
with fewer neutrons than 111
Cd release energy when they decay by bþ
or EC decay.
Now let us consider the case of the even A nuclei with A ¼ 112. We calculate that
ZA ¼ 48.29. Plotting the actual masses of the A ¼ 112 nuclei vs. Z (Fig. 2.7) gives us
two parabolas, one for the even–even nuclei and one for the odd–odd nuclei,
40 NUCLEAR PROPERTIES
displaced from one another by the energy 2d. Since all nuclei on the upper parabola
(the odd–odd nuclei) can decay to a nucleus on the lower parabola (the even–even
nuclei), we conclude that there are no stable odd–odd nuclei. (The only known
exceptions to this rule occur in the light nuclei where nuclear structure effects
make 2
H, 6
Li, 10
B, and 14
N stable). Note that some odd–odd nuclei can thus
decay by both b2
or bþ
emission. Note that double b decay is energetically possible
Figure 2.6 Mass excesses of the known nuclei with A ¼ 111.
–40
–50
M-A(MeV)
–60
–70
–80
–90
42 44 46 48 50 52 54 56
Z
A = 112
odd-odd
even-even
Figure 2.7 Mass excesses of the known nuclei with A ¼ 112.
2.7 SEMIEMPIRICAL MASS EQUATION 41
in some cases (112
Pd ! 112
Cd þ 2b2
þ 2ne). This mode of decay has only been
observed for 130
Te and 82
Se, and the half-lives for this mode of decay are very
long (t1/2 ¼ 1020
–1021
y). Note that we can have more than one stable isotope for
a given A and that all of them will be even–even.
This parabolic dependence of the nuclear mass upon Z for fixed A can be used to
define a nuclear mass surface (Fig. 2.8). The position of the minimum mass for each
A value defines what is called the valley of b stability. b decay is then depicted as
falling down the walls of the valley toward the valley floor.
2.8 NUCLEAR SIZES AND SHAPES
We ask how big are nuclei? Our zero-order answer is that the radii of nuclei are in
the range of 1–10 fm. Our first-order answer to this question begins by assuming the
nucleus is spherical with a uniform density out to some sharp cutoff radius, that is,
the nucleus has the shape and density distribution of a billiard ball. This density
distribution is shown in Figure 2.9.
One can parameterize this distribution by saying that the nuclear radius R can be
written as
R ¼ r0A1=3
(2:10)
where the nuclear radius constant can be taken to be 1.2 fm for the “charge radius”
and 1.4 fm for the “matter radius.” What do we mean by this dichotomy? What we
Figure 2.8 Plot of the nuclear mass excesses vs. neutron number N and atomic number Z for
the light nuclei showing the nuclear mass surface and the valley of b stability from Halliday,
et al., 1992; reprinted by permission of John Wiley & Sons, Inc.
42 NUCLEAR PROPERTIES
mean is that when one measures the nuclear radius by scattering high-energy electrons
from the nucleus or when one measures the radius by scattering low-energy
ions from the nucleus, one gets slightly different answers for the nuclear size. The
electron probes the charge distribution via the electromagnetic force, that is, the
distribution of the protons while other particles probe the matter distribution or
the range over which nuclear forces are acting. Which value of r0 should one use
in calculations? The answer depends upon the nuclear property being calculated
and whether it is sensitive to the distribution of charge or matter.
A somewhat more sophisticated approach to the problem of defining the nuclear
size and density is to assume the nuclear density distribution, r(r), assumes the form
of a Fermi distribution, that is,
r(r) ¼
r0
1 þ e(rÀR)=a
(2:11)
where r0 is the density in the interior of the nucleus (r0 ¼ 0.172 nucleons/fm3
), a is
a measure of the diffuseness of the nuclear surface, and R is the half-density radius of
the nucleus (Fig. 2.10). The half-density radius is given by the expression
R ¼ r0 A1=3
where r0 ¼ 1.12 fm. The thickness of the nuclear skin, t, indicated in Figure 2.10,
can be related to the diffuseness parameter as t ¼ 4a ln 3 4.4a. Most nuclei
show a skin thickness t of 2.4–2.5 fm. The meaning of this value of t can be ascertained
by calculating the fraction of the nucleons that lies in the skin region of the
nucleus as a function of the nuclear size (Table 2.2).
Thus, the lighter nuclei are mostly “skin” and the heaviest nuclei still have substantial
“skin” regions. These approximate models for the nuclear size and density distribution
compare favorably to the measured distributions for typical nuclei (Fig. 2.11).
Up to this point, we have assumed that all nuclei are spherical in shape. That is not
true. There are regions of large stable nuclear deformation in the chart of nuclides, that
is, the rare earths (150 , A , 180) and the actinides (220 , A , 260). We shall
discuss these cases in more detail later in this chapter when we discuss the electric
moments of nuclei.
Figure 2.9 Schematic diagram of a sharp cutoff, constant density model for nuclei.
2.8 NUCLEAR SIZES AND SHAPES 43
Another question we might pose to ourselves is whether the neutron and proton
distributions in nuclei are the same? Modern models for the nuclear potential
predict the nuclear skin region to be neutron-rich. The neutron potential is predicted
to extend out to larger radii than the proton potential. Extreme examples of this behavior
are the halo nuclei. A halo nucleus is a very n-rich (or p-rich) nucleus (generally
with low A) where the outermost nucleons are very weakly bound. The density distribution
of these weakly bound outermost nucleons extends beyond the radius expected
from the R / A1=3
rule. Examples of these nuclei are 11
Be, 11
Li, and 19
C. The most
well-studied case of halo nuclei is 11
Li. Here the two outermost nucleons are so
weakly bound (a few hundred keV each) as to make the size of 11
Li equal to the
size of a 208
Pb nucleus (see Fig. 2.12).
2.9 QUANTUM MECHANICAL PROPERTIES
2.9.1 Nuclear Angular Momenta
It is well known and, in fact, an essential underlying part of chemical behavior that
the electron has an intrinsic angular momentum, s ¼ 1
2
hÀ
. That is, the electron
TABLE 2.2 Fraction of Nucleons in Nuclear “Skin”
Nucleus
Fraction of Nucleons
in the “Skin”
12
C 0.90
24
Mg 0.79
56
Fe 0.65
107
Ag 0.55
139
Ba 0.51
208
Pb 0.46
238
U 0.44
Figure 2.10 Nuclear density distribution: (a) in a schematic view and (b) in an artist’s
conception from R. Mackintosh, J. Al-Khalili, B. Jonson and T. Pena, Nucleus: A Trip into
the Heart of Matter. Copyright # 2001 by The Johns Hopkins University Press, 2001;
reprinted by permission of Johns Hopkins. (Figure also appears in color figure section.)
44 NUCLEAR PROPERTIES
behaves as if it is rotating or spinning about an internal axis. The electron spin
angular momentum provides an important criterion for assigning quantum
numbers to atomic electrons through the Pauli principle and thus has far-reaching
consequences. The electrons occupy quantum mechanical states or orbitals that
are labeled by the principal quantum number, N, which is one more than the
number of radial nodes in the atomic wave function and the angular momentum
quantum number, l, the number of angular nodes in the wave function. The electrons
distribute themselves among those states with degenerate energies so that their spinangular
momenta, s, are aligned (Hund’s rules). The resulting atomic energy level
can be characterized by a single total angular momentum, J, that is made up from
the total orbital motion of all the electrons, L, and a total intrinsic spin, S. These
values of L and S are calculated by two separate vector couplings of the two
0
0.10
Chargedensity(efm−3)
r (fm)
0.10
4He
12C
40Ca
48Ca
58Ni
124Sn
208Pb
0.10
0.10
0.10
0.10
EXP
MEAN FIELD THEORY
0.10
0.08
0.06
0.04
0.02
0
2 4 6 8 10
2 4 6 8 10
4 6 8 10
4 6 8 10
4 6 8 10
6 8 10
6 8 10
Figure 2.11 Nuclear ground-state charge distributions as measured for a sample of nuclei
throughout the periodic table from B. Frois, Proc. Int. Conf. Nucl. Phys., Florence, 1983,
eds. P. Blasi and R.A. Ricci (Tipografia Compositori Bologna) Vol. 2, p. 221.
2.9 QUANTUM MECHANICAL PROPERTIES 45
types of angular momenta of the electrons. The electrons in all but the heaviest
atoms exhibit such “LS” coupling.
The neutron and the proton also have an intrinsic angular momentum, s ¼ 1
2
hÀ
,
and so each appears as if it is spinning about an internal axis. Thus, we can
expect that a large nucleus, which contains some number of neutrons and protons,
will have a total intrinsic angular momentum, or a nuclear spin, I, just from the combination
of the intrinsic spins of the neutrons and protons. Similarly, we can imagine
that neutrons and protons will occupy discrete states in the nucleus, and some of
these states will have orbital angular momenta in a manner similar to the orbital
angular momenta of electronic states (l ¼ 1, p-states, etc.) While there are such similarities,
the fact that the potential well for nucleons is dramatically different from the
central Coulomb potential for electrons introduces several important differences in
the concepts used to describe nuclear states and levels. A detailed discussion of the
quantum mechanical structure of nuclei is presented in Chapter 6. At this point we
only need to address the gross features.
The orbital angular momenta of the nuclear (and atomic) states are all integer
multiples of hÀ
starting with zero. Nucleons exhibit a strong coupling of the
orbital and spin angular momenta of individual nucleons such that j ¼ l þ s is the
appropriate quantum number for a nucleon. We can immediately see that the combination
of the intrinsic spins of the nucleons with their orbital motion will always
give half-integer values for the total spin, I, of any odd-A nucleus and integer values
for any even-A nucleus:
odd-A nuclei I ¼
1
2
,
3
2
,
5
2
, . . .
even-A nuclei I ¼ 0, 1, 2, . . .
Figure 2.12 Schematic representation of the relative sizes of the halo nucleus 11
Li and
208
Pb. (Figure also appears in color figure section.)
46 NUCLEAR PROPERTIES
The numerical value will depend on the filling of the nuclear states with angular
momenta j and on the coupling of all of those angular momenta. At first glance
we might expect that a large nucleus might have a very large intrinsic angular
momentum. However, recall that the nuclear force has a short range and that the
nucleons are more strongly bound when they are in close proximity. Two nucleons
will be in the closest proximity when they are in the same orbital. If the two nucleons
in the same orbital are both neutrons or both protons, then their spins must be
opposed in order to satisfy the Pauli principle and each have a unique set of
quantum numbers. So we find that the nuclear force tends to put pairs of nucleons
into the same orbitals and their orbital angular momenta and intrinsic spins
cancel, summing to zero. (This behavior is opposite from that of atomic electrons.)
Thus, the angular momenta of the ground states of nuclei tend to be small, even for
nuclei with hundreds of nucleons in states with high angular momenta.
Parity, as used in nuclear science, refers to the symmetry properties of the wave
function for a particle or a system of particles. If the wave function that specifies the
state of the system is C(r, s) where r represents the position coordinates of the
system (x, y, z) and s represents the spin orientation, then C(r, s) is said to have positive
or even parity when
C(r, s) ¼ þC(Àr, Às) (2:12)
where the minus sign indicates the sign of the spatial coordinates has been reversed
as well as the direction of the spin. When
C(r, s) ¼ ÀC(Àr, Às) (2:13)
the system is said to have negative or odd parity. For a central potential [V ¼ V(r)],
that is, the potential energy depends only on the distance and not the spatial
orientation (q, f), the parity, denoted as p, is given as
p ¼ (À1)l
(2:14)
where l is the orbital angular momentum of the system (s, d orbitals have positive
parity while p or f orbitals have negative parity). The spin and parity of a given
nuclear state are usually used as labels for that state. Thus, a state with I ¼ 7
2 and
negative parity is referred to as a 7
2 À state.
2.10 ELECTRIC AND MAGNETIC MOMENTS
2.10.1 Magnetic Dipole Moment
The magnetic moments of nuclei are measures of the distribution of electric currents
in the nucleus while the electric moments are measures of the distribution of electric
charges. Because the magnetic moment may not be a familiar concept, we will begin
2.10 ELECTRIC AND MAGNETIC MOMENTS 47
by discussing a simple example of a “classical” magnetic moment. Consider an electron
moving with a velocity v in a circular orbit of radius r about a point (Fig. 2.13).
The magnetic dipole moment is defined as the product of the area of the loop made
by the electron, A, and the current, i. The area of the circle is pr2
and the current i is
(electron charge)/(time to make a loop) or e/(2pr/v). Thus, we have
m



 ¼ iA ¼
ev
2pr
 
pr2
À Á
¼
evr
2
(2:15)
Remember that the angular momentum of the electron moving in a circle, l, is m0vr.
Thus,
m



 ¼
evr
2
Â
m0
m0
¼
el
2m0
(2:16)
The gyromagnetic ratio g is defined as jmj/l ¼ e/2m0.
In quantum mechanics, the projection of the angular momentum l is mlhÀ
. We,
therefore, expect the magnetic dipole moment due to the orbital motion of the
electron to be given by
m ¼ (e=2m0)mlhÀ
¼ mlmB (2:17)
where mB is called the Bohr magneton (¼ehÀ
/2m0) and has a magnitude of
5.78 Â 1025
eV/tesla or 9.27 Â 10221
erg/gauss. The electron has an intrinsic
spin, ms ¼ 1
2. It also has a component of the magnetic moment due to this spin.
Extending these ideas to nucleons, we can define the nuclear magneton, mN
as (ehÀ
/2mp), which has the numerical value of 3.15 Â 1028
eV/tesla or
Figure 2.13 Classical magnetic dipole moment.
48 NUCLEAR PROPERTIES
5.50 Â 10224
erg/gauss:
mproton
l ¼ mlmN (2:18)
Note that the nuclear magneton is smaller than the Bohr magneton by the factor of
the ratio of the proton to electron masses, 1840.
It is traditional to rewrite the definition of the nuclear magnetic moment in terms
of magnetons and include a constant of proportionality called the gyromagnetic ratio
or simply g factor:
m ¼ glmlmN (2:19)
By adding a constant of proportionality we are anticipating that the magnetic
moment will be the net result of a complicated cancellation process. For example,
we would expect gl ¼ 1 for the orbital motion of a proton due to its charge and
gl ¼ 0 for a neutron because they are uncharged.
Both the neutrons and the protons have an intrinsic spin, and so by extension we
can expect additional contributions to the magnetic moment of the form:
m ¼ gsmsmN (2:20)
where the projection ms is 1
2 for fermions like the proton and neutron. The spin g
factor, gs ¼ 2.0023, as calculated with the relativistic Dirac equation for electrons,
including known higher order correction terms, is in very good agreement with
measurements. However, the measured values of gs for both the proton and the
neutron are surprisingly large:
Proton gs ¼ 5:5856912(22)
Neutron gs ¼ À3:8260837(18)
Notice that the neutron with exactly zero net charge has a nonzero magnetic
moment. Thus, both the proton and the neutron do not appear to be elementary
particles. Rather they both seem to have internal moving constituents. It was
noted some time ago that the magnetic moment of the proton is larger than the
expected value of “2” and that of the neutron is smaller than its expected value of
“0” by about 3.6 units. Older models of the nuclear force attributed these differences
to “clouds” of mesons surrounding the nucleons. In the modern theory of quantum
chromodynamics the nucleons are made up from three quarks each with their own
magnetic moments and electronic charges.
The presence of a magnetic dipole moment in many nuclei that have an intrinsic
spin has found enormous application in nuclear magnetic resonance (NMR) and
magnetic resonance imaging (MRI). NMR is extensively used in chemical
laboratories to identify the structural and chemical environments of the nuclei in
molecules, whereas MRI uses a tomographic technique to locate specific molecules
2.10 ELECTRIC AND MAGNETIC MOMENTS 49
on a microscopic scale. Both techniques rely on the splitting of the energies of the
magnetic substates by a (strong) magnetic field. NMR measures tiny shifts in the
relative energies of the magnetic substates due to induced magnetization of
the local electron density to provide information on the chemical environment.
These states have a fine structure or splitting due to the presence of neighboring
magnetic nuclei that provides information on the structure of the molecule. MRI
applies a spatially varying magnetic field to detect the resonance of a single type
of nucleus, usually the hydrogen nuclei in water and aliphatic compounds, and to
measure the concentration in a three-dimensional space. Both techniques are
nondestructive and can be applied to living systems. The concentration of water
molecules varies widely in tissues and other biological media and can provide
detailed microscopic images for medical purposes.
2.10.2 Electric Quadrupole Moment
Imagine the nucleus is an extended charged object as sketched in Figure 2.14. Consider
trying to calculate the potential energy at some point P, which is at a distance D
from the center of the charged object (nucleus). Suppose we evaluate the potential,
dF, at point P due to a charge at a distance r from the center of the charge object.
Assume further the line from the center of the object to the charge makes an angle u
with the line connecting the center of the object with the point P (Fig. 2.14). If the
density of charge in the object is r (u, f, r), then the total charge at the indicated
point is r dt or r (r2
dr sin u du df). We can write
dF ¼ r dt=d ¼ r dt ½D2
þ r2
À 2Dr cos uŠÀ1=2
(2:21)
Factoring out D and substituting the Legendre polynomials into the equation
where
P1( cos u) ¼ cos u (2:22)
e
z
r
D
Nuclear surface
z-axis
= (D2
- 2Dr cos q + r2
)
1
2
q
P
l
Figure 2.14 Potential at a point due to an extended charge object from B. G. Harvey,
Introduction to Nuclear Physics and Chemistry, 2nd Edition. Copyright # 1969 by
Prentice-Hall, Inc. Reprinted by permission of Pearson Education.
50 NUCLEAR PROPERTIES
and
P2( cos u) ¼ 3
2 cos2
u À 1
2 (2:23)
we get
dF ¼
r dt
D
1 þ
r
D
P1( cos u) þ
r
D
 2
P2(cosu) þ Á Á Á
!
(2:24)
If we set up the integrals over the entire volume of the charged object
(nucleus), we get
V ¼
1
D
ð
volume
rdt
!
þ
1
D2
ð
volume
rrcosudt
!
þ
1
D3
ð
volume
rr2 3
2 cos2
uÀ1
2
À Á
dtþÁÁÁ
!
(2:25)
The first term in the square bracket in this equation is the electric monopole
moment, which is equal to the nuclear charge, Ze. The second term in the
square bracket is the electric dipole moment while the third term in the
square bracket is the electric quadrupole moment. For a quantum mechanical
system in a well-defined quantum state, the charge density r is an even function,
and because the dipole moment involves the product of an even and an
odd function, the corresponding integral is identically zero. Therefore, there
should be no electric dipole moment or any other odd electric moment for
nuclei. For spherical nuclei, the charge density r does not depend on u, and
thus the quadrupole moment Q is given by
Q¼
ððð
r2
r(r) 3
2cos2
uÀ1
2
À Á
r2
drsinududf (2:26)
The quadrupole moment will differ from zero only if the nucleus is not
spherical. Thus, the quadrupole moment is a measure of the nonsphericity or
shape of the nucleus. We can further elaborate on this by making a simple
model (Fig. 2.15) for nonspherical nuclei. We shall assume such nuclei are
Figure 2.15 Sketch of the a spheroidal nucleus.
2.10 ELECTRIC AND MAGNETIC MOMENTS 51
spheroids with a shape generated by rotating an ellipse about one of its axes.
We can define a semiminor axis of the ellipse, c, and a semimajor axis, a (a
is the axis about which the ellipse rotates). If a is the long axis, we have a
prolate spheroid (the shape of an American football). If a is the short axis
(pancake shape), we have an oblate spheroid. We can show that
Q¼2
5Ze(a2
Àc2
) (2:27)
Since we also know the square of the mean radius R of the spheroid is
given by
R2
¼1
2(a2
þc2
)¼(r0A1=3
)2
(2:28)
we can solve for a and c, the two axes of the spheroid. Thus, the quadrupole
moment gives us a direct measure of the shape of nuclei. Note further that Q
has the dimensions of charge x area. It is common to tabulate Q/e, which has
the dimension of area. The nuclear dimension of area is the barn, which is
equal to 10224
cm2
. Hence quadrupole moments are frequently given in
barns. A plot of the experimental values of the electric quadrupole moments
is shown in Figure 2.16. Note (Fig. 2.16) that the rare earth and actinide
0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
10 20 30 40
40 50 82 12628148
50 60 70
Z or N
80 90 100 110 120 130 140 150
+
+
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
++
+
+
+
+
+
+
+
+
+
+
+
+
++
++
+ ++++++
+
+
+++ +
+
++
++ ++
+
+
+
+++++++
++
++
+
Plotted against N
Plotted against Z
Figure 2.16 Experimental values of the electric quadrupole moment of nuclei. The lines are
drawn through the data to emphasize the trends from M. A. Preston, Physics of the Nucleus.
Copyright # 1962 by Addison-Wesley Publishing Company. Reprinted by permission of
Pearson Education, Inc.
52 NUCLEAR PROPERTIES
nuclei have prolate shapes (Q ¼ þ) while there are other nuclei with oblate
shapes (Q ¼ 2).
Example Problem For 177
Hf, Q ¼ þ3.0 e-barns. Calculate the ratio of the
semimajor to semiminor axes of this prolate nucleus.
Solution
Q=e ¼ 2
5 Z(a2
À c2
)
a2
À c2
¼
Q=e
2
5 Ã Z
¼
þ3:0 Â 10À24
2
5 (72)
¼ 1:042 Â 10À25
R2
¼ 1
2 (a2
þ c2
) ¼ (r0A1=3
)2
a2
þ c2
¼ 2(r0A1=3
)2
¼ 2(1:2 Â 10À13
à 1771=3
)2
¼ 9:079 Â 10À25
2a2
¼ 1:012 Â 10À24
a ¼ 7:11 Â 10À13
cm c ¼ 6:34 Â 10À13
cm
a=c ¼ 1:12
PROBLEMS
1. Define or describe the following terms or phenomena in your own words:
nuclear surface energy, parity, asymmetry energy, packing fraction, nuclear
magneton, Schmidt limits, mass defect, magnetic dipole moment,
2. The total nuclear binding energies of 27
Mg, 27
Al, and 27
Si are 244.2667,
246.8741, and 241.6741 MeV, respectively. Determine the values of the
Coulomb energy and asymmetry energy coefficients of the semiempirical
mass equation using these data.
3. The ground-state quadrupole moment of 152
Eu is þ3.16 Â 102
fm2
. Deduce the
ratio of semimajor to semiminor axes for 152
Eu.
4. Some nuclei can decay by either b2
or bþ
emission. Show that such nuclei must
have even A, odd N.
5. For 181
Ta, Q/e ¼ 4.20 barns. Calculate the ratio of the semimajor to semiminor
axes of this nucleus.
6. Calculate the electric quadrupole moment of a charge of magnitude Ze distributed
over a ring of radius R with an axis along the z axis.
7. Use the semiempirical mass equation to compute, for given A, the relation
between Z and N for a nucleus that has Sn ¼ 0 (the neutron “drip line”).
Compute N/Z for A ¼ 100.
8. Find the electric dipole moment and electric quadrupole moment of two positive
point charges þ q, one at z ¼ a/2 and the other at z ¼ 2a/2.
PROBLEMS 53
9. Show that the quadrupole moment Q of a uniformly charged spheroid about the
axis of symmetry is 2
5 Z(b2
À a2
), where a, b are the semiaxes, b being along the
axis of symmetry. Show that the quadrupole moment about an axis making an
angle b with the axis of symmetry is ½3
2 cos2
b À 1
2ŠQ.
10. The quadrupole moments of 176
Lu and 127
I are 7.0 and 20.6 barns, respectively.
Assume that 176
Lu and 127
I are ellipsoids of revolution obtained by deforming
(without volume change) a sphere of radius R ¼ 1.4 A1/3
fm. Calculate the ratio
a/b of the semimajor to semiminor axes.
11. Explain why we expect that there are no stable odd–odd nuclei. What are the
exceptions to this rule?
12. Use the semiempirical mass equation to derive an expression for the
energy released in a decay. For fixed Z, how should the energy released
depend on A?
13. Assume that av ¼ 15.835, as ¼ 18.33, aasym ¼ 23.20, and ac ¼ 0.714 in the
semiempirical mass equation. Show the binding energy per nucleon reaches a
maximum for Z  26 (iron). Assume Z ¼ N ¼ A/2. Neglect pairing.
14. The red giant stars, which are cooler than the sun, produce energy from
reactions such as
9
Be þ 1
H À! 6
Li þ 4
He þ energy
From the masses tabulated in the Appendices, calculate the energy release for
this reaction and the percentage of the initial mass of the reactants converted
to energy.
15. Consider the nuclei 15
C, 15
N, and 15
O. Which of these nuclei is stable? What
types of radioactive decay would the other two undergo? Calculate the
binding energy difference between 15
N and 15
O. Assuming this difference
comes from the Coulomb term in the semiempirical binding energy equation,
calculate the nuclear radius.
16. By computing the appropriate parabolas, predict the types of decay
and their energies for the isobars of A ¼ 180. Which isobar or isobars are
stable?
17. Assume that all the sun’s energy is produced by the reaction
41
H À! 4
He þ 2bþ
The sun yields 2 cal/min-cm2
at the surface of Earth. The distance of Earth
from the sun is 1.49 Â 106
km. How much helium does the sun produce per
year?
54 NUCLEAR PROPERTIES
18. Calculate the electric monopole, dipole, and quadrupole moments of the
following arrangements of charge:
−e
−e
−e
+e
+e
+e
−d
d
d
d
2d
19. Suppose that the density of nucleons r in a nucleus varies with radial distance r
from the center of the nucleus as shown below. What fraction of the nucleons lie
in the surface region in the nuclei 28
Si, 132
Sn, and 208
Pb if r0 ¼ 0.17 nucleons/
fm3
, c ¼ 1.2 A1/3
fm, and a ¼ 2.4 fm?
r
ρ0
ρ0
2
c
a
20. Use the semiempirical mass equation to calculate the percentage contribution to
the average binding energy per nucleon of the volume energy, the surface
energy, the Coulomb energy, and the asymmetry energy for A ¼ 60 and
A ¼ 240.
21. Explain why in the decay of 238
U to 206
Pb by successive a and b2
decays, one
sees one or two successive a decays followed by b2
decays, and so on. Why are
there no bþ
or EC decays in this chain?
22. (a) What regions of the periodic table are characterized by large permanent
prolate nuclear deformations?
(b) What nuclei in the periodic table have the highest binding energy per
nucleon?
PROBLEMS 55
REFERENCES
Resnick, R., D. Halliday, and K. S. Krane. Physics, 4th ed., Wiley, New York, 1992.
Meyerhof, W. E. Elements of Nuclear Physics, McGraw-Hill, New York, 1967.
Myers, W. D. and W. J. Swiatecki. Nucl. Phys. 81, 1 (1966).
BIBLIOGRAPHY
Evans, R. D. The Atomic Nucleus, McGraw-Hill, New York, 1955.
Hasse, R. W. and W. D. Myers. Geometrical Relationships of Macroscopic Nuclear Physics,
Springer, Berlin, 1988.
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1988.
Mackintosh, R., J. Al-Khalili, B. Jonson, and T. Pena. Nucleus: A Trip to the Heart of Matter,
Johns Hopkins University Press, Baltimore, 2001.
Wong, S. S. M. Introductory Nuclear Physics, 2nd ed., Wiley, New York, 1998.
56 NUCLEAR PROPERTIES
CHAPTER 3
RADIOACTIVE DECAY KINETICS
The number of nuclei in a radioactive sample that disintegrate during a given time
interval decreases exponentially with time. Because the nucleus is insulated by the
surrounding cloud of electrons, this rate is essentially independent of pressure, temperature,
the mass action law, or any other rate-limiting factors that commonly affect
chemical and physical changes.1
As a result, this decay rate serves as a very useful
means of identifying a given nuclide. Since radioactive decay represents the transformation
of an unstable radioactive nuclide into a more stable nuclide, which may
also be radioactive, it is an irreversible event for each nuclide.
The unstable nuclei in a radioactive sample do not all decay simultaneously.
Instead, the decay of a given nucleus is an entirely random event. Consequently,
studies of radioactive decay events require the use of statistical methods. With
these methods, one may observe a large number of radioactive nuclei and predict
with fair assurance that, after a given length of time, a definite fraction of them
will have disintegrated but not which ones or when.
1
In the case of electron capture and internal conversion, the chemical environment of the electrons
involved may affect the decay rate. For L-electron capture in 7
Be (t1/2 ¼ 53.3 d), the ratio of tBeF2
1=2 =tBe
1=2
is 1.00084. Similarly, a fully stripped radioactive ion cannot undergo either EC or IC decay, a feature
of interest in astrophysics.
57
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
3.1 BASIC DECAY EQUATIONS
Radioactive decay is what chemists refer to as a first-order reaction; that is, the rate
of radioactive decay is proportional to the number of each type of radioactive nuclei
present in a given sample. So, if we double the number of a given type of radioactive
nuclei in a sample, we double the number of particles emitted by the sample per unit
time.2
This relation may be expressed as follows:
Rate of
particle emission
;
rate of
disintegration of
radioactive nuclei
0
@
1
A /
number of
radioactive nuclei
present
0
@
1
A
Note that the foregoing statement is only a proportion. By introducing the decay
constant, it is possible to convert this expression into an equation, as follows:
Rate of
disintegration of
radioactive nuclei
0
@
1
A ¼
decay
constant
 
Â
number of
radioactive
nuclei present
0
@
1
A (3:1)
The decay constant l represents the average probability per nucleus of decay occurring
per unit time. Therefore, we are taking the probability of decay per nucleus, l,
and multiplying it by the number of nuclei present so as to get the rate of particle
emission. The units of rate are (disintegration of nuclei/time) making the units of
the decay constant (1/time), that is, probability/time of decay.
To convert the preceding word equations to mathematical statements using
symbols, let N represent the number of radioactive nuclei present at time t. Then,
using differential calculus, the preceding word equations may be written as
À
dN
dt
/ N
À
dN
dt
¼ lN
(3:2)
Note that N is constantly reducing in magnitude as a function of time. Rearrangement
of Equation (3.2) to separate the variables gives
dN
N
¼ Àl dt (3:3)
2
In order to make this statement completely correct, we should say that as we double the number of nuclei
present, we double the rate of particle emission. This rate is equal to the number of particles emitted per
unit time, provided that the time interval is small.
58 RADIOACTIVE DECAY KINETICS
If we say that at time t ¼ 0 we have N0 radioactive nuclei present, then integration of
Equation (3.3) gives the radioactive decay law
N ¼ N0eÀlt
(3:4)
This equation gives us the number of radioactive nuclei present at time t. However,
in many experiments, we want to know the counting rate that we will get in a
detector as a function of time. In other words, we want to know the activity of
our samples.
Still, it is easy to show that the counting rate in one’s radiation detector, C, is
equal to the rate of disintegration of the radioactive nuclei present in a sample, A,
multiplied by a constant related to the efficiency of the radiation measuring
system. Thus
C ¼ 1A ¼ 1 À
dN
dt
 
¼ 1lN (3:5)
where 1 is the efficiency. Substituting into Equation (3.4), we get
C ¼ C0eÀlt
(3:6)
where C is the counting rate at some time t due to a radioactive sample that gave counting
rate C0 at time t ¼ 0. Equations (3.4) and (3.6) are the basic equations governing
the number of nuclei present in a radioactive sample and the number of counts
observed in one’s detector as a function of time. Equation (3.6) is shown graphically
as Figure 3.1. As seen in Figure 3.1, this exponential curve flattens out and asymptotically
approaches zero. If the same plot is made on a semilogarithmic scale (Fig. 3.2),
the decay curve is a straight line, with a slope equal to the value of 2(l/2.303).
Figure 3.1 Linear decay curve.
3.1 BASIC DECAY EQUATIONS 59
The half-life (t1/2) is another representation of the decay constant. The half-life of
a radionuclide is the time required for its activity to decrease by exactly one-half.
Thus, after one half-life, 50% of the initial activity remains. After two half-lives,
only 25% of the initial activity remains. After three half-lives, only 12.5% is yet
present and so forth. Figure 3.3 shows this relation graphically.
The half-life for a given nuclide can be derived from Equation (3.6) when the
value of the decay constant is known. In accordance with the definition of the
term half-life, when A/A0 ¼ 1/2, then t ¼ t1/2. Substituting these values into
Equation (3.6) gives
A
A0
¼
1
2
¼ eÀlt1=2
ð3:7Þ
Hence
t1=2 ¼
ln 2
l
¼
0:693
l
(3:8)
Figure 3.2 Semilogarithmic decay curve.
Figure 3.3 Relation between half-life and radioactivity.
60 RADIOACTIVE DECAY KINETICS
Note that the value of the expression for t1/2 has the units of 1/l or dimensions of
(time).
The half-lives for different nuclides range from less than 1026
s to 1010
y. The
half-life has been measured for all the commonly used radionuclides. When an
unknown radioactive nuclide is encountered, a determination of its half-life is
normally one of the first steps in its identification. This determination can be done
by preparing a semilog plot of a series of activity observations made over a
period of time. A short-lived nuclide may be observed as it decays through a
complete half-life and the time interval observed directly (Fig. 3.4).
Example Problem Given the data plotted below for the decay of a single radionuclide,
determine the decay constant and the half-life of the nuclide.
Solution
0 50 100 150 200
Time (min)
6
5
4
3
2
1
0
InR(Rincounts/s)
The data is plotted above. The slope (2l) is given as
Àl ¼ À(6:06 À 0)=(220 min À 0)
l ¼ 0:0275 minÀ1
t1=2 ¼ ln 2=l ¼ 0:693=0:0275 ¼ 25:2 min
What nuclide might this be?
3.1 BASIC DECAY EQUATIONS 61
It is difficult to measure the half-life of a very long-lived radionuclide. Here variation
in disintegration rate may not be noticeable within a reasonable length of time.
In this case, the decay constant must be calculated from the absolute decay rate
according to Equation (3.2). The absolute number of atoms of the radioisotope
present (N) in a given sample can be calculated according to
N ¼
6:02 Â 1023
(Avogadro’s number)
atomic weight radionuclide
 mass of the radionuclide (3:9)
The total mass of the radioisotope in the given sample can be determined once
the isotopic composition of the sample is ascertained by such means as mass
spectrometry. When the decay constant is known, the half-life can then be readily
calculated. A table of the half-lives of a number of the known nuclei can be
found in Appendix B.
Although the half-life of a given radionuclide is a defined value, the actual
moment of disintegration for a particular atom can be anywhere from the very beginning
of the nuclide’s life to infinity. The average or mean life of a population of
nuclei can, however, be calculated. The mean life t is naturally related to the
decay constant and is, in fact, simply the reciprocal of the decay constant:
t ¼
1
l
(3:10)
Figure 3.4 Direct graphic determination of half-life.
62 RADIOACTIVE DECAY KINETICS
or the mean life can be expressed in terms of the half-life:
t ¼ 1:443t1=2 (3:11)
One can understand the preceding relationship by recalling that the decay constant,
l, was defined as the average probability of decay per unit time, so the 1/l is the
average time between decays. The concept of average life allows us to calculate
of the total number of particles emitted during a defined decay period. This
number is essential in determining total radiation dose delivered by a radioisotope
sample, as in medical research and therapy. During the time equal to one mean
life, t, the activity falls to 1/e of its original value. For a sample of N0 nuclei
with lifetimes ti, we can write for the mean life t
t ¼
PN0
1 ti
N0
¼ À
1
N0
ðt¼1
t¼0
t dN ¼
1
N0
ð1
0
tlN dt ¼ l
ð1
0
teÀlt
dt
¼
Àlt þ 1
l
eÀlt
!t
0
¼
1
l
(3:12)
The average or mean life is also of fundamental physical significance because
it is the time to be substituted in the mathematical statement of the Heisenberg
uncertainty principle, that is,
DE † Dt ! hÀ
In this expression relating the uncertainty in energy of a system, DE, to its lifetime
Dt, t ; Dt.
DE ¼
hÀ
t
¼
0:658 Â 10À15
eV
t (s)
The quantity DE is called the width, G.
The natural unit of radioactivity is disintegrations/time, such as disintegration
per second (dps) or disintegrations per minute (dpm), and so on. The SI (International
System) unit of radioactivity is the Becquerel (Bq) where
1 Becquerel (Bq) ; 1 disintegration=s
Counting rates in a detection system are usually given in counts per second (cps),
counts per minute (cpm), and so on, and differ from the disintegration rates by a
factor representing the detector efficiency, 1. Thus
(dpm) 1 ¼ (cpm)
3.1 BASIC DECAY EQUATIONS 63
An older unit of radioactivity that still finds some use is the curie (Ci). It is
defined as
1 curie (Ci) ¼ 3:7 Â 1010
Bq ¼ 3:7 Â 1010
dis=s
The curie is a huge unit of radioactivity and is approximately equal to the activity
of one gram of radium. The inventories of radioactivity in a nuclear reactor upon
shutdown are typically 109
Ci, whereas radiation sources used in tracer experiments
have activities of mCi and the environmental levels of radioactivity are nCi or pCi.
Note also that because radionuclides, in general, have different half-lives, the
number of nuclei per curie will differ from one species to another. For
example, let us calculate how many nuclei are in 1 MBq (27 mCi) of tritium
(t1/2 ¼ 12.33 y). We know that
N ¼
( À dN=dt)
l
¼
106
=s
l
But
l ¼
0:693
t1=2
¼
(0:693)
(12:33 y)(p  107 s=y)
¼ 1:789 Â 10À9
sÀ1
Thus
N ¼
A
l
¼
(106
=s)
(1:789)(10À9=s)
¼ (5:59)(1014
) nuclei
The same calculation carried out for 14
C (t1/2 ¼ 5730 y) would give 2.60 Â 1017
nuclei/MBq. It is also interesting to calculate the mass associated with 1 MBq of
tritium. We have
M ¼
(N)(atomic weight)
(Avogadro’s number)
¼
(5:59)(1014
)(3)
(6:02)(1023)
¼ 2:78 Â 10À9
g
In other words, 1 MBq of tritium contains about 3 ng of tritium. Thus, an important
feature of radionuclides becomes apparent—we routinely work with extremely
small quantities of material. Pure samples of radioisotopes are called “carrier free.”
Unless a radionuclide is in a carrier-free state, it is mixed homogeneously with
the stable nuclides of the same element. It is, therefore, desirable to have a simple
expression to show the relative abundances of the radioisotope and the stable isotopes.
This specification is readily accomplished by using the concept of specific
activity, which refers to the amount of radioactivity per given mass or other
similar units of the total sample. The SI unit of specific activity is Bq/kg. Specific
activity can also be expressed in terms of the disintegration rate (Bq or dpm), or
64 RADIOACTIVE DECAY KINETICS
counting rate (counts/min, cpm, or counts/s, cps), or curies (or mCi, mCi) of the
specific radionuclide per unit mass of the sample.
3.2 MIXTURE OF TWO INDEPENDENTLY DECAYING
RADIONUCLIDES
Where two or more radioisotopes with different half-lives are present in a sample
and one does not or cannot distinguish the particles emitted by each isotope, a composite
decay rate will be observed. The decay curve, in this situation, drawn on a
semilogarithmic plot, will not be a straight line. The decay curves of each of the isotopes
present usually can be resolved by graphical means if their half-lives are sufficiently
different and if not more than three radioactive components are present. In
the graphic example shown in Figure 3.5, line C represents the total observed
activity. Only the activity of the longer-lived component A is observed after the
shorter-lived component B has become exhausted through decay. Extrapolation of
this long-time portion of the curve back to zero time gives the decay curve for component
A and the activity of component A at t ¼ 0. The curve for component B is
obtained by subtracting out, point by point, the activity values of component A
from the total activity curve. If the half-lives of the two components in such
samples are not sufficiently different to allow graphic resolution, a differential detection
method may be applicable. If the radiation characteristics of the nuclides in the
mixture are suitably distinct, that is, emission of different particles or g rays, it may
be possible to measure the activity of one component without interference from the
radiation emitted by the other component. A case in point would be where one
Figure 3.5 Graphic resolution of a composite decay curve.
3.2 MIXTURE OF TWO INDEPENDENTLY DECAYING RADIONUCLIDES 65
nuclide was a pure b emitter, while the other emitted both b and g rays. In the case
where the half-lives of the components are known but are not sufficiently different
to allow graphical resolution of the decay curve, computer techniques that utilize
least-squares fitting to resolve such a case are also available.
Example Problem Given the following decay data, determine the half-lives
and initial activities of the radionuclides present:
t (h) A (cpm)
0.1 270
0.5 210
1.0 170
1.5 130
2.0 110
2.5 90
3 80
4 65
5 55
7 44
10 34
15 22
100
10
1
0 2 4 6 8 10 12 14
Time(h)
Activity(cpm)
Solution From the graph, we see
t1=2 (b) ¼ 8:0 h A0(b) ¼ 80 cpm
t1=2(c) ¼ 0:8 h A0(c) ¼ 190 cpm
66 RADIOACTIVE DECAY KINETICS
3.3 RADIOACTIVE DECAY EQUILIBRIUM
When a radionuclide decays, it does not disappear but is transformed into a new
nuclear species of higher binding energy and often differing Z, A, J, p, and so on.
The equations of radioactive decay discussed so far have focused on the decrease
of the parent radionuclides but have ignored the formation (and possible decay)
of daughter, granddaughter, and so forth, species. It is the formation and decay of
these “children” that is the focus of this section.
Let us begin by considering the case when a radionuclide 1 decays with decay
constant l1, forming a daughter nucleus 2, which in turn decays with decay constant
l2. Schematically, we have
1 ! 2 !
We can write terms for the production and depletion of 2, that is,
Rate of change of 2 ¼ rate of production 2 rate of decay of
nuclei present of 2 2
at time t
dN2
dt
¼ l1N1 À l2N2 (3:13)
where N1 and N2 are the numbers of 1 and 2 present at time t.
Rearranging and collecting similar terms
dN2 þ l2N2 dt ¼ l1N1dt (3:14)
Remembering that
N1 ¼ N0
1 eÀl1t
(3:15)
we have
dN2 þ l2N2 dt ¼ l1N0
1 eÀl1t
dt (3:16)
This is a first-order linear differential equation and can be solved using the method of
integrating factors that we show below. Multiplying both sides by el2t
, we have
el2t
dN2 þ l2N2el2t
dt ¼ l1N0
1 e(l2Àl1)t
dt (3:17)
The left-hand side is now a perfect differential
d(N2el2t
) ¼ l1N0
1 e(l2Àl1)t
dt (3:18)
3.3 RADIOACTIVE DECAY EQUILIBRIUM 67
Integrating from t ¼ 0 to t ¼ t, we have
N2el2t

t
0
¼
l1N0
1 e(l2Àl1)t
l2 À l1




t
0
(3:19)
N2el2t
À N0
2 ¼
l1
l2 À l1
N0
1 (e(l2
Àl1)t
À 1) (3:20)
Multiplying by eÀl2t
and rearranging gives
N2(t) ¼
l1
l2 À l1
N0
1 eÀl1t
À eÀl2t
À Á
þ N0
2 eÀl2t
(3:21)
where N0
2 is the number of species 2 present at t ¼ 0. The first term in Equation
(3.21) represents the growth of the daughter due to the decay of the parent,
whereas the second term represents the decay of any daughter nuclei that were
present initially. Remembering that A2 ¼ l2N2, we can write an expression for
the activity of 2 as
A2 ¼
l1l2
l2 À l1
N0
1 eÀl1t
À eÀl2t
À Á
þ A0
2eÀl2t
(3:22)
These two equations, (3.21) and (3.22), are the general expressions for the number of
daughter nuclei and the daughter activity as a function of time, respectively.
The general behavior of the activity of parent and daughter species, as predicted
by Equation (3.22), is shown in Figure 3.6. As one expects qualitatively for the case
Activity
Time t
t Dmax
dD
dt
=0
r
D
Figure 3.6 Illustration of the conditions at ideal equilibrium, as would apply to the case of a
fairly long-lived parent nuclide.
68 RADIOACTIVE DECAY KINETICS
with N0
2 ¼ 0, the initial activity of the daughter begins at zero, rises to a maximum,
and if one waits long enough, eventually decreases.
Thus, there must be a time when the daughter activity is the maximum. We can
calculate this time by noting the condition for a maximum in the activity of 2 is
dN2
dt
¼ 0 (3:23)
Taking the derivative of Equation (3.21) and simplifying,
l1eÀl1t
¼ l2eÀl2t
(3:24)
Solving for t,
tmax ¼
ln (l2=l1)
l2 À l1
(3:25)
All of this development may seem like something that would be best handled by a
computer program or just represents a chance to practice one’s skill with differential
equations. But that is not true. It is important to understand the mathematical foundation
of this development to gain insight into practical situations. Let us consider
some cases that illustrate this point.
Consider the special case where l1 ¼ l2. Plugging into Equation (3.21) or (3.22)
or a computer program based upon them leads to a division by zero. Does nature
therefore forbid l1 from equaling l2 in a chain of decays? Nonsense! One simply
understands that one must redo the derivation [Equations (3.13) through (3.21)] of
Equations (3.21) and (3.22) for this special case (see Problems).
Let us now consider a number of other special cases of Equations (3.21) and
(3.22) that are of practical importance. Suppose the daughter nucleus is stable
(l2 ¼ 0). Then we have
dN2
dt
¼ l1N1 (3:26)
dN2 ¼ l1N1dt ¼ l1N0
1 eÀl1t
dt (3:27)
N2 ¼
l1N0
1
Àl1
eÀl1t
À Á
t
0
¼ N0
1 1 À eÀl1t
À Á
(3:28)
These relations are shown in Figure 3.7. They represent the typical decay of many
radionuclides prepared by neutron capture reactions, the type of reaction that
commonly occurs in a nuclear reactor.
In Figure 3.8, we show the activity relationships for parent and daughter [as predicted
by Equation (3.22)] for various choices of the relative values of the half-lives
of the parent and daughter nuclides. In the first of these cases, we have t1/2
(parent) , t1/2 (daughter), that is, the parent is shorter lived than the daughter.
3.3 RADIOACTIVE DECAY EQUILIBRIUM 69
This is called the no equilibrium case because the daughter buildup (due to the decay
of the parent) is faster than its loss due to decay. Essentially, all of the parent
nuclides are converted to daughter nuclides, and the subsequent activity is due to
the decay of the daughters only. Thus, the name “no equilibrium” is used. Practical
examples of this decay type are 131
Te ! 131
I, 210
Bi ! 210
Po, and 92
Sr ! 92
Y. This
situation typically occurs when one is very far from stability and the nuclei decay by
b decay toward stability.
A second special case of Equations (3.21) and (3.22) is called transient equilibrium
(Figs. 3.8c and 3.9). In this case, the parent is significantly (10Â) longer
lived than the daughter and thus controls the decay chain. Thus
l2 . l1 (3:29)
In Equation (3.21), as t ! 1,
eÀl2t
( eÀl1t
N0
2 eÀl2t
! 0
(3:30)
and we have
N2 %
l1
l2 À l1
N0
1 eÀl1t
(3:31)
Substituting
N1 ¼ N0
1 eÀl1t
(3:32)
0
16
14
12
10
8
6
4
2
0
Activityoramount/arbitraryunits
Time/r 5.25 y
60
Co
60
Ni
1 2 3 4 5 6
Figure 3.7 Decay of 60
Co (period 5.25 y) and the growth of 60
Ni (stable).
70 RADIOACTIVE DECAY KINETICS
we have
N1
N2
¼
l2 À l1
l1
(3:33)
At long times, the ratio of daughter to parent activity becomes constant, and
both species disappear with the effective half-life of the parent. The classic
examples of this decay equilibrium are the decay of 140
Ba (t1/2 ¼ 12.8 d) to 140
La
(t1/2 ¼ 40 h) or the equilibrium between 222
Rn (t1/2 ¼ 3.8 d) and its short-lived
decay products.
Activity
Time t
t Dmax t Dmax
t Dmaxt Dmax
r
r
r
r
D
D
D
D
tP < tD
Activity
Time t
(a) (b)
(c) (d )
tP > tD
tP >> tD
Activity
Time t
tP >>t >> tD
Activity
Time t
Figure 3.8 Activity curves for various parent–daughter relationships: (a) short-lived parent
(tP , tD); (b) long-lived parent (tP . tD); (c) very long-lived parent (tP ) tD); (d) almost
stable, or constantly replenished, parent (tP ) t ) tD).
3.3 RADIOACTIVE DECAY EQUILIBRIUM 71
A third special case of Equations (3.21) and (3.22) is called secular equilibrium
(Figs. 3.8d, and 3.9). In this case, the parent is very much longer lived (104
Â) than
the daughter or the parent is constantly being replenished through some other
process. During the time of observation, there is no significant change in the
number of parent nuclei present, although several half-lives of the daughter may
0
0.1
0.2
0.3
0.4
0.6
0.8
1.0
2
3
4
6
8
10
20
3.2 6.4 9.6 12.8
Activity/104/Cmin-1
Time/d
(a)
(b)
0
0.1
0.2
0.3
0.4
0.6
0.8
Activity/104Cmin-1
1.0
2
3
4
6
8
10
20
64 128 192 256 320 384 448 512 576 640
Time/h
90Y (pure)
90Y (in pure 90Sr)
90Sr (pure)
90Sr (initially pure) + 90Y
16.0 19.2 22.4 25.6
140Ba
140Ba (pure)
140La (pure)
140La in pure
140Ba (initially pure) + 140La
Figure 3.9 Typical cases of (a) transient and (b) secular equilibrium.
72 RADIOACTIVE DECAY KINETICS
occur. In the previous case of transient equilibrium, we had
N1
N2
¼
l2 À l1
l1
ð3:34Þ
Since we now also have
l1 ( l2 (3:35)
we can simplify even more to give
N1
N2
¼
l2
l1
(3:36)
l1N1 ¼ l2N2 (3:37)
A1 ¼ A2
In short, the activity of the parent and daughter are the same, and the total activity of
the sample remains effectively constant during the period of observation.
The naturally occurring heavy element decay chains (see below) where
238
U ! 206
Pb, 235
U ! 207
Pb, and 232
Th ! 208
Pb and the extinct heavy element
decay series 237
Np ! 209
Bi are examples of secular equilibrium because of the
long half-lives of the parents. Perhaps the most important cases of secular equilibrium
are the production of radionuclides by a nuclear reaction in an accelerator,
a reactor, a star, or the upper atmosphere. In this case, we have
Nuclear reaction À! (2) À! (3:38)
which produces the radionuclide 2 with rate R. If the reaction is simply the decay of
a long-lived nuclide, then R ¼ l1N1
0
and N2
0
¼ 0. Substitution into Equation (3.21)
gives the expression
N2 ¼
l1
l2 À l1
N0
1 eÀl1t
À eÀl2t
À Á
(3:39)
If the reaction is slower than the decay or
l1 ( l2 (3:40)
It is most appropriate to say (since l1% 0)
N2 %
l1
l2
N0
1 1 À eÀl2t
À Á
(3:41)
or in terms of the activities
A2 ¼ l2N2 ¼ R 1 À eÀl2t
À Á
(3:42)
Equation (3.42) is known as the activation equation and is shown in Figure 3.10.
3.3 RADIOACTIVE DECAY EQUILIBRIUM 73
Initially, the growth of the product radionuclide activity is nearly linear (due to
the behavior of 1 À eÀl2t
À Á
for small values of lt), but eventually the product
activity becomes “saturated” or constant, decaying as fast as it is produced. At an
irradiation time of one half-life, half the maximum activity is formed; after two
half-lives, three fourths of the maximum activity is formed, and so on. This situation
gives rise to the rough rule that irradiations that extend for periods that are greater
than twice t1/2 of the desired radionuclide are usually not worthwhile.
Equation (3.21) may be generalized to a chain of decaying nuclei of arbitrary
length by using the Bateman equations (Bateman, 1910). If we assume that at
t ¼ 0, none of the daughter nuclei are present, N0
2 ¼ N0
3 ¼ , . . . , N0
n ¼ 0, we get
(1) À! (2) À! (3) , . . . , (n) À!
Nn ¼ C1eÀl1t
þ C2eÀl2t
þ C3eÀl3t
þ , . . . , CneÀlnt
where
C1 ¼
l1l2, . . . , lnÀ1
(l2 À l1)(l3 À l1), . . . , (ln À l1)
N0
1
C2 ¼
l1l2, . . . , lnÀ1
(l1 À l2)(l3 À l2), . . . , (ln À l2)
N0
1
Cn ¼
l1l2, . . . , lnÀ1
(l1 À ln)(l2 À ln), . . . , (lnÀ1 À ln)
N0
1
(3:43)
0
0
0.25
Initialslope=Y
0.50
0.75
1.00
T½ 2T½ 3T½ 4T½ 5T½ 6T½
Accumulation time t
Activity(normalized)
tD =1.44 T½
lD D = YtD (1-e-l2t)
Figure 3.10 Growth of the activity of a primary reaction product induced by a constant
bombardment.
74 RADIOACTIVE DECAY KINETICS
These equations describe the activities produced in new fuel in a nuclear reactor. No
fission or activation products are present when the fuel is loaded, and they grow in as
the reactions take place.
Example Problem Consider the decay of a 1 mCi sample of pure 222
Rn (t1/2 ¼
3.82 d). Use the Bateman equations to estimate the activity of its daughters (218
Po,
214
Pb, 214
Bi, and 214
Po) after a decay time of 4 h.
The decay sequence is
222
Rn ÀÀÀ!
a 218
Po ÀÀÀ!
a 214
Pb ÀÀÀ!
bÀ
214
Bi ÀÀÀ!
bÀ
214
Po ÀÀÀ!
a
t1/2 3.82 d 3.1 m 26.8 m 19.9 m 164 ms
activity A ÀÀÀ! B ÀÀÀ! C ÀÀÀ! D ÀÀÀ! E ÀÀÀ!
l (1024
s) 0.021 37.3 4.31 5.81 4.3 Â 107
Solution
A ¼ A0eÀlAt
¼ 1mCi exp À
ln 2 † 4
24 † 3:82
 
A ¼ 0:97 mCi
B ¼ lB C1eÀl1t
þ C2eÀl2t
À Á
C1 ¼
lAN0
A
lB À lA
¼
A0
lB À lA
C2 ¼
lAN0
A
lA À lB
¼
A0
lA À lB
B ¼ lB
A0eÀlAt
lB À lA
þ
A0eÀlBt
lA À lB
 
B ¼ 37:3
0:97
37:3 À 0:021
þ
exp
À ln 2 † 4
3:1=60
0:021 À 37:3
0
B
B
@
1
C
C
A
B ¼ 0:97 mCi
(Actually B=A ¼ 1:00056)
The reader should verify that for C, D, and E, the only significant term is the
term multiplying eÀlAt
as it was for B. Thus, for D/A, we have
D
A
¼
lB
lB À lA
†
lC
lC À lA
†
lD
lD À lA
¼ 1:0091
The reader should, as an exercise, compute the quantities of C and E present.
3.4 BRANCHING DECAY
Some nuclides decay by more than one mode. Some nuclei may decay by either bþ
decay or electron capture; others by a decay or spontaneous fission; still others by
3.4 BRANCHING DECAY 75
g-ray emission or internal conversion, and so on. In these cases, we can characterize
each competing mode of decay by a separate decay constant li for each type of
decay where the total decay constant l is given by the sum
l ¼ l1 þ l2 þ Á Á Á ¼
XN
i¼1
li (3:44)
Corresponding to each partial decay constant li, there is a partial half-life ti
1=2 where
ti
1=2 ¼
0:693
li
(3:45)
and the total half-life, t1/2, is the sum of the reciprocals
1
t1=2
¼
1
t1
1=2
þ
1
t2
1=2
þ Á Á Á ¼
XN
i¼1
1
ti
1=2
(3:46)
The fraction of decays proceeding by the ith mode is given by the obvious
expression
fi ¼
li
Sli
¼
li
l
(3:47)
By analogy, the energy uncertainty associated with a given state, DE, through the
Heisenberg uncertainty principle can be obtained from the lifetime contributed by
each decay mode. If we use the definition DE ; G, the level width, then we can
express G in terms of the partial widths for each decay mode Gi such that
G ¼ G1 þ G2 þ G3 þ Á Á Á ¼
XN
i¼1
Gi (3:48)
where
Gi ¼
1
ti
(3:49)
where ti is the partial mean life associated with each decay mode. This approach is
especially useful in treating the decay of states formed in nuclear reactions in which
a variety of competing processes such as a emission, p emission, n emission, and so
on, may occur as the nucleus de-excites. In such cases, we can express the total
width as
G ¼ Ga þ Gp þ Gn (3:50)
76 RADIOACTIVE DECAY KINETICS
Example Problem Consider the nucleus 64
Cu (t1/2 ¼ 12.700 h). 64
Cu is known
to decay by electron capture (61%) and b2
decay (39%). What are the partial
half-lives for EC and b2
decay? What is the partial width for EC decay?
Solution
l ¼ ln 2=12:700 h ¼ 5:46 Â 10À2
hÀ1
l ¼ lEC þ lb ¼ lEC þ (39=61) lEC
lEC ¼ 3:329 Â 10À2
hÀ1
tEC
1=2 ¼ ( ln 2)=lEC ¼ 20:8 h
lb ¼ (39=61)lEC ¼ 2:128 Â 10À2
hÀ1
tb
1=2 ¼ ( ln 2)=lb ¼ 32:6 h
tEC
¼ tEC
1=2= ln 2 ¼ 30:0 h ¼ 108131 s
GEC ¼ hÀ
=tEC
¼ 6:582 Â 10À22
Me V Á s=108131 s ¼ 6:1 Â 10À27
MeV
All naturally occurring radioactive nuclei have extremely small partial widths. Did
you notice that 64
Cu can decay into 64
Zn and 64
Ni? This is unusual but can occur for
certain odd–odd nuclei (see Chapter 2).
3.5 NATURAL RADIOACTIVITY
There are approximately 70 naturally occurring radionuclides on Earth. Most of
them are heavy-element radioactivities present in the natural decay chains, but
there are several important light-element activities, such as 3
H, 14
C, 40
K, and so
forth. These radioactive species are ubiquitous, occurring in plants, animals, the
air we breathe, the water we drink, the soil, and so forth. For example, in the
70-kg “reference man,” one finds 4400 Bq of 40
K and 3600 Bq of 14
C, that is,
about 8000 dis/s due to these two radionuclides alone. In a typical U.S. diet, one
ingests 1 pCi/day of 238
U, 226
Ra, and 210
Po. The air we breathe contains 0.15
pCi/L of 222
Rn, the water we drink contains .10 pCi/L of 3
H while the Earth’s
crust contains 10 and 4 ppm of the radio elements Th and U, respectively.
One should not forget that the interior heat budget of planet Earth is dominated
by the contributions from the radioactive decay of uranium, thorium, and potassium.
The naturally occurring radionuclides can be classified as: (a) primordial, that is,
nuclides that have survived since the time the elements were formed, (b) cosmogenic,
that is, shorter-lived nuclides formed continuously by the interaction of
cosmic rays with matter, and (c) anthropogenic, that is, a wide variety of nuclides
introduced into the environment by the activities of humans, such as nuclear
weapons tests, the operation (or misoperation) of nuclear power plants, and the
like. The primordial radionuclides have half-lives greater than 109
y or are the
decay products of these nuclei. This class includes 40
K (t1/2 ¼ 1.277 Â 109
y),
87
Rb (t1/2 ¼ 47.5 Â 109
y), 238
U (t1/2 ¼ 4.467 Â 109
y), 235
U (t1/2 ¼ 0.704 Â 109
y),
3.5 NATURAL RADIOACTIVITY 77
and 232
Th (t1/2 ¼ 14.05 Â 109
y) as its most important members. (Additional
members of this group are 115
In, 123
Te, 138
La, 144
Nd, 147
Sm, 148
Sm, 176
Lu, 174
Hf,
187
Re, and 190
Pt.)
40
K is a b2
-emitting nuclide that is the predominant radioactive component of
normal foods and human tissue. Due to the 1460-keV g ray that accompanies the
b2
decay, it is also an important source of background radiation detected by
g-ray spectrometers. The natural concentration in the body contributes about
17 mrem/y to the whole body dose. The specific activity of 40
K is approximately
855 pCi/g potassium. Despite the high specific activity of 87
Rb of 2400 pCi/g,
the low abundance of rubidium in nature makes its contribution to the overall
radioactivity of the environment small.
There are three naturally occurring decay series. They are the uranium
(A ¼ 4n þ 2) series, in which 238
U decays through 14 intermediate nuclei to form
the stable nucleus 206
Pb, the actinium or 235
U (A ¼ 4n þ 3) series in which 235
U
decays through 11 intermediate nuclei to form stable 207
Pb, and the thorium
(A ¼ 4n) series in which 232
Th decays through a series of 10 intermediates to
stable 208
Pb (Fig. 3.11).
92
U0
91
Pa
90Th
89
Ac
88
Ra
87
Fr
86Rn
85
At
84Po
83
Bi
82
Pb
238
UI
UXI
AcU
Io
234
Th
226
Ra
222
Rn
218
At
218
Po
214
Bi
210
Bi 215
Bi
208
Pb
211
Pb
215
Po
210
Po
219
At
215
At
219
Rn
223
Fr
223
Ra
231
Th
227
Ac
227
Th
211
Bi
207
Pb
212
Pb
216
Po
212
Po
208
Pb
212
Bi
211
Po214
Po
214
Pb
210
Pb
RaA
RaC RaE
AcB
AcC
AcA
An
AcK
AcX
UY
RdAc RdTh
ThX
Tn
MsTh1
RaF AcC'
ThB
ThA
ThC
ThD
ThC'
AcDstable
stable stable
RaC'
RaDRaB
230
Th
234
UII
235U
231
Pa
4.5 × 10
9
y
24.1 d 7.5 × 104
y
7.04 × 10
6
y
232
Th
228
Ra
224
Ra
220
Rn
228
Th
1.41 × 10
10
y
3.28 × 10
4
y
1.6 × 103
y
3.82 d
1.5 s
164 µs
3.10 m
26.8 m
19.9 m 5.01 d
138.4 d
7.6 m
0.9 m
36.1 m
2.19 m
0.10 ms
1.78 ms
3.96 s
21.8 m
11.4 d
25.5 h
18.7 d 1.91 y
5.75 y
55.6 s
3.83 d
21.8 y
0.516 s 0.15 s
60.6 m
0.3 µs
10.64 h22.2 y
2.45 × 105
y
Figure 3.11 The decay series of U238
, U235
, and Th232
. Not shown are several intermediate
daughter products of little significance in geochemical applications. For the sake of
completeness, old notations still referred to frequently in present-day texts, e.g., RaA for
Po218
, Io for Th230
, are given in the scheme.
78 RADIOACTIVE DECAY KINETICS
Because the half-lives of the parent nuclei are so long relative to the other
members of each series, all members of each decay series are in secular equilibrium,
that is, the activities of each member of the chain are equal at equilibrium if the
sample has not been chemically fractionated. Thus, the activity associated with
238
U in secular equilibrium with its daughters is 14Â the activity of the 238
U. The
notation 4n þ 2, 4n, 4n þ 3 refers to the fact that the mass number of each
member of a given chain is such that it can be represented by 4n, 4n þ 2, 4n þ 3
where n is an integer. (There is an additional decay series, the 4n þ 1 series, that
is extinct because its longest lived member, 237
Np, has a half-life of only
2.1 Â 106
y, a time that is very short compared to the time of element formation.)
The uranium series contains two radionuclides of special interest, 226
Ra
(t1/2 ¼ 1600 y) and its daughter, 38 d 222
Rn. 226
Ra (and its daughters) are responsible
for a major fraction of the radiation dose received from internal radioactivity.
Radium is present in rocks and soils, and as a consequence in water, food, and
human tissue. The high specific activity and gaseous decay products of radium
also make it difficult to handle in the laboratory.
226
Ra decays by a emission to 222
Rn. This latter nuclide is the principal culprit in
the radiation exposures from indoor radon. Although radon is an inert gas and is not
trapped in the body, the short-lived decay products are retained in the lungs when
inhaled if the 222
Rn decays while it is in the lungs. Indoor radon contributes about
2 mSv/y (200 mrem/y) to the average radiation exposure in the United States,
that is, about two-thirds of the dose from natural sources. Under normal circumstances,
radon and its daughters attach to dust particles and are in their equilibrium
amounts. These dust particles can also deposit in the lungs. It has been estimated that
in the United States, 5000–10,000 cases of lung cancer (6–12% of all cases) are due
to radon exposure.
The second class of naturally occurring radionuclides is the cosmogenic nuclei,
produced by the interactions of primary and secondary cosmic radiation with
nuclei in the stratosphere. The most important of these nuclei are 3
H (tritium),
14
C, and 7
Be. Less importantly, 10
Be, 22
Na, 32
P, 33
P, 35
S, and 39
Cl are also produced.
These nuclei move into the troposphere through normal exchange processes and are
brought by Earth’s surface by rainwater. Equilibrium is established between the
production rate in the primary cosmic ray interaction and the partition of the
radionuclides among the various terrestrial compartments (atmosphere, surface
waters, biosphere, etc.) leading to an approximately constant specific activity of
each nuclide in a particular compartment. When an organism dies after being in
equilibrium with the biosphere, the specific activity of the nuclide in that sample
will decrease since it is no longer in equilibrium. This behavior allows these nuclides
to act as tracers for terrestrial processes and for dating.
14
C (t1/2 ¼ 5730 y) is formed continuously in the upper atmosphere by cosmic
rays that produce neutrons giving the reaction
n (slow) þ 14
N À! 14
C þ p
or, in a shorthand notation, 14
N(n, p)14
C. 14
C is a soft b2
emitter (Emax  158 keV).
3.5 NATURAL RADIOACTIVITY 79
This radiocarbon (14
C) reacts with oxygen and eventually exchanges with the stable
carbon (mostly 12
C) in living things. If the cosmic ray flux is constant, and the
terrestrial processes affecting 14
C incorporation into living things are constant,
and there are no significant changes in the stable carbon content of the atmosphere,
then a constant level of 14
C in all living things is found (corresponding to 1 atom of
14
C for every 1012
atoms of 12
C or about 227 Bq/kg C). When an organism dies, it
ceases to exchange its carbon atoms with the pool of radiocarbon, and its radiocarbon
content decreases in accord with Equation (3.6). Measurement of the specific
activity of an old object allows one to calculate the age of the object (see below).
14
C reaches the Earth’s surface at the rate of 2.3 atoms/cm2
/s after production
by cosmic ray interaction in the atmosphere, corresponding to a total production of
1.4 Â 1015
Bq/y. 14
C is also formed by the 14
N(n, p) reaction by atmospheric tests
of nuclear weapons. About 2.2 Â 1017
Bq were made in the atmospheric test “spike”
of the 1950s and 1960s that has been primarily transferred to the oceans and the biosphere.
This means that 14
C is the most significant fallout nuclide from the point of
view of population dose. Nuclear power plants also release 14
C as part of their
normal operation contributing 0.1 Â 1015
Bq/y.
Tritium (3
H) is produced naturally through atmospheric cosmic ray interactions
via the reaction
n (fast) þ 14
N À! 12
C þ 3
H
Tritium is also produced in ternary fission and by neutron-induced reactions with 6
Li
and 10
B. Tritium is a very low energy b2
emitter with a half-life of 12.33 y. The global
inventory of naturally produced tritium is 9.6 Â 1017
Bq. Tritium is readily incorporated
in water and is removed from the atmosphere by rain or snow. Its residence time
in the stratosphere is 2–3 y; after reaching the troposphere it is removed in 1–2
months. The “natural” concentration of 3
H in streams and freshwater is 10 pCi/L.
The nuclear weapons tests of the late 1950s and early 1960s also injected a huge
spike of tritium into the atmosphere along with 14
C. The tritium levels in the troposphere
increased by a factor of 100 at this time. Estimates of 2.4 Â 1020
Bq for this
spike have been made. Assuming that there will not be more atmosphere testing of
nuclear weapons, the tritium from fallout should decrease with a half-life of 12.3 y.
At present the fallout tritium in surface waters is approximately equal to that generated
from nuclear power plant operation (as a ternary fission product or from n reactions
with 10
B). (Nuclear plant operation generates 1016
Bq/y.) As a result of all of
these developments, the current tritium content of surface waters is 10Â the
“natural” level.
The third principal component of environmental radioactivity is that due to the
activities of humans, the anthropogenic radionuclides. This group of nuclides
includes the previously discussed cases of 3
H and 14
C along with the fission products
and the transuranium elements. The primary sources of these nuclides are nuclear
weapons tests and nuclear power plant accidents. These events and the gross
nuclide releases associated with them are shown in Table 3.1. Except for 14
C and
80 RADIOACTIVE DECAY KINETICS
3
H (T), the anthropogenic contributions from nuclear weapons testing or use (which
is the most significant source of man-made environmental exposure) are negligible
compared to other sources of natural radioactivity. (The principal component of
these large releases of radioactivity was shorter-lived fission products such as
131
I, which have decayed, leaving 137
Cs, 90
Sr, and the Pu isotopes as the nuclides
of most concern. For further descriptions of these events and their environmental
consequences, the reader is referred to the material in the Bibliography.)
3.6 RADIONUCLIDE DATING
An important application of the basic radioactive decay law is that of radionuclide
dating. From Equation (3.6), we have
N ¼ N0eÀlt
(3:51)
We can solve this equation for t:
t ¼
ln N0=Nð Þ
l
(3:52)
where N0 and N are the number of radionuclides present at times t ¼ 0 and t ¼ t and
l is the decay constant. The quantity t is the age of the object, and it can be determined
from a knowledge of the nuclear decay constant (t1/2) and the number of
radioactive nuclei present in the object now, N, and initially, N0. Clearly, N can
be determined by counting the sample (A ¼ lN), but the trick is to determine N0.
One obvious approach is to recognize that for a decay of parent P to daughter D,
the total number of nuclei is constant:
D(t) þ P(t) ¼ P(t0) ; P0
(3:53)
TABLE 3.1 Events Leading to Large Injections of Radionuclides into Atmosphere
(From Choppin, Rydberg and Liljenzin)
Source Country Time Radioactivity (Bq) Important Nuclides
Hiroshima &
Nagasaki
Japan 1945 4 Â 1016
Fission Products
Actinides
Atmospheric
weapons tests
USA
USSR
1963 2 Â 1020
Fission Products
Actinides
Windscale UK 1957 1 Â 1015 131
I
Chelyabinsk
(Kysthym)
USSR 1957 8 Â 1016
Fission Products
90
Sr, 137
Cs
Harrisburg USA 1979 1 Â 1012
Noble gases, 131
I
Chernobyl USSR 1986 2 Â 1018 137
Cs
Source: From Choppin et al. (1995).
3.6 RADIONUCLIDE DATING 81
and
P(t) ¼ P0
eÀlt
(3:54)
so that
t ¼
1
l
ln 1 þ
D(t)
P(t)
 
(3:55)
Thus, by measuring the current ratio of daughter to parent atoms (D(t)=P(t)) one can
deduce the age of the sample. (This assumes, of course, that there are no daughter
atoms present at t ¼ 0, that they are all due to the parent decay, and that none
have been lost.)
Example Problem In a rock, one finds a nuclidic ratio of 206
Pb to 238
U of 0.60.
What is the age of the rock?
Solution
t ¼
1
l
ln 1 þ
D(t)
P(t)
 
¼
1
ln 2=ð4:5 Â 109
yÞ
ln (1 þ 0:60)
t ¼ 3:1 Â 109
y
If we want to relax this latter condition that no daughter atoms were present at
t ¼ 0 [D (t ¼ 0) ¼ 0], then we need an additional term:
D(t) þ P(t) ¼ D0
þ P0
(3:56)
and we need to make an estimate of D0
. Suppose there is another isotope of the
daughter element that is stable and is not formed in the decay of anything else.
We can assume that
Ds(t) ¼ D0
s ; Ds (3:57)
where Ds is the number of such stable atoms. Then, dividing by Ds
D(t)
Ds
þ
P(t)
Ds
¼
D0
Ds
þ
P0
Ds
(3:58)
Substituting P0
¼ Pelt
and rearranging,
D(t)
Ds
¼
D0
Ds
þ
P(t)
Ds
elt
þ 1
À Á
(3:59)
Thus, if we plot a set of measurements of D(t)=Ds vs: P(t)=Ds, we will get a straight
line with the intercept D0
=Ds and a slope of (elt
2 1). Figure 3.12 shows such a plot
82 RADIOACTIVE DECAY KINETICS
of a set of meteorite samples using the 87
Rb ! 87
Sr decay as a chronometer
(t1/2 ¼ 4.75 Â 1010
y).
Other geochronometers that can be used in a similar manner involve the decay of
1.277 Â 109
y 40
K to 40
Ar (K/Ar dating) or the decay of 235
U or 238
U to their 207
Pb
and 206
Pb daughters. Each chronometer poses special problems with regard to the
loss of daughter species over geologic time by diffusion, melting, or chemical processes.
The “normalizing” stable nuclide in the case of the uranium decay series is
204
Pb, and in the case of K/Ar dating it is 36
Ar. (See Problems at end of chapter for a
further discussion of these methods.)
The dating methods discussed up to now have been based on the use of long-lived
radionuclides that are present in nature. Dating is also possible using “extinct radionuclides,”
that is, nuclei whose half-lives are so short that if they existed at the time
of formation of our solar system, they would have decayed away essentially completely
by now. The nuclides 129
I (t1/2 ¼ 1.57 Â 107
y) and 244
Pu (t1/2 ¼ 8.08 Â 107
y)
are noteworthy examples of this type of nuclide.
The decay of extinct radionuclides is measured by measuring anomalies in the
isotopic abundance of their stable daughters. For example, 129
I decays to 129
Xe
and its decay will lead to an anomalously high concentration of 129
Xe in the mass
spectrum of Xe isotopes found in a rock system. What is dated is the “formation
0.1
Initial Ratio = 0.7003 ± 0.004 (2s)
Both Furnace
UNSHOCKED HYPERSTENE CHONDRITES
OBSERVED FALLS
Rb87/Sr86
Sr87/Sr86
l(Rb87) = 1.39 ¥ 10-11 yr-1
7900
7800
7700
7600
7500
7400
7300
7200
7100
7000
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Age = 4.54 ¥ 109 yrs ± 0.12 ¥ 109 yrs (2s)
Knyahinya
Kyushu
Bruderheim
Homesleod
Modoc
Figure 3.12 87
Rb–87
Sr evolution diagram for six hypersthene chondrite meteorites. The
data can be interpreted as showing that 4.54 billion years ago all of these rocks had the
same 87
Sr/86
Sr ratio of 0.7003 (from Wetherill, 1975).
3.6 RADIONUCLIDE DATING 83
age” of the rock, that is, the time interval between the isolation of the solar system
material from galactic nucleosynthesis and the time at which the rock cooled enough
to retain its Xe. Formally, this formation age, D, may be calculated as from the
isotopic ratios in a fashion similar to that of Equation (3.59):
D ¼
1
l
ln
129
I=127
I
À Á
0
129XeÃ
=127Ið Þ
(3:60)
where 129
XeÃ
is the excess Xe attributed to the decay of 129
I, 127
I is the concentration
of stable, nonradiogenic 127
I, l is the decay constant for 129
I, and (129
I/127
I)0 is the
ratio of the abundance of the iodine isotopes at the time of isolation from galactic
nucleosynthesis. This latter ratio is derived from theories of nucleosynthesis and
is 1024
.
The decay of extinct 244
Pu is deduced from excess abundances of the nuclides
136
Xe, 134
Xe, and 132
Xe, produced by the spontaneous fission of 244
Pu. Uncertainties
arise because there is no stable isotope of Pu that can be used in the way that 127
I
is used in Equation (3.60) and the use of other heavy nuclides 238
U or 252
Th as
“substitutes” leads to difficulties due to differences in primordial production and
chemistry.
By far the most important dating method involves the decay of 14
C
(t1/2 ¼ 5730 y). As indicated previously, 14
C is formed continuously by the
cosmic ray induced 14
N(n, p) 14
C reaction in the upper atmosphere. This radiocarbon
(14
C) exchanges with stable carbon (12
C) in living things leading to the existence of
a constant level of 14
C in living systems as indicated schematically in Figure 3.13.
When an organism dies, it will cease to exchange its carbon atoms with the pool of
radiocarbon and its radiocarbon will decay. Measurement of the specific activity
dpm À 14
C=g 12
C
À Á
of an old object allows the determination of the age. When
organic matter has decayed for 10 or more half-lives of 14
C, it is no longer possible
to directly measure the 14
C radioactivity of an object. In these cases, one can use
accelerator mass spectrometry (AMS) to count the atoms of 14
C directly. An accelerator,
such as a cyclotron or tandem Van de Graaff, is used as a mass spectrometer
to separate the 14
C atoms from the more prevalent 12
C or 13
C. Another difficulty is
the separation of 14
C from the ubiquitous 14
N isobar and various molecular ions;
thus, accelerators are used to provide energetic ions that can be identified with standard
nuclear techniques. Using this technique, it has been possible to determine ages
as long as 100,000 y.
Example Problem Consider a sample of organic material that contains 1 mg of
C. Suppose it has a 14
C/12
C atom ratio of 1.2 Â 10214
.
a. How many 14
C atoms are present?
b. What would be the expected 14
C disintegration rate for this sample?
c. What is the age of this sample?
84 RADIOACTIVE DECAY KINETICS
Figure 3.13 Artist’s conception of how 14
C is generated and incorporated into living things
(reprinted by permission from Taylor, 2000).
3.6 RADIONUCLIDE DATING 85
Solution
a.
N ¼
10À3
g
12g=g-atom
 6:02  1023
atoms=g-atom ¼ 5:02 Â 1019
atoms
# 14
C atoms % (1:2 Â 10À14
)(5:02 Â 1019
) ¼ 6:02 Â 105
b.
A ¼ lN ¼
ln 2
(5730 y)(3:15 Â 107
s=y)
(6:02 Â 105
)
¼ 2:3 Â 10À6
Bq ¼ 0:2 dis=d
c. Note that a typical AMS facility would collect several thousand of these
14
C atoms in one hour. The assumed constant specific activity of 14
C in
nature in the prenuclear era is 227 Bq/kgC or 227 Â 1026
Bq/mg. From
Equation (3.52), the age would be
Age ¼
ln 227 Â 10À6
=2:3 Â 10À6
À Á
ln 2=5730 y
¼ 38,000 y
0 10,000 20,000 30,000
0
10,000
U/Thage(yrBP)
20,000
30,000
14C age (yr BP)
Figure 3.14 Correction to radiocarbon ages for coral samples based upon age estimates
made for the same samples using 234
U/230
Th ages.
86 RADIOACTIVE DECAY KINETICS
As noted earlier, the fundamental assumption in radiocarbon dating is that the
specific activity of 14
C in nature dpm 14
C=g12
C
À Á
is and has remained constant.
This assumes the cosmic ray flux that generates the 14
C has been constant, and
there are no sources of 14
C or 12
C that would change its equilibrium specific activity.
Neither of these assumptions is strictly true, and corrections must be used to obtain
correct ages from radiocarbon dating. In Figure 3.14, we show the typical magnitude
of these corrections. The primary cosmic ray flux is moderated by fluctuations in
solar activity or the Earth’s magnetic field over time. Since the Industrial
Revolution, the global carbon cycle is out of balance due to fossil fuel burning
(of “old” nonactive fossil carbon). This has caused a 1–3% dilution of the prehistoric
14
C/12
C ratio. As noted earlier, atmospheric testing of nuclear weapons
contributed a spike to the global 14
C inventory that perturbed the 14
C/12
C ratio
by a factor of 2 in the opposite direction. Continued operation of nuclear power
plants also contributes an amount that is 10% of the “natural” 14
C production
rate. Similarly, a dating scheme for water containing objects, such as wines,
based upon the equilibrium production of tritium (3
H) and its decay has been
similarly perturbed by an injection of thousands of times the natural levels due to
atmospheric testing.
PROBLEMS
1. Calculate the expected activity in Bq and in Ci for the following radionuclides
(see Appendix for nuclear data):
a. 1.0 g 239
Pu
b. 1.0 g 14
C
c. 1.0 g 137
Cs
d. spontaneous fission activity for 1.0 g 252
Cf
e. 1 g 226
Ra
2. Consider the decay sequence 239
U ! 239
Np ! 239
Pu ! . If you start with
1 mCi of initially pure 239
U, what is the activity of 239
Pu after (a) 1 day,
(b) 1 month, and (c) 1 year?
3. Calculate the time necessary to reduce the activities of the following nuclei to
1% of their initial values:
a: 131
I b: 3
H c: 137
Cs d: 14
C e: 239
Pu
4. What is the mass (g) of the following activities:
a. 1 mCi 241
Am
b. 1 pCi 239
Pu
c. 5000 Bq 252
Cf
5. What is the partial half-life for decay by spontaneous fission for 252
Cf?
PROBLEMS 87
6. If 222
Rn is initially purified from its daughters, how long does it take for them to
grow back to 50% of their values at secular equilibrium?
7. What are the partial half-lives of 22
Na for decay by (a) EC and (b) bþ
emission?
8. Calculate the relative mass ratios of 238
U, 226
Ra, and 222
Rn in an old
uranium ore.
9. Consider the decay of 140
Ba to 140
La. At what time does the 140
La activity reach
a maximum?
10. Consider a reactor in which the production rate of 239
U via the 238
U (n, g) 239
U
reaction is 105
atoms/s. Calculate the activity of 239
Pu after an irradiation of
(a) 1 day, (b) 1 month, and (c) 1 year.
11. What is the probability of a 222
Rn atom decaying in our lungs? The atmospheric
concentration of 222
Rn may be assumed to be 1 pCi/L. In an average breath we
inhale 0.5 L of air and exhale it 3.5 s later.
12. Consider a radionuclide (decay constant l) with activity A Bq at time t1.
Calculate the number of nuclei that decay between times t1 and t2.
13. Consider the following decay scheme (Evans, 1955):
A
B
C
λ1
λ2
λ3
λ4
a. Derive expressions for the activity of B and C as a function of time if at
t ¼ 0, A ¼ A, B ¼ C ¼ D ¼ 0.
b. What happens when the cross over transition l3 ¼ 0?
14. If one “milks” a sample of 99
Mo to remove the daughter nuclide 99
Tc, how long
does it take before the 99
Mo “cow” has an equilibrium amount of 99
Tc present?
15. Consider the case where A ! B ! C and where lA ¼ lB. Derive an expression
for the activity of B as a function of time. Calculate the time tm when the activity
of B reaches a maximum. Show that tm % (tAtB)1/2
where tA, tB are the mean
lives for A and B.
16. A uranium mineral was found to contain the Pb isotopes 204
Pb, 206
Pb, and 207
Pb
in the ratio of 1 : 1000 : 400. Estimate the age of this mineral (Choppin et al.,
1995).
88 RADIOACTIVE DECAY KINETICS
17. What was the rate of production of 24
Na in a 30-m reactor irradiation of 23
Na if
the activity of 24
Na was found to be 1.0 mCi 3 h after the end of irradiation?
18. Calculate the heat generated per kilogram of natural uranium by the 238
U and
the 235
U in secular equilibrium with their decay products. Assume all emitted
radiation is absorbed.
19. Given the following data:
Sample Rb/Sr Weight Ratio 87
Sr/86
Sr Atom Ratio
1 1.06 0.7597
2 3.51 0.8248
3 6.61 0.9085
4 9.33 0.9796
5 10.67 1.0200
Determine the age of the rock and the initial 87
Sr/86
Sr ratio.
REFERENCES
Choppin, G., J. Rydberg, and J. O. Liljenzin. Radiochemistry and Nuclear Chemistry,
Butterworth, Oxford, 1995.
Evans, R. D. The Atomic Nucleus, McGraw-Hill, New York, 1955.
Taylor, R. E. Fifty Years of Radiocarbon Dating, Am. Scient. 88, 60 (2000).
Wetherill, G. W. Radiometric Chronolgy of the Early Solar System, Ann. Rev. Nucl. Sci. 25,
283 (1975).
BIBLIOGRAPHY
Arnikar, H. J. Essentials of Nuclear Chemistry, 2nd ed., Wiley, New York, 1982.
Ehmann, W. D. and D. E. Vance. Radiochemistry and Nuclear Methods of Analysis, Wiley,
New York, 1991.
Friedlander, G., J. W. Kennedy, E. S. Macias, and J. M. Miller. Nuclear and Radiochemistry,
Wiley, New York, 1981.
Heyde, K. Basic Ideas and Concepts in Nuclear Physics, IOP, Bristol, 1994.
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1988.
Monographs and Specialized Articles
Eisenbud, M. Environmental Radioactivity, 3rd ed., Academic, Orlando, 1987.
Lal, D. and H. Suess. The Radioactivity of the Atmosphere and Hydrosphere, Ann. Rev. Nucl.
Sci. 18, 407 (1968).
UNSCEAR 93, Sources and Effects of Ionizing Radiation, UN, New York, 1993.
BIBLIOGRAPHY 89
CHAPTER 4
RADIOTRACERS
4.1 INTRODUCTION
The basic idea behind the use of radiotracers is that all the isotopes of a given
element will behave the same chemically. Thus, the atoms of radioactive 24
Na
will behave in the same way as those of stable 23
Na in a chemical system except
for the effects due to the small difference in mass. So what? The point is that it is
easier to follow the radioactive 24
Na atoms than it is to detect the nonradioactive
23
Na atoms. Using conventional chemical analysis techniques, one can typically
detect nanogram to microgram amounts of a substance. For 23
Na, these amounts
would correspond to needing 1013
–1016
atoms to get an analytical response. For
a radioactive atom, such as 24
Na, one detected disintegration corresponds to the
decay of a single atom. Routine radioanalytical techniques will allow detection of
quantities that are 105
times smaller than those needed for chemical analysis. This
high sensitivity is of great importance in any number of studies. For example, the
specific activity of pure tritium is 30 Ci/mmol. Thus, one can tolerate a dilution
factor of 1012
and still detect tritium-labeled compounds. It is thus possible to detect
the occurrence of metabolic substances that are normally present at such low
concentrations to defy the most sensitive chemical methods of identification.
One can measure the activity of enzymes by following the rate of disappearance
of a labeled substrate or the rate of appearance of a labeled product. Biological
compounds, such as vitamins or hormones, which are normally in such low
91
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
concentrations as to make detection difficult, can be measured using radiotracer
techniques such as radioimmunoassay (RIA).
The unique advantages of radiotracer experiments include their high sensitivity,
their simplicity, and small expense (compared to competing technologies such as
mass spectrometry). In a well-designed experiment, the presence of radiotracers
does not affect the system under study and any analysis is nondestructive. Interference
from other species that may be present is not important (as compared to
conventional methods of analysis where interferences may thwart the analysis).
Perhaps the most outstanding advantage of the use of radioisotopes is the opportunity
offered to trace dynamic mechanisms. Such biological phenomena as ion
transport across cell membranes, turnover, intermediary metabolism, or translocation
in plants could, before the advent of radiotracer methods, be approached only
indirectly.
The use of the isotope effect to study rate-determining steps in a sequence of
chemical reactions represents an additional advantage of radiotracer methodology.
The term isotope effect (to be discussed more fully later) refers to the influence
on a reaction rate of the difference in the masses of isotopes. This effect may
create significant problems in the use of radioisotopes as tracers but can, nevertheless,
be used to advantage in a limited number of cases in order to understand the
kinetics of certain chemical reactions.
4.2 DESIGN OF A RADIOTRACER EXPERIMENT
4.2.1 Basic Design Criteria
The use of radiotracers is dependent on certain basic assumptions being fulfilled.
The first assumption, mentioned above, is that the radioactive isotopes of a given
element behave identically as the stable isotopes of the same element. Actually,
this assumption is not exactly true. The difference in masses between radiotracer
nuclei and stable nuclei can cause a shift in the reaction rate or equilibria (the
isotope effect). It is true, however, that in most cases the isotope effect does not significantly
affect the utility of the radioisotope method. Since the degree of chemical
bond stability due to vibrational motion is directly related to the square root of the
masses of the isotopes involved, it is apparent that an isotope effect will be of significance
only for elements of low atomic weight (at wt 25).
The isotopes of hydrogen present the extreme case. Thus, 1
H, 2
H(D), and 3
H(T)
could scarcely be expected to act as the same substances chemically since the relative
mass differences are so great. Hence, tritium cannot be employed uncritically as
a tracer for hydrogen in regard to reaction rates, although, of course, its use in determining
hydrogen location in an organism is not precluded.
In the case of radioactive 14
C and stable 12
C, there is a 15% difference in mass
that may affect the outcome of some studies. In general, one should note that the
so-called isotopic effect, in reality, should be considered from the standpoint of
the two basic types: the intramolecular and the intermolecular isotopic effects.
92 RADIOTRACERS
In regard to the first type, the decarboxylation (loss of CO2 from the carbonyl group)
of malonic acid (HOO14
C2212
CH22212
COOH) is a good example. Here the pyrolytic
decarboxylation of malonic acid can occur at either end of the molecule and gives
rise to CO2, and acetic acid. Consequently, the reaction is concerned with the
relative bond strength of 12
C2214
C and 12
C2212
C. Since the former bond is relatively
more stable (owing to the greater mass of the 14
C), we would expect that the CO2
resulting from the reaction would be comparatively enriched in 12
C and that the
acetic acid would be comparatively enriched in 14
C from the 22COOH. Such is
the case. Inasmuch as the competitive reactions occur within the same molecule,
the isotopic effect will be observed even if the reaction goes to completion.
In the case of the intermolecular isotope effect, the decarboxylation of benzoic
acid–7–14
C can be cited as an example. Here, one is usually dealing with the decarboxylation
of two compounds: C6H52214
COOH and C6H52212
COOH. Again,
since the 12
C2214
C bond is relatively more stable, we would expect that during
the initial phase of the decarboxylation reaction there would be an enrichment of
12
CO2 compared to the specific activity of the labeled carbon in the benzoic acid.
As the reaction approaches completion, however, the reactant (benzoic acid) will
be relatively enriched with the 14
C-labeled variety, and, consequently, the decarboxylation
product (CO2) will have a specific activity higher than the labeled carbon of
the original starting material. When the reaction is driven to completion, the overall
specific activity of CO2 will naturally be the same as the labeled carbon atom in the
benzoic acid; that is, no isotope effect can be observed. This is an important concept
for it indicates that the significance of the isotope effect can be minimized or ignored
in radiotracer studies if the intramolecular type of reaction is not involved.
The second basic assumption is that the radioactivity does not change the chemical
and physical properties of the experimental system. It is essential that the radiation
dose from the tracer does not elicit a response from the experimental system that
would distort the experimental results. The amount of activity employed should be
restricted to the minimum necessary to permit reasonable counting rates in the
samples to be assayed. The possibility of excessive concentration of the tracer compound
in certain biological tissues and the degree of radiosensitivity of these tissues
must also be carefully considered, especially when a- or b2
-emitting tracers are
used. Fortunately, the excellent sensitivity of most radioactivity assay methods minimizes
the need to employ tracer doses of such a magnitude that any detectable radiation
damage occurs. The possibilities of interference due to physiological response
to radiation are further minimized because most studies are short term and thus completed
before any latent radiation effects appear. In physical tracer studies, the radiation
damage produced by the decaying nuclei must not so destroy the crystal
structure and similar features as to modify the experimental results.
In addition, one must remember that as the result of radioactive decay, the daughter
atom is usually a different chemical element than the mother atom. One must be
sure that the presence of these “foreign species” and any related equilibria does not
affect the observations. (For example, if both the mother and daughter atoms are
b2
emitters, then one might see a increase in the count rate due to the inability to
distinguish the b2
particles from the tracer and its daughter.)
4.2 DESIGN OF A RADIOTRACER EXPERIMENT 93
A third basic assumption, for biological studies, is that there is no deviation from
the normal physiological state. If, to produce the required tracer activity, the chemical
level of the compound given to an organism greatly exceeds the normal physiological
or chemical level, the experimental results are open to question. The specific
activity of the tracer compound must be high enough for the total chemical level to
be administered to be within the normal range. As an example, 36
Cl might be quite
useful for biological investigations, but the maximum specific activity obtainable in
the inorganic form is about 100 mCi=g Cl. This situation is in contrast to the specific
activities of 14
C (as Ba14
CO3) of up to 2.2 Ci/g of carbon.
A fourth basic assumption is that the chemical and physical form of the radionuclide-labeled
compound is the same as the unlabeled variety. Herein we deal
with certain subtle effects related to the low concentration of the tracer species.
What are these effects? One involves the question of adsorption on surfaces, such
as the walls of the container for the system. For example, it is easy to show that
to cover the surface of a 1-L flask may require 10À7
mole. For a 1-L solution containing
1mCi of a radioactive species with a half-life of 30 days, one quickly calculates
the number of moles present is 10À13
, and, thus, one could think that all the
radioactive atoms might be adsorbed on the container walls (if no other atoms were
present). This adsorption can be prevented by adding (or already having present)
enough molecules of the nonradioactive compound to fill all the adsorption sites,
that is, a carrier. Similarly, the presence of a significantly larger concentration of
nonradioactive species is necessary to prevent the tracer from co-precipitating or
being adsorbed on the surface of any precipitates that form. (Because the number
of tracer atoms is frequently too small to satisfy the Ksp for any precipitate, the
tracer can only be precipitated in the presence of a large number of nonradioactive
species.)
Most importantly, the radionuclide and the stable nuclide must undergo isotopic
exchange. In practice, this means that the tracer and the stable atom must be in the
same redox state. By heating or using redox cycles, the experimenter must assure
this to be true. Anomalous experimental results have frequently been traceable to
the chemical form of the administered radiotracer. Since reactor production of radionuclides
often results in side reactions (see Chapter 10), various oxidation states may
be present when the sample is produced. In one case involving phosphate–32
P
uptake in plants, the unexpected experimental results were explained by the fact
that a large percentage of the tracer dose was actually in the form of phosphite–32
P.
The radiochemical purity of a compound cannot be assumed. The presence of
other radioactive species in low chemical concentration but high specific activity
is frequently encountered. This situation is particularly true in the labeling of compounds
with 3
H by means of the Wilzbach direct-exposure method (discussed
below). Thus, for example, direct hydrogenation of a double bond with 3
H during
the Wilzbach operation may give rise to a small amount of impurity (saturated
form of the compound in question) having a specific activity many times higher
than the 3
H-labeled compound derived from the recoil-labeling operation.
The problem of radiochemical purity with respect to the chemical state of aged
tritium or 14
C-labeled compounds is still more acute. Because of the short range
94 RADIOTRACERS
of the low-energy b particles associated with these two isotopes, the sizable radiation
dose delivered to such compounds by their own radiation leads to selfdecomposition
(radiolysis) and hence a variable concentration and a number of
labeled products.
The fifth basic assumption is that only the labeled atoms are traced. Never
assume that the appearance of the radioactive label in a given sample indicates
the presence of the administered compound. It is the labeled atoms that are being
followed, not the intact compound. Not only may metabolic reactions involve the
cleavage of the labeled atom from the original compound, but exchange reactions
may also occur, thus removing labile atoms from the labeled compound. Such
chemical exchanges particularly plague many experiments with tritium-labeled
compounds. The extent of chemical exchange is strongly dependent on the molecular
species involved, the position of the label in the molecule, and the environmental
factors (such as the pH of a biological fluid).
4.2.2 Practical Considerations
The feasibility of radiotracer experiments is usually dependent on certain practical
matters. These factors include:
Availability of the Radiotracer A primary factor is whether a radioisotope of
the element to be traced is available with the proper characteristics (half-life, particle
energy, etc.). For example, although radioisotopes of oxygen and nitrogen would be
highly desirable in many investigations, the longest-lived radionuclides available of
these elements have half-lives of 2 and 10 min, respectively. Clearly, such short
half-lives severely limit the use of such isotopes for many tracer experiments. On
the other hand, for some elements, a choice of usable radioisotopes may be available
such as 22
Na or 24
Na and 57
Co or 60
Co. Of equal importance is the available specific
activity of a given radionuclide. There are radionuclides, such as 36
Cl, that cannot
readily be made with desirable specific activity. Ideally, the radiotracer should
have a half-life that is several times the duration of the experiment to lessen or
avoid corrections for decay, but short enough to not cause long-term contamination
or disposal issues. If possible, the emitted radiation should be relatively easy to
detect. A second factor is whether the tagged compound desired is commercially
available or can be easily synthesized. The number of labeled compounds available
is large indeed, and most radiochemical suppliers will attempt custom syntheses of
unstocked compounds. In some cases, however, it is not economically feasible or
even possible to introduce a given radioactive atom into the molecular structure
under consideration. Furthermore, the available specific activity may be too low
for the proposed experimental use of the tagged compound.
Calculation of the Amount of Tracer Needed for the Experiment Radiotracer
experiments should, in general, involve easily detected quantities of radioactivity.
There is little excuse for doing a tracer experiment where the result is
uncertain due to a lack of observed counts or one that requires the ultimate in
4.2 DESIGN OF A RADIOTRACER EXPERIMENT 95
low-level counting systems. As an example of how such calculations are done, let us
consider the laboratory exercise to determine the solubility product constant of silver
iodide.
To begin with, we must have some idea of the basic chemistry/science that we
are attempting to measure. Let us assume we do not know the Ksp of AgI but do
know the Ksp for the analogous compounds, AgCl and AgBr. Thus, we have
AgCl ! Agþ
þ ClÀ
AgBr ! Agþ
þ BrÀ
with Ksp ¼ 1:8 Â 10À10
and 5:4 Â 10À13
, respectively. From this we will guesstimate
the Ksp of AgI as 10215
. (The actual value is 8:5 Â 10À17
, but that sort of
error in experimental design must be tolerated.) From this, we can calculate the
solubility of silver iodide, S, as
S ¼ (Ksp)1=2
¼ (10À15
)1=2
¼ 3 Â 10À8
M
What we will do in the experiment is to take a solution containing labeled I2
and
precipitate the AgI from the solution. We will measure the activity of the AgI
precipitate, suspend it in a known volume of water, and measure the activity of the
water. From this measurement, we can calculate the solubility of AgI and thus Ksp.
What iodine radiotracer should we use? Considering that we will have to assay
the activity of a liquid under a suitable counting geometry, we need a tracer that
emits energetic photons (to minimize absorption corrections). This consideration
as well as the others outlined above, causes us to choose 131
I (t1=2 ¼ 8 d) as the
tracer. (Solutions of 131
I-labeled sodium iodide are available commercially.)
Let us assume we precipitate a convenient amount of AgI for counting and handling,
say 20 mg. The fraction iodine in this precipitate is 54% % 10:8 mg. Upon
equilibrating this precipitate with water (2 mL), one will have (0:002)(3 Â 10À8
) ¼
6 Â 10À11
mole of iodine in solution or 7:6 Â 10À9
g.
Assume we count this water solution using a NaI well detector (see Chapter 18)
that has an efficiency of 20% for the 0.365-MeV photons from 131
I. Assuming one
wants to collect 103
counts in a 10-min count (and 81% of the 131
I decays result in a
0.365-MeV photon), one estimates a radioactivity in the solution as
D ¼ (counts=time)(branching ratio)(detection efficiency)
¼ (103
=10)(1=0:81)(1=0:2) % 600 dpm 10 Bq
The fraction of the iodine in the precipitate that dissolved is (7:6 Â 10À9
=
10:8 Â 10À3
) ¼ 7 Â 10À7
. Therefore the activity of the precipitate must be
400 mCi  15 MBq. Since the initial precipitation of AgI is quantitative, this is
the nominal amount of tracer needed in the solution. However, it is common to
96 RADIOTRACERS
build in a “safety factor” of 5–10 in the amount of tracer used (to account for the
misestimates such as we made in the Ksp of AgI).
The exact amount of 131
I in the original precipitate must be known, but the
activity of this precipitate is so high as to preclude its direct measurement in
the well detector. Therefore, what one does is to prepare a standard dilution of
the original I2
solution that can be measured in the well detector.
Evaluation of Hazard The first item to be considered is the possibility of harm
to the experimenter or to co-workers. In the great majority of radiotracer experiments,
the hazard from direct external radiation does not pose a serious problem.
However, there are situations where such is not the case, for example, where high
levels (millicuries) of g-ray emitters are utilized. For instance, 10 mCi of 24
Na
will deliver a dose of about 204 milliroentgens per hour (mR/h) (at a 1-ft distance).
One should also be quite cautious about the radiation dose delivered to the hands and
fingers while handling radioactive materials. Another item of concern in the use of
a- or b2
-emitting tracers is the possibility of ingestion of the labeled compounds,
particularly those known to have a long turnover time in the human body. This
problem is made acute where the sample is in the form of an aerosol or a dry
powder at some stage of the experiment.
Radiation damage to a biological system under study may occur at two levels: the
physiological and the histological. In general, higher radiation doses are required to
elicit the latter type of damage. Whenever it is suspected that radiation damage is
influencing the physiological response of the organism, it is advisable to repeat
the experiment with lower levels of radioactivity, while maintaining the same
total chemical level of the administered compound. Biological effects of radiation
from radiotracer doses have been reported to occur at the following dose levels:
0:045 mCi 131
I=g of body weight in mice, 0:8 mCi 32
P=g of body weight in mice,
47 mCi 24
Na=g body weight in mice and rats, 0:5 mCi 89
Sr=g body weight in mice
and rats, 0:05 mCi 32
P=mL of rearing solution for mosquito larvae, and
2 mCi 32
P=L of nutrient solution for barley plants.
Attention must also be given to the disposal of radioactive wastes resulting
from the experiment, such as excreta, carcasses, or large volumes of solutions.
The possible method of disposal will depend on the specific radioisotope present,
its concentration and activity, and the nature of the waste.
4.3 PREPARATION OF RADIOTRACERS AND THEIR COMPOUNDS
There are several hundred radionuclides that have been used as radiotracers. A
partial list of the properties of these nuclides and their production methods are
shown in Table 4.1. The three common production mechanisms for the primary
radionuclides are (n,g) or (n,p) or (n,a) reactions in a nuclear reactor (R),
charged-particle-induced reactions usually involving the use of a cyclotron (C),
and fission product nuclei (F), typically obtained by chemical separation from
irradiated uranium. The neutron-rich nuclei are generally made using reactors or
4.3 PREPARATION OF RADIOTRACERS AND THEIR COMPOUNDS 97
as fission products, while the proton-rich nuclei are produced in cyclotrons. [Not
shown in Table 4.1 are the short-lived positron emitters, 11
C, 13
N, 15
O, and 19
F
commonly used in positron emission tomography (PET), that are produced in
cyclotrons.]
In certain experiments, the primary radionuclides may be used directly, but
usually the investigator wants to secure a specific labeled compound for use in radiotracer
experiments. Before considering the details of the production of these labeled
compounds, let us discuss the nomenclature and rules used in referring to them.
TABLE 4.1 Commonly Used Tracers
Nuclide
Method of
Productiona
Half-Life
Tracer Radiations/
Energy (MeV)
3
H(T) R 12.33 y b2
0.018
14
C R 5730 y b2
0.156
22
Na C 2.60 y bþ
, g 1.274
24
Na R 15.0 h g 1.369
32
P R 14.3 d b2
1.71
33
P R 25.3 d b2
0.249
35
S R 87.4 d b2
0.167
36
Cl R 3.0 Â 105
y b2
0.71
45
Ca R 162.6 d b2
0.257
47
Ca R 4.54 d b2
1.99; g 1.297
51
Cr R 27.7 d g 0.320
54
Mn R 312 d g 0.835
55
Fe R 2.73 y EC
59
Fe R 44.5 d g 1.292, 1.099
57
Co C 271.7 d g 0.122
60
Co R 5.27 y g 1.173, 1.332
63
Ni R 100.1 y b2
0.067
65
Zn C, R 244.3 d g 1.116
75
Se R 119.8 d g 0.265, 0.136
86
Rb R 18.6 d b2
1.77
85
Sr R, C 64.8 d g 0.514
99
Mo/99
Tcm
F 65.9 h/6.01 h g 0.143
106
Ru F 373.6 d b2
0.039
110
Agm
R 249.8 d b2
3.0
109
Cd C 461 d g 0.088
111
In C 2.80 d g 0.171
125
I R 59.4 d g 0.035
131
I R 8.02 d b2
0.606, g 0.365
137
Cs F 30.1 y g 0.662
153
Gd R 240.4 d g 0.103
201
Tl C 72.9 h g 0.167
210
Pb R 22.3 y b2
0.017, 0.064
a
R, reactor; C, cyclotron; F, fission product.
98 RADIOTRACERS
1. The position of a single labeled atom in a molecule is shown following the
chemical name of the compound. Thus, acetic–1–14
C acid is CH3
14
COOH,
whereas acetic–2–14
C acid is 14
CH3COOH.
2. Certain terms are used to indicate the distribution of material with more than
one labeled atom. These terms and their meanings are as follows:
a. Specifically labeled. Chemicals are designated as specifically labeled when
all labeled positions are included in the name of the compound and 95% or
more of the radioactivity of the compound is at these positions. Thus,
specifically labeled aldosterone-1, 2-3
H implies that !95% of the tritium
label is in the 1 and 2 positions.
b. Uniformly labeled (U). Uniformly labeled compounds are labeled in all
positions in a uniform or nearly uniform pattern. Thus, L-valine-14
C (U)
implies that all of the carbon atoms in L-valine are labeled with approximately
equal amounts of 14
C.
c. Nominally labeled (N). This designation means that some part of the label
is at a specific position in the material, but no further information is available
as to the extent of labeling at other positions. Thus, cholestrol-7-3
H
(N) implies that some tritium is at position 7, but it may also be at other
positions in the molecule.
d. Generally labeled (G). This designation is for compounds (usually tritium
labeled) in which there is a random distribution of labeled atoms in the
molecule. Not all positions in a molecule are necessarily labeled.
Since the greatest number of labeled compounds are 14
C labeled, our discussion
on the preparation of labeled compounds deals mainly with them. However, many of
the general principles can be applied to other nuclides and molecules as well.
4.3.1 Chemical Synthesis
A 14
C label may be introduced into a wide variety of compounds by the standard
synthetic procedures of organic chemistry. In addition, some new methods have
been devised to conserve the radionuclides being used. When chemical synthesis
is at all possible, it is usually the method of choice. Synthetic methods give the greatest
control over yield, position of the label, and purity of the product. For all syntheses
involving 14
C, Ba14
CO3 is usually taken as the starting material. Quite often this
is converted to 14
CO2 for the synthesis.
Chemical synthesis of labeled compounds suffers from some limitations and
problems, though. One limitation concerns the amount and cost of the radioactive
starting material. This factor necessitates devising synthetic routes to the desired
compounds in which the radiolabel can be introduced near the end of the sequence
of reactions, so as to secure as high an overall yield of labeled material as possible.
At present, numerous labeled compounds are available commercially as starting
materials for syntheses. Still, in planning a new synthetic route, it is necessary to
consider its compatibility with the specific starting material available.
4.3 PREPARATION OF RADIOTRACERS AND THEIR COMPOUNDS 99
Another disadvantage of chemical synthesis is that when it is used to produce
certain biologically important compounds, such as amino acids, a racemic
mixture of D and L isomers results. Since organisms, by and large, metabolize the
L-form selectively, as in the case of amino acids, the use of such racemates in
biological investigations is somewhat unphysical and may lead to undesirable
confusion. Methods for the resolution of racemic mixtures are available. Most of
these are tedious and not suited for small-scale operation.
4.3.2 Biosynthesis
Living organisms, or active enzyme preparations, offer a biochemical means of synthesizing
certain labeled compounds that are not available by chemical synthesis. These
include both the macromolecules (proteins, polysaccharides, nucleic acids, etc.) and
many simpler molecules (vitamins, hormones, amino acids, and sugars). The
successful use of biosynthesis for the production of a given labeled compound
depends on several factors. First, an organism must be selected that will synthesize
and accumulate practical quantities of the desired compound. Culture conditions
must be established so as to provide optimal yields of high specific activity. Last
and most important, you must plan to isolate and purify the labeled compound, as
well as determining the distribution pattern of the label, if a specific labeling is desired.
Photosynthetic methods offer the advantage of using the relatively cheap 14
CO2
(from Ba14
CO3) as the starting material. Carbon-14-labeled starch, glucose,
fructose, and sucrose can be isolated in good yields from green leaves or algal suspensions
that have been exposed to 14
CO2 and illuminated for a prolonged period.
Microorganisms or enzyme systems prepared from them have been used to
produce organic acids labeled with 14
C, either by direct synthesis or transformation
of labeled substrates. Several species of microorganisms have been used to produce
higher fatty acids by condensation.
In general, biosynthetic procedures are likely to be laborious and limited to smallscale
operations. One often encounters purification problems when attempting to
isolate specific biological compounds in a typical system.
4.3.3 Tritium Labeling
Compounds may be labeled with tritium by several methods. The classic synthetic
methods utilizing labeled intermediates have the advantage of yielding products that
have predictable specific activities, are specifically labeled, and have a minimum of
aged by-products. Among the methods of tritium labeling are:
1. By Reduction of Unsaturated Precursors The method of choice for labeling
with tritium is the reduction of a suitable unsaturated precursor (containing a double
bond, carbonyl group, etc.) with carrier-free tritium gas or tritiated metal hydrides.
The major limitation of this method is the availability of a suitable unsaturated
precursor of the desired compound. It is essential to carry out the synthesis in a nonhydroxylic
solvent (dioxane, ethyl acetate, etc.). Reductions carried out in alcohol or
water will lead to almost complete exchange of the tritium gas with the solvent.
100 RADIOTRACERS
2. By Exchange Reactions Random tritium labeling may be secured by simple
exchange methods, with or without catalytic action. Although high specific activities
may be obtained by this method, some of the introduced tritium may be labile.
Removal of this labile tritium and purification of the product are necessary.
3. By Gas Exposure In the mid-1950s, Wolfgang and Rowland described
tritium recoil labeling of organic compounds. Wilzbach, in 1957, first described the
simplified approach to random labeling with tritium that has come to be called the
Wilzbach gas exposure method. In this method, the compound to be labeled is
exposed to Curie amounts of carrier-free tritium gas in a sealed reaction vessel for
a period of a few days to several weeks. The energy released in the disintegration
of tritium and absorbed by the system provides the activation energy necessary to
effect labeling. Compounds labeled by the Wilzbach method are “generally labeled
(G).” Specific activities of 1–125 mCi/g of purified compound have been reported.
Unfortunately, Wilzbach labeling is often accompanied by the formation of
tritiated by-products of high specific activity. As in the case of exchange labeling,
a considerable portion of the tritium in the labeled compound is often labile. This
formation of labeled by-products is the major problem of the gas exposure
method since the specific activity of the by-products may be several orders of magnitude
greater than the desired compound. In general, because of the magnitude of
the purification procedures required and the random nature of the labeling, we
suggest that all other synthetic routes be explored before the gas exposure
method is chosen.
4.3.4 Radiolysis of Labeled Compounds
In many situations, the experimenter will prefer to buy labeled compounds from
commercial suppliers rather than attempt to synthesize them. The radiochemical
purity of such purchased compounds cannot be assumed. Radiation-induced selfdecomposition
(radiolysis) can result in the formation of a variety of labeled degradation
products, which must be removed before experimental use of the compounds.
The extent of radiolysis depends on the nature of the labeled compound, how long it
has been stored, and the manner of storage. Radiolysis is most significant with
low-energy b2
emitters (especially tritium) since the decay energy is dissipated
almost entirely with the compound itself. Furthermore, impurities involving other
radionuclides may be present.
4.4 TRACING OF PHYSICAL PROCESS
In many cases in which radiotracers are used, the chemical identity of the tracer is
not important. These applications can be referred to as tracing physical processes.
For example, consider those experiments that seek to locate an object in some
system by labeling it with radioactivity and then measuring the position of the
radioactivity in the system. Quite often a tracer that decays by g-ray emission is
4.4 TRACING OF PHYSICAL PROCESS 101
selected so that the radiation from the source will penetrate large masses of tissue,
pipe, earth, and the like. Mixing studies are frequently carried out using radiotracers.
Here the objective is usually to see if proper mixing has taken place
between two components of a system. Generally, one of the components is
labeled with a radiotracer and its distribution in the system is monitored as a function
of time. A short-lived tracer is often used so that it can quickly decay away at
the conclusion of the experiment and leave an essentially “nonradioactive” mixed
system for further use.
A form of isotope dilution (see below) is frequently used to measure the volume
of an inaccessible container. A small volume V1 of tracer solution is assayed to give
its activity A. The tracer is added to the liquid in the container and mixed, and a
sample of size V1 is removed and assayed to show its activity A2. The volume of
the original container is given as
V ¼ V1
A1
A2
À 1
 
(4:1)
Leak testing can also be done using radiotracers. Here the basic idea is simple—
namely, to inject radiotracer into a pipe, flask, or whatever is suspected to be leaking
and look for activity that appears outside the container. Suitable caution must be
exercised, of course, to be sure that the “leaked radioactivity” is not a hazard.
For studies involving water, it may only be important to assure that the tracer
remains fluid-bound. One can use tritium or almost any metal atom that can be
complexed with EDTA (ethylenediaminetetraacetic acid) or DTPA (diethylenetriaminepeutacetic
acid) (with a large stability constant).
4.5 CHEMICAL APPLICATIONS OF TRACERS
One of the most important uses of radiotracers in chemistry has been to test separation
procedures in analytical chemistry. Tracers furnish a specific, easyto-apply,
quick method of following the path of a given material in a chemical
separation. Physical chemical data or separation parameters can also be determined.
An example of this type of application is the work of Sunderman and
Meinke (1957) in studying separations by precipitation. The scavenging efficiency
of Fe(OH)3 was evaluated by seeing how many other radiolabeled ions would coprecipitate
with Fe(OH)3 and the quantitative extent to which they were incorporated
in the Fe(OH)3 precipitate. A famous example of the use of radiotracers in
evaluating separation procedures is the work of Kraus and Nelson (1957) in which
they studied the pH dependence, eluant volume and similar factors in separation
of metal ions by ion exchange. Tracers have also been used to locate the position
of a particular fraction in column, thin-layer, and paper chromatography. When
used in thin-layer chromatography (TLC), paper chromatography, and electrophoresis,
autoradiography is frequently utilized to locate the position of the
activity.
102 RADIOTRACERS
The use of radiotracers is an excellent technique for measuring the solubility
product constant of sparingly soluble salts or for making other studies of substances
present in low concentrations. Another very important and classic example of the
use of radiotracers is that of studying the occurrence and properties of isotopic
exchange reactions—reactions of the type
AXÃ
þ BX () BXÃ
þ AX
where X, XÃ
represent stable and radioactive atoms, respectively, of the same
element.
Perhaps the most significant of the numerous applications of radiotracers in
chemistry has been the study of chemical reaction mechanisms. In fact, most of
the proposed reaction mechanisms have been “verified” by means of a radiotracer
study. One of the simplest mechanistic experiments using radiotracers is to test
the equivalence of various atoms in molecules in chemical reactions. An example
of this type of study is the work of Volpin et al. (1959) on the equivalence of the
seven carbon atoms in the tropylium ring. Volpin et al. reacted labeled diazomethane
with benzene and brominated the cyclohepatriene product to form a labeled
tropylium bromide, as shown below:
C6H6 ÀÀÀÀÀÀÀ!
14
CH2N2
C7H8 ÀÀÀÀ!
Br2 14
C7H7Br (4:2)
Then the tropylium bromide was subjected to a Grignard reaction and the product
oxidized to give a labeled benzoic acid. Thus, one had
14
C7H7Br ÀÀÀÀÀÀÀ!
C6H5MgBr 14
C7H7C6H5 ÀÀÀÀÀ!
HNO3
C6H5
14
COOH (4:3)
The specific activity of the labeled benzoic acid was found to be one-seventh that
of the initial labeled diazomethane, thus showing the equivalence of the seven
carbon atoms of the tropylium ring.
Another popular use of radiotracers in studying chemical reaction mechanisms
is the study of molecular rearrangments. An example of this class of reactions
that illustrates the use of radiotracers is the cyclization of v-phenoxyacetophenone
(I) to 2-phenylbenzofuran (II). Two possible mechanisms are shown
below:
O
CH2
14
COC6H5
O
14
C6H5
−
O C6H5
14
(I) (II)
(4:4)
4.5 CHEMICAL APPLICATIONS OF TRACERS 103
Or
+
+
O
14
COC6H5
CH2
OH
CH2
C6H5
14
C O
–H+
O
OH
14
C
C6H5
C6H5
CH2
O
(II)
14
(I) (4:5)
Note that the two possible mechanisms can be distinguished by the position of
the labeled atom in the 2-phenylfuran.
Two problems that are common in tracer studies of the reaction mechanism
occurred in this study. The first was the synthesis of the labeled starting material
(I) for the reaction. A chemical synthesis, shown below, was used to produce (I):
14
CH3COBr
Freidel
Craft
Reaction
C6H5
14
COCH3
Br2 in
acetone
C6H5
14
COCH2Br
Phenol in
acetone
with
Na2CO3
C6H5
14
C CH2 O C6H5
O
(I)
(4:6)
After obtaining the labeled starting material and carrying out the cyclization
reaction, the second problem was to degrade the product (II) to reveal the position
of the label. This was done by using the following steps:
O C6H5
OCH3
Metallic Na
+CH3
I in ethanol
CH CH C6H5 Oxidation with
permanganate
CH2 CH2
OCH3
COOH + C6H5COOH
OCH3
C6H5 N-bromosuccinimide
and triethylamine
(4:7)
Note that the reaction above the two carbon atoms in the five-membered ring of II
end up in different reaction products. Thus, one can check to see which product
contains the label and decide on the correct mechanism.
4.6 ISOTOPE EFFECTS
Up to now in our discussion of radiotracers, we have assumed that all isotopes of a
given element, stable or radioactive, would behave alike chemically and physically.
We will now examine this point more critically to see how different isotopes behave
and how this difference in behavior (the isotope effect) can be detected and used to
104 RADIOTRACERS
our advantage. The difference in behavior between different isotopes of the same
element is due to the different masses. This mass difference will affect the kinetic
energy of the molecules (giving rise to physical isotope effects) or will change the
vibrational and rotational properties of molecules (giving rise to chemical isotope
effects).
Some examples of physical isotope effects follow:
1. Gaseous Effusion Graham’s law states that
R ¼
ffiffiffiffiffiffi
m1
m2
r
(4:8)
where R is the relative rate of effusion of two isotopes of mass m1 and m2,
respectively, through a hole. Thus, a labeled molecule will effuse at a different
rate than an unlabeled molecule.
2. Distillation For a given temperature, the velocity of a light isotope will be
greater than that of a heavy isotope, so that the lighter isotope will have a
greater vapor pressure.
Chemical isotope effects are divided into two classes—those affecting the
position of the equilibrium in a chemical reaction and those affecting the rate of a
chemical reaction. Equilibrium isotope effects have their origin in the fact that the
extent to which any chemical reaction “goes” is governed by the number of possible
ways it can proceed (the phase space available). The more equally probable reaction
paths available, the more likely the reaction will go. To illustrate this point, consider
the exchange reaction
AX þ BXÃ
() AXÃ
þ BX
The equilibrium constant K for the reaction is given by
K ¼
fAXÃ fBX
fAX fBXÃ
(4:9)
where fI is the partition function for the ith species. ( fI reflects the probability of
occurrence of the ith species and is formally the probability of occurrence of a set
of vibrational, rotational, and translational energy levels for that species.) The
partition functions can be calculated by using statistical mechanics, and they may
depend in a complicated way on the relative masses of AX, AXÃ
, and so on.
Table 4.2 shows some typical values of K for various exchange reactions and
TABLE 4.2 Typical Equilibrium Isotope Effects
Reacting System K
H2
H(g) þ H2O(l) , H2(g) þ 2
HOH(l) 3.2
14
CO2(g) þ 12
COCl2(g) , 14
COCl2(g) þ 12
CO2(g) 1.0884
[Co(NH3)4
12
CO3]þ
þ 14
CO3
22
, 12
CO3
22
þ [Co(NH3)4
14
CO3]þ
0.8933
4.6 ISOTOPE EFFECTS 105
illustrates the point that chemical equilibrium is shifted when a radioisotope is
substituted for another isotope. (Note that when X ¼ X*, K ¼ 1). Table 4.2 also
shows the important fact that the greater the mass difference between isotopes,
the larger the equilibrium isotope effects are. In general, it has been found that
such effects can be neglected when the atomic number is greater than 10.
Kinetic isotope effects are very important in the study of chemical reaction mechanisms.
The substitution of a labeled atom for an unlabeled one in a molecule will
cause a change in reaction rate for Z , 10, and this change can be used to deduce the
reaction mechanism. The change in reaction rate due to changes in the masses of the
reacting species is due to differences in vibrational frequency along the reaction
coordinate in the transition state or activated complex.
Experimentally, it is relatively straightforward to measure the existence and magnitude
of kinetic isotope effects. Consider the reaction shown below, proceeding
through a reaction intermediate, AB, as
A þ B À! AB À! products
Let S0 and Sg be the specific activities of B at time t ¼ 0 and after a fraction g of the
reaction has been completed, respectively. If B is the only labeled reactant, we have
d½BŠ
dt
¼ k½AŠa
½BŠb
(4:10)
And assuming BÃ
is present as a tracer
d½B*Š
dt
¼ kÃ
½AŠa
½B*Š½BŠbÀ1
(4:11)
where a and b are the reaction order with respect to A and B. Dividing the previous
two equations, we have
d½BŠ
d½B*Š
¼
k½BŠ
kýB*Š
(4:12)
Separation of variables and integration gives
log10
Sg
S0
 
¼
kÃ
k
À 1
 
log10 (1 À g) (4:13)
Thus, plotting log10(Sg/S0) versus log(1 2 g) gives a straight line of slope (kÃ
/k 2 1).
If there was no isotope effect, k ¼ kÃ
, then the slope will be zero. Any finite slope in
the preceding plot will give kÃ
/k. This effect is shown in Figure 4.1 for the study of the
Cannizzaro reaction by Downes and Harris (1952).
106 RADIOTRACERS
4.7 BIOLOGICAL APPLICATIONS
Today the largest number of applications of radiotracers is in biology and medicine.
Because of the large number of applications, it is beyond the scope of this work to
review them in any detail. Instead, we shall focus on three specific applications, one
that is very old, one that is “middle-aged,” and one that is “adolescent’ in its scope, if
not in its development. We refer to autoradiography, radioimmunoassay, and DNA
(deoxyribonucleic acid) analysis, respectively.
1. Autoradiography Autoradiography is the oldest method of detecting radioactivity.
In autoradiography, a radioactive sample is placed on a photographic emulsion.
After a period of time, the film is developed and the precise location of the
radioactive matter in the sample is determined from the pattern of darkening on
the film. Thus, autoradiography is used to locate radionuclides in a tissue section,
gross sample, or chromatogram. Special techniques (Tolgyessy, 1972) are used to
get optimum spatial resolution in the image and to prevent artifacts (non-radiationinduced
darkening of the film). Optical or electron microscopes can be used to
examine the images since the grain size of film is of the order of microns.
Figure 4.1 Plot of log(Sg/S0) as a function of log (1 2 g) for the Cannizzaro reaction,
showing the isotope effect. [From Downes and Harris (1952).]
4.7 BIOLOGICAL APPLICATIONS 107
2. Radioimmunoassay (RIA) Radioimmunoassay is a highly sensitive method
of determining the amounts of hormones, drugs, vitamins, enzymes, viruses,
tumor antigens, and serum proteins in biological samples. It is based on the immunological
reaction of antibodies and antigens. One starts with either an antigen or antibody
labeled with a radiotracer such as 3
H, 14
C, or 125
I. In a radioimmunoassay for
antigen, one uses the idea that if a limited amount of antibody is available, the
antigen molecules will compete for binding sites on the antibody. If one
starts with a certain amount of radiolabeled antigen, any additional antigen added
will displace some the radiolabeled antigen and will not allow it to bind to the
antibodies.
Thus, the procedure might be to mix antibody and radiolabeled antigen together
and let them bind to each other. Precipitate the complex from the solution. Measure
the activity of the supernatant. This is the amount of unbound antigen. Now mix
the same amounts of antibody and radiolabeled antigen together along with the
unknown stable antigen sample. The stable antigen will compete with the radiolabeled
antigen for binding sites on the antibody molecules. Some of the radiolabeled
antigen will not be able to bind as it did before. Precipitate the complexes
from solution. Measure the activity of the supernatant. This activity is a measure
of the amount of stable, unlabeled antigen present in the sample. By constructing
a calibration curve that shows the amount of radioactivity present in the supernatant
after adding known amounts of unlabeled antigen, one can determine the amount of
antigen present in a sample. Thus, RIA is a special form of isotope dilution analysis
(IDA) discussed below.
More than 107
immunoassays are performed in medicine and biochemistry
in the United States per year. The important advantages of this technique are
high sensitivity and high specificity. In some cases, picogram quantities can be
measured.
3. DNA Analysis DNA analysis is widely discussed in the news media because
of its use in identifying criminals, establishing paternity, detecting genetic diseases,
and so on. The DNA in each living cell can be used as a fingerprint to identify individuals
in a large population. To obtain a DNA fingerprint of an individual, one
extracts the DNA from a sample of blood, skin, hair, semen, and the like
(Fig. 4.2). (The cell walls are destroyed by osmosis or other techniques and the
double-stranded DNA is decomposed into single-stranded pieces, which are collected.)
This DNA is then cut into pieces using enzymes that cut either side of a
repeated sequence. The result is a DNA mixture of segments of differing size. Electrophoresis
is used to sort the fragments by size spatially. The spatially separated
fragments are allowed to react with radiolabeled “gene probes.” These gene
probes contain radiolabeled specific fragments of DNA that bind only to those
DNA segments containing a nucleotide sequence that is complementary to its
own (a sequence that would be its matching strand in the DNA double helix).
The original DNA fragments are then identified by the radiolabeled DNA that
has reacted with them, usually by autoradiography. The physical pattern on the
autoradiograph is a pattern of the DNA sequences and sizes.
108 RADIOTRACERS
4.8 ENVIRONMENTAL APPLICATIONS
In recent years a great deal of applied research has centered on the study of problems
related to the environment and environmental processes. In some of these studies,
radiotracers have been used as primary tools to measure the dynamics of many physical
and biological processes. In the best studies, the use of radiotracers to measure
flow patterns, dispersion, and similar features is closely coupled to tests of theoretical
models of the processes involved. This modeling is important because in
environmental studies the experimental conditions are difficult to control and, in
general, only a few of the many possible conditions in a given experiment will be
sampled. Therefore, it is important to have some way (i.e., a model) to correlate
experimental results measured under special conditions to general statements
regarding an environmental process.
Radiotracers have been employed in studies of physical and biological processes
in the atmosphere and the hydrosphere. Among the quantities that have been
measured in atmospheric studies are the natural airflow patterns in large- and
small-scale investigations, the dispersion of atmospheric pollutants from various
sources, and the identification of the sources of various pollutants. In studies of
Figure 4.2 Schematic view of DNA fingerprinting. (Figure also appears in color figure
section.)
4.8 ENVIRONMENTAL APPLICATIONS 109
the hydrosphere, radiotracers have been utilized to measure general water circulation
patterns and various features of the hydrologic cycle, including precipitation, runoff,
and stream flow; total water inventories; infiltration; groundwater problems, such as
the origin and age of water, its flow velocity and direction, evaporative transport, and
aerosol production. Many studies of a biological nature, such as pollutant dispersal,
uptake, and concentration in the ecosystem, have involved the use of radiotracers.
The controlled use of radioactivity to study processes occurring in our environment
is a well-established and respected technique. Radioactive tracers have
several significant advantages over conventional tracers for environmental studies:
1. The detectability of the tracer is not influenced by the physico-chemical nature
of the environment (i.e., factors such as watercolor and pH).
2. Because nuclear radiation, particularly g-radiation, is highly penetrating, the
tracer can be detected while part of a living organism or when deeply buried in
the ground.
3. Because only a small number of atoms are necessary to give a significant disintegration
rate, there is a much better detection sensitivity in radiotracer
experiments compared to conventional tracer experiments. This point is
particularly important for environmental studies where high dilution factors
are commonly encountered.
4. Because tracers of short half-life can be used that will rapidly disappear after
the completion of an experiment, experiments can be repeated several times
without damaging the environment or getting erroneous results due to persistence
of tracers from previous experiments in the environment.
5. In many instances,radiotracers are the cheapest method of tracing pollutant flow.
The principal disadvantage of using radiotracers in environmental studies is the
actual (or imagined) problem of nuclear safety. Public concern over possible
harmful effects of ionizing radiation has increased in recent years. This fact, together
with the lack of control over experimental conditions found in many environmental
studies, has caused many regulatory agencies to establish extremely strict rules concerning
the use of radiotracers in environmental research. Many research workers
now find it difficult to demonstrate that the radiotracer concentrations will not
exceed the regulatory agency’s maximum permissible radionuclide concentration
limits at all times and places. One must be especially aware, in this connection,
of “concentration effects” present in food chains. At a minimum, the experimenter
can usually look forward to a significant amount of red tape before performing
environmental studies with radiotracers.
The criteria for choosing a particular radionuclide as a tracer for environmental
studies are similar to those encountered in radiotracer laboratory experiments. Such
items as the nature of the radiation emitted, the half-life of the radionuclide, the ease
of obtaining it, the ease of detecting its radiations, and its cost play important roles.
Tables 4.3a, 4.3b, and 4.3c show some typical radiotracers that have been used in
environmental studies.
110 RADIOTRACERS
Many examples of the application of radiotracers to environmental problems
exist, and so we have selected only a few of the more interesting ones to discuss.
Barry (1971) used 41
Ar, a short-lived gas found in reactor stack effluents, to trace
the dispersion of stack effluent from the Chalk River nuclear reactor and relate it
to conventional dispersion models. The 41
Ar concentrations in the air at various
distances from the emitting stack were measured by circulating the air through
a counter consisting of layers of plastic scintillator. The b particles emitted in
the decay of 41
Ar were detected by the plastic scintillators. The detectors were
TABLE 4.3a Radioactive Tracers for Gaseous Material
Nuclide Half-Life Radiation of Interest (MeV) Chemical Form
35
S 87 d b: 0.167 (100%) H2S
41
Ar 110 min g: 1.37 Gas
76
As 26.5 h g 0.55–2.02 AsH3
82
Br 36 h g 0.55–1.32 CH3Br
85
Kr 10 y b 0.7 Gas
g 0.54
133
Xe 5.27 d b: 0.34 Gas
g: 0.03, 0.08
Source: Radioisotope Tracers in Industry and Geophysics. Vienna: International Atomic Energy
Agency, 1967.
TABLE 4.3b Radioactive Tracers for Solid Material
Nuclide Half-Life Radiation of Interest Chemical Form
24
Na 15 h g 1.37 (100%) Na2CO3 in polypropylene balls
g 2.75 (100%)
46
Sc 84 d g 0.89 (100%) Sc2O3
g 1.48 (100%)
51
Cr 27.8 d g 0.325 (9%) Absorbed on quartz
64
Cu 12.8 b 0.57 (38%) CuO
g: 0.51 (19%)
65
Zn 274 d g: 1.11 (49%) ZnO
82
Br 36 h g 0.55 (70%) CaBr in polyethylene containers
g 1.32 (27%)
110 m
Ag 253 d g 0.66 (100%) Absorbed on solid grains
g 1.50 (13%)
140
La 40 h g: 0.33–2.54 La2O3 polypropylene balls
144
Ce 285 d b: 3.1 (98%) Ce2O3
182
Ta 115 d g: 1.19 Ta2O3
g: 1.12
198
Au 2.7 d g: 0.41 (91%) AuCl3 absorbed on powder
Source: Radioisotope Tracers in Industry and Geophysics. Vienna: International Atomic Energy Agency,
1967.
4.8 ENVIRONMENTAL APPLICATIONS 111
connected to automatic recording equipment that could monitor the stack effluent
dispersion continuously.
A typical example of a radiotracer study in an aqueous system is the study of the
concentration dynamics of soluble material in the Eshkol Reservoir of Israel by
Gilath and Stuhl (1971). The Eshkol Reservoir is a shallow lake (depth, 27 in.;
volume, 3.5 Â 106
m3
) with 106
m3
of water flowing through the reservoir per
day. As is typically done in studies of this type, a “spike” of radiotracer was injected
into the reservoir and its dispersal was measured. 82
Br was chosen as the tracer
because of its high maximum permissible concentration limit (100 mCi/m3
), its
low detection limit (2 Â 1026
mCi/mL), and its general solubility in water. A
spike of 15 mCi 82
Br was injected into the water over a 1-h period, allowing the
tracer concentration to be measured for 3 to 4 times the typical residence time
of the reservoir (70 h). The 82
Br activity was measured by using NaI detectors stationed
at the reservoir outlet and in probes lowered into the water from boats.
An interesting set of radiotracer studies of the deposition of pesticides was made
by Atkins and Eggleton (1971). The pesticides were labeled with 14
C, which was
assayed using liquid scintillation counting. Results of the study showed that direct
TABLE 4.3c Radioactive Tracers for Organic Materials
Nuclide Half-Life Radiation of Interest Chemical Form
3
H 12.26 y b: 0.018 (100%) Various organic compounds
14
C 5568 y b: 0.155 (100 %) Various organic compounds
24
Na 15 h g: 1.37 (100%) Naphthenate
g: 2.75 (100%) Salicylate
35
S 97 d b: 0.167 (100%) Various organic compounds
38
Cl 0.3 min g: 1.60 (31%) Chlorobenzene
g: 2.15 (47%)
59
Fe 44.5 d g: 1.1 (57%) Ferrocene
g 1.29 (43%) Dicyclopentadienyl-iron
60
Co 5.3 y g 1.17 (100%) Naphthenate
g 1.33 (100%)
64
Cu 12.8 h g 0.51 (19%) Naphthenate
65
Ni 2.56 h g: 0.37, 1.11, Stearate
1.49 Oxalate
77
Ge 11 h g: 0.21–2.02 Various organic compounds
82
Br 36 h g: 0.55–1.48 Bromobenzene
Paradibromo-benzene
124
Sb 60 d g 0.61 (99%) Triphenylstibine
g: 0.72 (14%)
131
I 8.04 d g 0.36 (80%) I-Kerosene
g 0.64 (9%) Iodobenzene
140
La 40 h g: 0.33–2.54 Naphthenate
198
Au 2.7 d g: 0.41 (99%) Sodium cyanide solution
Source: Radioisotope Tracers in Industry and Geophysics. Vienna: International Atomic Energy
Agency, 1967.
112 RADIOTRACERS
absorption on soil or vegetation was a more effective way of removing the pesticides
from the atmosphere than deposition in rain. Furthermore, the mean residence times
of the pesticides in the atmosphere were shown to be long enough to allow global
distribution of a pesticide from any given location.
4.9 INDUSTRIAL USE OF RADIOTRACERS
There are a large number of industrial uses for radiotracers. Table 4.4 shows some
typical industrial uses of radiotracers.
Several industrial uses involve the determination of flow rates and liquid
volumes. In the flow rate applications, one introduces the tracer into the liquid
stream by either pulse or continuous injection and measures the activity as a function
of distance. (One must be careful to begin sampling beyond the mixing region,
where tracer and fluid mix. This is done by beginning sampling at a distance of
100Â the stream diameter from the point of injection.) When a pulse with activity
A Bq is injected into a stream with flow rate F, then F is given simply as
F ¼
A
Ð1
0 r(t) dt
(4:14)
where r is the efficiency-corrected counting rate in a downstream detector.
Another interesting and important use of radiotracers in industry is in wear and
corrosion studies. In studies of wear (tribology) one labels the part under study
with a radionuclide. The radioactivity is concentrated on the surface undergoing
wear by plating, diffusion, or ion implantation with a low-energy accelerator. The
labeled part is put in service and, typically, one measures the radioactivity released
in the lubricant as a measure of wear. Calibrations of the technique can be done to
get absolute measures of wear.
4.10 NUCLEAR MEDICINE
The most rapidly expanding area of tracer use is in nuclear medicine. Nuclear medicine
deals with the use of radiation and radioactivity to diagnose and treat disease.
The two principal areas of endeavor, diagnosis and therapy, involve different
methods and considerations for radiotracer use. (As an aside, we note that radiolabeled
drugs that are given to patients are called radiopharmaceuticals.) A list of
radionuclides commonly used in diagnosis is shown in Table 4.5. Most nuclear
medicine procedures (.90%) use either 99
Tcm
or one of the iodine isotopes. Most
diagnostic use of radiotracers is for imaging of specific organs, bones, or tissue.
Typical administered quantities of tracer are 1–30 mCi for adults. Nuclides used
for imaging should emit photons with an energy between 100 and 200 keV, have
small decay branches for particle emission (to minimize radiation damage), have
4.10 NUCLEAR MEDICINE 113
TABLE 4.4 Industrial Uses of Radionuclides
Nuclide Application
3
H Self-luminous aircraft and exit signs
Luminous dials, gauges, wrist watches
Luminous paint
24
Na Location of pipeline leaks
Oil well studies
46
Sc Oil exploration tracer
55
Fe Analysis of electroplating solutions
Defense power source
60
Co Surgical instrument and
medicine sterilization
Safety and reliability of oil burners
63
Ni Detection of explosives
Voltage regulators and current
surge protectors
Heat power source
85
Kr Home appliance indicator lights
Gauge thickness of various thin materials
Measurement of dust and pollutant levels
90
Sr Survey meters
109
Cd XRF of metal alloys
124
Sb Oil exploration tracer
126
Sb Oil exploration tracer
131
I Petroleum exploration
136
Cs Oil exploration tracer
137
Cs Measure and control liquid flow in pipes
Measure of oil well plugging by sand
Measure fill level of consumer products
140
Ba Oil exploration tracer
147
Pm Used in electric blanket thermostats
Gauge thickness of thin materials
Oil exploration tracer
151
Sm Heat source
156
Eu Oil exploration tracer
192
Ir Pipeline, boiler and aircraft
weld radiography
Oil exploration tracer
198
Au Oil exploration tracer
204
Tl Thickness gauge
210
Po Reduction of static charge
229
Th Extend life of fluorescent lights
230
Th Coloring and fluorescence in
glazes and glass
232
Th With W, electric arc welding rods
234
U Natural color, brightness in dentures
(continued)
114 RADIOTRACERS
a half-life that is 1.5Â the duration of the test procedure, and be inexpensive and
readily available. 99
Tcm
is used in more than 80% of nuclear medicine imaging
because its 143-keV g rays produce excellent images with today’s g cameras, it
has a convenient 6-h half-life and it can be obtained “generators” involving a transient
equilibrium with 99
Mo.
Most people are familiar with the medical or dental use of X-rays for providing
images of tissue or bones where an external radiation source is used to do the
imaging. To improve contrast, agents such as barium sulfate, which attenuate the
X-rays, are frequently administered to the patient. In the 1970s, a significant
improvement in medical imaging occurred with the advent of computerized tomography
(CT). In this technique, photographic plates are replaced by one or more
radiation detectors, and an apparatus is used to move the source of imaging radiation
relative to the patient with a digital computer system with appropriate software to
provide on-line images from observed changes in counting rates as the source–
patient geometry changes. [Tomography is from the Greek words, “to cut or
section” (tomos) and “to write” (graphein). Tomography shows slices of the body
with typical resolution of ,1 mm.] A simple diagram of such apparatus is shown
in Figure 4.3.
Tomography can involve images generated by the transmission of radiation
through the body (Fig. 4.3) or by incorporating radionuclides into the body and
detecting the emitted radiation (emission tomography). For emission tomography,
the imaging techniques can involve PLANAR (see below) images where a twodimensional
view of an organ is obtained (Fig. 4.4a), SPECT (single-photon
emission computerized tomography) (Fig. 4.4b) where a three-dimensional computer
reconstructed image is obtained, or PET. In PET (Fig. 4.5), positron-emitting
nuclides, such as 18
F, 11
C, 15
O, or 13
N, are introduced into a region to be
studied. The two 0.511-MeV photons, produced when the bþ
annihilates, emerge
in opposite directions and define a line passing through the point where the
decay occurred. The two photons are detected in coincidence by an array of
scintillation detectors. After the observation of many decays, computer techniques
are used to reconstruct a three-dimensional image of the area where the decays
occurred.
TABLE 4.4 Continued
Nuclide Application
235
U Nuclear reactor fuel
Fluorescent glassware, glazes, and wall tiles
238
Pu Radioisotope thermal generator
241
Am Smoke detectors
244
Cm Analysis of pit mining and drilling slurries
252
Cf Luggage inspection for explosives
Soil moisture content
4.10 NUCLEAR MEDICINE 115
TABLE 4.5 Commonly Used Diagnostic Radionuclides
Nuclide Application
11
C PET brain scans
14
C Radiolabeling
13
N PET scans
15
O PET scans of cerebral blood flow
18
F PET brain scans
32
P Bone disease diagnosis
33
P Radiolabeling
35
S Heart disease diagnosis
Nucleic acid labeling
47
Ca Cell function and bone formation
46
Sc Blood flow studies
47
Sc Cancer diagnosis
51
Cr Red blood cell survival studies
Intestinal blood loss
51
Mn Myocardial localizing agent
52
Mn PET scans
59
Fe Bone marrow scanning
Iron metabolism studies
57
Co Scanning of various organs
58
Co Tracer for pernicious anemia
64
Cu PET scans
67
Cu Cancer diagnosis
67
Ga Tumor and inflammatory lesion imaging
68
Ga Thrombosis and atherosclerous studies
72
Se Brain imaging
75
Se Protein studies
Liver and pancreas imaging
81
Krm
Lung imaging
82
Rb Myocardial localizing agent
85
Sr Measurement of bone metabolism
99
Tcm
Brain, heart, lung, thyroid, gall bladder, skin,
lymph node, bone, liver, spleen,
and kidney imaging
Blood flow studies
109
Cd Cancer detection
Pediatric imaging
Heart disease diagnostics
111
In Detection of heart transplant rejections
Imaging of abdominal infections
Imaging of metastatic melanoma
123
I Thyroid disorders
125
I Osteoporosis detection
Tracer for drugs
131
I Thyroid disorders
Brain biochemistry in disease
(continued)
116 RADIOTRACERS
Most imaging is of the PLANAR type in which a stationary g-ray detector is
used. Typically, a single picture is taken of a patient’s liver, heart, and the like to
determine the presence and distribution of the radionuclide. Sometimes, multiple
images are taken over a short time to study the dynamic behavior of an organ
through its radionuclide uptake. SPECT is used mostly for brain and cardiac
imaging with typical resolutions of 3–5 mm. The radionuclides used are 99
Tcm
,
201
Tl, 67
Ga, 111
In, and 123
I, all single-photon emitters. PET imaging is used primarily
for dynamic studies of the brain, heart, and lungs but its use is still expanding.
11
C-labeled glucose has been used extensively for the study of brain metabolism.
Other nuclides, such as 15
O, can be used to study blood flow and volume.
Figure 4.6 shows a set of PET pictures using both 11
C and 15
O to study the use of
TPA (tissue plasminogen activator), a clot-busting drug, in a patient with an acute
myocardial infarction.
99
Tmm
is the most widely used radionuclide for diagnostic purposes. It is used in
10 million procedures per year in the United States and 20 million/per year worldwide.
As discussed previously, its single 142.7-keV photon is ideal for imaging, and
its 6-h half-life will accommodate most procedures with excessive radiation dose to
TABLE 4.5 Continued
Nuclide Application
127
Xe Lung imaging
Neuroimaging for brain disorders
133
Xe Lung ventilation studies
169
Yb Gastrointestinal tract diagnosis
191
Irm
Cardiovascular angiography
195
Ptm
Pharmacokinetic studies of antitumor agents
DETECTOR ARRAY
TRANSLATE-ROTATE
MOTION MOTION
CONTROL
DATA
ACQUISITION
SYSTEM
COMPUTER
SYSTEM
DISPLAY
Figure 4.3 Schematic diagram of a CT system.
4.10 NUCLEAR MEDICINE 117
the patients. It is easily obtained from a Mo/Tc generator (“cow”) and thus is
available for continuous use at a reasonable cost.
How does a 99
Mo/99
Tcm
generator work? 99
Mo [which can be produced as a
fission product or from the 98
Mo(n, g) reaction] decays to 99
Tcm
as follows:
99
Mo ÀÀÀÀÀ!
bÀ
99
Tcm
ÀÀÀÀÀÀÀÀÀÀ!
IT(142:7 keVg) 99
Tc
The decay of 99
Mo goes about 91% of the time to the isomeric state of 99
Tc. This
state decays to the ground state of 99
Tc (t1=2 ¼ 2:1 Â 105
g) by the emission of a
single 142.7-keV photon. 99
Mo, as a reaction product, is purified and dissolved in
acid media to form the anionic species molybdate (MoO4
22
) and paramolybdate
(Mo7O24
6).
The molybdate anions are adsorbed on an aluminum oxide column.
This column can be “milked” at will to extract the [99
Tcm
O4
2
] ion formed by the
RECTILINEAR
MOTION
to ELECTRONICS
to ELECTRONICS
PMT
CRYSTAL
LEAD
COLLIMATOR
COLLIMATOR
SOURCE DISTRIBUTION
PM TUBE
LIGHT GUIDE
Na(T1) CRYSTAL
(a) (b)
Figure 4.4 (a) Schematic diagram of PLANAR imaging and (b) schematic diagram of
SPECT.
Figure 4.5 Schematic drawing of the elements of a PET system.
118 RADIOTRACERS
decay of 99
Mo. The daughter 99
Tcm
is eluted from the column with 0.9% NaCl. The
99
Mo remains bound to the column as it is insoluble in 0.9% NaCl (see Fig. 4.7).
After separation from the molybdenum, the technetium is converted to a suitable
complex prior to use in a patient.
The 99
Mo/99
Tcm
system represents a case of transient equilibrium. We can use
Equations (3.6) and (3.22) to trace out the activity of the 69.5 h 99
Mo and 6.0 h
99
Tcm
as a function of time (Fig. 4.8). The 99
Tcm
activity grows in after each
milking of the cow, with a maximum amount being present approximately 22 h
after separation. If all the 99
Mo decayed to 99
Tcm
, then the activity of 99
Tcm
would exceed that of the 99
Mo after equilibrium. Since only 91% of the 99
Mo
decays to 99
Tcm
, then there is slightly less 99
Tcm
than 99
Mo.
As stated earlier, the nuclides used most for PET are 11
C, 13
N, 15
N, and 18
F. 15
O is
used for studies of blood volume and flow in the form of 15
O-labeled carbon monoxide
and carbon dioxide. These gases are administered by inhalation with CO binding
to the hemoglobin in the blood. It is also possible to use 15
O-labeled H2O. The most
widely used 18
F-labeled radiotracer is [18
F] 2-deoxy-2-fluoro-D-glucose (FDG).
When taken into tissue, the fluoroglucose is converted in fluoroglucose-6-phosphate,
which cannot be metabolized further and is trapped in the tissue. The trapped tracer is
used then for imaging those organs that metabolize glucose most rapidly.
The therapeutic uses of radiation and radioactivity are no less important than the
diagnostic uses. One is most familiar with external sources of radiation being used to
destroy diseased tissue. A problem with these radiation therapies is similar to that
Figure 4.6 PET pictures of the heart of a patient with acute myocardial infarction treated
with a thrombolytic agent. Top row shows scans after administration of water containing
15
O to trace blood flow. Bottom row shows tomograms obtained after administration of
acetate containing 11
C to trace the heart’s metabolism, that is, its rate of oxygen use. The
defects are clearly visible on day 1, both in the impaired blood flow (top left) and the
impaired metabolic use of oxygen (bottom left). Recovery of blood circulation has taken
place on day 2 and is maintained. (Figure also appears in color figure section.)
4.10 NUCLEAR MEDICINE 119
encountered in chemotherapy, how to kill the diseased cells without killing so many
normal cells that the organism does not survive.
One approach to this problem is to use an internal source of radiation in the form
of a physically or chemically implanted radionuclide. 131
I, for example, has been
used to treat disorders of the thyroid because of the ability of this gland to concentrate
iodine. A promising avenue is the development of monoclonal antibodies that
seek out particular cancer cells and bind to them. If one can radiolabel these
antibodies with nuclides such as 211
At, 131
I, 186,188
Re, 125
I, or 90
Y, then one can
deliver a large dose to the cancer cells with reduced damage to the normal tissue.
An alternative approach, using external radiation, is to deposit, by various means,
a large amount of energy into the tumor cell with as little loss of energy elsewhere
as possible. One straightforward way to do this is to deposit a radiation-absorbing
compound preferentially in the cancer cell and to irradiate the organism, thus
localizing the dose. One such material is boron, which can undergo the 10
B(n, a)
reaction, splitting into two large fragments with short ranges in tissue. This
therapy is being developed at present. Another approach, that of using protons to
irradiate the diseased tissue, takes advantage of the fact that the energy deposition
PRODUCTION
OF PARENT
RADIOCHEMICAL
SEPARATION
and
CHEMICAL
TREATMENT
0.9%
NaCl
“COW”
ADDITION TO
ALUMINA COLUMN
ELUTION WITH
PHYSIOLOGICAL
SALINE
ELUTED 99mTcO4
-
99mTcO4
-
99mTcO4
-
99MoO4
2-
99MoO4
2-
99Mo
235U FISSION 98Mo (n, g )
b DECAY
of 99Mo
to 99mTc
Figure 4.7 Schematic diagram of a Mo/Tc cow. [Reprinted from Ehmann and Vance (1991).]
120 RADIOTRACERS
of protons in matter is concentrated near the end of its range in matter (the Bragg
peak). In theory, carefully choosing the energy of the incident proton, one could
localize the damage to the tumor tissue. It turns out (Fig. 4.9) that the sizes and
positions of real tumors are much larger than the typic beam dimensions so that
the proton beams have to be smeared to match the tumor size.
Figure 4.8 Decay of an initially pure sample of 99
Mo.
Figure 4.9 Relative dose at various depths in tissue for X-rays, for 190-MeV protons, for
protons with a smeared energy distribution, and from 60
Co g rays.
4.10 NUCLEAR MEDICINE 121
4.11 ISOTOPE DILUTION ANALYSIS
There are certain analytical methods that can be applied to all fields of tracer use.
Foremost among these is isotope dilution analysis (IDA). In this section we summarize
the variants of this analytical method.
4.11.1 Direct IDA
The basic idea of isotope dilution analysis is to measure the changes in specific
activity of a substance upon incorporation into a system containing an unknown
amount of that substance. There are several types of IDA. We begin by considering
direct IDA.
In direct IDA, we are faced with the problem of determining the amount of some
inactive material A in a system. Let us define this unknown amount as x grams. To
the system containing x grams of inactive A, we add y grams of active material AÃ
of
known activity D. Thus, we know the specific activity of the added active material,
S1. That is,
S1 ¼
D
y
(4:15)
After thoroughly mixing the active material AÃ
with the inactive A in the system,
one isolates, not necessarily quantitatively, and purifies a sample of the mixture of A
and AÃ
and measures its specific activity, S2. Clearly, conservation of material says that
S2 ¼
D
x þ y
(4:16)
Since
y ¼
D
S1
(4:17)
we can substitute for y, obtaining
S2 ¼
D
x þ (D=S1)
(4:18)
Rearranging, we get
x ¼
D
S2
À
D
S1
¼
D
S1
S1
S2
À 1
 
¼ y
S1
S2
À 1
 
(4:19)
This is the basic equation of direct isotope dilution analysis. The unknown amount x
of material A is given in terms of the amount y of added labeled material AÃ
and the
two measured specific activities S1 and S2.
Example Problem Let us consider a practical problem to illustrate the use of
this technique. A protein hydrolysate is to be assayed for aspartic acid. Exactly
5.0 mg of aspartic acid, having a specific activity of 0.46 mCi/mg, is added to
122 RADIOTRACERS
the hydrolysate. From the hydrolysate, 0.21 mg of highly purified aspartic acid,
having a specific activity of 0.01 mCi/mg, can be isolated. How much aspartic
acid was present in the original hydrolysate?
Solution We say that
x ¼ number of mg aspartic acid in original hydrolysate
y ¼ 5:0 mg
S1 ¼ 0:46 mCi=mg
S2 ¼ 0:01 mCi=mg
x ¼ y
S1
S2
À 1
 
¼ (5:0)
0:46
0:01
À 1
 
x ¼ 225 mg aspartic acid
Thus, by isolating a small fraction of the added aspartic acid and measuring the
diminution in its specific activity, the aspartic acid content of the original sample can
be determined. Note this example involved a large change in specific activity upon
dilution. Poor experimental design or other circumstances may lead to a small
change in specific activity upon dilution. In such cases, the results obtained from
IDA involve a small difference between two large numbers and are quite uncertain.
4.11.2 Inverse IDA
Inverse IDA is a simple variant on the basic direct IDA. In inverse IDA, we measure
the change in specific activity of an unknown radioactive material AÃ
after diluting it
with inactive A. Specifically, let us assume that we have q milligrams (where q is
unknown) of a radioactive substance AÃ
whose specific activity is known (i.e.,
Sq ¼ D=q). (Sq can be measured by isolating a small portion of AÃ
, weighing it,
and measuring its activity.) Let us add r milligrams of inactive A to AÃ
and thoroughly
mix the A and AÃ
. Suppose that we then isolate and purify some of the mixture
and measure its specific activity Sr. Note that Sr ¼ D=(q þ r). And so, we have
Sr ¼
D
q þ r
¼
qSq
q þ r
(4:20)
by substitution. Rearranging, we have
Sr(q þ r) ¼ qSq (4:21)
r
q
¼
Sq
Sr
À 1 (4:22)
q ¼
r
(Sq=Sr) À 1
(4:23)
The above equation is the basic equation of inverse isotope dilution analysis and
indicates that the unknown amount q of active material AÃ
can be deduced by adding r
4.11 ISOTOPE DILUTION ANALYSIS 123
grams of inactive material A to AÃ
and measuring the specific activities before
and after the addition Sq and Sr, respectively.
4.11.3 General Comments
Certain general comments can be made about the experimental techniques used in
isotope dilution analysis. First, reagents and tracers of high purity are necessary.
They must not contain any spurious activity or any unknown compounds, for the
presence of either could affect the specific activity of substances being analyzed.
Although pure reagents and tracers are generally available commercially, it would
be wise to check for contaminants before use.
One of the key steps in any isotope dilution analysis concerns the isolation and
purification of the diluted activity, plus the measurement of its specific activity.
Two techniques are usually preferred for the separation: precipitation and solvent
extraction. As a purification step, precipitation has the advantage that the precipitate
can easily be weighed at the time of separation, thereby allowing a quick determination
of the specific activity. The main problem with the use of precipitation techniques
involves the occurrence of co-precipitation phenomena, in which unwanted
materials are precipitated along with the desired substance, thus altering the sample
specific activity. Precipitation techniques are used for the isolation of inorganic
components.
Solvent extraction is a frequently employed technique in isotope dilution analysis.
It gives very clean separations, resulting in high-purity samples. It has the
disadvantage of requiring further chemical processing to determine the mass of
material isolated and the specific activity.
One must be aware of the possible occurrence of certain problems in isotope
dilution analysis. One of these is incomplete isotopic exchange, in which the
active and inactive atoms do not mix. This lack of exchange can be due to differing
physical and chemical states of tracer and inactive materials. Steps must be taken to
ensure complete exchange. One must also be sure that the labeled position in any
compound is relatively inert. If the atom in question is very labile, one can get a
reduction in specific activity without any dilution having taken place. To compare
specific activities, all samples must be counted under identical conditions with
proper corrections for self-absorption in samples of varying mass.
In summary, we can say that isotope dilution analysis is a highly sensitive, selective
analytical method capable of high precision. It offers the opportunity to determine
the amount of material present in a system without the need for a quantitative
separation of the material from the system. The applications of isotope dilution
analysis cited in the literature are myriad. Perhaps the best summary of these applications
is the book by Tolgyessy, Braun, and Krys (1972).
4.11.4 Special IDA Techniques
Substoichiometric isotope dilution analysis was first developed by Ruzicka and
Stary (1968) as another variation on the basic IDA technique. The basic idea of
124 RADIOTRACERS
substoichiometric IDA is to isolate equal but substoichiometric amounts of both the
diluted and the undiluted substance being analyzed and count these samples. Since
the mass of the samples is the same, the specific activities in Equations (4.3) and
(4.7) can be replaced by the activities. In this way, the sometimes tricky task of
measuring the specific activities is avoided. The key to the technique is obviously
whether the analyst can isolate exactly equal quantities of both diluted and undiluted
samples.
Another variant on the basic isotope dilution technique is that of double isotope
dilution, or variants thereof, as first proposed by Block and Anker (1948). It is used
in reverse IDA where the specific activity of the original unknown radioactive
material AÃ
cannot be measured for some reason. Hence a second dilution is
made to determine the specific activity of the original sample.
Consider a system containing an unknown amount q of some active substance AÃ
whose specific activity AÃ
cannot be measured. Take two equal aliquots of this
unknown substance AÃ
. Add r milligrams of inactive A to one aliquot and p milligrams
of inactive A to the other aliquot. Measure the specific activities of the two
aliquots, Sr and Sp, respectively. For the first sample of specific activity Sr, we have
q ¼
r
(Sq=Sr) À 1
Sq ¼ Sr þ
r
q
Sr (4:24)
For the second sample we have
q ¼
p
(Sq=Sp) À 1
Sq ¼ Sp þ
p
q
Sp (4:25)
Setting the two expressions for Sq equal and rearranging yields
q ¼
rSr À pSp
Sp À Sr
(4:26)
A major difficulty with double isotope dilution analysis is that, because of the
double dilution, the specific activities involved become low and therefore more
uncertain.
4.12 RADIOMETRIC TECHNIQUES
The central idea in all radiometric techniques of analysis is to have a radioactive
reagent RÃ
of known activity combine quantitatively with some unknown amount
of material U to form a radioactive addition product RÃ
U. By measuring the activity
of the product RÃ
U, the original amount of unknown material U is deduced. The
advantages of such techniques are the high sensitivity due to the use of radioactivity
and the requirement that the product RÃ
U need not be chemically pure. All that is
required is the RÃ
U not contain any spurious radioactivity. The disadvantages of
these techniques are that the reaction between RÃ
and U must be quantitative, and
4.12 RADIOMETRIC TECHNIQUES 125
there must not be another species in the system besides U that will react with RÃ
.
Several discussions of radiometric techniques are in the literature, including the
monograph on radiometric titrations by Braun and Tolgyessy (1948) and the
survey of radiometric techniques by Bowen (1969).
One of the radiometric techniques is precipitation with a radioactive reagent.
For example, the halides can be precipitated with 110
Ag, silver precipitated with
131
I, the sulfates and chromates precipitated with 212
Pb or 131
Ba, while Al, Be, Bi,
Ga, In, Th, U, Zr, and the rare-earth elements can be precipitated with 32
PO3À
4 . This
method suffers from the lack of selectivity and the need to make careful corrections
for self-absorption in counting the samples.
One of the most popular radiometric techniques is that of radiometric titrations.
In a radiometric titration, the unknown is titrated with a radioactive reagent, and the
radioactivity of the product or supernate, formed by the chemical reaction of the
reagent and the unknown material, is monitored as a function of titrant volume to
determine the endpoint. This means that the radioactive component being followed
must be isolated during the titration and its activity measured—that is, a discontinuous
titration.
The classic example of a radiometric titration concerns the titration of some
unknown material with a radioactive reagent to give a radioactive precipitate. In
this case, the activity of the supernatant or the precipitate can be followed as a function
of titrant volume, as shown in Figure 4.10. In this type of titrations, the tracer
must have a long half-life and must emit high-energy b2
or g rays so as to minimize
self-absorption corrections (assuming, as is common practice, that the supernatant or
precipitate is removed from the system and counted in an external sample counter
after the addition of each volume of titrant).
Figure 4.10 Radiometric titration of 110
AgNO3 with NaCl.
126 RADIOTRACERS
PROBLEMS
1. Compute the amount of a radionuclide necessary to perform an experiment with a
sample count rate of 1000 cpm, a detector efficiency of 33%, a sample aliquot for
counting consisting of 10% of the total isolated sample and where the percent
incorporation of the nuclide into the total isolated sample was 0.5%.
2. Isotope X, with a half-life of 5 d, is to be used in an experiment that includes the
following factors: (a) sample count rate of 100 cpm, (b) detector efficiency of
10%, (c) assume the sample with the lowest count rate will represent a 0.5%
incorporation, and (d) assume all samples will represent only 5% of the total
isotope administered. What amount of X must be used?
3. Three tracers, 90-y 151
Sm (0.076 MeV b2
, 100% of the disintegrations and 0.022
MeV X-ray, 4%), 244.3-d 65
Zn (0.33 MeV b2
, 1.7% of disintegrations and 0.511
MeV g rays, 3.45), and 14.3-d 32
P (1.71 MeV b2
, 100% of the disintegrations)
will be used simultaneously in a multitracer experiment. Suppose you wish to
measure the uptake of these three elements in the blood of a rat and the loss of
these elements to the rest of the rat’s organs and tissue from the blood. What
levels of the tracer will you inject into the rat? Why? (Assume that you will withdraw
0.1-mL blood volumes every hour for 24 h. The total blood volume of an
adult rat is about 15 mL.)
4. A 10-mL sample of blood is withdrawn from a patient, and the red cells are
labeled with 51
Cr, a 27-d g emitter. One milliliter of the labeled blood diluted
to 15 mL with water gave a net counting rate of 33,000 cpm (background corrected).
The remaining labeled blood is injected back into the patient, and after
several hours 10 mL of blood is withdrawn and counted as before. The net counting
rate (background corrected) was 500 cm. What is the total volume of the
patient’s blood?
5. Isotope dilution analysis is applied to the following analysis. Calculate the
amount of the compound Y present in the sample and express your answer as
percent by weight. A 1-g sample is analyzed for compound Y, molecular
weight of 150. A derivative is formed of compound Y and the added radioactive
Y (1.5 mCi at a specific activity of 3 mCi/mmol). The derivative, molecular
weight of 150 (1 mol of compound Y per mole of derivative), is recrystallized
until pure. It has a specific activity of 4.44 Â 103 dpm/mg.
6. A 5-kg batch of crude penicillin was assayed by isotope dilution analysis: To a
1-g sample of the batch was added 10 mg of pure penicillin having an activity
of 10,500 cpm; only 1.40 mg of pure penicillin having an activity of 290 cpm
was recovered. What is the penicillin content of the batch?
7. Isotope dilution analysis permits one to determine the purity of a radiochemical.
Compound X, molecular weight of 150 (specific activity 1.0 mCi/mmol), was
checked for purity by carefully weighing 1.5 mg of the radiochemical and
mixing with 1000 mg of unlabeled compound X and recrystallizing until a
PROBLEMS 127
constant specific activity. Radioassay gave a value of 2500 dpm/mg. What was
the purity of the radiochemical in percent?
REFERENCES
Atkins, D. H. F. and A. E. J. Eggleton. In Nuclear Techniques in Environmental Pollution,
IAEA, Vienna, 1971, pp. 521–534.
Barry, P. J. Nuclear Techniques in Environmental Pollution, IAEA, Vienna, 1971,
pp. 241–255.
Block, K. and H. S. Ancker. Science 107, 228 (1948).
Bowen, H. J. M. Chemical Applications of Radioisotopes, Methuen, London, 1969.
Downs, A. M. and G. M. Harris. J. Chem. Phys. 20, 196 (1952).
Ehmann, W. D. and D. E. Vance. Radiochemistry and Nuclear Methods of Analysis, Wiley,
New York, 1991.
Gilath, C. and Z. Stuhl. In Nuclear Techniques in Environmental Pollution, IAEA, Vienna,
pp. 483–496.
Kraus, K. A. and F. Nelson. Ann. Rev. Nucl. Sci. 7, 31 (1957).
Ruzicka, J. and J. Stary. Substoichiometry in Radiochemical Analysis, Pergamon, Oxford,
1968.
Sunderman, D. N. and W. W. Meinke. Anal. Chem. 29, 1578 (1957).
Tolgyessy, J., T. Braun, and T. Kyrs. Isotope Detection Analysis, Pergamon, Oxford, 1972.
Volpin, M. E. et al. Zh. Obshch. Khim. 29, 3711 (1959).
BIBLIOGRAPHY
Braun, T. and J. Tolgyessy. Radiometric Titrations, Pergamon, Oxford, 1967.
Choppin, G. R., J. Rydberg, and J. O. Liljenzin. Nuclear Chemistry, 2nd ed., An excellent
treatment that emphasizes the practical aspects of nuclear chemistry.
Collins, C. J. and O. K. Nevill. J. Am. Chem. Soc. 73, 2471 (1951).
Downs, J. and K. E. Johnson. J. Am. Chem. Soc. 77, 2098 (1955).
Groh, J. and G. Hevesy. Ann. Phys. 65, 216 (1921).
Lieser, K. H. Nuclear and Radiochemistry, VCH, New York, 1997. An excellent account of
the modern aspects of radiochemistry.
Petti, P. L. and A. J. Lenox. Ann. Rev. Nucl. Sci. 44, 155 (1994). Excellent survey of radio-
therapy.
Wang, C. H., D. L. Willis, and W. Loveland. Radiotracer Methodology in the Biological,
Environmental and Physical Sciences, Prentice-Hall, Englewood Cliffs, NJ, 1975. Much
of this chapter has been summarized from this book.
Welch, M. J. J. Chem. Ed. 71, 830 (1994). Up-to-date survey of nuclear medicine.
128 RADIOTRACERS
CHAPTER 5
NUCLEAR FORCES
5.1 INTRODUCTION
In Chapter 1, we discussed the four forces of nature, the electromagnetic, the strong
(nuclear), the weak, and the gravitational force. In dealing with the structure, reactions,
and decay of nuclei, we shall be dealing with the electromagnetic, strong, and weak interactions.
The principal force we shall concern ourselveswith is thestrong or nuclear force.
In this chapter, we shall summarize some important features of the nuclear force.
One basic characteristic of all the fundamental forces is their exchange character.
They are thought to operate through the virtual exchange of particles that act as force
carriers. What do we mean by the term virtual? We mean that the exchange particles
only exist for a short time consistent with the Heisenberg uncertainty principle and
cannot be detected experimentally.
How is this possible? Consider the familiar electromagnetic interaction. Two
charged particles can be imagined to interact electromagnetically by the emission
of virtual photons that are continuously emitted and absorbed by the particles
(i.e., exchanged). The Heisenberg uncertainty principle tells us that
DE † Dt ! hÀ
or that we can “violate” the law of conservation of energy by an amount of energy
DE for a time Dt such that
Dt % hÀ
=DE
129
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
(The emission of a virtual photon by a charged particle violates the law of conservation
of energy by the photon energy DE.) If this photon is traveling at the
speed of light, it can travel a distance R such that
R ¼ c Dt % hÀ
c=DE ¼ hÀ
c=En
where En is the photon energy.
If the exchanged particle is not a photon but has mass m, its minimum energy is
its rest mass mc2
, so
Dt hÀ
=mc2
and the range R of the interaction is
R hÀ
=mc
The exchange particles are the graviton for the gravitational force, the pion for
the strong interaction between nucleons, the photon for the electromagnetic force,
and the W+
and Z bosons for the weak interaction. For an exchange particle of
zero mass (the photon), the range of the force is essentially infinite. In the case of
the strong interaction between nucleons, the range of the force is less than 1.4 fm,
so mexchange ! 140 MeV/c2
. In the case of the weak interaction, the exchange
particles, the W+
and Z bosons have masses m % 90 GeV/c2
, so R % 1023
fm.
When dealing with atoms and molecules and their interactions, one is dealing
primarily with the electromagnetic interaction, which is well known. In principle,
the problems of atomic and molecular structure are thus soluble, albeit sometimes
with a great deal of mathematical complexity. For the nuclear or strong interaction
that is not the case. Although we know much about nucleons and their interactions,
there are some features of the nuclear force that are poorly understood even today.
Since each nucleon is a composite particle, it is not surprising that the interaction
between nucleons is complicated. Nonetheless, an exploration of some of the features
of the nuclear force will greatly aid us in understanding nuclear phenomena.
5.2 CHARACTERISTICS OF THE STRONG FORCE
As discussed earlier, the range of the nuclear force R is thought to be short with
R 1.4 fm. What evidence do we have for this? The fact that the strong force
plays no role in atomic or molecular structure restricts its range to less than the
nuclear radius. In our discussion of the semiempirical binding energy equation,
we showed that nuclear forces “saturate” and that nucleons only interact with
their nearest neighbor. Thus, the range of the nucleon–nucleon interaction must
be of the order of the size of a nucleon, that is, a few femtometers (10215
m).
We know that the nuclear force is strongly attractive, binding nucleons
together to form a densely packed nucleus. Experiments involving the scattering
130 NUCLEAR FORCES
of high-energy particles from nuclei have shown the nuclear force also has a repulsive
core. What we mean by this statement is that below some value of the separation
between nucleons (0.5 fm), the nuclear force becomes repulsive instead of attractive.
(This feature, due to the quark substructure of the nucleon, prevents the nucleus
from collapsing on itself.)
The simplest bound nuclear system, the deuteron, consists of a neutron and a
proton. The deuteron is known to have a quadrupole moment, 0.00286 barns,
which tells us that the deuteron is not perfectly spherical and that the force between
two nucleons is not spherically symmetric. Formally, we say the force between
two nucleons has two components, a spherically symmetric central force and an
asymmetric tensor force that depends on the angles between the spin axis of each
nucleon and the line connecting them.
The deuteron has only one bound state, a triplet angular momentum state, in
which the spins of the neutron and proton are parallel, adding to make a I ¼ 1
state. The singlet 1
S state in which the nucleon spins are antiparallel is unbound.
Thus, the nuclear force is spin dependent. Also we shall see that the nuclear force
depends on the coupling of the nucleon spin and nucleon orbital angular momentum.
The deuteron magnetic moment, 0.857 mn, is close to the sum of the neutron
(21.913) and proton magnetic moments (2.793). Detailed studies show a small
portion (4%) of the time, the neutron and proton are in a 3
D state (L ¼ 2,
S ¼ 1, I ¼ 1) rather than ground state 3
S configuration (L ¼ 0, S ¼ 1, I ¼ 1).
Using the relationship between force and potential energy discussed earlier, we
can represent the nuclear force in terms of a simple plot of the nuclear potential
energy as a function of distance to the center (Fig. 5.1). Since low-energy particles
+V0
d
d << R
r = R
RangeæR
B/AæV0
r
-V0
0
V(r)
Figure 5.1 Schematic representation of the radial dependence of the nuclear potential energy.
5.2 CHARACTERISTICS OF THE STRONG FORCE 131
cannot probe the interior of nucleons or the nucleus, we can usually ignore the repulsive
core in most problems involving low-energy nuclear structure and just use a
square well potential (V ¼ 2V0 for r , R, V ¼ 0 for r . R). Occasionally, the
Yukawa form of the potential is used where V ¼ 2V0 exp(2r/R)/(r/R) or the
Woods–Saxon form where V ¼ V0={1 þ exp½(r À R)=aŠ}. The typical values of R
for these potentials are 1.5–2 fm with V0 ¼ 30–60 MeV. Important additional
components of the nuclear force are discussed as they become important in our
discussions of nuclear structure in Chapter 6.
5.3 CHARGE INDEPENDENCE OF NUCLEAR FORCES
The nuclear force between two nucleons is charge independent. By this we mean
that the strong interaction between two protons or two neutrons or a neutron and
a proton is the same. (Of course, there will be differing electromagnetic forces in
these cases.) Evidence for the charge independence of nuclear forces can be
found in nucleon–nucleon scattering and in the binding energies of light mirror
nuclei (Table 5.1). (Mirror nuclei are isobars where the number of protons in one
nucleus is equal to the number of neutrons in the other nucleus and vice versa.)
In Table 5.1, we tabulate the total nuclear binding energy of some light mirror
nuclei, the difference in Coulomb energy between the nuclei, and the resulting net
“nuclear” binding energy. The latter quantity is remarkably similar for these
mirror nuclei, supporting the idea of charge independence of nuclear forces.
Example Problem Consider the mirror nuclei 25
Mg and 25
Al. What is the
energy difference between their ground states?
Solution Note the “conversion” of 25
Mg into 25
Al will involve the change of
one neutron into one proton. The neutron and proton have slightly different
masses, of course. The extra proton will interact electromagnetically with the
other 12 protons giving a second term in the energy difference:
DE ¼ E(A, Z þ 1) À E(A, Z) ¼ DEcoul À (mn À mH)c2
DEcoul ¼ 6
5 Ã e2
=R Ã Z ¼ 6
5 (1:44=1:2 Â 251=3
)12
(mn À mH)c2
¼ 0:782 MeV
DE ¼ 5:910 À 0:782 ¼ 5:128 MeV
So we would expect the ground state of 25
Al to be 5.128 MeV above the ground
state of 25
Mg.
The observation of the masses of mirror nuclei suggests the strong or nuclear
force between a neutron and a proton is the same. This equivalence leads naturally
to considering the neutron and the proton as corresponding to two states of the same
particle, the nucleon. (A similar situation holds for the p meson, where the p0
, pþ
,
132 NUCLEAR FORCES
and p2
mesons show the same strong force behavior.) To express this idea, we say
there is a quantum number T for the nucleon (or the p meson) called the isospin. In
analogy to spin angular momentum, we say that for the nucleon T ¼ 1
2, and in this
hypothetical isospin space there are two projections of T, T3 ¼ þ 1
2 (the proton)
and T3 ¼ À 1
2 (the neutron). (An alternate notation system refers to the isospin projection
as Tz .) For a system with isospin T, there are 2T þ 1 members of the isospin
multiplet. In a nucleus of N neutrons and Z protons,
T3 ¼ (Z À N)=2
For even nuclei, 0 T A/2, while for odd nuclei, 1
2 T A=2.
Isospin is a useful concept in that it is conserved in processes involving the strong
interaction between hadrons. The use of isospin can help us to understand the
structure of nuclei and forms the basis for selection rules for nuclear reactions
and nuclear decay processes. While a detailed discussion of the effects of isospin
upon nuclear structure, decay, and reactions is reserved for later chapters, a few
simple examples will suffice to demonstrate the utility of this concept.
Consider the A ¼ 14 isobars, 14
C, 14
N, and 14
O. 14
C and 14
O are mirror nuclei and
have ground states with T3 ¼ +1. As such they must be part of an isospin triplet
with T ¼ 1 (T3 ¼ 0, +1). Thus, in the T3 ¼ 0 nucleus, 14
N, there must be a state
with T ¼ 1, T3 ¼ 0 that is the analog of the T3 ¼ 0 ground states of 14
C and 14
O.
(See Problems section for further details.) We expect the three members of this
multiplet to have approximately the same energy levels after correction for the
Coulomb effect and the neutron–proton mass difference.
In heavy nuclei, the Coulomb energy shift between members of an isospin
multiplet can be large due to the large number of protons in the nucleus. Thus,
the isobaric analog of the ground state of one member of an isospin multiplet
may lie at several MeV excitation. When Fox et al. (1964) were doing routine excitation
function measurements for the 89
Y(p,n) 89
Zr reaction, which essentially
converts a neutron in the traget nucleus into a proton, they observed two sharp
peaks in the neutron yields near Ep ¼ 5 MeV, as shown in Figure 5.2. This
observation was unexpected, as the reaction was populating levels in the 90
Zr
TABLE 5.1 Properties of Light Nuclei
A Nucleus
Total Binding Energy
(MeV)
Coulomb Energy
(MeV)
Net Nuclear
Binding Energy (MeV)
3 3
H 28.486 0 28.486
3
He 27.723 0.829 28.552
13 13
C 297.10 7.631 2104.734
13
N 294.10 10.683 2104.770
23 23
Na 2186.54 23.13 2209.67
23
Na 2181.67 27.75 2209.42
41 41
Ca 2350.53 65.91 2416.44
41
Sc 2343.79 72.84 2416.63
5.3 CHARGE INDEPENDENCE OF NUCLEAR FORCES 133
compound nucleus at an excitation energy of 10 MeV where the spacing between
levels was small and no states were known that produced such large resonances.
Angular distributions quickly showed the (J, p) of these states were 22
and 32
.
It was pointed out that the ground and first excited state of 90
Y had 22
and 32
and were separated by 200 keV. Calculations of the Coulomb energies showed
these resonances to correspond to the isobaric analogs of the ground state and
first excited state of 90
Y. Their yields were enhanced because they represent
particularly simple nuclear configurations in contrast to the normal states found at
excitation energies of 10 MeV.
PROBLEMS
1. Make a table such as Table 5.1 showing the total binding energy, the Coulomb
energy, and the net nuclear binding energy for 14
C, 14
O, and the 2.31-MeV
level of 14
N.
4.25
1
2
NumberofNeutronsper100mC(x1,000)
3
4
5
6
7
8
9
10
4.50
.588 Threshold
Incident Proton Energy (MeV)
4.75 5.00 5.25 5.50
Y89(p,n)
.391 Threshold
Sr88(p,n)
Figure 5.2 Neutron yields vs. proton energy for the reactions shown. Reprinted from J. D.
Fox, C. F. Moore, and D. Robson, Phys. Rev. Lett. 12, 198 (1964). Copyright 1964 by the
American Physical Society.
134 NUCLEAR FORCES
2. If the difference in energy of the ground state of 14
C and the T ¼ 1 analog
of the 14
C ground state in 14
N is 2.15 MeV is due to the Coulomb energy
difference between the nuclei, calculate an average radius R for these
A ¼ 14 nuclei.
3. For a Yukawa nuclear potential with V0 ¼ 40 MeV and r0 ¼ 1.5 fm, calculate the
ratio between the nuclear and Coulomb potential for r ¼ 1, 2, 4, 8, and 16 fm.
REFERENCE
1. Fox, J. D., C. F. Moore, and D. Robson. Phys. Rev. Lett. 12, 198 (1964).
REFERENCE 135
CHAPTER 6
NUCLEAR STRUCTURE
Nuclei have a very regular structure with many general and simple properties that
are predicted by quantum mechanical treatment of particles moving in a potential
well. This situation is very similar to descriptions of the electrons in atoms. The
potential energy well due to the Coulomb force (atoms) is substantially different
from that due to the nuclear force (nuclei). The previous chapter contained a discussion
of the basic properties of the nuclear force and how we have been able to determine
its features. In summary, the exact form of the nuclear or strong force is
unknown, but the force is known to be short ranged (%l fm) with a repulsive core
and is known to saturate. That is, the force acts primarily between nearest neighbors
to hold them together without letting them interpenetrate one another. We use these
features to form the basis of our study of nuclear structure.
We can also learn a great deal about the basic features of nuclear structure and the
nature of the force that holds the nucleus together if we simply look carefully at the
lightest stable and unstable nuclei. The building blocks of nuclei are the nucleons,
protons, and neutrons, of course. The proton is stable and is usually found as a
hydrogen atom bound to a single electron. The mass of the electron is small compared
to that of the proton (511 keV/939,000 keV or 1/1800) and the binding
energy is even smaller (13 eV/939,000,000 eV or 1028
). The electrons are
almost always carried along by the nuclei, so it is most convenient to imagine building
nuclides up from hydrogen atoms, 1
H, rather than bare protons. On the other
hand, the free neutron is unstable and decays with a half-life of 10 min into a
proton, an electron, and an antineutrino. Thus, imagining that we will construct
137
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
nuclei from these constituents, we should not expect to be able to make arbitrary
heavy isotopes of any given chemical element because eventually the neutrons
will decay as if they were independent.
If we now bring together two nucleons, we find a rather important and interesting
fact, only one combination produces a stable (bound) nucleus. One proton and one
neutron will combine to form a deuteron, or one hydrogen atom plus one neutron
will form a deuterium atom with its atomic electron. Both of the other combinations,
two protons that can be labeled 2
He and two neutrons, are unbound and come apart
almost as rapidly as the constituents come together. It is easy to see that the diproton,
or 2
He, is more unstable than the dineutron due to the Coulomb repulsion between
the two positively charged protons. Thus, we find a preference for equal numbers of
neutrons and protons even in the smallest nucleus.
If we look more carefully at the deuteron, we expect that there should be two
possible combinations of the spins of the two nucleons. Both the proton and
neutron have S ¼ 1
2, and we can have the parallel combination Sp þ Sn ¼ 1 and
the antiparallel combination Sp þ Sn ¼ 0. Both of these states exist in a deuterium
nucleus, and the S ¼ 1 state is the ground state (lowest energy), and the S ¼ 0
state is an excited state and is, in fact, unbound. Therefore, the alignment of the
spins of the two unlike nucleons has an important effect on the total binding
energy. This provides part of the explanation as to why the dineutron is unbound.
Notice that the intrinsic spins of two neutrons in an l ¼ 0, or s, state must be
paired (according to the Pauli principle). However, the nuclear force prefers the parallel
alignment. In order to align the spins in the same direction, the neutrons have to
be in a (l ¼ 1, or p, state, which requires more relative energy. In addition, the fact
that the deuteron is not spherical having an intrinsic electric quadrupole moment
tells us that there is a noncentral component of the nuclear force.
We can continue our survey of the lightest nuclei with A ¼ 3. Only the combinations
of two protons and one neutron, 3
He, and one proton with two neutrons,
3
H, are bound, while the combinations of three protons, 3
Li, and three neutrons
are unbound. Again we see a balance between the numbers of neutrons and
protons with the extreme cases being unbound. The nuclear spins of both bound
A ¼ 3 nuclei are 1
2 indicative of a pair of nucleons plus one unpaired nucleon;
three unpaired nucleons would have had a total spin of 3
2. In the A ¼ 3 system the
more neutron-rich nucleus, tritium, 3
H, is very slightly less stable than 3
He and, it
decays by b2
emission with a 12.3-y half-life.
Only one combination of four nucleons is bound, 4
He, with two protons and two
neutrons. All other combinations of four nucleons are unbound. Moreover, 4
He, or
the a particle, is especially stable (very strongly bound), and the nucleons are paired
to give a total spin S ¼ 0. Interestingly, if we add a nucleon of either type to the a
particle, we produce an unbound nucleus! Thus, there are no stable nuclei with
A ¼ 5 as both 5
He and 5
Li break apart very rapidly after formation. This creates a
gap in the stable masses and poses a problem for the building up of the elements
in stars, which is discussed in Chapter 12. There are two bound nuclei with
A ¼ 6, 6
He and 6
Li, with the helium isotope decaying into the lithium isotope, the
others are unbound. Continuing on, between mass 6 and 209, all mass numbers
138 NUCLEAR STRUCTURE
have at least one bound nucleus although there are no stable nuclei at A ¼ 8. There is
generally one stable nucleus for each odd mass number, and in heavy nuclei there are
often two stable nuclei for each even mass number. There are at most three stable
isobars for a given mass number.
We can summarize our observations about light nuclei and the nuclear force as
follows: The nuclear force acts between nucleons in a uniform way, protons have
an additional Coulombic repulsion that can destabilize proton-rich nuclei, but
very neutron-rich nuclei are also unstable. The symmetric nuclei with equal
numbers of neutrons and protons are favored (at least in light nuclei); and finally
the nuclear force depends on the spin alignment of the nucleons. Because the underlying
nature of the nuclear force is unknown at present, several parameterizations of
an effective force have been developed. A detailed discussion of these effective
forces or equivalently the nucleon potentials is beyond the scope of this book.
Now imagine the complexity of describing a nucleus in which each nucleon is interacting
with its nearest neighbors through the nuclear force, and at the same time all
the protons are pushing on each other with the Coulomb force! This problem and the
closely related problem of molecular motion in a liquid drop have not been solved in
detail yet, and so we will present models of the average behavior of the nucleons in
effective energy potentials.
6.1 NUCLEAR POTENTIALS
The combined interactions of the neutrons and protons can be described in terms of a
“nuclear potential well.” Because the protons are charged particles, we generally
treat the neutrons and protons as if they move inside separate potential wells (superimposed
on one another). It is useful to imagine in a very schematic and simple way
the forces that would act on a neutron as it is brought up to a nucleus. At large distances
(.few femtometers) then there will be no force (no change in the potential
energy). When the neutron reaches the “surface” of the nucleus (or comes within
the range of the nuclear force 1 fm from the “edge”), there will be an attraction
from the nearest-neighbor nucleons, and the neutron will be pulled into the
nucleus. This attraction will increase rapidly in the surface region as the nucleon
comes in contact with other nucleons until it is surrounded by nucleons and is in
the interior of the nucleus. The potential energy will stay approximately constant
if the neutron is moved inside the nucleus and is not near the edge. This behavior
is summarized in the potential energy function shown as a function of distance
from the center of the nucleus shown on the left side of Figure 6.1.
On the other hand, if we bring a proton up to the same nucleus, we will have a
slightly different behavior. At first, the nucleus will repel the proton due to the
long-range Coulomb force. Then, as we bring the proton very near to the surface,
the same nuclear attraction will begin to overcome the repulsion. The nuclear attraction
will increase until the proton is surrounded by nucleons as in the neutron case,
but there will always be a net repulsion from the other protons. The repulsion
decreases the overall attraction, and the proton potential energy well will not be
6.1 NUCLEAR POTENTIALS 139
as deep as the neutron well. The models that we will describe later in this chapter
will rely on the ideas behind these simple schematic potentials.
Before going on to describe the models of nuclear structure in detail, it is useful to
make a short comparison of the characteristics of the atomic and the nuclear potential
energies. The atomic potential is in some sense, easier to describe because it is
created by a central nucleus that can be ignored in almost all atomic calculations.
The nucleus supplies the overall attraction for the electrons, but it does not interact
with the electrons. In the nuclear case the potential is created by the nucleons
themselves, and, if we disturb the nucleons (add or subtract one), then the overall
potential will have to be readjusted. Fortunately, the changes in the potential
energy for a large nucleus are often relatively small and the general behavior of
the whole nucleus remains the same. There can be, of course, major disruptions
that cannot be described with a simple potential energy.
6.2 SCHEMATIC SHELL MODEL
With a general understanding of the form of nuclear potentials, we can begin to solve
the problem of the calculation of the properties of the quantum mechanical states
that will “fill” the energy well. One might imagine that the nucleons will have
certain finite energy levels and exist in stationary states or orbitals in the nuclear
well similar to the electrons in the atomic potential well. This interpretation is
Potential
energy
Center
of nucleus +
n
Distance –
0
Potential well
Nucleus with
radius = R
R
P
Distance
Figure 6.1 Schematic representations of a general neutron–nucleus potential and a proton–
nucleus potential as a function of radius.
140 NUCLEAR STRUCTURE
quite valid and forms the basis of the “shell model” of the nucleus. The potential
well for nucleons has a very different shape from that for atomic electrons and so
we should expect that the energy levels and their filling patterns would be different.
As a very first approximation we could model the nucleus as a spherical rigid
container (also called a square-well potential). The potential energy is assumed to
be exactly zero when the particle is inside the walls of the container and the walls
are so strong and high that the particle can never get out. An analogy would be a
gaseous atom inside a very small spherical balloon. The energy levels for a particle
in such a potential well are shown in Figure 6.2. We could compare these energy
levels to the known nuclei, but the potential is so unrealistic that we would not
expect to have much success. For example, notice that this potential goes to infinity
at the edge of the nucleus, but the nuclear potential felt by a neutron goes to zero at
the edge.
A much more useful potential is the harmonic oscillator potential, which has a
parabolic shape. As indicated in Figure 6.2, this potential also has steep sides that
continue upward and will be useful only for the low-lying energy levels. The harmonic
oscillator potential has the feature of equally spaced energy levels. This potential
does not “saturate,” rather it has a rounded bottom and so will not be very good for
large nuclei with large central volumes. Nevertheless, the harmonic oscillator potential
is used extensively for light nuclei, and harmonic oscillator wave functions are
often used in reaction calculations. The harmonic oscillator states are labeled by
their total angular momentum starting at 0. Each principal quantum number level
is said to form a shell of orbitals. The energy gap between each shell will be
exactly the same, and all the sublevels with a given principal quantum number
will be degenerate. The number of orbitals is given by the expression 2N þ 1
where N ¼ 0, 1, 2, . . . . The Pauli principle states that the number of nucleons
(fermions) needed to fill each orbital is 2, as for electrons in atomic orbitals, so
the number of nucleons needed to fill the shells are 2 [2N þ 1] ¼ 2, 6, 10, . . . .
This filling agrees with the enhanced stability of the lightest nuclei (4
He, 16
O),
taking the neutrons and protons in separate orbits, but does not agree with that of
heavier nuclei.
A dramatic improvement was made to the simple harmonic oscillator potential by
the addition of a spin–orbit correlation. It is known that relativistic particles have a
tendency to align their orbital and intrinsic angular momenta (spins). This alignment
is the basis of the familiar change in the chemistry of the bottom-row elements in the
periodic table. For example, thallium favors the 1þ oxidation state even though
thallium is the heaviest member of Group 13 or IIIA. This comes about because
the three atomic p states separate (or split apart) into two groups in energy according
to the alignment of the orbital (l ¼ 1) and intrinsic spin (s ¼ 1
2 ) angular momentum.
The p1/2 state with the spin and angular momentum coupled in opposite directions
comes lower in energy and holds two electrons, while the third electron lies in the
p3/2 state and is easily ionized.
The addition of the spin–orbit term to the nuclear harmonic oscillator potential
causes a separation or removal of the degeneracy of the energy levels according
to their total angular momentum ( j ¼ l þ s). In the nuclear case, the states with
6.2 SCHEMATIC SHELL MODEL 141
the parallel coupling and larger total angular momentum values are favored and
move lower in energy than those with smaller total spin values for a given combination.
The ordering of the energy levels from a spin–orbit/harmonic oscillator
shell model is shown in Figure 6.3 with their spectroscopic notation. Each total
> >
O
–Vo
[168]
[112]
[70]
[40]
[20]
[8]
[2]
Harmonic
Oscillator
Square
Well with
Rounded
Edges
1s (2)
1p (6)
1d (10)
2s (2)
1f (14)
2p (6)
1g (18)
2d (10)
3s (2)
1h (22)
2f (14)
3p (6)
1i (26)
2g (18)
3d (10)
4s (2) 4s1/2 (2)
3d3/2 (4)
2g7/2 (8)
3d5/2 (6)
1i11/2 (12)
2g9/2 (10)
[126]
[82]
[50]
[28]
[20]
[8]
[2]
3p1/2 (2)
2f5/2 (6)
1i13/2 (14)
3p3/2 (4)
2f7/2 (8)
1h9/2 (10)
3s1/2 (2)
1h11/2 (12)
2d3/2 (4)
2d5/2 (6)
1g7/2 (8)
1g9/2 (10)
2p1/2 (2)
2p3/2 (4)
1f5/2 (6)
1f7/2 (8)
2s1/2 (2)
1d3/2 (4)
1d5/2 (6)
1p1/2 (2)
1p3/2 (4)
1s1/2 (2)
With
Spin-Orbit
Coupling
–Vo
O
r r
6 hw
5 hw
3 hw
2 hw
1 hw
0 hw
4 hw
Figure 6.2 Energies of single-particle orbitals in harmonic-oscillator and “rounded squarewell”
potentials, the latter with and without spin–orbit coupling. Numbers in parentheses
indicate orbital capacities and those in square brackets give cumulative capacity up to the
given point. [Reproduced by permission from Gordon and Coryell (1967).]
142 NUCLEAR STRUCTURE
angular momentum state has 2j þ 1 suborbitals or projections of the angular momentum
(mj ¼ 2j, . . . , 0, . . . , þ j) just like the l values of atomic electrons. Recall that
we always have separate neutron states and proton states, and the Pauli principle will
put a maximum of two neutrons or protons into each orbital.
4s
3p
2f
1h
3s
2d
1g
(16) [184] 184
[126]
[100]
126
[82]
[64]
(10)
(2)
(6)
(4)
(8)
(4)
(2)
(6)
(2)
(4)
(2) [2]
[6]
[8]
[14]
[16]
[20]
[28]
[38]
[40]
[50]
20
28
50
8
2
82
(4)
(2)
(8)
(12)
(6)
(10)
(14)
(2)
(4)
(6)
(8)
(10)
(12)
(2)
(4)
(6)
(8)
1j
15/2
3d3/2
4s
1/2
3p
1/2
3s1/2
3p
3/2 2f
5/2
2d3/2
2d
5/2
1g
7/2
1g
7/2
2p
1/2
2p
1/2
1f
5/2
1f
7/2
1d3/2
2s
1/2
1d
5/2
1p
1/2
1p
3/2
1s
1/2
1s
1p
0
1d
2s
1f
2p
2f
7/2
1h9/2
1h11/2
2g7/2
2g
9/2
1i11/2
1i13/2
3d5/2
3d
2g
1i
6 hw
even
5 hw
odd
4 hw
even
3 hw
odd
2 hw
even
1 hw
odd
even
–
Figure 6.3 Energy level pattern and spectroscopic labeling of states from the schematic shell
model. The angular momentum coupling is indicated at the left side and the numbers of nucleons
needed to fill each orbital and each shell are shown on the right side. From M. G. Mayer and
J. H. D. Jenson, Elementery Theory of Nuclear Shell Structure, Wiley, New York, 1955.
6.2 SCHEMATIC SHELL MODEL 143
Let us consider placing nucleons into these shell model states. The lowest level is
called the 1s1/2, s for l ¼ 0, and j ¼ l þ s ¼ 1
2. This level has only 2l þ 1 ¼ 1 m value
and can hold only two protons in the proton well and two neutrons in the neutron
well. The next levels are the 1p3/2 and 1p1/2 pair in the next highest shell
(N ¼ 1hÀ
v). Thus, 4
He represents the smallest nucleus with exact filling of both
N ¼ 0 harmonic oscillator shells for neutrons and protons and might be expected
to have an enhanced stability. The next shell filling occurs when the N ¼ 0hÀ
v
and N ¼ 1hÀ
v shells are filled. This requires eight protons and eight neutrons, so
16
O should be an especially stable nucleus. The other shell closures occur at 20,
28, 50, 82, and 126 nucleons. These values correspond to places in the nuclidic
table with unusually large numbers of isotopes and isotones due to their enhanced
stability. A few stable nuclei have both closed neutron and proton shells and are
very strongly bound (relative to their neighbors), such as 4
He, 16
O, 40
Ca, 48
Ca,
and 208
Pb. A few doubly closed shell nuclei have been produced outside the
range of stable nuclei such as 56
Ni, 100
Sn, and l32
Sn, and others have been sought
such as 10
He and 28
O but have been shown to be unbound.
Example of Shell Model Filling: 7
Li
1. Place the three protons into the lowest available orbital. The protons in the
1s1/2 state must be paired according to the Pauli principle, so we have a
configuration (1s1/2)2
(1p3/2)1
.
2. Place the four neutrons into their lowest available orbitals. The neutrons
should be paired in the partially filled orbital (i.e., in contrast to the case for
atomic electrons), giving a configuration of (1s1/2)2
(1p3/2)2
.
Prediction All nucleons are paired except for the 1p3/2 proton. Therefore, the spins
and angular momenta will cancel except for this proton. The nuclear spin should be
3
2
hÀ
and the nuclear parity should be negative corresponding to the parity of a p state
(odd l value).
Question What would this model predict for an excited state of 7
Li? Two possibilities
should be apparent. We could promote the p3/2 proton to the p1/2 state or we could
uncouple the p3/2 neutrons giving three unpaired neutrons in the p3/2 level. 7
Li has
only one bound excited state, and it corresponds to promotion p3/2 ! p1/2 of the
proton. The breaking of pairs has a significant energy cost and causes the nucleus
to become unbound.
Notice that the light nuclei are extremely fragile due to the large level spacing and
relatively small number of levels. The small numbers of nucleons are very sensitive
to small changes in the configurations and have relatively few excited states. Heavy
nuclei are much more “resilient” due to the large number of nearby energy levels
with slightly different configurations, and these nuclei almost always have very
large numbers of bound excited states.
144 NUCLEAR STRUCTURE
The reality of this scheme of assigning nucleons to various simple shell model
states can be checked very directly by nuclear reactions that give or take a
nucleon from the nucleus. The (p, 2p) reaction is such a reaction that removes a
proton from the nucleus. The energy required to remove a given proton is thus
a measure of the energy of the corresponding nuclear state. In Figure 6.4, we
show the results of such a study of the 16
O(p, 2p) reaction. Three peaks in the
cross section are seen corresponding to the removal of protons from the 1p1/2,
1p3/2, and 1s1/2 orbitals.
The energy level diagram for the schematic shell model, shown in Figure 6.3,
allows us to make a large number of predictions about the ground states of broad
ranges of nuclei. First, the strong pairing of nucleons in the individual orbitals
tells us immediately that the (net) spin of all nuclei with both even numbers of
protons and even numbers of neutrons will be zero. Also the parities of the wave
functions of all these nuclei will be positive. Thus, the ground-state spin and
parity of all even–even nuclei is 0þ
. These predictions are exactly correct, and
the fact that all even–even nuclei have no net nuclear spin is the reason why relatively
few nuclei can be used in NMR studies. Second, we expect that the ground
states of odd-A nuclei, those with an even number of one kind of nucleon and an
odd number of the other kind, will be described by the spin and parity of that
single odd nucleon. These predictions are often correct, particularly if we recognize
that single vacancies or holes in subshells will give the same angular momentum and
parity as a single particle in the same subshell. This equivalence of “particles” and
“holes” can be shown by detailed angular momentum coupling calculations that we
will not go into here. However, recall that a completely filled subshell will couple to
a spin of 0, so by symmetry if we add one particle to get a given j value, we should
expect to get the same spin value when we take one particle from the completely full
subshell.
60
0
50 30
16O
1
2
=38.7°
s
L =0
L
19 MeV 12.4 MeV
=1
43 MeV
Energy (MeV)
10
20
d1σ/dE2dΩ2(µbMeV–2θr–1)
40
θ
−
3
2p−
1
2p−
3
2p−
1
2s−
1
2p−
Figure 6.4 Energy spectrum of emitted protons from the 16
O(p, 2p) reaction, showing the
single-particle states. [From Tyren et al. (1958).]
6.2 SCHEMATIC SHELL MODEL 145
The shell model can also be used to predict the ground-state spins and parities of
odd-proton/odd-neutron nuclei by combining the individual jp values of the two
unpaired particles. Notice that two combinations will always be possible, and we
will need a way to decide which of the two alignments of the total-nucleon
angular momenta will be lower in energy (i.e., be the ground state). The ground
state of the deuteron with its single proton and single neutron provides the key to
this selection. The spin angular momenta of the neutron and proton are aligned in
the deuteron ground state, thus for the ground state of an odd–odd nucleus we
should couple the total j values so that the intrinsic spins of the odd particles are
aligned. We can do this by inspection of the angular momenta or by applying a
set of rules based on the systematics of the shell model orbitals. Brennan and
Bernstein (1960) have summarized these data in the form of three rules. When
the odd nucleons are both particles or holes in their respective subshells, rule 1
states that when j1 ¼ l1 + 1=2 and j2 ¼ l2 + 1=2, then J ¼ j j1 À j2j. Rule 2 states
that when j1 ¼ l1 + 1=2 and j2 ¼ l2 + 1=2, then J ¼ j j1 + j2j. Rule 3 states that
for configurations in which the odd nucleons are a combination of particles and
holes, such as 36
Cl, J ¼ j1 þ j2 þ 1.
Example Problem Consider the odd–odd nuclei, 38
Cl, 26
Al, and 56
Co. Predict
the ground-state spin and parity for these nuclei.
Solution
a. 38
Cl has 17 protons and 21 neutrons. The last proton is in a d3/2 level while the
last neutron is in an f7/2 level. (Fig. 6.3).
jp ¼ 2 À 1
2 , jn ¼ 3 þ 1
2
J ¼ 7
2 À 3
2



 ¼ 2
p ¼ À
b. 26
Al has 13 protons and 13 neutrons. The last proton and the last neutron are
in d5/2 hole states, that is, jp ¼ jn ¼ 2 þ 1
2.
J ¼ 5
2 þ 5
2



 ¼ 5
p ¼ þ
c. 56
Co has 27 protons and 29 neutrons. The last proton is in f7/2 hole state and
the last neutron is in a p3/2 state (1 þ 1
2 ).
J ¼ 7
2 þ 3
2 À 1 ¼ 4
p ¼ þ
The simple shell model is very robust and is even successful in describing nuclei
at the limits of stability. For example, 11
Li is the heaviest bound lithium isotope. The
shell model diagram for this nucleus is indicated in Figure 6.5. Notice the prediction
146 NUCLEAR STRUCTURE
of two filled neutron shells. The binding energy is only 300 keV for the whole
nucleus so it is very fragile. It is also known that 10
Li, which does not have a
filled p shell, is unbound. This again emphasizes the importance of pairing in
nuclei. The two neutrons in the highest energy level that is very close to zero are
alone in a pl/2 state and the empty sl/2 state is very close in energy. This nucleus
has an unusually large interaction radius (or size) and a high probability to dissociate
into 9
Li þ 2n that have been attributed to a large physical extent of the very weakly
bound neutrons in the highest energy level. In fact, there is some debate in the
literature as to the relative ordering of the s and p states.
Another nuclear parameter that can be determined experimentally that depends on
nuclear structure is the magnetic moment. The magnetic moment of a nucleus is a
measure of the response of that nucleus to an external magnetic field and is made
up from the net effect of the motion of the protons plus the intrinsic spins of the
protons and neutrons. The magnetic moment, mi, of one particle can be written as:
mi ¼ glLi þ gsSi (6:1)
where Li is the angular momentum and Si is the intrinsic spin of particle i. The
gyromagnetic ratios, gl and gs, are
gl ¼ lm0 gs ¼ 5:5845m0 for protons
and
gl ¼ 0 gs ¼ À3:8263m0 for neutrons
where m0 is the nuclear magneton:
m0 ¼ ehÀ
=2mpc
Due to the large amount of cancellation of the spins and angular momenta due to the
strong coupling of nucleons in matching orbitals and pairing of spins, we should
Figure 6.5 Energy level pattern and filling for the exotic nucleus 11
Li in the schematic shell
model.
6.2 SCHEMATIC SHELL MODEL 147
expect that the magnetic moments would be small and strongly dependent on the
number and orbits of any unpaired particles. A relatively simple formula for the
magnetic moments of nuclei with single unpaired nucleons, called the Schmidt
limit, contains two forms depending on the relative orientation of the angular
momentum and the spin:
For j ¼ l þ s,
m ¼ lgl þ 1
2 gs
For j ¼ l À s,
m ¼ ( j=j þ 1) (l þ 1)gl À 1
2 gs
 Ã
(6:2)
or equivalently
m ¼ j gl + 1
2 l þ 1
À Á
(gs À gl)
 Ã
(6:3)
where the þ sign is for j ¼ l þ 1
2 and the 2 sign for j ¼ l À 1
2.
The measured magnetic moments of the odd-mass nuclei are similar in magnitude
to the Schmidt limits as shown in Figure 6.6. Notice that the measured
values fall into two groups at approximately 60% of the predicted values. The
fact that the magnetic moments are less than those expected for single particles
indicate that the nuclear wave function is not completely dominated by one particle.
(If we were to show only the magnetic moments of nuclei that have one particle
more than a closed-shell configuration, we would see better agreement with the
Schmidt limits.) Also there is a large amount of variation in the magnetic
moments that indicates the complexity of the underlying structure and that the
cancellation effect of paired particles is not as complete as we might hope.
Up to this point we have concentrated on the properties of the ground states of
nuclei predicted by the schematic shell model. However, we can use these energy
levels to construct excited states by the promotion of particles and the appropriate
coupling of odd (unpaired) particles. First of all, this model has already shown
that odd–odd nuclei always have two possible couplings of the angular momenta
of the odd particles. One coupling leads to a high spin J ¼ j1 þ j2 and a low total
spin J ¼ j1 2 j2. We have already described how to decide which state will lie
lower in energy, but notice the other state will always be present. This state will
be an isomer that will decay to the ground state by g-ray emission (usually with a
relatively long half-life due to the large change in angular momentum between
the states). The relative energy splitting of the two levels decreases as the mass
increases due to the dilution effect of more and more nucleon–nucleon interactions.
Examples of isomeric pairs of levels and excited states in the simple shell model are
given in the accompanying examples.
148 NUCLEAR STRUCTURE
Examples of Isomers
1. 26
Al is a nucleus with 13 protons and 13 neutrons. If we fill in the shell model
energy level diagram from the bottom, we find the following configurations:
Protons (1s1=2)2
(1p3=2)4
(1p1=2)2
(1d5=2)5
Neutrons (1s1=2)2
(1p3=2)4
(1p1=2)2
(1d5=2)5
Figure 6.6 Magnetic moments of the odd-proton (A) and of the odd-neutron nuclei plotted as
a function of the nuclear spin, j. The Schmidt limits are shown by the solid lines. The data
generally fall inside the limits and are better reproduced as 60% of the limits.
6.2 SCHEMATIC SHELL MODEL 149
and recall that a 1d5/2 level is filled by six particles. Therefore, the net configuration
contains a proton hole coupled to a neutron hole in 1d5/2 states.
This is written simply as p(1d5=2)À1
 v(1d5=2)À1
. Coupling the proton and
neutron angular momenta, we expect jp + jn ¼ 0 and 5hÀ
for the nuclear
spins. The Brennan–Bernstein rules predict that the high spin isomer has
the lower energy for identical orbitals, in agreement with observation. The
parities of both orbitals are positive (N ¼ 2hÀ
v shell) so the parities of both
coupled states are positive.
2. 198
Au is a nucleus with 79 protons and 119 neutrons. Filling in the shell model
energy level diagram we should find that the highest partially filled orbitals are
p(1h11=2)9
v(1i13=2)7
both of which are partially filled subshells near major shell closures. If we
make the simplest assumption that all the neutrons and protons are paired
except the last odd particles, than we would expect a configuration:
p(1h11=2)  v(1i13=2) with jp + jn ¼ 1 and 12hÀ
for the nuclear spins. The
parities of these orbitals are negative (N ¼ 5hÀ
v) and positive (N ¼ 6hÀ
v),
respectively, making the product negative. Notice that we could add
or remove a pair of neutrons from this configuration, making 200
Au
and 196
Au, and we would leave the odd neutron in the same orbital.
Therefore, we would make the same predictions for their ground and isomeric
states.
An interesting subset of nuclei is those nuclear pairs in which the numbers of
protons and neutrons are interchanged, for example, 3
He and 3
H. These sets of
nuclei are called mirror pairs, and the schematic shell model predicts that they
will have identical ground and excited states, after correcting for the (small)
upward shift of the proton levels by the Coulomb force and the difference in mass
of a neutron and a proton. This shift caused by increasing the nuclear charge by
one unit while keeping the mass constant can be readily calculated from the
Coulomb energy inside a uniformly charged sphere:
Ec ¼
3
5
Ze2
R
(6:4)
where Z is the atomic number and R is the radius. The Coulomb energy difference
between a mirror pair, where Z refers to the higher atomic number, is then
DEc ¼
3
5
e2
R
Z2
À (Z À 1)2
 Ã
¼
3
5
e2
R
(2Z À 1) %
Ze2
R
(6:5)
This shift is an overestimate as it assumes the nuclei are rigid spheres but, nonetheless,
is straightforward to calculate. A large number of mirror pairs have been studied,
and the agreement between the energy levels in the mirrors is dramatic.
150 NUCLEAR STRUCTURE
An example of the energy level matching in the mirror pair 17
F, 17
O is shown
in Figure 6.7. The agreement of the levels is quite remarkable and can be taken
as strong evidence for the charge independence of the nuclear force, that is,
the protons and neutrons move in essentially identical but separate orbitals in the
nucleus.
After all these successes of the very simple shell model, we should be careful to
note that there are a number of other well-established and simple properties of nuclei
that it cannot describe. For example, the energy levels of essentially all nuclei, and
particularly the even–even nuclei with all paired particles, have series of states that
are arranged in groups (or bands) with energy spacings and state-to-state transitions
that are characteristic of a collective vibration and/or rotation of the entire nucleus.
Specifically, even–even nuclei have low-lying 2þ and 4þ excited states that are
very strongly related to the 0þ ground state that, once excited, cascade rapidly
back to the ground state by g-ray emission. Examples of such collective states are
shown in Figure 6.8. These states correspond to macroscopic vibration of the
entire nucleus around the spherical ground-state shape.
Another example of collective motion that is outside the shell model is found in
the rare-earth and actinide elements. These nuclei lie between the major shell closures
in the shell model and the filling of the midshell high-spin orbitals causes
the nuclei to be deformed (stretched like a U.S. football) in the ground state. The
orbitals that are being filled in these regions have relatively large l values, for
example, g and h states. The angular part of these orbitals is relatively concentrated
in space (due to the large number of angular nodes in the wave function) and each
suborbital is relatively planar. Recall that s orbitals are spherically symmetric, and
orbitals with larger l values are divided by more and more planar nodes. Thus, the
midshell nucleons fill relatively nonspherical suborbitals. As we have already discussed,
the simple shell model was developed with a spherically symmetric potential.
We should expect that the energy levels would shift if the shape of the potential
were changed. We will consider the effects of just such a change later in this chapter.
Figure 6.7 Energy levels of the ground state and first few excited states of the mirror pair 17
F,
17
O are shown. The states are labeled by their intrinsic spin and parity. The matching of these
mirror states is remarkable and strongly supports the idea of the neutrons and protons moving
in identical orbitals.
6.2 SCHEMATIC SHELL MODEL 151
6.3 INDEPENDENT PARTICLE MODEL
A more detailed model can be constructed for the nucleons in terms of a central
potential that holds all the nucleons together plus a “residual potential” or “residual
interaction” that lumps together all of the other nucleon–nucleon interactions. Other
such important one-on-one interactions align the spins of unlike nucleons (p-n) and
cause the pairing of like nucleons (p-p, n-n). The nucleons are then allowed to move
independently in these potentials, that is, the Schro¨dinger equation is solved for the
5+ 0
5.2 yr.
60
27Co
60
28Ni
94
41Nb
94
42Mo
Q = 2823.9
Q = 2045.1
99.025% 4+ 2505.77
2+ 1332.52
stable
stable
.871
.702
1332
1173
98.1% 4+ 1573.72
2+ 871.10
0+ 0
2.0x104 yr.
0+ 0
(6)+ 0
0.057%
(a)
(b)
Figure 6.8 (a) Energy level diagram showing the first (lowest energy) 2þ
and 4þ
states in
60
Ni. The high-spin ground state, 5þ
, of 60
Co b decays primarily to the 4þ
state and
initiates a well-known g-ray cascade to the 2þ
state and then the 0þ
ground state. (b) For
comparison, the energy level diagram showing the first (lowest energy) 2þ
and 4þ
states in
94
Mo. The high-spin ground state, 6þ
, of 94
Nb also primarily feeds the 4þ
state initiating a
g-ray cascade. (Figure also appears in color figure section.)
152 NUCLEAR STRUCTURE
combined interaction to provide the energy levels and wave functions for the
individual particles. Once again there will be a large amount of cancellation of
the effects of the independent nucleons, and the overall properties of the nucleus
can be determined by the last (few) unpaired nucleons or holes.
The central potential can be a simple harmonic oscillator potential f(r)  kr2
or
more complicated such as a Yukawa function f(r)  (eÀar
=r)À1
or the Woods–
Saxon function that has a flat bottom and goes smoothly to zero at the nuclear
surface. The Woods–Saxon potential has the form
U(r) ¼
U0
1 þ exp½(r À R0)=aŠ
þ
Uls
r2
0
1
r
d
dr
1
1 þ exp½(r À R0)=aŠ
 
l † s (6:6)
where R0 ¼ r0A1/3
, with r0 ¼ 1.27 fm, a ¼ 0.67 fm, and the potentials are given by
U0 ¼ ½À51 þ 33(N À Z)=AŠ MeV
Uls ¼ À0:44U0
The spin–orbit strength (second term) is peaked on the nuclear surface as shown in
Figure 6.9.
A residual interaction that is also quite simple has been developed and applied
with good results. Recall that the nucleon–nucleon force is attractive and very
short ranged, so one might image that the nucleons must be in contact to interact.
Thus, the simplest residual interaction is an attractive force that only acts when
the nucleons touch or a d interaction (in the sense of a Kronecker d from
quantum mechanics). This can be written as V(r1, r2) ¼ a d12, where a is the strength
of the interaction, and the d function only allows the force to be positive when
the nucleons are at exactly the same point in space. In practice, the strength of
the potential must be determined by comparison to experimental data. Notice,
U(r) r
dU(r)
dr
Figure 6.9 Radial dependence of the strength of the spin–orbit potential.
6.3 INDEPENDENT PARTICLE MODEL 153
however, such models have a very small number of parameters to be adjusted to give
an overall or average agreement with the data. The “best-fit” values are then used to
calculate the properties of other nuclei and their excited states.
6.4 COLLECTIVE MODEL
As we have seen, the nucleons reside in well-defined orbitals in the nucleus that can
be understood in a relatively simple quantum mechanical model, the shell model. In
this model, the properties of the nucleus are dominated by the wave functions of the
one or two unpaired nucleons. Notice that the bulk of the nucleons, which may even
number in the hundreds, only contribute to the overall central potential. These core
nucleons cannot be ignored in reality and they give rise to large-scale, macroscopic
behavior of the nucleus that is very different from the behavior of single particles.
There are two important collective motions of the nucleus that we have already
mentioned that we should address: collective or overall rotation of deformed
nuclei and vibrations of the nuclear shape about a spherical ground-state shape.
Rotational motion is characteristic of nonspherical nuclei, and the deformation
can be permanent (i.e., the ground state remains deformed) or it can be induced
by centrifugal stretching of a nucleus under rapid rotation. The nuclei with
masses in the region 150 , A , 190 and A . 220 lie between the major shells
and generally have permanent deformations. On the other hand, the rapid rotation
of a nucleus can be dynamically induced by nuclear reactions. It is common to
create rapidly rotating nuclei in compound nuclear reactions that decay by g-ray
emission, eventually slowing down to form spherical ground states.
The deformation can be very complicated to describe in a single-particle framework,
but a good understanding of the basic behavior can be obtained with an
overall parameterization of the shape of the whole nucleus in terms of quadrupole
distortions with cylindrical symmetries. If we start from a (solid) spherical nucleus,
then there are two cylindrically symmetric quadrupole deformations to consider.
The deformations are indicated schematically in Figure 6.10 and give the nuclei
ellipsoidal shapes (an ellipsoid is a three-dimensional object formed by the rotation
of an ellipse around one of its two major axes). The prolate deformation in which
one axis is longer relative to the other two produces a shape that is similar to that
of a U.S. football but more rounded on the ends. The oblate shape with one axis
shorter than the other two becomes a pancake shape in the limit of very large
deformations.
The surface of the ellipsoid can be written in terms of the expansion:
R(u, q) ¼ Ravg 1 þ bY20(u, q)½ Š (6:7)
where Ravg, is the average radius of the three major axes, b is the dimensionless
measure of the deformation, and Y20 is the spherical harmonic function. Formally
b ¼
4
3
ffiffiffiffi
p
5
r
b À a
Raverage
154 NUCLEAR STRUCTURE
where b and a are the semimajor and semiminor axes of the ellipsoid and Raverage is
the average radius, R2
av ¼ 1
2 (a2
þ b2
). The deformation parameter can be positive
(prolate shapes) or negative (oblate shapes) and is generally a small number. For
example, the superdeformed prolate shape with an axis ratio of 2 : 1 has b 0.6.
The energy levels from the quantum mechanical solution of the rotation of a rigid
body have the characteristic feature of increasing separation with angular momentum.
The energy levels are given by the expression:
Erot ¼
J(J þ 1)hÀ 2
2=
(6:8)
where J is the rotational quantum number describing the amount of rotation and = is
the moment of inertia of the rigid body. The moment of inertia of a solid sphere with
Figure 6.10 Schematic representations of the prolate and oblate deformations of a uniform
sphere. A prolate deformation corresponds to the stretching of the distribution along only
one axis while the distribution shrinks equally along the other two axes. An oblate
deformation corresponds to the compression of the distribution along one axis with
increases along the other two axes.
6.4 COLLECTIVE MODEL 155
mass m is =rigid ¼ 2mR2
=5. Substituting in constants and using R ¼ 1.2A1/3
fm, we
find that the rotational energy levels of a sphere at Erot ¼ 36.29 J(J þ 1)/A5/3
MeV
for J in hÀ
units. Note the large power of A in the denominator, which causes the
expression for the rotational constant, hÀ 2
=2=, to be on the order of keV-s. This
expression is called the rigid-body limit, and spherical rigid-body values for
rotational energies are generally smaller than those observed, meaning that the
real nuclear moment of inertia is smaller. These spherical moment of inertia
expressions can be readily extended to nuclei with static or rigid deformations by
substituting the appropriate moment of inertia, thus, = ¼ 2mR2
avg=5(1 þ 0:31b).
The result is similar in that the deformed rigid-body estimate of the moment of
inertia is too large and the rotational energy is too small.
We have already seen that nuclei have some properties that are similar to those
of a liquid drop; in fact, the overall binding energy is well represented in these
terms. The moment of inertia for the rotation of the liquid in a rigid deformed container,
for example, a large water balloon with a negligible mass wall, is
=irro ¼ 9
8 p
À Á
mR2
b2
. This moment of inertial is smaller than that of a rigid body
because the liquid can “flow” inside the container to follow the motion of the
walls and the moment of inertia goes to zero at b ¼ 0 as expected by symmetry.
This rotational behavior is called irrotational flow. The irrotational flow moment
of inertia gives a value that is usually smaller than the experimental value,
leading to rotational energies that are larger than the experimental data. Thus,
we have the situation that:
=irro , =exp , =rigid
which allows us to bracket the experimental value with numerical estimates.
Example of a Rotational Constant The ground-state rotational band of 152
Gd is
shown in Figure 6.11. Use the energy separation between the 2þ and 0þ levels to
estimate the rotational constant in keV, the moment of inertia in amu-fm2
, and then
compare your result to that obtained to the rigid-body result with a deformation
parameter of b ¼ 0.2. Finally, evaluate the irrotational flow moment of inertia for
this nucleus.
Erot ¼
J(J þ 1)hÀ 2
2=
DErot(2 ! 0) ¼ (6 À 0)
hÀ 2
2=
¼ 344:3 keV
hÀ 2
2=
¼ 57:3 keV =exp ¼ 364:7 amu-fm2
=rigid ¼
2
5
mR2
avg(1 þ 0:31b)
=rigid ¼ 2494 amu-fm2
½1 þ 0:31(0:2)Š ¼ 2648 amu-fm2
156 NUCLEAR STRUCTURE
=exp
=rigid
¼
1
7:3
=irro ¼
9
8p
mR2
avgb2
=irro ¼ 2232 amu-fm2
(0:2)2
¼ 89:3 amu-fm2
Different rotational bands in a given nucleus can have different effective moments
of inertia. This could reflect a larger deformation or a change in the number of paired
nucleons or a different alignment of a pair of nucleons of high spin. The result is that
each band can have a different pattern of energy vs. spin (Fig. 6.12a). If one plots
2==hÀ 2
vs. the rotational frequency hÀ 2
v2
for a given nucleus, then one observes a
kink or “backbend” in the plot corresponding to the region where the two bands
cross (Fig. 6.12b).
A special class of quantum rotors are the superdeformed nuclei. The moments of
inertia, after scaling by A5/3
, are all similar due to the fact that the shape of these
nuclei is largely independent of mass with an axis ratio of 2 : 1 due to shell stabilization
effects discussed below.
Another interesting case of nuclear rotation occurs in the spherical nuclei. The
observation of equally spaced g-ray transitions implies collective rotation, but such
bands have been observed in near spherical 199
Pb. It has been suggested that these
bands arise by a new type of nuclear rotation, called the “shears mechanism.” A few
0
0.344
0.755
1.227
1.747
2.300
2.884
3.499
4.143
0+
2+
4+
6+
8+
10+
12+
14+
16+
Figure 6.11 Low-lying energy levels of 152
Gd, which clearly fit the pattern of a rotational
band. The rotational constant hÀ 2
=2= ¼ 68:9 keV can be extracted.
6.4 COLLECTIVE MODEL 157
valence neutron and proton holes couple to form “long” angular momenta, jn and jp,
which couple to give the total spin j. By varying the angle between these “blades of
the shears,” states of differing spin are created. This gives rise to a magnetic
moment, and the radiation associated with the g-ray transition between the states is M1.
The other important macroscopic motions of nuclei are the vibrations of the
nuclear volume around the spherical ground state. Recall that the great majority
of nuclei have spherical ground states, but they also can behave like liquid drops;
so we might imagine that the surface of the nucleus could be caused to vibrate
harmonically, back and forth, around the spherical ground state. In this picture we
could parameterize the shape vibrations, also called surface oscillations, in terms
of the spherical harmonic functions with their characteristic multipolarities. We
should also be careful to differentiate between the characteristic motion labeled
by the multipolarity, or “shape symmetry,” of the mode and the number of
quanta, or phonons, in each vibrational mode. One might imagine multiple excitation
of a single mode, single excitation of several symmetries simultaneously,
or any other combination.
The lowest order macroscopic vibration is a swelling/compression of the whole
nucleus with l ¼ 0. This is sometimes called the “breathing” mode. The next
macroscopic vibration, labeled l ¼ 1, is a dipole motion. However, such a motion
of the entire surface, first in one direction and then back in the other, simply corresponds
to translation of the nucleus and not internal vibration. This motion would
have to be caused by a “restoring force” that was outside the nucleus and so there
is no intrinsic dipole motion of a (whole) nucleus. The next order vibration,
labeled l ¼ 2, is the quadrupole motion in which the nucleus symmetrically
stretches in and then out without moving its center of mass. This is clearly a
vibrational motion with a “restoring force” generated by the nuclear potential.
The third-order vibration, labeled l ¼ 3, is the octupole motion in which the
nucleus asymmetrically expands on one end while pinching on the other. This
vibration creates pear-shaped figures and requires significantly more energy to
excite compared to the more symmetric quadrupole shapes.
EI
w2
1
1
2
2
l
(a) (b)
Figure 6.12 Schematic picture of two intersecting bands with different moments of inertia,
=1 and =2, and the corresponding backbending plot.
158 NUCLEAR STRUCTURE
Recall that the energy levels of the quantum mechanical harmonic oscillator are
all equally spaced, having energies EN ¼ (N þ 1/2)hÀ
v0, N ¼ 0, 1, 2 (see Fig. 6.13).
The fundamental frequency of the oscillation, v0, is equal to the square root of the
force constant divided by the effective mass. Considering even–even nuclei with 0þ
ground states, single excitation of quadrupole motion with l ¼ 2 will require an
N ¼ 2 state as N ¼ 1 is not allowed because it would break the symmetry of the
nuclear wave function. This N ¼ 2 excitation gives rise to a 2þ
state with two
hÀ
v0 of energy. We would expect the subsequent multiple excitation of this mode
would create a 4þ
state with twice the excitation energy and so on. Because there
are three ways to couple two quadrupole phonons together (Jp
¼ 4þ
, 2þ
, and 0þ
),
the two-phonon state is triply degenerate. The three-phonon state at an energy of
3hÀ
v0 will include 6þ
, 4þ
, 3þ
, 2þ
, and 0þ
states. From a global perspective, we
observe that the ratios of the energy of the 4þ
to 2þ
states in even–even nuclei
are approximately 2 : 1 with two strong deviations. First, the deformed rotational
nuclei have 4þ
/2þ
ratios of 10 : 3 as discussed above. And, second, when the
number of neutrons or protons are close to the magic numbers for closed spherical
shells, the nucleus becomes more resistant to oscillation, and the energies of the 2þ
and 4þ
states increase dramatically as well as their ratio.
It is interesting to note that the vibrational model of the nucleus predicts that
each nucleus will be continuously undergoing zero-point motion in all of its
modes. This zero-point motion of a quantum mechanical harmonic oscillator is a
formal consequence of the Heisenberg uncertainty principle and can also be seen
in the fact that the lowest energy state, N ¼ 0, has the finite energy of hÀ
v/2.
Figure 6.13 Schematic vibrational energy-level diagrams of medium-mass even–even
nuclei. [From Eichler (1964).]
6.4 COLLECTIVE MODEL 159
From another standpoint, the superposition of all of these shape oscillations can be
viewed as a natural basis for the diffuseness of the nuclear surface.
The energy of rotational states built on vibrations is given by
E ¼
hÀ 2
2=
J(J þ 1) À K2
 Ã
(6:9)
where K is the projection of J on the nuclear symmetry axis. For b(l ¼ 2, K ¼ 0)
vibrations, Jp
¼ 0þ
, 2þ
, 4þ
, . . . for g (l ¼ 2, K ¼ 2) vibrations, Jp
¼ 2þ
, 3þ
,
4þ
, . . . A typical sequence of states is shown in Figure 6.14.
6.5 NILSSON MODEL
Up to now, we have discussed two extremes of nuclear structure, those aspects that
can be explained by the properties of single or individual particles moving in a
spherically symmetric central potential and those aspects corresponding to largescale
collective motions of groups of nucleons away from spherical symmetry.
Additional insight into the structure of nuclei can be obtained by considering the
states of single particles moving in a deformed nuclear potential. S. G. Nilsson
extensively studied this problem, and the resulting model of nuclear structure is
referred to as the Nilsson model.
Using a deformed harmonic oscillator potential, one can make several useful
observations about the nuclear structure of deformed nuclei. In Figure 6.15, we
show the energies of single-particle states of such a potential as a function of the
deformation of the potential. At spherical symmetry, one observes the gaps in
the level spacings corresponding to the major harmonic oscillator shells that we
have already discussed. But as the deformation changes, the levels move in
energy and new magic numbers (shell gaps) occur when the ratio of the semimajor
4+
2+
0+5+
4+
3+
2+
8+
6+
4+
2+
0+
β-vib
K=0
γ-vib
K=2
g
Figure 6.14 Positive parity levels of a typical deformed nucleus.
160 NUCLEAR STRUCTURE
to the semiminor axes of the nucleus is a simple whole number. Thus, nuclei with
axes ratios of 2 : 1 have special stability (the superdeformed nuclei). In addition,
each spherical shell model state, for example, an f7/2 state, is split into (2j þ 1)/2
levels that can be labeled with a new quantum number V defined as the projection
of the single-particle angular momentum on the nuclear symmetry axis (Fig. 6.16).
For prolate deformation, states of highest V lie the highest in energy.
The angular momentum of an odd A deformed nucleus, J, is the vector sum of the
angular momentum of the last unpaired nucleon and the rotational angular momentum,
R, of the core of remaining nucleons, as shown schematically in Figure 6.17.
The projection of the total nuclear angular momentum J upon the nuclear symmetry
axis is given the symbol K. For axially symmetric nuclei, the direction of R is
perpendicular to the symmetry axis and J ¼ V ¼ K. Each Nilsson single-particle
level may be the ground state of a rotational band. For the ground state of such
Figure 6.15 Single particle levels of a deformed harmonic oscillator as a function of
the deformation of the potential. The ratios of the semimajor to semiminor axes of the
oscillator are shown also.
6.5 NILSSON MODEL 161
bands, J ¼ V ¼ K. When J ¼ 3
2 or greater, the allowed nuclear spins of the members
of the band are J0, J0 þ 1, J0 þ 2, etc. The energies of the members of the band are
given as
E ¼
hÀ 2
2=
½J(J þ 1) À J0(J0 þ 1)Š (6:10)
The Nilsson model is able to predict the ground state and low-lying states of
deformed odd A nuclei. Figure 6.18 is a more detailed picture of how the energies
of the Nilsson levels vary as a function of the deformation parameter b2 for the
1/2
1/2
3/2
3/25/2
5/2
7/2
β < 0 (oblate) β > 0 (prolate)
β = 0
7/2
W W
Figure 6.16 Schematic diagram of the splitting of the f7/2 spherical shell model level as the
potential deforms. Positive deformations correspond to prolate shapes while negative
deformations correspond to oblate shapes.
R
K
W
I
Figure 6.17 Addition of angular momenta in a deformed odd A nucleus. V is the projection
of the total angular momentum of the odd nucleon. It is added vectorially to the rotational
angular momentum of the core, R, to give the total angular momentum J whose projection
on the symmetry axis is K.
162 NUCLEAR STRUCTURE
first 20 nucleons in the nucleus. Let us consider the nuclei 19
F, 19
Ne, 21
Ne, and 23
Na.
According to the simple shell model (Fig. 6.3), the last odd nucleon in these nuclei
should be in a d5/2 state giving rise to a ground-state spin and parity Jp
of 5
2
þ
. One
observes Jp
for these nuclei to be 1
2
þ
, 1
2
þ
, 3
2
þ
, and 3
2
þ
, respectively. The Nilsson
model allows us to understand these observations. From the quadrupole moment
of these nuclei, 0.1 barns, we can deduce b ¼ 0.1. Thus, from the levels shown in
Figure 6.18, we expect the 9th particle to have Jp
¼ 1
2
þ
, while the 11th particle
will have Jp
¼ 3
2
þ
in agreement with the observations. The low-lying excited
states of many nuclei can also be explained by considering the positions of the
nearby levels with small deformations in the Nilsson diagrams.
6.6 NUCLEUS AS A FERMI GAS
The preceding discussion of nuclear structure and models was mostly aimed at
explaining the detailed properties of the ground states and small excitations of
-0.6
2.50
3.00
E/hw0
3.50
4.00
-0.4 -0.2 0
A ~ 23
e
0.2 0.4 0.6
(1/2)-
(1/2)-
(1/2)+
(3/2)+
(1/2)+
(1/2)+
(5/2)+
(3/2)+
N = 2
N = 1
p1/2
One nucleon
outside the
closed shell “8”
(3/2)-
20
d3/2
d5/2
s1/2
2
8
p3/2
Figure 6.18 Energy of a single nucleon in a deformed potential as a function of deformation
1. The is diagram pertains to either Z , 20 or N , 20. Each state can accept two nucleons.
6.6 NUCLEUS AS A FERMI GAS 163
nuclei. These nuclei are produced and take part in nuclear reactions that are usually
very complicated dynamical situations compared to the (simple) situation of
nucleons confined in a static central potential. Detailed calculations with wave functions
in dynamical potentials associated with scattering and absorption have only
been carried out in a few cases. On the other hand, a very wide variety of nuclear
reactions have been studied experimentally and often exhibit amazing simplicity.
Models have been developed that treat the average behavior of the large number
of nucleons in a nucleus on a statistical basis. An important statistical model developed
to describe the average behavior of medium and large nuclei, particularly
useful in reactions, treats the nucleus as a fluid of fermions. This approximation,
called the Fermi gas model, uses the now familiar concept of confining the nucleons
to a fixed spherical shape with a central potential, but in this case the nucleons are
assumed to be all equivalent and independent. This situation loosely corresponds to
an ideal gas confined to a fixed volume with the addition of Fermi–Dirac statistics
and Pauli blocking to prevent spatial overlap of the particles.
An important feature of the Fermi gas model is that it allows us to describe the
average behavior of a nucleus with thermodynamical functions. The fact that the
energy levels in a (large) nucleus with a finite excitation energy are so closely
spaced allows us to use entropy to predict the evolution of the system. (This idea
is somewhat opposite to the ground-state situation that is dominated by the wave
functions of individual particles.) The concept of thermodynamic entropy is
closely linked to a thermodynamic nuclear temperature. Nuclear reactions are
often described in terms of the imagined temperature of the internal particles, and
excited nuclei emit light particles and g rays as they lose their excitation energy
and “cool” as they approach the ground state.
The first step in developing the Fermi gas model is to determine the highest level
that is occupied by nucleons. Next the average energies and momenta are calculated
because we will assume that all the lower levels are exactly filled. The nucleons are
confined to a fixed total volume and are assumed to have a uniform density. When
quantum mechanical particles are confined in a rigid container, then they occupy
fixed states that can be labeled with appropriate quantum numbers, that is, nx, ny,
nz for a rectangular box with three dimensions, Lx, Ly, Lz. The particle will have a
specific momentum in each state so, alternatively, we could label the states by
their momenta, px, py, pz, or by their wavenumbers, kx, ky, kz where ki ¼ (nip/Li),
which explicitly incorporates the dimensions of the box. We would like to know
what is the highest quantum number, the largest momentum, pf, or the wavenumber,
kf, of the highest filled level, called the Fermi level. The Fermi level wavenumber
can be written as:
k2
f ¼ k2
x þ k2
y þ k2
z (6:11)
or in terms of the quantum numbers:
k2
f L2
p2
¼ n2
x þ n2
y þ n2
z (6:12)
164 NUCLEAR STRUCTURE
The number of different combinations of the positive integer quantum numbers that
fulfill this equality is given by the volume of one-octant of a sphere:
Nstates ¼
1
8
 
4p
3
kf L
p
 3
(6:13)
Remember that the Pauli principle allows us to put particles with two spins
(up/own) into each level, and if the nucleons are all in their lowest possible
states, the number of filled states can be assumed to be equal to the number of
each type of nucleon. Thus, the Fermi wavenumber for protons is
kf ¼
p
L
2Nstates
3p
 1=3
¼
p
L
2Z
3p
 1=3
¼
p
r0
2Z
3pA
 1=3
(6:14)
and similarly for neutrons where we have taken L to the nuclear radius, r0A1/3
.
Notice that we have obtained an expression that depends only on Z/A (or N/A)
and the radius constant, r0, so the value of the Fermi energy will be similar for
most nuclei because the variation of Z/A is small for stable nuclei and enters via
the cube root. The Fermi energy for nucleons in those nuclei with Z=A ¼ 1
2 taking
r0 ¼ 1.2 fm is
Ef ¼
k2
f hÀ 2
2m
% 32 MeV (6:15)
If the number of neutrons is greater than the number of protons, as in heavy
nuclei, then the Fermi energies will be slightly different for the two kinds of
particles. An approximate representation of the Fermi energy for protons and
neutrons is
Eprotons
f ffi 53
Z
A
 2=3
MeV
Eneutrons
f ffi 53
A À Z
A
 2=3
MeV (6:16)
The average kinetic energy of the nucleons in the well can be shown to be 3
5 Ef , or
approximately 20 MeV. Notice that the nucleons are moving rapidly inside the
potential well but not extremely fast.
Example Problem What is the deBroglie wavelength of a neutron moving with
the average Fermi energy in a 208
Pb nucleus? You can assume that the neutron is
nonrelativistic and use r0 ¼ 1.2 fm.
6.6 NUCLEUS AS A FERMI GAS 165
Solution
kf ¼
p
r0
2Z
3p A
 1=3
¼ 1:145 fmÀ1
Ef ¼
k2
f hÀ 2
2m
¼ 27 MeV
Eavg ¼ 3
5 Ef ¼ 16 MeV
l ¼
h
p
¼
2p
kf
¼ 5:487 fm
Note that this wavelength is similar to the lead radius R  1.2A1/3
¼ 7.1 fm.
A schematic version of the Fermi gas potential energy well for a large nucleus is
shown in Figure 6.19. Recall that nucleons are bound by approximately 8 MeV, on
average, so the uppermost filled energy level (Fermi level) should be approximately
at 28 MeV. The lowest level is then approximately 32 MeV below this, which
makes the Fermi gas potential energy well relatively shallow. The levels between
the Fermi level and zero potential energy are assumed to be completely empty in
the ground state and become occupied when the nucleus absorbs excitation energy.
As a nucleus absorbs energy nucleons are promoted from the filled levels into the
unfilled region between the Fermi level and zero potential energy. Each promotion
leads to a specific excitation energy, and combinations of multiple excitations can
lead to the same or similar energies. At high excitations the number of combinations
of different possible promotions for a specific excitation energy grows dramatically.
The tremendous growth of the number of energy levels with excitation energy is
Neutrons Protons
Ec
Uo
0
B
Fermi level
EF,p
EF,n
Figure 6.19 A schematic version of the potential energy well derived from the Fermi gas
model. The highest filled energy levels reach up to the Fermi level of approximately
32 MeV. The nucleons are bound by approximately 8 MeV, so the potential energy
minimum is relatively shallow.
166 NUCLEAR STRUCTURE
one of the interesting features of nuclei. The number of levels is so large that we can
describe the system by an average level density, r(EÃ
, N), which is simply the
number of levels per unit excitation energy, EÃ
, for a fixed number of nucleons,
N. The fact that excited nuclei, even with a finite number of particles, have very
dense and nearly continuous distributions of levels is the feature that allows us to
describe their deexcitation with statistical techniques.
The connection between the microscopic description of any system in terms of
individual states and its macroscopic thermodynamical behavior was provided by
Boltzmann through statistical mechanics. The key connection is that the entropy
of a system is proportional to the natural logarithm of the number of levels available
to the system, thus:
S(E, N) ¼ kB ln G(E, N) ¼ kB ln ½r(E, N)DEŠ (6:17)
where G is the total number of levels. The entropy of an excited nucleus is thus
proportional to the level density in some energy interval, DE, and goes to zero as
the excitation energy goes to zero. (Recall that there is only one nuclear ground
state.) The thermodynamic temperature can be calculated from the entropy as:
1
T
¼
@S(EÃ
, N)
@E
¼ kB
@ ln r(EÃ
, N)
@E
(6:18)
In statistical mechanics the Boltzmann constant, kB, with dimensions of energy per
degree is included in expressions so that the temperatures can be given in degrees
Kelvin. The numerical values of nuclear temperatures in Kelvin are very large,
for example 109
K, so the product of kB and T is usually quoted in energy units
(MeV) and the Boltzmann factor is often not written explicitly.
At this point we have not distinguished between nuclear systems and macroscopic
systems on the basis of their size. There is, however, at least one important
difference between the two. The difference is the way the entropy S(EÃ
, N) should
be evaluated. In statistical mechanics one has different physical situations (ensembles)
for evaluating thermodynamic quantities: fixed energy and particle number
(microcanonical ensemble), fixed temperature and particle number (canonical
ensemble), and fixed temperature and chemical potential (grand canonical ensemble).
In the evaluation of thermodynamic quantities for macroscopic systems,
each of these approaches provides essentially the same result. Thus, the entropy
may be evaluated by calculating any of the following: Smicrocanonical, Scanonical, or
Sgrandcanonical. This is not the case for nuclear systems because the only appropriate
ensemble is the microcanonical ensemble of isolated systems. The fundamental
definition of nuclear temperature should be written
1
T
¼
@Smicrocanonical(EÃ
, N)
@E
(6:19)
6.6 NUCLEUS AS A FERMI GAS 167
and it is not correct to substitute an entropy obtained with a different ensemble into
this expression.
Standard procedures permit the evaluation of the entropy of a Fermi gas under the
conditions of a grand canonical ensemble, which we will have to adjust to obtain the
microcanonical entropy. For low excitation energies, EÃ
, the entropy is
Sgrandcanonical(EÃ
, N) ¼ 2(aEÃ
)1=2
(6:20)
where a is a constant proportional both to the number of particles and to the density
of the single-particle levels of the Fermi gas at the Fermi energy, Ef . If Sgrandcanonical
is used to replace Smicrocanonical, one obtains T ¼ (E/a)1/2
as the link between temperature
and excitation energy. This result would be appropriate for macroscopic
systems, but as we said it must be modified for isolated nuclear systems. For
small systems
Smicrocanonical ¼ Sgrandcanonical þ DS (6:21)
where DS becomes vanishingly small compared to Sgrandcanonical as the number of
particles or the excitation energy becomes large. An approximate expression for
DS for a Fermi gas at relatively low energy is
DS % Àgln (EÃ
) (6:22)
with g being a number of the order of unity, ranging from 1 to 2 depending on
whether isospin and angular momentum are explicitly considered in the labeling
of the states. When the appropriate Smicrocanonical is used to evaluate the nuclear
temperature, one finds
1
T
¼
@Sgrandcanonical
@E
þ
@DS
@E
(6:23)
For the moderately low energies this provides
1
T
%
a
EÃ
 1=2
À
g
EÃ
 
(6:24)
as the link between excitation energy and nuclear temperature. For large excitation
energies, EÃ
, and large particle number, the correction term proportional to g
vanishes and EÃ
’ aT2
.
The density of nuclear states can then be written as:
r(EÃ
) /
a
aEÃð Þg exp 2(aEÃ
)1=2
 Ã
(6:25)
The factor a here is called the level density parameter and is adjusted to correspond
to level densities measured at low excitation energies. The analyses of data over a
broad mass range suggest that a is proportional to the mass of the nuclear system
A being a % A/8.5 MeV21
. The level densities can be corrected for angular momentum
by including preexponential statistical factors and subtracting the collective
energy that is involved in rotation. The rotational energy is often included with an
effective moment of inertia, a parameter adjusted to match experimental spectra
and yields.
168 NUCLEAR STRUCTURE
We can extend the Fermi gas level density analysis to predict the relative probability
of various decay modes of excited nuclei if we make the assumption that the
nuclei are in full thermal equilibrium. That is, we assume that all of the energy levels
corresponding to a given excitation energy are fully populated. It is not possible for a
single nucleus to be in many states simultaneously, it can only be in one. So the
thermal equilibrium that we require must apply to a set of nuclei created in many
(identical) reactions. This is, of course, how chemical reactions take place when
Avogadro’s number of atoms or molecules with various kinetic energies but one
temperature follow a path from reactant to products based on a specific reaction
mechanism. Nuclear reactions are usually detected by producing large numbers of
nuclei, )103
, and then observing various reaction products and determining the
probabilities of each process.
Excited nuclei that have attained statistical equilibrium will decay into different
products in proportion to the number of states available to the whole system after the
decay. The different decays are often called channels, and we speak of the probability
to decay into a given channel. A very schematic representation of the
energy levels and the energies involved in the decay of an excited nucleus into
various channels is shown in Figure 6.20. The total sum of the probabilities for
decay into all channels is, of course, one. We can simply count the number of
states available for a decay channel and obtain a general expression for the relative
probability, P(1, n), for an excited nucleus to emit a portion with size n, requiring an
energy 1. The expression is
P(1, n) / G(1, n) † G(E À 1, N À n) (6:26)
where G(E, N) is the number of states in the vicinity of energy E for a system of mass
number N. The first factor on the right-hand side is contributed by the states in the
Figure 6.20 A representation of the branching decays from a highly excited compound
nucleus. In the statistical model, the relative probability for the excited nucleus to decay
into a specific channel is proportional to the number of possibilities or statistical weight of
that channel divided by the sum of all of the statistical weights of all of the channels.
6.6 NUCLEUS AS A FERMI GAS 169
emitted piece, and the second is contributed by the states in the (large) daughter
nucleus. The number of internal states can be taken to be the log of the level
densities used to define the entropy, above, and we will need to include a term for
the kinetic energy of the emitted piece. However, we need to integrate the emission
rates over the whole course of the nuclear reaction to obtain the total yields that can
be measured in the laboratory.
Focusing on comparisons to measurable quantities, the relative probability of a
reaction (exit) channel can be written as the ratio of the cross section for that
channel, si, to the total reaction cross section, sr. The ratios are labeled as the relative
decay widths, Gi, in a notation that is, unfortunately, easy to confuse with the
number of states discussed above. The sum of the decay widths is the total width
of the state and can be used to calculate the lifetime of the excited state. Thus,
si=sT ¼ Gi=GT (6:27)
and notice that the double ratio of the relative probability of two decay channels does
not depend on the total reaction cross section:
si=sj ¼ Gi=Gj (6:28)
The width for the emission of a particle with a binding energy of Bn and no internal
states has been shown to have the form:
Gi / G(E, N)
ðEÀBi
0
1r(E À Bi À 1) d1 (6:29)
with E the excitation of the excited parent nucleus. Therefore, the relative intensities
of the channels change because the binding energies of the emitted particles change
the density of states through the exponential dependence of the level density. This
can be extended to the case of fission decay leading, in the simplest approximation,
to a slightly different integral:
Gf / G(E, N)
ðEÀEf
0
r(E À Ef À 1) d1 (6:30)
in terms of the fission barrier, Ef. The ratio Gn/Gf is very important in determining
the survival of the very heaviest elements when they are synthesized in nuclear reactions.
Notice that if the nucleus emits a neutron to remove excitation, it retains its
large atomic number; however, if it fissions, then it is converted into two nuclei
with much smaller atomic numbers. The integrals in the above expressions can be
evaluated in the Fermi gas approximation with the following approximate result
(Vandenbosch and Huizenga, 1973):
Gn
Gf
%
2TA2=3
K0
exp
Ef À Bn
T
 
(6:31)
170 NUCLEAR STRUCTURE
where K0 ¼ hÀ 2
/2mr0
2
 15 MeV and T is the nuclear temperature created by the initial
reaction. The exponential function contains the difference between the fission barrier
and the neutron separation energy. Therefore, this ratio is only near one when these
two values are nearly equal. If there is a large difference between the fission barrier
and the separation energy, then the ratio will be very large or very small.
Example Problem In a certain nuclear reaction, a beam of 18
O was combined
with 233
U nuclei to form a compound nucleus of 256
Fm. The nuclei were produced
with an excitation energy of 95 MeV. Calculate the nuclear temperature
assuming that g ¼ 1, and then the relative probability of neutron to fission
decay of the excited system.
Solution
1
T
%
a
EÃ
 1=2
À
1
EÃ
 
with a ¼ A/8.5 ¼ 256/8.5 ¼ 30.1 MeV21
1
T
%
30:1
95
 1=2
À
1
95
 
T % 1:8 MeV
We need to find the neutron separation energy and the fission barrier for this
nucleus in order to evaluate the ratio:
Gn
Gf
%
2TA2=3
K0
exp
Ef À Bn
T
 
The neutron separation energy is 6.38 MeV, and the fission barrier is 5.90 MeV.
Gn
Gf
%
2 Â 1:8 Â 2562=3
15
exp
5:90 À 6:38
1:8
 
Gn
Gf
% 9:68 exp (À0:266) ¼ 7:4
PROBLEMS
1. Predict the ground-state spins and parities for 41
K, 44
Ca, and 60
Co.
2. An odd A nucleus has a Jp
¼ 7
2
þ
ground state. (a) What is J, K for the first two
excited states? (b) If the energy of the first excited state is 100 keV, what is
the energy of the second excited state?
3. Define or describe the following terms or phenomena: island of isomerism, spin–
orbit coupling, b vibration, Schmidt limits, and Nilsson states.
PROBLEMS 171
4. What is the energy of the 2þ
! 0þ
g-ray transition in 172
Hf assuming 172
Hf is a
perfect rotor with its moment of inertia I given as 2
5 mR2
?
5. The a decay of 241
Am (t1/2 ¼ 420 y, Jp
¼ 5
2
þ
) populates members of at least
two rotational bands (A and B) in 237
Np shown in the table below. (a) Using
the collective model, predict the energies of the 9
2
þ
, 9
2
À
, and 11
2
À
levels. (b)
Calculate the effective moment of inertia of 237
Np.
Band J, p E (keV)
B 11
2
À
?
B 9
2
À
?
B 7
2
À
103
A 9
2
þ
?
B 5
2
À
59.6
A 7
2
þ
33.2
A 5
2
þ
0
6. Calculate the energy of the 4þ
, 6þ
, 8þ
, and 10þ
members of the ground-state
rotational band of an even–even nucleus if the energy of the 2þ
member of
the band is 0.044 MeV above the ground state.
7. 237
Np has levels at 0.033, 0.060, 0.076, 0.103, and 0.159 MeV. Which of these, if
any, would you expect to be members of a rotational band whose first member is
the 5
2
þ
state of 237
Np?
8. Analyze the following level schemes in terms of the collective and Nilsson
models:
0:425 7
2
À
6:000 4þ
5:220 3þ
0:129 7
2
þ
4:230 2þ
0:117 5
2
þ
4:113 4þ
0:005 3
2
þ
1:369 2þ
0 1
2
þ
0 0þ
171
Tm, b ¼ 0:2À0:3 24
Mg
9. A deformed even–even nuclide has energy levels characterized by the following
values of spin, parity, and K value. You will note that not all of the information is
given for each level. Fill in the blanks with the required values. In the appropriate
space, assign each of the levels to a particular mode of excitation, for example,
vibrational. Assume all bands are characterized by the same value of the moment
of inertia.
172 NUCLEAR STRUCTURE
Energy J p K Assignment
__ __ __ __ __
0.400 1 2 0
0.376 3 þ __
__ 4 þ 0
0.349 6 þ 0
0.310 2 þ 2
__ 2 þ 0
0.200 0 þ 0
0.166 4 þ __
__ __ þ 0
0 __ __ __
10. Using the shell model, calculate the ground-state spins, parities, and magnetic
moments for 32
S, 33
S, and 41
K. Predict the following characteristics of the
ground states of 25
Mg and 63
Cu: the state of the odd nucleon, the total
nuclear angular momentum, nuclear magnetic dipole moment, the sign of the
nuclear quadrupole moment, and the parity. Explain the probable cause of
any important discrepancies between your predictions and the following
measured values:
25
Mg I ¼
5
2
m ¼ À0:96 Q ¼ þ0:2
63
Cu I ¼
3
2
m ¼ þ2:22 Q ¼ À0:1
11. The energies (MeV) and spins of the lowest excited states of 182
W are for
J ¼ 2, 4, 6, we have E ¼ 0.100, 0.329, and 0.680. Do these values agree with
a rotational model?
12. For the nucleus 235
U at an excitation energy of 30 MeV, what is the ratio of the
density of levels of spin J to the total density of levels at that excitation energy?
13. 249
Bk is known to have the following level scheme. Fill in the missing energies
and J, p values.
93:7 ?
82:6 7
2
À
41:8 9
2
þ
39:6 ?
8:8 3
2
À
0 7
2
þ
14. Given the following shell model state, k17/2, show qualitatively how it
might split as a function of increasing prolate deformation. Label each
PROBLEMS 173
state as to its V value and indicate the maximum number of particles in each V
state.
15. Show that the Brennan–Bernstein rules forbid the existence of odd nuclei with
ground states 0þ
or 12
. Find some exceptions.
16. Given the following level scheme for 110
Cd, predict the character of each state.
1:783 2þ
1:542 ?
1:473 0þ
0:656 2þ
0 0þ
17. 121
Sb has a spin of 5
2 and a magnetic moment of 3.36 nm. What is the state of
the 51st proton? What would the shell model predict?
REFERENCES
Brennan, M. H. and A. M. Bernstein. Phys. Rev. 120, 927 (1960).
Eichler, E. Rev. Med. Phys. 36, 809 (1964).
Gordan, G. E. and C. D. Coryell. J. Chem.–Ed. 44, 636 (1967).
Heyde, K. Basic Ideas and Concepts in Nuclear Physiscs, 2nd ed., IOP, Bristol, 1999.
Tyren, H., P. Hillman, and Th. Maris. Nucl. Phys. 7, 10 (1958).
Vandenbosch, R. and J. R. Huizenga. Nuclear Fission, Academic, New York, 1973.
BIBLIOGRAPHY
General
Most, if not all, textbooks in nuclear science have a chapter discussing nuclear
structure. Among the favorites of the authors are:
Das, A. and T. Ferbel. Introduction to Nuclear and Particle Physics, Wiley, New York, 1994.
Enge, H. A. Introduction to Nuclear Physics, Addison-Wesley, Reading, MA, 1966.
Friedlander, G., J. W. Kennedy, E. S. Macias, and J. M. Miller. Nuclear and Radiochemistry,
Wiley, New York, 1981.
Hodgson, P. E., E. Gadioli, and E. Gadioli-Erba. Introductory Nuclear Physics, Clarendon,
Oxford, 1997.
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1987.
More Specialized, Advanced Treatments
Bohr, A. and B. R. Mottelson. Nuclear Structure, Vols. I and II, Benjamin, New York, 1969.
Casten, R. F. Nuclear Structure from a Simple Perspective, Oxford, New York, 1990.
174 NUCLEAR STRUCTURE
de Shalit, A. and H. Feshbach. Theoretical Nuclear Physics, Vol. I. Nuclear Structure, Wiley,
New York, 1974.
Nilsson, S. G. and I. Ragnarsson. Shapes and Shells in Nuclear Structure, Cambridge,
Cambridge, 1995.
BIBLIOGRAPHY 175
CHAPTER 7
a DECAY
In a series of seminal experiments Ernest Rutherford and his collaborators
established the important features of a decay. The behavior of the radiations from
natural sources of uranium and thorium and their daughters was studied in magnetic
and electric fields. The least penetrating particles, labeled “a rays” because they
were the first to be absorbed, were found to be positively charged and quite
massive in comparison to the more penetrating negatively charged “b rays” and
the most penetrating neutral “g rays.” In a subsequent experiment the a rays from
a needlelike source were collected in a very small concentric discharge tube, and
the emission spectrum of helium was observed in the trapped volume. Thus,
a rays were proven to be energetic helium nuclei. The a particles are the most ionizing
radiation emitted by natural sources (with the extremely rare exception of the
spontaneous fission of uranium) and are stopped by as little as a sheet of paper or
a few centimeters of air. The particles are quite energetic (Ea ¼ 4–9 MeV) but
interact very strongly with electrons as they penetrate into material and stop
within 100 mm in most condensed materials.
Understanding these features of a decay allowed early researchers to use the
emitted a particles to probe the structure of nuclei in scattering experiments and
later, by reaction with beryllium, to produce neutrons. In an interesting dichotomy,
the a particles from the decay of natural isotopes of uranium, radium, and their
daughters have sufficient kinetic energies to overcome the Coulomb barriers of
light elements and induce nuclear reactions but are not energetic enough to
induce reactions in the heaviest elements.
177
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
a Particles played an important role in nuclear physics before the invention of
charged particle accelerators and were extensively used in research. Therefore,
the basic features of a decay have been known for some time. The process of a
decay is a nuclear reaction that can be written as:
A
z (Z)N À! AÀ4
z À 2(X)2À
NÀ2 þ 4
2He2þ
2 þ Qa (7:1)
where we have chosen to write out all of the superscripts and subscripts. Thus the
a decay of 238
U can be written
238
U À! 234
Th2À
þ 4
He2þ
þ Q (7:2)
The Qa value is positive (exothermic) for spontaneous a decay. The helium nucleus
emerges with a substantial velocity and is fully ionized, and the atomic electrons on
the daughter are disrupted by the sudden change, but the whole process conserves
electrical charge. We can rewrite the equation in terms of the masses of the
neutral atoms:
A
z (Z)N À! AÀ4
zÀ2 (X)NÀ2 þ 4
2He2 þ Qa (7:3)
and then calculate the Qa value because the net change in the atomic binding energies
(65.3Z7/5
2 80Z2/5
eV) is very small compared to the nuclear decay energy.
What causes a decay? (or, what causes Qa to be positive?) In the language of the
semiempirical mass equation, the emission of an a particle lowers the Coulomb
energy of the nucleus, which increases the stability of heavy nuclei while not affecting
the overall binding energy per nucleon because the tightly bound a particle has
approximately the same binding energy/nucleon as the original nucleus.
Two important features of a decay are that the energies of the a particles are
known to generally increase with the atomic number of the parent, but yet the
kinetic energy of the emitted particle is less than that of the Coulomb barrier in
the reverse reaction between the a particle and the daughter nucleus. In addition,
all nuclei with mass numbers greater than A % 150 are thermodynamically unstable
against a emission (Qa is positive), but a emission is the dominant decay process
only for the heaviest nuclei, A ! 210. The energies of the emitted a particles can
range from 1.8 MeV (144
Nd) to 11.6 MeV (212
Pom
) with the half-life of 144
Nd
being 5 Â 1029
times as long as that of 212
Pom
. Typical heavy-element a decay
energies are in the range from 4 to 9 MeV, as noted earlier.
In general, a decay leads to the ground state of the daughter nucleus so that the
emitted particle carries away as much energy as possible and as little angular momentum
as possible. The ground-state spins of even–even parents and daughters (including
the a particle, of course) are zero, which makes l ¼ 0 a-particle emission the most
likely process for these nuclei. Small branches are seen to higher excited states, but
such processes are strongly suppressed. Some decays of odd-A heavy nuclei populate
low-lying excited states that match the spin of the parent so that the orbital angular
178 a DECAY
momentum of the a particle can be zero. For example, the strongest branch (83%) of
the a decay of 249
Cf goes to the 9th excited state of 245
Cm because this is the lowest
lying state with the same spin and parity as that of the parent. a Decay to several
different excited states of a daughter nucleus is called fine structure; a decay from
an excited state of a parent nucleus to the ground state of the daughter nucleus is
said to be long-range a emission because these a particles are more energetic and
thus have longer ranges in matter. The most famous case of long-range a emission
is that of 212
Pom
where a 45-s isomeric level at 2.922 MeV decays to the ground
state of 208
Pb by emitting a 11.65-MeV a particle.
We will consider the general features of a emission, and then we will describe
them in terms of a simple quantum mechanical model. It turns out that a emission
is a beautiful example of the quantum mechanical process of tunneling through a
barrier that is forbidden in classical mechanics.
7.1 ENERGETICS OF a DECAY
As we have seen in the overview of the nuclear mass surface in Chapter 2, the a particle,
or 4
He nucleus,is an especially strongly bound particle. This combined with the factthat
the binding energy per nucleon has a maximum value near A % 56 and systematically
decreases for heavier nuclei creates the situation that nuclei with A ! 150 have positive
Qa values for the emission of a particles. This behavior can be seen in Figure 7.1.
Figure 7.1 Variation of the a-particle separation energy as a function of mass number. [From
Valentin, Subatomic physics: nuclei and particles. Copyright # 1981 by North-Holland
Publishing Company. Reprinted by permission of North-Holland Publishing Company.]
7.1 ENERGETICS OF a DECAY 179
For example, one of the heaviest naturally occurring isotopes, 238
U (with a mass defect,
D of þ47.3070 MeV) decays by a emission to 234
Th (D ¼ þ40.612 MeV) giving a
Qa value of:
Qa ¼ 47:3070 À (40:612 þ 2:4249) ¼ 4:270 MeV
Note that the decay energy will be divided between the a particle and the heavy recoiling
daughter so that the kinetic energy of the a particle will be slightly less. (The kinetic
energy of the recoiling 234
Th nucleus produced in the decay of 238
U is 0.070 MeV.)
Conservation of momentum and energy in this reaction requires that the kinetic energy
of the a particle, Ta, is
Ta ¼
234
238
Qa ¼ 4:198 MeV
The kinetic energies of the emitted a particles can be measured very precisely so we
should be careful to distinguish between the Qa value and the kinetic energy, Ta. The
very small recoil energy of the heavy daughter is very difficult to measure, but it is
still large compared to chemical bond energies and can lead to interesting chemistry.
For example, the daughter nuclei may recoil out of the original a source. This can
cause serious contamination problems if the daughters are themselves radioactive.
TheQa valuesgenerallyincreasewithincreasingatomicnumber,but thevariationin
the mass surface due to shell effects can overwhelm the systematic increase (Fig. 7.2).
Figure 7.2 Variation of a-decay energies indicating the effect of the N ¼ 126 and Z ¼ 82
shell closures along with the N ¼ 152 subshell.
180 a DECAY
The sharp peaks near A ¼ 214 are due to the effects of the N ¼ 126 shell. When 212
Po
decays bya emission, the daughternucleus isdoubly magic—208
Pb (very stable) with a
large energy release. The a decay of 211
Pb and 213
Po will not lead to such a large Qa
because the products are not doubly magic. Similarly, the presence of the 82 neutron
closed shell in the rare-earth region causes an increase in Qa, allowing observable adecay
half-lives for several of these nuclei (with N ¼ 84). Also one has observed
short-lived a emitters near doubly magic 100
Sn, including 107
Te, 108
Te, and 111
Xe.
And, in addition, a emitters have been identified along the proton dripline above
A ¼ 100. For a set of isotopes (nuclei with a constant atomic number) the decay
energy generally decreases with increasing mass. These effects can be seen in
Figure 7.2. For example, the kinetic energy of a particles from the decay of uranium
isotopes is typically 4 to 5 MeV, those for californium isotopes are %6 MeV, and
those for rutherfordium isotopes are %8 MeV. However, the kinetic energy from the
decay of 212
84 Po to the doubly magic 208
82 Pb daughter is 8.78 MeV.
The generally smooth variation of Qa with Z, A of the emitting nucleus and the
two-body nature of a decay can be used to deduce masses of unknown nuclei. One
tool in this effort is the concept of closed decay cycles (Fig. 7.3). Consider the a and
b decays connecting 237
93 Np, 241
95 Am, 241
94 Pu, and 237
92 U. By conservation of energy, one
can state that the sum of the decay energies around the cycle connecting these nuclei
must be zero (within experimental uncertainty). In those cases where experimental
data or reliable estimates are available for three branches of the cycle, the fourth can
be calculated by difference.
Even though the energies released by the decay of a heavy nucleus into an a particle
and a lighter daughter nucleus are quite substantial, the energies are paradoxically
small compared to the energy necessary to bring the a particle back into
nuclear contact with the daughter. The electrostatic potential energy between the
233
91
233
90
237
92
237
93
241
94
241
95
245
96
245
Pa
4.95
a
a
a
a
a
a
1.25 0.52
b b
- b - b -
0.021 EC 0.81
4.23
5.64
5.14
6.45
5.62
Th
Np Am
Pu
Bk
CmU
Figure 7.3 Decay cycles for part of the 4n þ 1 family. Modes of decay are indicated over
the arrows; the numbers indicate total decay energies in MeV.
7.1 ENERGETICS OF a DECAY 181
two positively charged nuclei, called the Coulomb potential, can be written as:
VC ¼
2Z
R
e2
4p10
(7:4)
whereZistheatomicnumberofthedaughterandRistheseparationbetweenthecenters
of the two nuclei. (As pointed out in Chapter 1, e2
=4p10 is 1:440 MeV Á fm.) To obtain
a rough estimate of the Coulomb energy, we can take R to be 1.2(A1/3
þ 41/3
) fm,
where A is the mass number of the daughter. For the decay of 238
U we get
VC ¼
(2)(90)(1:440 MeV Á fm)
1:2 2341=3
þ 41=3
À Á
fm
%
259 MeV Á fm
9:3 fm
¼ 28 MeV (7:5)
which is 6 to 7 times the decay energy. This factor is typical of the ratio of the Coulomb
barrier to the Q value. If we accept for the moment the large difference between the
Coulomb barrier and the observed decay energy, then we can attribute the two
general features of increasing decay energy with increasing atomic number Z and
decreasing kinetic energy with increasing mass among a set of isotopes to
the Coulomb potential. The higher nuclear charge accelerates the products apart, and
the larger mass allows the daughter and a particle to start further apart.
Example Problem Calculate the Qa value, kinetic energy Ta, and the Coulomb
barrier VC, for the primary branch of the a decay of 212
Po to the ground state of
208
Pb.
Solution Using tabulated mass defects we have
Qa ¼ À10:381 À (À21:759 þ 2:4249) ¼ 8:953 MeV
Ta ¼
208
212
Q ¼ 8:784 MeV
and
VC ¼
(2)(84)(1:440 MeV Á fm)
1:2 2081=3
þ 41=3
À Á
fm
% 27 MeV
The 212
Po parent also decays with a 1% branch to the first excited state of 208
Pb at
an excitation energy of 2.6146 MeV. What is the kinetic energy of this a particle?
Qa ¼ 8:953 À 2:6146 ¼ 6:339 MeV
Ta ¼
208
212
† 6:1339 ¼ 6:22 MeV
As discussed previously, many heavy nuclei (A ! 150) are unstable with
respect to a decay. Some of them also undergo b2
decay. In Chapter 3, we discussed
the natural decay series in which heavy nuclei undergo a sequence of b2
and a
decays until they form one of the stable isotopes of lead or bismuth, 206,207,208
Pb
182 a DECAY
or 209
Bi. We are now in a position to understand why a particular sequence occurs.
Figure 7.4 shows a series of mass parabolas (calculated using the semiempirical
mass equation) for some members of the 4n þ 3 series, beginning with 235
U. Each
of the mass parabolas can be thought of as a cut through the nuclear mass surface
at constant A. 235
U decays to 231
Th. 231
Th then decays to 231
Pa by b2
decay. This
nucleus, being near the bottom of the mass parabola, cannot undergo further b2
decay, but decays by a emission to 227
Ac. This nucleus decays by b2
emission to
227
Th, which must a decay to 223
Ra, and so forth.
7.2 THEORY OF a DECAY
The allowed emission of a particles could not be understood in classical pictures of
the nucleus. This fact can be appreciated by considering the schematic potential
energy diagram for 238
U shown in Figure 7.5. Using simple estimates we have
drawn a one-dimensional potential energy curve for this system as a function of
radius. At the smallest distances, inside the parent nucleus, we have drawn a flatbottomed
potential with a depth of 230 MeV (as discussed in Chapter 6). The
potential rapidly rises at the nuclear radius and comes to the Coulomb barrier
height of VC % þ28 MeV at 9.3 fm. At larger distances the potential falls as 1=r
according to Coulomb’s law.
Starting from a separated a particle and the daughter nucleus, we can determine
that the distance of closest approach during the scattering of a 4.2-MeV a particle
will be 62 fm. This is the distance at which the a particle stops moving toward
the daughter and turns around because its kinetic energy has been converted into
potential energy of repulsion. Now the paradox should be clear: The a particle
A=219
40
30
20
10
84 86 88 90 92 94
A=223
A=227
A=231
Atomic Number
MassDefect(MeV)
Figure 7.4 Mass parabolas for some members of the 4n þ 3 natural decay series. The main
decay path is shown by a solid line while a weak branch is indicated by a dashed line.
7.2 THEORY OF a DECAY 183
should not get even remotely near to the nucleus; or from the decay standpoint, the a
particle should be trapped behind a potential energy barrier that it cannot get over.
The solution to this paradox was found in quantum mechanics. A general property
of quantum mechanical wave functions is that they are only completely confined
by potential energy barriers that are infinitely high. Whenever the barrier
has a finite size, the wave function solution will have its main component inside
the barrier plus a small but finite part inside the barrier (generally exponentially
decreasing with distance) and another finite piece outside the barrier. This phenomenon
is called tunneling because the classically trapped particle has a component of
its wave function outside the potential barrier and has some probability to go through
the barrier to the outside without going over the top. The details of these calculations
are discussed in Appendix E and in many quantum mechanics textbooks. Some features
of tunneling should be obvious: The closer the energy of the particle to the top
of the barrier, the more likely that the particle will get out. Also, the more energetic
the particle is relative to a given barrier height, the more frequently the particle will
“assault” the barrier and the more likely that the particle will escape.
It has been known for some time that half-life for a decay, t1/2, can be written in
terms of the square root of the a-particle decay energy, Qa, as follows:
log t1=2
À Á
¼ A þ
B
ffiffiffiffiffiffi
Qa
p (7:6)
Figure 7.5 A (reasonably accurate) one-dimensional potential energy diagram for 238
U
indicating the energy and calculated distances for a decay into 234
Th. Fermi energy
%30 MeV, Coulomb barrier %28 MeV at 9.3 fm, Qa 4.2 MeV, distance of closest approach
62 fm. (Figure also appears in color figure section.)
184 a DECAY
where the constants A and B have a Z dependence. This relationship, shown in
Figure 7.6, is known as the Geiger–Nuttall law of a decay (Geiger and Nuttall,
1911, 1912) due to the fact that Geiger and Nuttall found a linear relationship
between the logarithm of the decay constant and the logarithm of the range of a particles
from a given natural radioactive decay series. This simple relationship
describes the data on a decay, which span over 20 orders of magnitude in decay
constant or half-life. Note that a 1-MeV change in a-decay energy results in a
change of 105
in the half-life. A modern representation of this relationship due to
Hatsukawa, Nakahara, and Hoffman has the form
log10(t1=2) ¼ A(Z) Â
Ad
ApQa
!1=2
 arccos
ffiffiffiffi
X
p
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X(1 À X)
ph i
À 20:446 þ C(Z, N)
(7:7)
where C(Z, N) ¼ 0 for ordinary regions outside closed shells and
C(Z, N) ¼ ½1:94 ¼ À0:020(82 À Z) À 0:070(126 À N)Š
for 78 Z 82, 100 N 126, and
C(Z, N) ¼ ½1:42 À 0:105(Z À 82) À 0:067(126 À N)Š
for 82 Z 90, 110 N 126. In these equations, Ap, Z refer to the parent
nuclide, Ad, and Zd refer to the daughter nuclide, and X is defined as
X ¼ 1:2249(A1=3
d þ 41=3
) Â
Qa
2Zde2
1015
1010
105
10-5
4 5 6 7 8 9 10
92
Alpha-decay energy Qα, Mev
Half-lifet½,sec
92
96
100
88
ZP = 84
1
Figure 7.6 Geiger–Nuttall plot.
7.2 THEORY OF a DECAY 185
This relationship is useful for predicting the expected a-decay half-lives for
unknown nuclei.
The theoretical description of a emission relies on calculating the rate in terms of
two factors. The overall rate of emission consists of the product of the rate at which
an a particle appears at the inside wall of the nucleus times the (independent) probability
that the a particle tunnels through the barrier. Thus, the rate of emission, or
the partial decay constant la, is written as the product of a frequency factor, f, and a
transmission coefficient, T, through the barrier:
la ¼ fT
Some investigators have suggested that this expression should be multiplied by an
additional factor to describe the probability of preformation of an a particle
inside the parent nucleus. Unfortunately, there is no clear way to calculate such a
factor, but empirical estimates have been made. As we will see below, the theoretical
estimates of the emission rates are higher than the observed rates, and the preformation
factor can be estimated for each measured case. However, there are other uncertainties
in the theoretical estimates that contribute to the differences.
The frequency with which an a particle reaches the edge of a nucleus can be estimated
as the velocity divided by the distance across the nucleus. We can take the
distance to be twice the radius (something of a maximum value), but the velocity
is a little more subtle to estimate. A lower limit for the velocity could be obtained
from the kinetic energy of emitted a particle, but the particle is moving inside a
potential energy well, and its velocity should be larger and correspond to the well
depth plus the external energy. Therefore, the frequency can be written as:
f ¼
v
2R
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 V0 þ Qð Þ=m
p
2R
(7:8)
where we have assumed that the a particle is nonrelativistic, V0 is the well depth indicated
in Figure 7.5 of approximately 30 MeV, m is the reduced mass, and R is the
radius of the daughter nucleus (because the a particle needs only to reach this distance
before it is emitted). We use the reduced mass because the a particle is moving inside
the nucleus, and the total momentum of the nucleus must be zero. The frequency of
assaults on the barrier is quite large, usually on the order of 1021
/s.
The quantum mechanical transmission coefficient for an a particle to pass
through a barrier is derived in Appendix E. Generalizing the results summarized
in Equation (E.48) to a three-dimensional barrier shown in Figure 7.5, we have
T ¼ eÀ2G
(7:9)
where the Gamow factor (2G) can be written as:
2G ¼
2
hÀ
ðb
R
2m
Za ZD e2
r
À Qa
 !1=2
dr (7:10)
186 a DECAY
where
Qa ¼
Za ZD e2
b
(7:11)
and the classical distance of closest approach, b, is given as
b ¼
Za ZD e2
Qa
(7:12)
In these equations, e2
¼ 1.440 MeV-fm, Qa is given in MeV, Za, ZD are the atomic
numbers of the a particle and daughter nucleus, respectively.
Rearranging we have
2G ¼
2
hÀ
ffiffiffiffiffiffiffiffiffiffiffiffi
2mQa
p ðb
R
b
r
 1=2
dr (7:13)
This can be integrated to give
2G ¼
2b
hÀ
ffiffiffiffiffiffiffiffiffiffiffiffi
2mQa
p
cosÀ1
ffiffiffi
R
b
r
À
ffiffiffi
R
b
r ffiffiffiffiffiffiffiffiffiffiffiffi
1 À
R
b
r !
(7:14)
Substituting back for b and simplifying, we have
2G ¼ 2
ffiffiffiffiffiffiffiffiffiffiffi
2m
hÀ 2
Qa
s
Za ZD e2
À Á
cosÀ1
ffiffiffi
R
b
r
À
ffiffiffi
R
b
r ffiffiffiffiffiffiffiffiffiffiffiffi
1 À
R
b
r !
(7:15)
For thick barriers,
R
b
( 1
Qa
VC
( 1
 
we can approximate
cosÀ1
ffiffiffi
R
b
r
%
p
2
À
ffiffiffi
R
b
r
(7:16)
We get
2G ¼ 2
ffiffiffiffiffiffiffiffiffiffiffi
2m
hÀ 2
Qa
s
Za ZDe2
À Á p
2
 
(7:17)
where B is the “effective” Coulomb barrier, that is,
B ¼
ZaZDe2
ra þ RD
(7:18)
7.2 THEORY OF a DECAY 187
Typically, the Gamow factor is large (2G 60–120), which makes the transmission
coefficient T extremely small (10255
–10227
). Combining the various equations,
we have
t1=2 ¼
ln 2
l
¼
ln 2
fT
¼
ln 2
½2(V0 þ Qa)=mŠ1=2
2R
eÀ2G
(7:19)
or
log t1=2 ¼ a þ
b
ffiffiffiffiffiffi
Qa
p (7:20)
that is, we get the Geiger–Nuttall law of a decay, where a þ b are constants that
depend on Z, and so forth.
This simple estimate tracks the general behavior of the observed emission rates
over the very large range in nature. The calculated emission rate is typically one
order of magnitude larger than that observed, meaning that the observed half-lives
are longer than predicted. This has led some researchers to suggest that the probability
to find a “preformed” a particle inside a heavy nucleus is on the order of
1021
or less. One way to obtain an estimate of the “preformation factor” is to
plot, for even–even nuclei undergoing l ¼ 0 decay, the ratio of the calculated
half-life to the measured half-life. This is done in Figure 7.7. The average preformation
factor is 1022
.
Figure 7.7 Plot of the ratio of the calculated partial a-decay half-life for ground-state l ¼ 0
transitions of even–even nuclei to the measured half-lives. The calculations were made using
the simple theory of a decay.
188 a DECAY
Example Problem Calculate the emission rate and half-life for 238
U decay
from the simple theory of a decay. Compare this to the observed half-life.
l ¼ f T
where
f ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2(V0 þ Q)=m
p
2R
R ¼ r0 A1=3
1 þ A1=3
1
 
¼ 1:2 41=3
þ 2341=3
À Á
¼ 9:3 fm
Solution Note that since we previously calculated b % 62 fm, R/
b ¼ 8:63=62 ( 1:
m ¼ 4 Ã
234
238
¼ 3:933 amu
f ¼
c Ã
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2(30 þ 4:2)=(3:933 Ã 931:5)
p
2 Ã 9:3
¼ (2:26 Â 1021
)=s
We know that
T ¼ eÀ2G
where
2G % 2
2m
hÀ 2
Qa
 1=2
(ZaZDe2
)
p
2
À 2
ffiffiffiffi
Q
B
r !
2
2m
hÀ 2
Qa
 
¼ 2
2m (amu) 931:5
MeV
amu
 
(hÀ
c)2
(MeV-fm)2
Qa (MeV)
0
B
B
@
1
C
C
A
1=2
¼ 2
(2)(3:933)(931:5)
(197:3)2
(4:27)
 1=2
¼ 0:420 (MeV-fm)À1
(ZaZDe2
) ¼ (2)(90)(1:440) ¼ 259:2 MeV-fm
p
2
À 2
ffiffiffiffi
Q
B
r !
¼
p
2
À 2
4:27
27:9
 1=2
¼ 0:788
T ¼ eÀ85:8
¼ 5:43 Â 10À38
l ¼ fT ¼ (2:26 Â 1021
)(5:43 Â 10À38
) ¼ 1:23 Â 10À16
sÀ1
t1=2 ¼
ln 2
l
¼ 5:65 Â 1015
s ¼ 1:8 Â 108
y
7.2 THEORY OF a DECAY 189
The observed half-life of 238
U is 4.47 Â 109
y, which is a factor of 25 times
longer than the calculated value. Note the qualitative aspects of this calculation.
The a particle must hit the border of the parent nucleus 1038
times before it can
escape. Also note the extreme sensitivity of this calculation to details of the
nuclear radius. A 2% change in R changes l by a factor of 2. In our example,
we approximated R as RTh þ Ra. In reality, the a particle has not fully separated
from the daughter nucleus when they exit the barrier. One can correct for this by
approximating R % 1.4A1/3
.
The theory presented above neglects the effects of angular momentum in that it
assumes the a particle carries off no orbital angular momentum (l ¼ 0). If a decay
takes place to or from an excited state, some angular momentum may be carried off
by the a particle with a resulting change in the decay constant. In quantum
mechanics, we say that the a particle has to tunnel through a barrier that is larger
by an amount called the centrifugal potential:
Vl ¼
l(l þ 1)hÀ 2
2mR2
(7:21)
where l is the orbital angular momentum of the a particle, m is the reduced mass,
and R is the appropriate radius. This centrifugal potential is added to the potential
energy V(r) resulting in a thicker and higher barrier, increasing the half-life
(Fig. 7.8).
One can evaluate the effect of this centrifugal potential upon a-decay half-lives
by simply adding this energy to the Coulomb barrier height. If we define
s ¼
centrifugal barrier height
Coulomb barrier height
(7:22)
0 R
V(r)+
V(r) =
l(l + 1) h2
zZe2
r
2Ma r2
r
Energy
Figure 7.8 Modification of the potential energy in a decay due to the centrifugal potential.
[From W. E. Meyerhof, Elements of Nuclear Physics, Copyright 1967 by McGraw-Hill Book
Company, Inc. Reprinted by permission of McGraw-Hill Book Company, Inc.]
190 a DECAY
we can say
s ¼
l(l þ 1)hÀ 2
2mR2
R
ZaZDe2
¼
l(l þ 1)hÀ 2
2mRZaZDe2
(7:23)
Then all we need to do is to replace all occurrences of B by B (1 þ s). A simple
pocket formula that does this is
ll=0 % ll¼0 exp½À2:027l(l þ 1)ZÀ1=2
AÀ1=6
Š (7:24)
This centrifugal barrier correction is a very small effect compared to the effect of Qa
or R upon the decay rate.
We should also note that conservation of angular momentum and parity during
the a-decay process places some constraints on the daughter states that can be populated.
Since the a particle has no intrinsic spin, the total angular momentum of the a
particle must equal its orbital angular momentum l, and the a particle parity must be
(21)l
. If parity is conserved in a decay, the final states are restricted. If the parent
nucleus has Jp
¼ 0þ
, then the allowed values of Jp
of the daughter nucleus are 0þ
(l ¼ 0), 12
(l ¼ 1), 2þ
(l ¼ 2), and so on. These rules only specify the required spin
and parity of the state in the daughter, while the energy of the state is a separate
quantity. Recall from Chapter 6 that the heaviest elements are strongly deformed
and are good rotors. The low-lying excited states of even–even nuclei form a
low-lying rotational band with spins of 2, 4, 6, and so forth, while odd angular
momenta states tend to lie higher in energy. Because of the decrease in the
energy of the emitted a particle when populating these states, decay to these
states will be inhibited. Thus, the lower available energy suppresses these decays
more strongly than the centrifugal barrier.
Example Problem 241
Am is a long-lived a emitter that is used extensively
as an ionization source in smoke detectors. The parent state has a spin and
parity of 5
2
À
and cannot decay to the 5
2
þ
ground state of 237
Np because that
would violate parity conservation. Rather, it decays primarily to a 5
2
À
excited
state (85.2%, EÃ
¼ 59.5 keV) and to a 7
2
À
higher lying excited state (12.8%,
EÃ
¼ 102.9 keV). Estimate these branching ratios and compare them to the
observed values.
Qa
5
2
ÀÀ Á
¼ 5:578 MeV Qa
7
2
ÀÀ Á
¼ 5:535 MeV
f 5
2
ÀÀ Á
¼ 2:29 Â 1021
=s f 7
2
ÀÀ Á
¼ 2:29 Â 1021
=s
G 5
2
ÀÀ Á
¼ 33:01 G 7
2
ÀÀ Á
¼ 33:84
l 5
2
ÀÀ Á
¼ 4:89 Â 10À8
=s l 7
2
ÀÀ Á
¼ 9:2 Â 10À9
=s
7.2 THEORY OF a DECAY 191
Solution Assuming that the branches to other states are small and do not contribute
to the sum of partial half-lives, we can write
Branching ratio
5
2À
%
l(5=2À
)
l(5=2À) þ l(7=2À)
¼ 0:84
Note that the observed half-life of 433 y is again significantly longer than the predicted
half-life of 3 y. This difference is attributed to the combined effects of
the preformation factor and the hindrance effect of the odd proton in the americium
parent (Z ¼ 95); see below.
7.3 HINDRANCE FACTORS
The one-body theory of a decay applies strictly to even–even a emitters only. The
odd-nucleon a emitters, especially in ground-state transitions, decay at a slower rate
than that suggested by the simple one-body formulation as applied to even–even
nuclei. Consider the data in Figure 7.9 that shows the a-decay half-lives of the
even–even and odd A uranium isotopes. The odd A nuclei have substantially
longer half-lives than their even–even neighbors do.
The decays of the odd A nuclei are referred to as hindered decays, and a
hindrance factor may be defined as the ratio of the measured partial half-life for
a given a transition to the half-life that would be calculated from the simple
one-body theory applied to even–even nuclides.
1012
109
106
103
100
4.0 4.5 5.0 5.5 6.0
Alpha Decay Energy (MeV)
T½(Years)
Figure 7.9 The a-decay half-lives of the even–even (squares) and odd A (diamonds)
isotopes of uranium. The measured values are connected by the solid line; the medians are
shown by the dashed line.
192 a DECAY
In general, these hindrances for odd A nuclei may be divided into five classes:
1. If the hindrance factor is between 1 and 4, the transition is called a “favored”
transition. In such decays, the emitted a particle is assembled from two
low-lying pairs of nucleons in the parent nucleus, leaving the odd nucleon
in its initial orbital.
To form an a particle within a nucleus, two protons and two neutrons must
come together with their spins coupled to zero and with zero orbital angular
momentum relative to the center of mass of the a particle. These four nucleons
are likely to come from the highest occupied levels of the nucleus. In odd A
nuclei, because of the odd particle and the difficulty of getting a “partner”
for it, one pair of nucleons is drawn from a lower lying level, causing the
daughter nucleus to be formed in an excited state.
2. A hindrance factor of 4–10 indicates a mixing or favorable overlap between
the initial and final nuclear states involved in the transition.
3. Factors of 10–100 indicate that spin projections of the initial and final states
are parallel, but the wave function overlap is not favorable.
4. Factors of 100–1000 indicate transitions with a change in parity but with
projections of initial and final states being parallel.
5. Hindrance factors of .1000 indicate that the transition involves a parity
change and a spin flip, that is, the spin projections of the initial and final
states are antiparallel, which requires substantial reorganization of the
nucleon in the parent when the a is emitted.
7.4 HEAVY PARTICLE RADIOACTIVITY
As an academic exercise, one can calculate the Q values for the emission of heavier
nuclei than a particles and show that it is energetically possible for a large range of
heavy nuclei to emit other light nuclei. For example, contours of the Q values for
carbon ion emission by a large range of nuclei are shown in Figure 7.10 calculated
with the smooth liquid drop mass equation without shell corrections. Recall that the
binding energy steadily decreases with increasing mass (above A  60), and several
light nuclei have large binding energies relative to their neighbors similar to the a
particle. As can be seen in Figure 7.10, there are many nuclei with positive Q
values for carbon ion emission. Such emission processes or heavy particle radioactivity
have been called heavy cluster emission.
We should also note that the double-shell closures at Z ¼ 82 and N ¼ 126 lead to
especially large positive Q values, as already shown in Figure 7.2. Thus, the emission
of other heavy nuclei, particularly 12
C, has been predicted or at least anticipated
for a long time. Notice also that 12
C is an even–even nucleus and s-wave emission
without a centrifugal barrier is possible. However, the Coulomb barrier will be
significantly larger for higher Z nuclei than that for a particles.
7.4 HEAVY PARTICLE RADIOACTIVITY 193
We can use the simple theory of a decay to make an estimate of the relative
branching ratios for a emission and 12
C emission from 220
Ra, a very favorable
parent that leads to the doubly magic 208
Pb daughter. In this case we find
Qa ¼ 7.59 MeV and QC ¼ 32.02 MeV. Using the simple theory and ignoring
differences in the preformation factor, the predicted half-life for 12
C emission is
only longer by a factor of 2!
220
Ra ) 216
Rn þ 4
He Q ¼ 7:59 lcalc ¼ 5:1 Â 104
s
220
Ra ) 208
Pb þ 12
C Q ¼ 32:02 lcalc ¼ 3:34 Â 104
s
The encouraging results from simple calculations such as these have spurred many
searches for this form of radioactivity.
It was relatively recently that heavy cluster emission was observed at a level
enormously lower than these estimates. Even so, an additional twist in the process
was discovered when the radiation from a 223
Ra source was measured directly in
a silicon surface barrier telescope. The emission of 14
C was observed at the rate
of 1029
times the a-emission rate, and 12
C was not observed. Thus, the very
large neutron excess of the heavy elements favors the emission of neutron-rich
light products. The fact that the emission probability is so much smaller than the
simple barrier penetration estimate can be attributed to the very small probability
Neutron number
Protonnumber
0 MeV
−20 MeV
+20 MeV
+40 MeV
60 80 100 120 140 160 180
50
60
70
80
90
100
110
Figure 7.10 Contours of the Q value for the emission of a 12
C nucleus as a function of
neutron and proton numbers calculated with the liquid drop model mass formula. The
contour lines are separated by 10 MeV. The dotted curve indicates the line of b stability
[Eq. (2.9)].
194 a DECAY
to “preform” a 14
C residue inside the heavy nucleus. This first observation has
been confirmed in subsequent measurements with magnetic spectrographs. The
more rare emission of other larger neutron-rich light nuclei have been reported in
very sensitive studies with nuclear track detectors.
7.5 PROTON RADIOACTIVITY
For very neutron-deficient (i.e., proton-rich) nuclei, the Q value for proton emission,
Qp, becomes positive. One estimate, based on the semiempirical mass equation, of
the line that describes the locus of the nuclei where Qp becomes positive for groundstate
decay is shown in Figure 7.11. This line is known as the proton-drip line.
Our ability to know the position of this line is a measure of our ability to describe
the forces holding nuclei together. Nuclei to the left of the proton dripline in
Figure 7.11 can decay by proton emission.
Proton decay should be a simple extension of a decay with the same ideas of
barrier penetration being involved. A simplification with proton decay relative to
a decay is that there should be no preformation factor for the proton. The situation
is shown in Figure 7.12 for the case of the known proton emitter 151
Lu. One notes
certain important features/complications from this case. The proton energies, even
for the heavier nuclei, are low (Ep  1–2 MeV). As a consequence, the barriers to
be penetrated are quite thick (Rout ¼ 80 fm), and one is more sensitive to the proton
energy, angular momentum changes, and so forth.
Figure 7.11 Locus of neutron and proton driplines as predicted by the liquid drop model.
7.5 PROTON RADIOACTIVITY 195
30
VCoul
Vp.tot(l = 5)
Vp.tot(l = 0)
Radius / fm
20 40 60 80 100
Ep = (1233 ± 3) keV
(85 ± 10) ms (Exp.)
T1/2l
0
2
5
3 ms
26 ms
81 ms
151Lu(11/2-) 150Yb(0+)
V/MeV
20
10
1.25
0
-10
-20
-30
-40
Figure 7.12 Proton–nucleus potential for the semiclassical calculation of the 151
Lu partial
proton half-life. [From S. Hofmann, In D. N. Poenaru (Ed.), Nuclear Decay Modes, Copyright
1996 by IOP Publishing. Reprinted by permission of IOP Publishing.]
Figure 7.13 (a) Energy spectrum obtained during the irradiation of a 96
Ru target with
261 MeV 58
Ni projectiles. (b) Expanded part of the spectrum showing the proton line from
151
Lu decay. [From S. Hofmann, In D. N. Poenaru (Ed.), Nuclear Decay Modes, Copyright
1996 by IOP Publishing. Reprinted by permission of IOP Publishing.]
196 a DECAY
The measurements of proton decay are challenging due to the low energies and
short half-lives involved. Frequently, there are interfering a decays (Fig. 7.13). To
produce nuclei near the proton dripline from nuclei near the valley of b stability
requires forming nuclei with high excitation energies that emit neutrons relative
to protons and a particles to move toward this proton dripline. This, along with
difficulties in studying low-energy proton emitters, means that the known proton
emitters are mostly in the medium mass—heavy nuclei. A review article by
Hofmann (1996) summarizes the details of proton decay.
PROBLEMS
1. Using the conservation of momentum and energy, derive a relationship between
Qa and Ta.
2. All nuclei with A . 210 are a emitters, yet very few emit protons spontaneously.
Yet both decays lower the Coulomb energy of the nucleus. Why is
proton decay not more common?
3. Use the Geiger–Nuttall rule to estimate the expected a decay half-lives of the
following nuclei: 148
Gd, 226
Ra, 238
U, 252
Cf, and 262
Sg.
4. Use the one-body theory of a decay to estimate the half-life of 224
Ra for decay
by emission of a 14
C ion or a 4
He ion. The measured half-life for the 14
C decay
mode is 1029
relative to the 4
He decay mode. Estimate the relative preformation
factors for the a particle and 14
C nucleus in the parent nuclide.
5. 212
Pom
and 269
110 both decay by the emission of high-energy a particles
(Ea ¼ 11.6 and 11.1 MeV, respectively). Calculate the expected lifetime of
these nuclei using the one-body theory of a decay. The observed half-lives
are 45.1 s and 170 ms, respectively. Comment on any difference between the
observed and calculated half-lives.
6. Consider the decay of 278
112 to 274
110. The ground-state Qa value is
11.65 MeV. Calculate the expected ratio of emission to the 2þ
, 4þ
, and 6þ
states of 110.
7. What is the wavelength of an a particle confined to a 238
U nucleus?
8. 8
Be decays into two a particles with Qa ¼ 0.094 MeV. Calculate the expected
half-life of 8
Be using one-body theory and compare this estimate to the
measured half-life of 2.6 Â 1027
s.
9. Calculate the kinetic energy and velocity of the recoiling daughter atom in the
a decay of 252
Cf.
10. Calculate the hindrance factor for the a decay of 243
Bk to the ground state of
239
Am. The half-life of 245
Bk is 4.35 h, the decay is 99.994% EC and 0.006%
a decay. Also, 0.0231% of the a decays lead to the ground state of 241
Am.
Qa for the ground state decay is 6.874 MeV.
PROBLEMS 197
11. Calculate Qa for gold. Why do we not see a decay in gold?
12. The natural decay series starting with 232
Th has the sequence abba. Show why
this is the case by plotting the mass parabolas (or portions thereof) for A ¼ 232,
228, and 224.
13. Using the semiempirical mass equation, verify that Qa becomes positive for
A ! 150.
14. Calculate the heights of the centrifugal barrier for the emission of a particles
carrying away two units of angular momentum in the decay of 244
Cm.
Assume R0 ¼ 1 Â 10213
cm. What fraction of the Coulomb barrier height
does this represent?
15. Use one-body theory to calculate the expected half-life for the proton decay of
185
Bi.
REFERENCES
Hofmann, S. Proton Radioactivity, In D. N. Poenaru (Ed.), Nuclear Decay Modes, IOP,
Bristol, 1996.
Meyerhof, W. Elements of Nuclear Physics, McGraw-Hill, New York, 1967, pp. 135–145.
BIBLIOGRAPHY
Textbook discussions of a decay that are especially good.
Evans, R. The Atomic Nucleus, McGraw-Hill, New York, 1953.
Heyde, K. Basic Ideas and Concepts in Nuclear Physics, IOP, Bristol, 1994, pp. 82–103.
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1988, pp. 246–271.
Rasmussen, J. O. Alpha Decay, In K. Siegbahn (Ed.), Alpha-, Beta-, and Gamma-Ray
Spectroscopy, North-Holland, Amsterdam, 1965, Chapter XI.
Wong, S. S. M. Introductory Nuclear Physics, 2nd ed., Wiley, New York, 1998.
198 a DECAY
CHAPTER 8
b DECAY
8.1 INTRODUCTION
We have seen that many thousands of nuclei can be produced and studied in the lab.
However, only less than 300 of these nuclei are stable; the rest are radioactive. We
have also seen that the degree of instability grows with the “distance” a given
nuclide is from the stable nuclide with the same mass number. In the previous
chapter we considered the process of a decay in which heavy nuclei emit a particles
to reduce their mass and move toward stability. The Coulomb barrier limits this
process to those regions where the Q value provides sufficient energy to tunnel
through the barrier. The vast majority of unstable nuclei lie in regions in which a
decay is not important and the nuclei undergo one or another form of b decay in
order to become more stable. In a certain sense, the stable nuclei have a balance
between the numbers of neutrons and protons. Nuclei are said to be unstable with
respect to b decay when these numbers are “out of balance.” In a very qualitative
way b decay “converts” a neutron into a proton (or vice versa) inside a nucleus,
which becomes more stable while maintaining a constant mass number. The b
decay process is more complicated than a emission, and we will provide an overview
and a discussion of its basic features in this chapter.
b Decay is named for the second most ionizing rays that were found to emanate
from uranium samples. The naturally occurring b rays were identified as fast moving
(negative) electrons relatively easily, but it took many years to obtain a full understanding
of the emission process. The difficulty lies in the fact that two particles are
199
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
“created” during the decay as compared to the “disruption” of a heavy nucleus in a
decay. In contrast to a decay, angular momentum plays a crucial role in understanding
the process. Let us consider the simplest form of b decay to illustrate the difficulties.
The proton and the neutron are the two possible isobars for A ¼ 1. We know
that the neutron has a larger mass than the proton and is thus unstable with respect to
the combination of a proton and an electron. A free neutron will undergo b decay
with a half-life of approximately 10 min. We might expect to write the decay
equation as:
1
0n ! 1
1 p þ 0
À1e þ Q (incomplete)
However, all three particles in this equation are fermions with intrinsic spins
S ¼ 1
2
hÀ
. Therefore, we cannot balance the angular momentum in the reaction as
written. The spins of the proton and the electron can be coupled to 0 or 1 hÀ
and
can also have relative angular momenta with any integral value from the
emission process. This simple spin algebra will never yield the half-integral value
on the left-hand side of the equation. Another fermion must be present among the
products.
Another feature of b decay that was puzzling at first but really pointed to the
incompleteness of the previous equation is that the b rays have a continuous
energy distribution. That is, electrons are emitted from a source with a distribution
of energies that extends from a maximum at the Q value down to zero. Recall that if
there are only two products from a reaction then they will precisely share the decay
energy according to conservation of momentum. We have clearly seen such sharp
energy spectra in a decay. (The continuous energy distribution is not an instrumental
artifact, nor does it come from electron scattering.) Quite dramatic pictures of the
tracks of charged particles from beta decay show events in which the particles
move in one direction in clear violation of conservation of linear momentum. The
way out of this mounting paradox with violations of very strongly held conservation
laws is to introduce another conservation law and recognize that another unseen
particle must be created and emitted. The conservation law is conservation of the
number of “particles” in a reaction, and the unseen particle is a form of neutrino,
literally “the little neutral one” in Italian.
8.2 NEUTRINO HYPOTHESIS
Enrico Fermi on his voyage to the new world postulated that a third particle was
needed to balance the emission of the electron in b decay. However, the existing
conservation laws also had to be satisfied, so there were a number of constraints
on the properties of this new particle. Focusing on the decay of a neutron as a
specific example, the reaction is already balanced with respect to electric charge,
so any additional particle must be neutral. The electrons were observed with energies
up to the maximum allowed by the decay Q value so the mass of the particle
must be smaller that the instrumental uncertainties. Initially, this instrumental
200 b DECAY
limit was ,1 keV, but this has been reduced to ,10 eV in recent work. Recent
experiments have shown that the neutrinos have a very small mass (Chapter 12).
The third constraint on the neutrino from the decay is that it must be an “antiparticle”
in order to cancel or compensate for the creation of the electron, a “particle.” The
fourth constraint is that the neutrino must have half-integral spin and be a
fermion in order to couple the total final angular momentum to the initial spin of 1
2
hÀ
Combining all of these constraints we can now rewrite the previous equation
properly as:
1
0n À! 1
1 p þ 0
À1e þ 0
0 ne þ Q
where we have used the notation of placing a bar over the Greek character nu to
indicate that the neutrino is an antiparticle and a subscript indicating the neutrino
is an electron neutrino (Chapter 1). As indicated in Chapter 1, the existence of antiparticles
and antimatter extends quite generally, and we produce and observe the
decays of antielectrons (positrons), antiprotons, antineutrons, and so forth, and
even combine positrons and antiprotons to make antihydrogen!
The spins of all of the final products can be combined in two ways and still couple
to the initial spin of the neutron. Focusing on the spins of the created particles, they
can vector couple to Sb ¼ 1 in a parallel alignment or to Sb ¼ 0 in an antiparallel
alignment. Both of these can combine with S ¼ 1
2 of the neutron for a resultant
vector of 1
2. The two possible relative alignments of the “created” spins are
labeled as Fermi (F) (Sb ¼ 0) and Gamow–Teller (GT) (Sb ¼ 1) decay modes
after the people that initially described the mode. Both modes are very often possible
and a source will produce a mixture of relative spins. In some cases, particularly the
decay of even–even nuclei with N ¼ Z (the so-called self mirror nuclei), the neutron
and protons are in the same orbitals so that 0þ
to 0þ
decay can only take place by a
Fermi transition. In heavy nuclei with protons and neutrons in very different orbitals
(shells), the GT mode dominates. In complex nuclei, the rate of decay will depend on
the overlap of the wave functions of the ground state of the parent and the state of the
daughter. The final state in the daughter depends on the decay mode. Notice that in
the example of neutron decay, the difference between the two modes is solely the
orientation of the spin of the bare proton relative to the spins of the other products.
The decay constant can be calculated if these wave functions are known. Alternatively,
the observed rate gives some indication of the quantum mechanical
overlap of the initial and final state wave functions.
The general form of b2
decay of a heavy parent nucleus, A
Z, can be written as:
A
ZN À! A
(Z þ 1)þ
NÀ1 þ eÀ
þ ne þ QbÀ
where we have written out the charges on the products explicitly. Notice that
the electron can be combined with the positive ion to create a neutral atom
(with the release of very small binding energy). This allows us to use the masses
of the neutral atoms to calculate the Q value, again assuming that the mass of the
8.2 NEUTRINO HYPOTHESIS 201
antineutrino is very small. Thus,
QbÀ ¼ M½A
ZŠ À M½A
(Z þ 1)Š
Up to this point we have concentrated on the b-decay process in which a neutron is
converted into a proton. There are a large number of unstable nuclei that have more
protons in the nucleus than the stable isobar and so will decay by converting a proton
into a neutron. We can write an equation for bþ
decay that is exactly analogous to
the previous equation:
A
ZN À! A
(Z À 1)À
Nþ1 þ eþ
þ ne þ Qbþ
where we have replaced both the electron and the electron antineutrino with their
respective antiparticles, the positron and the electron neutrino. Note in this case,
in contrast to b2
decay, the charge on the daughter ion is negative. This means
that there is an extra electron present in the reaction compared to that with a
neutral daughter atom. Thus, the Q value must reflect this difference:
Qbþ ¼ M½A
ZŠ À (M½A
(Z À 1)Š þ 2mec2
)
where me is the electron mass. Recall that particles and antiparticles have identical
masses. This equation shows that spontaneous bþ
decay requires that the mass difference
between the parent and daughter atoms be greater than 2mec2
¼ 1.022 MeV.
Nature takes this to be an undue restriction and has found an alternative process
for the conversion of a proton into a neutron (in an atomic nucleus). The process
is the capture of an orbital electron by a proton in the nucleus. This process, called
electron capture (EC), is particularly important for heavy nuclei. The reaction is
written:
A
ZN À! A
(Z À 1)Nþ1 þ ne þ QEC
where all of the electrons are implicitly understood to be present on the atoms. This
process also has the property that the final state has only two products so conservation
of momentum will cause the neutrino to be emitted with precise energies depending
on the binding energy of the captured electron and the final state of the daughter
nucleus.
To summarize, there are three types of decay, all known as b decay. They are
A
ZP À! A
Zþ1D þ bÀ
þ ne
A
ZP À! A
ZÀ1D þ bþ
þ ne
eÀ
þ A
ZP À! A
ZÀ1D þ ne
202 b DECAY
indicating b2
decay of neutron-rich nuclei, bþ
decay of proton-rich nuclei, and electron
capture decay of proton-rich nuclei. Neglecting the electron binding energies in
computing the decay energetics, we have
QbÀ ¼ (MP À MD)c2
Qbþ ¼ (MP À MD)c2
À 2mec2
QEC ¼ (MP À MD)c2
where M is the atomic mass of the nuclide involved and me is the electron mass.
Typical values of Qb2 near stability are 0.5–2 MeV, Qbþ  2–4 MeV, and
QEC  0.2–2 MeV.
As a final point in the introduction, it is interesting to note that the analogous
process of positron capture by neutron excessive nuclei should be possible in principle.
However, such captures are hindered by two important facts: First, the number
of positrons available for capture is vanishingly small in nature, and second, both the
nucleus and the positron are positively charged and will repel one another. Compare
this to the situation for electron capture in which the nucleus is surrounded by (negative)
electrons that are attracted to the nucleus, of course, and the most probable
position to find any s electrons is at the nucleus (r ¼ 0).
Example Problem Write the balanced equation for positron capture on the
b-unstable nucleus, 24
Na. Calculate the Q value for this process.
Solution On the left-hand side of the equation we assume that we have a
24
Na nuclide (with 11 electrons) and a single positron, which is an antilepton.
The conservation rules imply that the mass number of the product will be 24,
the atomic number will be Z ¼ 11 þ 1, the 11 electrons will carry over, and an
antilepton has to be created to conserve lepton number. Thus,
24
Na þ eþ
À! 24
Mg
þ
þ ne þ Qpc
We must be careful about the number of electrons on both sides of the equation
when we calculate the Q value. If we use mass defects rather than the masses and
assume a zero-mass neutrino, then
Qpc ¼ (D(24
Na) þ mec2
) À (D(24
Mg) À mec2
)
or
Qpc ¼ (D(24
Na) þ 2mec2
) À D(24
Mg)
Qpc ¼ (À8:418 þ 1:022) þ 13:933 ¼ 6:537 MeV
8.3 DERIVATION OF SPECTRAL SHAPE
b Decay is clearly a process that follows first-order kinetics, and the rate of decay
should be described by a single decay constant. Experimentally, b decay has been
8.3 DERIVATION OF SPECTRAL SHAPE 203
observed with a huge range of half-lives, from a few milliseconds (and no shorter)
to 1016
y. This large range is reminiscent of the range for a decay, and we
should expect that the nuclear structure of the parent, ground state, and the
available daughter states will play important roles in determining the half-life.
We should also recognize that the calculation of the rate will require a full
quantum mechanical approach because the decay process involves the creation of
two particles, and the kinetic energy spectrum is relativistic for the electron
because Qb  mec2
.
Fermi developed a quantum mechanical theory of b decay building on the foundation
of the theory for the spontaneous emission of photons by systems in excited
states. At first blush these may seem unrelated, but in both cases a system in a very
well-defined single state that has excess energy releases the energy spontaneously by
the creation of a particle (or particles). The decay constant for the emission of a
photon was shown in the appendix E to be given by the general expression:
l ¼
2p
hÀ
ð
CÃ
finalVpCinitial dt








2
r(Ef )
which is also called Fermi’s golden rule. The wave functions, C, represent the complete
initial and final states of the entire system, and Vp is a (very) small perturbative
interaction that stimulates the transition. The form and the strength of the perturbation
will have to be determined. Fermi assumed that the interaction responsible
for b decay is different from the gravitational, Coulomb, and nuclear forces. This
interaction between the nucleons, electron, and neutrino is called the weak interaction,
and a new constant expressing its strength, like e and G, can be defined.
This constant, g, has the numerical value of 0.88 Â 1024
MeV/fm3
, which is
approximately 1023
of the electromagnetic force constant. The last factor, r(Ef ),
is the density of states that are available to the system after the transition and is
often written as dn/dE where n is the number of states per unit energy interval.
In this case the final energy is the decay Q value. The initial wave function contains
only the parent nucleus, whereas the final wave function will have parts for all
the resultant particles. Specifically, for b decay Cinitial ¼ fgs(A
Z), the complete
wave function for the parent in its ground state. The final wave function will
have three parts, CÃ
final ¼ fÃ
j (A
Z)fÃ
(e)fÃ
(n), a part for the daughter nucleus in the
appropriate state j, a part for the traveling wave of the electron, and a part for the
corresponding traveling wave of the neutrino, all of which must be coupled so
that energy is conserved.
The quantum mechanical problem can be separated into two parts, the determination
of r(Ef ) and the matrix element
Ð
wÃ
finalVp winitialdt



2
, to make the calculation
tractable. The determination of the density of final states, dn/dE, is done using
quantum statistical mechanics. It is basically the problem of counting the number
of ways the decay energy can be divided among the electron and the neutrino,
neglecting for the moment, the recoiling daughter nucleus. Classically, the
number of states of a free electron with momentum between pe and pe þ dpe in a
volume V is (V4pp2
e dpe)=h3
. (This is the volume of a spherical shell in phase
204 b DECAY
space where the volume of a unit cell is h3
.) Similarly for the neutrino, the number of
states of the free neutrino with momentum between pn and pn þ dpn in a volume V is
(V4pp2
ndpn)=h3
. The total number of states is the product of these two factors:
dn ¼
16p2
V2
p2
e p2
n dpe dpn
h6
If we assume the neutrino has zero rest mass
pn ¼
Tn
c
¼
Q À Te
c
dpn ¼
dQ
c
Then, substituting, we get
dn ¼
16p2
V2
h6c3
(Q À Te)2
p2
e dpe dQ
dn
dQ
¼
16p2
V2
h6c3
(Q À Te)2
p2
e dpe
(One must understand that this equation expresses the variation of the number of
final states with changes in the Q value of the decay and does not represent differentiation
with respect to a constant Q.)
The electron and neutrino wave functions can be written as plane waves as:
fe(r) ¼ Aeiker
¼
1
ffiffiffiffi
V
p eiker
fn(r) ¼ Beiknr
¼
1
ffiffiffiffi
V
p eiknr
where we have applied a normalization condition to determine the constants A and
B. We can expand the exponentials for r  0 (the nuclear volume) as:
eikr
¼ 1 þ ikr þ Á Á Á ffi 1
Thus
fe(r  0) ffi
1
ffiffiffiffi
V
p
fn(r  0) ffi
1
ffiffiffiffi
V
p
The probability of emitting an electron with a momentum pe between pe and dpe
becomes
l(pe) dpe ¼
1
2p3hÀ 7
c3
jMifj2
g2
(Q À Te)2
p2
e dpe
8.3 DERIVATION OF SPECTRAL SHAPE 205
where jMifj2
is a nuclear matrix element representing the overlap between the initial
and final nuclear states. This matrix element must be evaluated with the detailed
nuclear wave functions, for example, those available from the shell model.
Collecting all constants for a given decay, the probability of a decay as a function
of the electron momentum is
l(pe) dpe ¼ (constants)(Q À Te)2
p2
e dpe
This form (even though it is mixed with a momentum part and an energy part for the
electron) clearly goes to zero at pe ¼ 0 and also at Te ¼ Q and thus has a maximum
in between. The shape of this function is shown in Figure 8.1. This function is often
called the statistical or phase space factor for the decay.
We should be sure to note that we have made a big approximation in ignoring the
charge on the emitted electron. Positively charged b particles (positrons) will be
repelled by the nucleus and shifted to higher energies, whereas negatively
charged b particles (electrons) will be attracted by the nucleus and slowed down.
These effects were incorporated by Fermi by using Coulomb-distorted wave functions
and are contained in a spectrum distortion expression called the Fermi
0.0
N(p)
1.0 2.0 3.0
pmax = 2.967 MeV/c
p (MeV/c)
0.0
N(Te)
1.0 2.0
(Te)max = Q
Te (MeV)
Figure 8.1 Shape of the statistical factor for b decay, which represents the expected shape
of the electron momentum distribution before distortion by the Coulomb potential.
206 b DECAY
function, F (ZD , pe), where ZD is the atomic number of the daughter nucleus. The b
spectrum thus has the form:
l(pe) dpe ¼ (constants)F(ZD, pe)p2
e(Q À Te)2
dpe
The effects of the Coulomb distortion can be seen in the measured spectra from the
decay of 64
Cu shown in Figure 8.2. This odd–odd nucleus undergoes both b2
and
bþ
decay to its even–even neighbors with very similar Q values.
Relaxing the restriction that the neutrino rest mass is zero, we get
(Heyde, 1999)
l(pe) dpe ¼
jMifj2
2p3hÀ 7
c3
g2
F(ZD, pe)p2
e(Q À Te)2
1 À
m2
nc4
(Q À Te)2
 1=2
dpe
8.4 KURIE PLOTS
We have seen that the b spectrum has an endpoint at the Q value, but the form of
equation for the spectrum does not allow us to easily identify the endpoint.
0
8
6
4
N(p)
2
0
0.5
p(MeV/c)
64Cu b-
pmax
pmax
1.0 0
N(Te)
0
5
4
3
2
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7
64Cu b-
(Te)max
0
8
6
4
N(p)
2
0
0.5
p(MeV/c)
64Cu b+
1.0 0
N(Te)
0
5
4
3
2
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7
64Cu b+
(Te)maxTe(MeV)
Te(MeV)
Figure 8.2 Momentum and energy spectra from the decay of 64
Cu for b2
and bþ
decay. The
Q values for these decays are 0.5782 and 0.6529 MeV, respectively.
8.4 KURIE PLOTS 207
Notice that with a little rearrangement this spectrum can be represented as:
l(pe)
p2
eF(ZD, pe)
 1=2
/ (Q À Te)jMifj2
If the nuclear matrix element does not depend on the electron kinetic energy, as we
have assumed so far, then a plot of the reduced spectral intensity, the left-hand side,
versus the electron kinetic energy will be a straight line that intercepts the abscissa at
the Q value. Such a graph is called a Kurie plot, and an example is shown in
Figure 8.3. This procedure applies to allowed transitions (see below). There are correction
terms that need to be taken into account for forbidden transitions.
8.5 b-DECAY RATE CONSTANT
The differential form of the spectrum can be integrated over all electron momenta to
obtain the total decay constant. The expression, for a constant nuclear matrix
element, to be integrated is
l ¼
g2
jMifj2
2p3hÀ 7
c3
ð pmax
0
F(ZD, pe)p2
e(Q À Te)2
dp
Note that an appropriate relativistic substitution for T in terms of the momentum is
still needed. This integral has been shown to only depend on the atomic number of
the daughter and the maximum electron momentum. The integral, called the Fermi
Figure 8.3 Example of a Kurie plot. (From Camp and Langer, 1963.)
208 b DECAY
integral, f(ZD, Q), is complicated but numerical expressions or tables of the solutions
are available. Note that the differential Fermi function, F(ZD, pe), contains the
momentum and the Fermi integral, f(ZD, Q), contains the Q value. The Fermi integral
is a constant for a given b decay and has been presented in many forms. For
example, curves of the Fermi integral are shown in Figure 8.4.
The decay constant is now reduced to an expression with the nuclear matrix
element, M(; jMifj), and the strength parameter, g, written:
l ¼
g2
jMj2
m5
ec4
2p3hÀ 7 f(ZD, Q)
or in terms of the half-life of the parent, t1/2
ft1=2 ¼ ln 2
2p3
hÀ 7
g2jMj2
m5
ec4
/
1
g2jMj2
The left-hand side of this equation is called the comparative half-life, or “ft value”
because this value can be readily measured in experiments and should only depend
on the nuclear matrix element and the b-decay strength constant. Recall that b decay
half-lives span many orders of magnitude so the ft values will span a similarly
large range. It is therefore convenient to use the common logarithm of the ft
value (with t1/2 in seconds) to characterize observed b decays.
Values of log ft may be calculated from the nomograph and curves in Figure 8.4,
which are due to Moszkowski (1951). Log ft values can be calculated for b2
, bþ
,
and EC decay. These ft values fall into groups that can be correlated with the
spin and parity change in the decay (see below) and can, then, be used to assign
spins and parities in nuclei whose structure is not known (see Figure 8.5).
Example Problem Using the graph of the Fermi integral in Figure 8.4, estimate
the log ft value for the decay of 32
P (t1/2 ¼ 14.28 d).
Solution
1. This is a neutron-rich nucleus and undergoes b2
decay, thus:
QbÀ ¼ M(32
P) À M(32
S) ¼ D(32
P) À D(32
S)
QbÀ ¼ (À24:305) À (À26:015) MeV ¼ þ1:71 MeV
2. From the figure, Z ¼ 15, Q ¼ 1.71 MeV, log( ft) ¼ log( f0t) þ log(C) ¼
7.8 þ 0.2 ¼ 8.0
The creation of relative angular momentum in b decay is even more difficult than
that in a decay and causes more severe “hindrance” for each unit of relative angular
momentum. The difficulty is easy to see with a simple calculation. We can write the
relative angular momentum for two bodies as the cross product L ¼ r  p where r is
the radius of emission and p is the momentum. Taking a typical nuclear radius of 5 fm
and a typical b-decay energy of 1 MeV, we find the maximum of the cross product to
8.5 b-DECAY RATE CONSTANT 209
Figure 8.4 Rapid method for determining log10( ft) values. (From Moszowski, 1951.) The
above figures permit the rapid calculation of log( ft) for a given type of decay, given
energy, branching ratio, etc. Notation: E0 for b+
emission is the maximum kinetic energy
of the particles in MeV; E0 for K electron capture is the Q value in MeV. When a bþ
emission and K electron capture go from and to the same level, E0 for the K capture ¼ E0
for bþ
emission þ 1.02 MeV. Z is the atomic number of the parent, t is the total half life,
and p is the percentage of decay occurring in the mode under consideration. When no
branching occurs, p ¼ 100. To obtain log( ft), obtain log( f0t) using part (a). Read off log(c)
from parts (b), (c), and (d) for b2
, bþ
, and K EC, respectively. Get D log(ft) from part (e)
if p , 100. For p ¼ 100, D log( ft) ¼ 0. log(ft) ¼ log( f0t) þ log(C) þ D log( ft).
210 b DECAY
be L ¼ 5 fm (1.4 MeV/c) ¼ 7.90 MeV fm/c or 0.035 hÀ
units. Log ft values increase
by an average of 3.5 units for each unit, of orbital angular momentum or degree of
forbiddenness. Such an increase in the lifetime indicates a hindrance of 3 Â 1024
for each unit of angular momentum. There is a large spread in the values, however,
due to the strong effect of the nuclear matrix element for each decay.
The quantum mechanical selection rules for b decay with no relative angular
momentum in the exit channel (l ¼ 0) are DI ¼ 0, 1 and Dp ¼ 0. The two values
Figure 8.4 Continued.
60
50
40
30
20
10
3 4 5 6 7 8 9 1110 12 13 14 15 16 17 18 19 20 21 22 23
log ft
Numberofcases
Superallowed
Allowed
First forbidden
Second forbidden
Third forbidden
Fourth forbidden
Figure 8.5 Systematics of experimental log ft values. (From Meyerhof, 1967.) Copyright #
McGraw-Hill Book Company, Inc. Reprinted by permission of McGraw-Hill Book Company.
8.5 b-DECAY RATE CONSTANT 211
for the spin change come directly from the two possible couplings of the spins of the
electron and neutrino. Some representative “allowed” b decays are described in
Table 8.1 along with their ft values and the character of the decay.
The decay of 14
O to the 0þ
excited state of 14
N can only take place by a Fermi
decay where the created spins couple to zero. This parent nucleus also has a weak
branch to the 1þ
ground state that takes place by a Gamow–Teller transition. In
contrast, the decay of 6
He to the ground state of 6
Li must take place by a
Gamow–Teller transition in order to couple the total resultant angular momentum
to zero. As mentioned earlier, the decay of the neutron into a proton can take
place with no change in angular momentum between the spin 1
2 particles, and the
angular momentum coupling rules allow both decay modes.
The decay of the neutron into the proton is an important example of decay between
mirror nuclei. In the b decay of mirror nuclei, the transformed nucleons (neutron !
proton or proton ! neutron) must be in the same shell and have very similar wave
functions. This gives rise to a large matrix element jMifj2
and a very small log ft
value. For the b decay of mirror nuclei to their partners, log ft values are about 3,
which is unusually small. Such transitions are called superallowed transitions.
When the initial and final states in b decay have opposite parities, decay by an
allowed transition cannot occur. However, such decays can occur, albeit with
reduced probability compared to the allowed transition. Such transitions are
called forbidden transitions even though they do occur. The forbidden transitions
can be classified by the spin and parity changes (and the corresponding observed
values of log ft) as in Table 8.2.
Remember that in b decay,
JP ¼ JD þ Lb þ Sb
pP ¼ pD(À1)Lb
TABLE 8.1 Representative Allowed b Decays
Parent Daughter Half-Life (s) Qb (MeV) log ft Character
6
He (0þ
) 6
Li (1þ
, gs) 0.808 3.5097 2.42 Gamow–Teller
14
O (0þ
) 14
N (0þ
, 2.313) 71.1 1.180 2.81 Fermi
n (1
2
þ
) p (1
2
þ
) 612 0.7824 20.27 Mixed
14
O (0þ
) 14
N (1þ
, gs) 1.16 Â 104
4.123 7.36 Gamow–Teller
TABLE 8.2 Classifications of b-Decay Transitions
Transition Type log ft Lb Dp Fermi DI Gamow–Teller DI
Superallowed 2.9–3.7 0 No 0 0
Allowed 4.4–6.0 0 No 0 0, 1
First forbidden 6–10 1 Yes 0, 1 0, 1, 2
Second forbidden 10–13 2 No 1, 2 1, 2, 3
Third forbidden .15 3 Yes 2, 3 2, 3, 4
212 b DECAY
where the subscripts P, D refer to the parent and daughter, Lb is the orbital angular
momentum carried away by the emitted electron and Sb is the coupled spin of the
electron–neutrino pair (Sb ¼ 0 for a Fermi transition and Sb ¼ 1 for a Gamow–
Teller transition).
8.6 ELECTRON CAPTURE DECAY
When the decay energy is less than 1.02 MeV (2mec2
), the b decay of a proton-rich
nucleus to its daughter must take place by electron capture (EC). For decay energies
greater than 1.02 MeV, EC and bþ
decay compete. In EC decay, only one particle,
the neutrino, is emitted with an energy MPc2
-MDc2
-Be where Be, is the binding
energy of the captured electron. The decay constant for electron capture can be
written, assuming a zero neutrino rest mass, as:
lEC ¼
g2
jMifj2
T2
n
2p2c3hÀ 3
jwK(0)j2
where we have assumed that the capture of a 1’s (K) electron will occur because the
electron density at the nucleus is the greatest for the K electrons. The K electron
wave function can be written as:
wK(0) ¼
1
ffiffiffiffi
p
p
Zmee2
4p10hÀ 2
 3=2
Thus
lKÀEC ¼
g2
Z3
jMifj2
T2
n
constants
Comparison of the decay constants for EC and bþ
decay shows
lK
lbþ
¼ (constants)
Z3
T2
n
f(ZD, Q)
Thus EC decay is favored for high Z nuclei. Of course, the decay energy must be
greater than 1.02 MeV for bþ
decay, a situation found mostly in low Z nuclei
where the slope of the walls of the valley of b stability is large (see Fig. 2.8) and
decay energies of .1.02 MeV occur.
Electron capture decay produces a vacancy in the atomic electron shells and
secondary processes that lead to filling that vacancy by the emission of X-rays
and Auger electrons occur. These X-rays permit the detection of EC decays.
8.6 ELECTRON CAPTURE DECAY 213
8.7 PARITY NONCONSERVATION
In Chapter 1, we introduced the concept of parity, the response of the wave function
to an operation in which the signs of the spatial coordinates were reversed. As we
indicated in our discussion of a decay, parity conservation forms an important selection
rule for a decay. Emission of an a particle of orbital angular momentum l
carries a parity change (À1)l
so that 1þ
! 0þ
or 2À
! 0þ
a decays are forbidden.
In general, we find that parity is conserved in strong and electromagnetic
interactions.
In the late 1950s, it was found (Wu et al., 1957) that parity was not conserved in
weak interaction processes such as nuclear b decay. Wu et al. (1957) measured the
spatial distribution of the b2
particles emitted in the decay of a set of polarized
60
Co nuclei (Fig. 8.6). When the nuclei decay, the intensity of electrons emitted
in two directions, I1 and I2, was measured. As shown in Figure 8.6, application
of the parity operator will not change the direction of the nuclear spins but will
reverse the electron momenta and intensities, I1 and I2. If parity is conserved,
we should not be able to tell the difference between the “normal” and “parity
reversed” situations, that is, I1 ¼ I2. Wu et al. (1957) found that I1 = I2, that is,
that the b particles were preferentially emitted along the direction opposite to
the 60
Co spin. (God is “left-handed.”) The effect was approximately a 10–20%
enhancement.
60Co
J
p
1
p
2
P
I1
I2
p1
p2
I2
J
Z
I1
Figure 8.6 Schematic diagram of the Wu et al. apparatus. (From H. Frauenfelder and E. M.
Henley, Subatomic Physics, 2nd Edition. Copyright 1991 by Prentice-Hall, Inc. Reprinted by
permission of Pearson Prentice-Hall.) A polarized nucleus emits electrons with momenta p1
and p2 that are detected with intensities I1 and I2. The left figure shows the “normal” situation
while the right figure shows what would be expected after applying the parity operator. Parity
conservation implies the two situations cannot be distinguished experimentally (which was
not the case).
214 b DECAY
8.8 NEUTRINOS
A number of studies have been undertaken of the interaction of neutrinos with
nuclei, to determine the neutrino mass, and to show that neutrinos and antineutrinos
are produced in bþ
and b2
decay, respectively. Neutrinos also provide important
information about stellar nuclear reactions because they have a very low probability
for interacting with matter and come directly out from the stellar interior.
Starting with the simple equation for the b2
decay of the neutron and the bþ
decay of the proton, we can write two closely related reactions that are induced
by neutrinos:
ne þ pþ
! n þ eþ
ne þ n ! pþ
þ eÀ
These reactions, called inverse b decay, were obtained by adding the antiparticle of
the electron in the normal b decay equation to both sides of the reaction. When we
did this we also canceled (or annihilated) the antiparticle/particle pair. Notice that
other neutrino-induced reactions such as ne þ n ! pþ
þ eÀ
do not conserve lepton
number because an antilepton, ne, is converted into a lepton, e2
. Proving that this
reaction does not take place, for example, would show that there is a difference
between neutrinos and antineutrinos. One difficulty with studying these reactions
is that the cross sections are extremely small, of order 10219
barns, compared to
typical nuclear reaction cross sections, of order 1 barn (10224
cm2
).
The combination of two studies of inverse b decay clearly showed that the neutrinos
emitted in b2
and bþ
decay were different. Both used nuclear reactors to
provide strong sources of antineutrinos. Recall that nuclear fission produces very
neutron-rich products that undergo a series of rapid b decays emitting antineutrinos.
In the first experiment, performed by Reines and Cowen (1953), a large volume of
liquid scintillator was irradiated, and protons in the organic solution were changed
into a neutron and a positron. The positron was rapidly annihilated with an electron
providing the first signal of an interaction. The neutron was captured within a few
microseconds by Cd nuclei that were added to the scintillator and provided a
second correlated signal. The flux of neutrinos from the reactor was sufficient to
produce a few events per hour in a 1-m3
volume of scintillator.
In the second study, Ray Davis and co-workers, irradiated a large volume
of liquid CCl4 with antineutrinos from a reactor. The putative reaction,
ne þ 37
Cl ! 37
Ar þ eÀ
, could be detected by periodic purging of the liquid, collection
of the noble gas, and then detection of the induced activity (37
Ar is unstable, of
course). The reaction was not observed to occur. Thus, they concluded that the
reactor emits antineutrinos and that lepton number is conserved in the reactions.
Example Problem Estimate the flux of antineutrinos from an operating nuclear
power reactor. For this estimate assume the power plant produces 1 GW of
thermal power, that fission produces 200 MeV per event, and that there are
approximately 6 rapid b decays per fission.
8.8 NEUTRINOS 215
Solution There is one antineutrino per b2
decay, of course, so this is really a
problem in dimensional analysis:
Rate ¼ 1 GW(106
J=s)=GW)(1 fission=200 MeV)
 (1 MeV=1:602  10À13
)(6ne= fission)
Rate ¼ 2 Â 1017
antineutrinos=s
8.9 b-DELAYED RADIOACTIVITIES
The central feature of b decay is that, for example, in the b2
direction, the decay
converts a neutron into a proton at a constant mass number. This conversion will
clearly change the number of pairs of like nucleons in the nucleus, and we have
already seen that unpaired nucleons influence the overall stability. b Decay in
even mass chains will convert odd–odd nuclei into the even–even isobar with
potentially large Q values due to a gain of twice the pairing energy. The large Q
values lead to high-energy b particles and rapid decays, but the relative stability
of the daughter may be less than that of the parent. The large Q values also allow
the population of higher lying states in the daughter. If the nuclei are far from the
(most) stable isobar, the decay may have sufficient energy to populate states in
the daughter that are above the separation energy.
90
Sr provides an example of a change in relative stability following b decay. This
even–even parent is an important fission product that has a 29-y half-life. It decays
to the odd–odd 90
Y, which then decays to the stable isobar 90
Zr with a half-life of
only 64 h. Thus, a pure preparation of 90
Sr will come into equilibrium with its
daughter after about a week, and the observed activity will be the sum of the two
decays. A chemical separation can be used to strip out the daughter activity. The
daughter will decay away in the separated sample and will grow back into the
parent sample. There are several examples of these parent–daughter pairs that
provide convenient sources of short-lived activities. For example, the 66-h 99
Mo
decays predominantly to a 6-h excited state in 99
Tc because the decay to ground
state would require a very large spin change. The daughter, 99
Tcm
, is used extensively
in nuclear medicine.
The natural decay chains have several examples of short-lived a activities that
are “delayed” by a longer-lived parent. In fact, the existence of these activities on
Earth is possible by the fact that the “head” of the chain has a half-life on the
order of the age of the Earth. Another more practical example near the end of the
4n chain is 212
Pb with a half-life of 10.6 h that decays to 212
Bi. The daughter
rapidly decays by a or b emission. The lead nucleus is also preceded by a shortlived
Rn parent, which can produce very thin sources of a particles by emanation.
The b decay of nuclei far from the bottom of the valley of b stability can feed
unbound states and lead to direct nucleon emission. This process was first recognized
during the discovery of fission by the fact that virtually all the neutrons are
216 b DECAY
emitted promptly, but on the order of 1% are delayed in time with respect to the
fission event. These delayed neutrons play a very important role in the control of
nuclear reactors. The fission products are very neutron rich and have large bdecay
energies. For example, 87
Br is produced in nuclear fission and decays with
a half-life of 55 s to 87
Kr with a Q value of 6.5 MeV. The decay populates some
high-lying states in the krypton daughter. Notice that 87
Kr has 51 neutrons, one
more than the magic number 50, and the neutron separation energy of 5.1 MeV is
less than the Q value. Thus, any states that lie above the neutron separation
energy will be able to rapidly emit a neutron and form 86
Kr.
Example Problem An important delayed neutron emitter in nuclear fission is
137
I. This nuclide decays with a half-life of 25 s and emits neutrons with an
average energy of 0.56 MeV and a total probability of approximately 6%. Estimate
the energy of an excited state in 137
Xe that would emit a 0.56-MeV neutron.
Solution First obtain the Q value for the neutron emission reaction. This is the
minimum amount of energy necessary to “unbind” the 83rd neutron and should
be negative, of course.
137
Xe ! n þ 136
Xe þ Qn
Qn ¼ D(137
Xe) À ½D(136
Xe) þ D(n)Š
Qn ¼ À82:218 À ½À86:425 þ 8:0714Š ¼ À3:86 MeV
The average energy of the excited state will be Qn plus the kinetic energies of the
particles, that is, the neutron plus the energy of the recoil. In this case the recoil
energy is very small and could have been ignored. The recoil energy is obtained
by conservation of momentum in the two-body decay.
EÃ
¼ ÀQn þ Tn þ Tn
1
137
À Á
¼ 3:86 þ 0:57 ¼ 4:43 MeV
Now as a check, obtain the Q value for the b decay and verify that it is more than
the excitation energy:
137
I ! 137
Xe þ n þ Qb
Qb ¼ D(137
I) À D(137
Xe) ¼ À76:72 À 82:21 ¼ 5:49 MeV
The population of high-lying unbound states by b decay is an important
feature of nuclei near the driplines. b-Delayed proton emission and b-delayed
neutron emission have been studied extensively and provide important insight
into the structure of exotic nuclei.
8.10 DOUBLE-b DECAY
The periodic variation of the mass surface caused by the pairing energy also
causes a large number of even–even nuclei to be unstable with respect to two
8.10 DOUBLE-b DECAY 217
successive b decays. This process is called double-b decay, and extensive
searches have been carried out for it. The difficulty is that the probability of a
double transition is extremely low. A gross estimate can be made by squaring
the rate constant obtained above, and the number of decays from even large
samples is at best one per day and at worst a few per year for the systems that
have been considered for study.
Two reactions have been studied as possible candidates for double-b decay. The
first reaction is simply two times the normal b decay process:
A
Z ! A
(Z À 2) þ 2eÀ
þ 2n
and thus follows the conservation laws. A second, more exotic reaction has been proposed
as a test of weak interaction theory and proceeds without creation of neutrinos:
A
Z ! A
(Z À 2) þ 2eÀ
Instrumental searches for this latter neutrinoless process have been made, but there is
no strong evidence for its existence. The former two-neutrino decay has been observed
with a variety of techniques that were carefully tuned to detect the rare products.
As an example of the process, the 86
Kr nucleus just mentioned above as the
daughter in delayed neutron emission is stable with respect to single b2
decay to
86
Rb having a Q value of 20.526 MeV. However, 86
Kr is unstable with respect to
the double-b decay to 86
Sr as it has a Q value of 1.249 MeV. In this case decay
to the intermediate state is energetically forbidden, and only the simultaneous
emission of two b particles can take place. To obtain the gross estimate, we can
rewrite the expression for the decay constant:
l ¼
mec2
hÀ
 
jMj2
m4
ec2
2p3hÀ 6
g2
f(ZD, Q)
 
The first term is the constant 8 Â 1020
/s and the second term reflects the details of
the decay. Using jMj ¼
ffiffiffi
2
p
for the decay from the 0þ
ground state, to the 0þ
ground
state of the daughter, the second term is 1.5 Â 10225
f. For this case, log(f)  1.5,
then taking the first term times the square of the second for double-b decay, we get
l  10226
/s, or 10219
per year! Given that a mole of this gas has 1024
atoms, we
expect about one decay per day in the entire sample.
The techniques used to observe double-b decay fall into three general categories:
geochemical, radiochemical, and instrumental. The geochemical studies rely on
assumptions that are similar to those used in geochemical dating (see Chapter 3). A
sample of an ore containing the parent nuclide is processed; the daughter atoms are
chemically extracted and then assayed, for example, with a mass spectrometer. The
number of daughter atoms is then compared to the number of parent atoms and
with an estimate of the lifetime of the ore, the double-b decay half-life can be calculated.
Difficulties with this technique are discussed in the Chapter 3. The radiochemical
searches for double-b decay relied on chemically separating and identifying a
radioactive daughter of the process. Such cases are relatively rare but the decay
238
U ! 238
Pu was observed by chemically separating a uranium ore and observing
218 b DECAY
the characteristic a decay of the plutonium isotope. The successful instrumental
searches for double-b decay have used time projection chambers in which samples
of the parent were introduced into the active volume of the detector. The tracks of
the two coincident b particles can be observed providing a clear signal for the
exotic process.
PROBLEMS
1. The b2
decay of 144
Ce is shown below.
0+
144
Ce
30%
70%
0–
1–
2+
0.134
0.081
0.034
0 144
Pr
a. What log ft value should we expect for the b decay to the 12
state of 141
Pr?
b. Why is there no b decay to the 2þ
level?
2. Sketch quantitatively the shape of the neutrino energy spectrum for the following
types of decay. Label all axes carefully and indicate the types of neutrinos
involved.
a. The electron capture decay of 207
Bi, QEC ¼ 2.40 MeV.
b. The bþ
decay of 22
Na, Qb ¼ 3.86 MeV.
c. The b2
decay of 14
C, Qb ¼ 0.156 MeV.
3. Suppose a state in a Bi isotope decays by EC to the 2þ
state of an even–even Pb
nucleus in which the three lowest states are the 0þ
, 2þ
, and 4þ
, with
EEC ¼ 1.0 MeV. Assume QEC ¼ 4 MeV, t1/2 ¼ 4 s. Calculate Jp for the initial
state of the Bi nucleus.
4. Given the b decay scheme shown below for the decay of a pair of isomers to three
excited states A, B, and C of the daughter nucleus. List the spins and parities of
the three levels A, B, and C.
1
2
–
+
A
B
C
E1
M1
log ft = 5
log ft = 9
β
–, log ft = 6
9
2
PROBLEMS 219
5. The results of some measurements with a b-ray spectrometer of the radiation
coming from a given radionuclide are shown below.
Number of
Electrons
Magnetic field strength
The two sharp peaks were labeled K and L by the experimenter. Explain what the
labels K and L mean. Which peak is the K peak? Why?
6. A 12
excited state of a Lu isotope decays to a 0þ
state of a Yb isotope with a
maximum bþ
energy of 4.6 MeV. Estimate t1/2 for the transition. Do not
neglect electron capture.
REFERENCES
Camp, D. C. and L. M. Langer. Phys. Rev. 129, 1782 (1963).
Meyerhof, W. E. Elements of Nuclear Physics, McGraw-Hill, New York, 1967.
Moszkowski, S. A. Phys. Rev. 82, 35 (1951).
Wu C. S. et al. Phys. Rev. 105, 1413 (1957).
BIBLIOGRAPHY
Evans, R. D. The Atomic Nucleus, McGraw-Hill, New York, 1956.
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1988.
Lamarsh, J. R. Introduction to Nuclear Reactor Theory, Addison-Wesley, Reading,
MA, 1967.
Moe, M. and P. Vogel. Ann. Rev. Nucl. Sci. 44, 247 (1994).
Remies, T. and C. L. Cowen, Jr. Phys. Rev. 92, 830 (1953).
Siegbahn, K. Alpha, Beta and Gamma Ray Spectroscopy, North-Holland, Amsterdam, 1966.
Wu, C. S. and S. A. Moszkowski. Beta Decay, Wiley, New York, 1966.
220 b DECAY
CHAPTER 9
g-RAY DECAY
9.1 INTRODUCTION
g-Ray decay occurs when a nucleus in an excited state releases its excess energy by
emission of electromagnetic radiation, that is, a photon. Thus, we have
A
XÃ
À! A
X þ g
where the symbol Ã
indicates an excited state of the nucleus. Note that there is no
change in Z or A during this type of decay, only the release of energy. One can also
get g-ray emission from a high-lying excited state to a lower-lying state of the same
nucleus. Thus, g-ray transitions do not have to go to the ground state of the nucleus.
Figure 9.1 depicts a typical situation in which a series of g rays deexcite the levels
of a nucleus with so-called crossover transitions also occurring (4 ! 1, 4 ! 2,
etc.). Also note that the g-ray energy spectrum shows discrete lines corresponding to
each transition. Note that the energies of the g rays can vary from a few keV to
many MeV. Any nucleus with bound excited states can decay by g-ray emission.
In some unusual cases a nucleus can have two configurations of nucleons that
have very similar low-lying energies but very different total angular momenta.
One of these states will be lower in energy, of course, but the transition between
the two states will be strongly hindered, due to the fact that the photon will have
to balance the large change in angular momentum. This hindered decay is similar
to the hindrance of the decay of triplet states in atomic and molecular systems.
These long-lived nuclear states are called isomeric states, and their g-ray decay is
221
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
called an isomeric transition, or simply IT decay. An example of an isomeric state is
shown in Figure 9.2 for 69
Zn. The ground state of the zinc nucleus is unstable with
respect to b decay with a half-life of 56 min. The lowest excited state of this nucleus
has an energy of only 439 keV, but it has a much larger spin and the opposite parity
compared to the ground state. The transition from the excited state to the ground
state is hindered by the large change in angular momentum, 4hÀ
, combined with a
change in parity (discussed below), which leads to an IT half-life of 14 h.
9.2 ENERGETICS OF g DECAY
Imagine a g transition between two nuclear states. Applying the law of conservation
of energy, we have
MÃ
0 c2
¼ M0c2
þ Eg þ Tr
Figure 9.1 Schematic diagram of the g-ray transitions for a nucleus and the resulting g-ray
energy spectrum.
Figure 9.2 Decay scheme for 69
Znm
.
222 g-RAY DECAY
where Eg is the photon energy, Tr the kinetic energy of the recoiling nucleus after
g-ray emission, and MÃ
0 and M0 are the masses of the higher and lower nuclear
states, respectively. Applying the law of conservation of momentum, we have
pg ¼ pr
where pg and pr are the momenta of the photon and recoiling nucleus, respectively.
The kinetic energy of the recoil is so small that nonrelativistic mechanics can be
used. Hence we have
Tr ¼ p2
r =2M0
Substituting
Tr ¼ p2
g=2M0
Tr ¼ E2
g=2M2
0
If Eg ¼ 2 MeV, and A ¼ 50, the recoil energy is about 40 eV, which is negligible
except for Mossbauer studies (see later).
Example Problem (a) Calculate the recoil energy for the IT decay of 69
Znm
to
the ground state of 69
Zn. Using the energy of the excited state from Figure 9.2, we
have
MÃ
0 À M0
À Á
c2
¼ Eg ¼ 0:439 MeV
Solution Recall that M0c2
¼ amuÃ
931.5 MeV/amu. Using the mass defect of
68.417 MeV found in the Wallet Cards (see Appendix B), the mass of 69
Zn is
68.297 amu.
Tr ¼
E2
g
2M0c2
¼
0:439 MeVð Þ2
2 68:297 Â 931:5ð Þ
Tr ¼ 1:50 Â 10À6
MeV ¼ 1:5 eV
(b) Calculate the recoil energy from the emission of a 15.1-MeV photon by an
excited 12
C nucleus. Recall that the mass defect of 12
C is 0 so that the mass of
12
C is 12.000 amu.
Tr ¼
E2
g
2M0c
¼
15:1 MeVð Þ2
2 12 Â 931:5 MeVð Þ
Tr ¼ 1:02 Â 10À2
MeV ¼ 10:2 keV
9.3 CLASSIFICATION OF DECAY TYPES
The conservation of angular momentum has provided an enormous amount of information
on the structure of nuclei and plays a controlling role in the g-ray decay
9.3 CLASSIFICATION OF DECAY TYPES 223
process. From a schematic viewpoint, a stationary nucleus in a definite quantum
mechanical state makes a transition to a lower energy state during g decay and
emits a single photon. Both the initial and final states of the nucleus will have a definite
angular momentum and parity, and so the photon must connect the two states
and conserve both parity and angular momentum. Photons each carry an exact
integer number of angular momentum units (hÀ
) and each has a definite parity.
The conservation of angular momentum and parity are different, of course, and conservation
of each has a different effect on the possible properties of the emitted
photon. The angular momenta of the initial and final states of final nucleus can be
labeled as Ii hÀ
and IfhÀ
and change in intrinsic angular momentum, DI(hÀ
) is, of
course, l ¼ D I ¼ j(Ii À If)jhÀ
. A photon must carry at least one unit of angular
momentum so that DI ¼ 0 is forbidden for single-photon emission. The emitted
photon should have a minimum intrinsic spin of lhÀ
units to connect the two
nuclear states. However, the standard coupling rules for angular momenta allow
the photon to carry away up to a maximum of l ¼ (Ii þ If)j jhÀ
units. Therefore,
given known values of the spins of initial and final states of the nucleus, the
angular momentum carried by the photon can take any value in the range:
Ii À Ifð Þ



 l Ii þ Ifð ÞhÀ
The multipolarity of the photon is a quantification of the amount of angular momentum
carried by the photon. The nomenclature is that a photon with l units of angular
momentum is called a 2l
-pole photon. (The nomenclature comes from the classical
radiation patterns of electromagnetic radiation and the design of the antennas used to
create those patterns.) For example, a photon with l ¼ 1 is called a dipole photon,
l ¼ 2 a quadrupole photon, and so on as indicated in Table 9.1. The transition
rate depends strongly on the angular momentum change so that the smallest value
of l ¼ Ii À Ifð Þ



hÀ
is usually observed although conservation of parity plays a role.
Transitions with the maximum change in the angular momentum of the nuclear
states are called stretched transitions.
To understand the parity of electromagnetic transitions, we need to recall that
each of the initial and final states of the nucleus undergoing the transition can be
TABLE 9.1 g-Ray Selection Rules and Multipolarities
Radiation Type Name l ¼ DI Dp
E1 Electric dipole 1 Yes
M1 Magnetic dipole 1 No
E2 Electric quadrupole 2 No
M2 Magnetic quadrupole 2 Yes
E3 Electric octupole 3 Yes
M3 Magnetic octupole 3 No
E4 Electric hexadecapole 4 No
M4 Magnetic hexadecapole 4 Yes
224 g-RAY DECAY
viewed as having a definite distribution of matter and charge. When the excited
nucleus makes a transition from the excited state to a lower energy state, the
distribution of matter and charge will change in some way. For example, a
nucleus that is spinning with a certain value of angular momentum will slow
down as it excites and reaches the ground state. Thus, the emission of the photon
can be associated with the change in the overall distribution of neutrons and
protons, but we can identify two different changes that are analogous to classical
antennas. A shift in the distribution of charge (e.g., the transition of a proton from
one orbital to another) will give rise to an electric field, but a shift in the distribution
of current in the nucleus (e.g., the shift of the direction of a proton orbital) will
give rise to a magnetic field. The parity of the photon depends on both the
angular momentum and the type (electric or magnetic) of transition indicated
in Table 9.1. Notice that electric and magnetic radiation with a given multipole
character have opposite parities.
With the list of properties of photons we can generalize the procedure to identify
the probable type of photon for a given transition between nuclear states. First, the
parity of the photon will be given by the difference in parities of the two nuclear
states. Then the angular momentum of the photon will be limited to be in the
range of Ii À Ifj j to Ii þ If. The combination of allowed angular momenta and
parity will determine the character of the electromagnetic radiation. For example,
the first excited state in 7
Li has spin and parity 1
2
À
and the ground state is 3
2
À
. Possible
electromagnetic transitions between the two states must have Dp ¼ No and
1 l 2. Consulting Table 9.1, we find that the only candidates are Ml (l ¼ 1,
Dp ¼ No) and E2 (l ¼ 2, Dp ¼ No) while the other combinations E1 and M2 are
ruled out by parity. As we will see in the next section, all of the allowed radiation
types will be emitted but at substantially different rates so that the overall radiation
usually has one predominant character.
Example Problem Use the electromagnetic selection rules to identity the character
of the crossover transitions that could link the second excited state at
2.080 MeV ( 7
2
þ
) in 23
Na with the ground state ( 3
2
þ
).
Solution First, we should note that Dp ¼ No. Then,
Ii À Ifj j l Ii þ If ) 2j j l 5
l Dp Type
2 No E2
3 No M3
4 No E4
5 No M5
As a final point on the topic of selection rules, we noted that Dl ¼ 0 is forbidden
for the emission of a single photon. The electric monopole distribution (E0)
9.3 CLASSIFICATION OF DECAY TYPES 225
corresponds to the static electric charge of the nucleus and is constant. Similarly, the
M0 distribution corresponds to the nonexistent magnetic monopole moment. Nonetheless,
there are a few examples of even–even nuclei that have first excited and
ground states that are both 0þ
. Once populated, these states decay by internal conversion
processes in which the atomic electrons, particularly s electrons with
significant penetration into the nucleus, or an electron–positron pair (if there is
sufficient energy) are directly emitted from the atom.
Example Problem Calculate the ratio of the wavelength of the 439-keV
photon emitted when the isomeric state of 61
Znm
makes an IT to the diameter
of this nucleus.
Solution For a photon,
Eg ¼ hn
ln ¼ c
l ¼
hc
Eg
¼
6:626 Â 10À34
js
À Á
3:0 Â 108
m=s
À Á
439 Â 103 eVð Þ 1:602 Â 10À19 J=eVð Þ
l ¼ 2:82 Â 10À12
m
Recall that R ¼ 1.2 Â A1/3
fm, so that
l
2R
¼
2:82 Â 10À12
m
2 Â 4:92 Â 10À15 m
¼ 287
Thus, the nucleus is not an effective antenna due to its small size compared to the
wavelength of the radiation. g Rays are in the long wavelength limit and are not
very sensitive to the detailed internal structure of the emitting nucleus.
9.4 ELECTROMAGNETIC TRANSITION RATES
Determining the rate at which an excited state will decay by the emission of a photon
is a very general quantum mechanical problem that is not limited to the world of
nuclei. The detailed derivation of the transition rate is beyond the scope of this
text, and we will only sketch out the results. The decay constant for the emission
of a photon by a very well defined single state that has excess energy is shown in
Appendix E to be given by the general expression:
l ¼
2p
hÀ
ð
wÃ
finalVp winitial dv








2
r Ef
À Á
which is also called Fermi’s golden rule. The wave functions, w, represent the complete
initial and final states of the entire system and Vp is a (very) small perturbative
226 g-RAY DECAY
interaction between the nuclear and electric fields that stimulates the transition.
The form and the strength of the perturbation will depend on the multipolarity of
the transition. The last factor, r(Ef) is the product of the density of nuclear and electromagnetic
states that are available to the system after the transition. The initial
wave function contains only the nuclear excited state, whereas the final wave function
will have parts for the electromagnetic wave and the daughter nuclear state.
After some extensive calculus and input from the theory of electromagnetism, we
come to an expression for the electromagnetic decay rate:
l l, Ii, p ! If, pð Þ ¼
8p l þ 1ð Þ
l 2l þ 1ð Þ!!½ Š2
k2lþ1
hÀ
B l, Ii, p ! If, pð Þ
where k is the photon wave number (k ¼ Eg=hÀ
c). The symbol !! calls for the double
factorial of its argument, which for the case of l ¼ 2 and 2l þ 1 ¼ 5 would be the
product of the odd integers: 5!! ¼ 5 Â 3 Â 1 ¼ 15. The reduced transition probability
B(l, Ii, p ! If, p) is the matrix element for the reduced nuclear wave
functions (i.e., summed over magnetic orientations) and the multipole operator
(either electric or magnetic in character),
B(l, Ii, p ! If, p) ¼
1
2Ii þ 1
kIfj Olk kIijl



2
in which the symbols j in the nuclear wave functions are meant to represent all the
other relevant quantum numbers. (As an aside, we should note that the two “types”
of electric and magnetic radiation are only different in terms of their parity and in the
orientation of their plane of polarization.) This expression is still somewhat complicated
and is difficult to evaluate. Victor Weisskopf derived a general expression for
the reduced transition probability with the assumption that the transition results from
the change of a single particle inside a nucleus with a uniform density with the
familiar radius function, R ¼ r0 A1/3
. His expression for electric multipole radiation,
called the Weisskopf single-particle limit, is
Bsp E, lð Þ ¼
1
4p
3
l þ 3ð Þ
!2
r0ð Þ2l
A2l=3
e2
fm2l
The single-particle limit for magnetic multipole radiation obtained by assuming that
the change in current is due to a single nucleon is
Bsp M, lð Þ ¼
10
p
3
l þ 3
!2
r0ð Þ 2lÀ2ð Þ=2
m2
n fm2lÀ2
One of the nagging features of these expressions is that the radial integral from the
multipole expansion introduces a factor of r2l
, and thus the dimensions of B(E, l) and
Bsp(E, l) depend on l.
Either of the single-particle limits for the reduced electric or magnetic transition
probability can be substituted into the expression for the transition rate to obtain
9.4 ELECTROMAGNETIC TRANSITION RATES 227
numerical estimates of the deexcitation rates under the assumption that one particle
was responsible for the change in electric charge distribution or electric current
associated with the change in nuclear states. The transition rates vary over an
enormous range, as shown in Figure 9.3 depending most strongly on the value
of l. Electric transitions are faster than magnetic transitions by about two orders
of magnitude. Looking back to the discussion of the fact that several different
types of photons can be associated with a given nuclear transition, we now see
that we expect the rates of emission to favor the lowest multipolarity. This fact
can be simply demonstrated by evaluating the expressions for the transition rate
for electric dipole, l ¼ 1, and electric quadrupole, l ¼ 2, radiation with a typical
nuclear radius parameter of r0 ¼ 1.2 fm. Combining the expressions for the transition
rate and the reduced transition probability for an E1 transition we get:
lsp E, lð Þ ¼
8p l þ 1ð Þ
l 2l þ 1ð Þ!!½ Š2
k2lþ1
hÀ
1
4p
3
l þ 3
!2
e2
(r0 fm)2l
A2l=3
Substituting in l ¼ 1,
lsp(E1) ¼
8p 2ð Þ
1 2l þ 1ð Þ!!½ Š2
Eg
hÀ
c
 
e2
4phÀ
3
1 þ 3
!2
1:2 fmð Þ2
A2=3
sÀ1
Figure 9.3 Weisskopf single-particle estimates of the transition rates for (a) electric
multipoles and (b) magnetic multipoles. From Condon and Odishaw, Handbook on
Physics, 2nd Edition. Copyright # 1967 by McGraw-Hill Company, Inc. Reprinted by
permission of McGraw-Hill Book Company, Inc.
228 g-RAY DECAY
Recall that hÀ
c ¼ 197:3 MeV-fm and e2
¼ 1.440 MeV-fm, so that
lsp(E1) ¼
16p
9
E(MeV)
197:3 MeV-fm
 3
1:440 fm
4phÀ
3
4
!2
(1:2 fm)2
A2=3
sÀ1
lsp(E1) ¼ 1:03 Â 1014
E3
g A2=3
sÀ1
Similar substitution into the expression for lsp(El) with l ¼ 2 for electric quadrupole
radiation will eventually yield
lsp(E2) ¼ 7:28 Â 107
E5
g A4=3
So we see that the rates depend very strongly on the energy of the photon and on the
size (mass number) of the emitting nucleus. If we consider the specific hypothetical
case of a 1-MeV transition in a medium mass nucleus, A ¼ 100, the ratio of transition
rates is
lsp(E1)
lsp(E2)
¼
1:03 Â 1014
E3
g A2=3
7:28 Â 107E5
g A4=3
¼ 1:41 Â 106
EÀ2
g AÀ2=3
lsp(E1)
lsp(E2)
¼ 6:54 Â 104
The formulas for the transition rates are summarized in Table 9.2 for the lowest five
multipoles of each character. The transition rates always increase with a high power
of the g-ray energy so that low-energy transitions, say below 100 keV, are much
slower than high-energy transitions, say above 1 MeV. Table 9.2 also shows that
in some cases, particularly in heavy nuclei, an l þ 1 electric transition can
compete favorably with an l magnetic transition.
The Weisskopf estimates are usually good to within a factor of 10, which is
remarkable given the large number of orders of magnitude that they span, and
provide important references for comparison to the observed transition rates.
Notice that if a transition occurs more rapidly than the single-particle rate then
TABLE 9.2 Weisskopf Single-Particle Transition Rates (Eg is
in MeV)
Multipole E M
l l (sÀ1
) l (sÀ1
)
1 1:03 Â 1014
A2=3
E3
g 3:15 Â 1013
E3
g
2 7:28 Â 107
A4=3
E5
g 2:24 Â 107
A4=3
E5
g
3 3:39 Â 101
A2
E7
g 1:04 Â 101
A4=3
E7
g
4 1:07 Â 10À5
A8=3
E9
g 3:27 Â 10À6
A2
E9
g
5 2:40 Â 10À12
A10=3
E11
g 7:36 Â 10À13
A8=3
E11
g
9.4 ELECTROMAGNETIC TRANSITION RATES 229
the transition is more collective, that is, more particles participate in the change. If
the transition is significantly slower than the Weisskopf estimate, then the nuclear
matrix element must be smaller than the single-particle limit, that is, the overlap
of the initial and final states is smaller. The ratio of the observed decay rate to the
Weisskopf estimated rate is often quoted in the literature as the transition rate in
Weisskopf units (W.u.).
Example Problem Use the electromagnetic selection rules to identity the character
of the isomeric transition from the first excited state at 0:439 MeV( 9
2
þ
) in
69
Znm
with the ground state ( 1
2
À
). Then calculate the Weisskopf single-particle
rates for the allowed transitions.
Solution First, we should note that Dp ¼ Yes. Then,
Ii À Ifj j l Ii þ If ) 4j j l 5
Thus, only M4 and E5 transitions are allowed. Using the expressions in Table 9.2:
l(M4) ¼ 3:27 Â 10À6
A2
E9
g (sÀ1
)
l(M4) ¼ 7:66 Â 10À6
(sÀ1
)
and
l(E5) ¼ 2:40 Â 10À12
A10=3
E11
g (sÀ1
)
l(E5) ¼ 3:77 Â 10À10
(sÀ1
)
so we expect that the transition will be predominantly M4 in character. This
example also shows that the observed transition occurs about twice as fast as
the single-particle estimate:
l ¼ ln 2=(14 h  3600 s=h) ¼ 1:37  10À5
(sÀ1
)
which indicates that the “current” from more than one particle contributes to the
magnetic transition.
It should be noted that E2 transitions are often enhanced by an order of magnitude
compared to the single-particle estimates. This enhancement of these specific transitions
stems from collective nuclear motion, and the enhancement is particularly
strong for nuclei that lie in between major shell closures. An example of a set of
E2 transitions to the ground state of 160
Dy and the first three excited (collective)
states is shown in Figure 9.4. The excited nucleus cascades down from the 6þ
level in a series of three E2 transitions with no crossover transitions. The lifetimes
of the states, indicated in the figure, were used to calculate the transition rates in
Weisskopf units, also indicated in the figure. Notice that the rate of emission in
this case ranges from 200 to 1100 times the single-particle rate. If we take a
230 g-RAY DECAY
closer look at the transition rate for electric quadrupole transitions, then we would
find that the reduced transition probability could be written in terms of the quadrupole
moment, Q0, as:
B E2, Ji ! Jfð Þ ¼
5
16p
e2
Q2
0kJi, K, 2, 0 Jf, Kj l2
in which the last term is a Clebsch–Gordon coefficient, which is a very general
normalization coefficient for the values of the spins and the spin projections that
can be found in standard reference tables. In the present case of transitions
between the states of a rotational band, K ¼ 0, Ji ¼ J, and Jf ¼ J 2 2, the normalization
coefficient only contains values of J:
kJi, K, 2, 0jJf, Kl2
¼ kJ, 0, 2, 0j J À 2ð Þ, 0l2
¼
3J(J À 1)
2(2J þ 1)(2J À 1)
so that the reduced transition probability is
B(E2, J ! J À 2) ¼
15
32p
e2
Q2
0
J(J À 1)
(2J þ 1)(2J À 1)
Thus, the experimental transition rate provides a measurement of the quadrupole
moment of the nucleus, and we should not be surprised that a strongly deformed
nucleus with a large quadrupole moment will have a larger E2 transition rate
because the whole nucleus can participate in the transition compared to a single
particle.
In the single-particle estimates of g-ray decay, one presumes a single nucleon
interacts with a photon. This means there is an isospin selection rule
DT ¼ 0 or 1
for g-ray decay between two pure isospin states. Also we note that E1 g transitions
cannot occur when DT ¼ 0 in a self-conjugate nucleus (N ¼ Z).
6+ (254 W.u.) 18.6 ps
4+ (358 W.u.) 103 ps
2+ (1098 W.u.) 2026 ps
160
Dy
0+
Figure 9.4 Schematic diagram of the ground-state rotational band transitions for 160
Dy.
(Figure also appears in color figure section.)
9.4 ELECTROMAGNETIC TRANSITION RATES 231
9.5 INTERNAL CONVERSION
Internal conversion (IC) is a competing process to g-ray decay and occurs when an
excited nucleus interacts electromagnetically with an orbital electron and ejects it.
This transfer of the nuclear excitation energy to the electron occurs radiationlessly
(without the emission of a photon). The energy of the internal conversion electron,
EIC, is given by
EIC ¼ Etransition À Eelectron binding energy
Thus, if a transition has Etransition ¼ 0.412 MeV, you would expect to see the spectrum
of emitted internal conversion electrons shown in Figure 9.5. Note the different
lines corresponding to the ejection of electrons from the K, L, and M shells. The
nucleus will interact more readily with the K electrons than with the L electrons,
Figure 9.5 Kinetic energy spectrum of internal conversion electrons for a 412-keV
nuclear transition in 198
Hg. Superimposed on this spectrum is the accompanying spectrum
of b2
particles from the b decay that feeds the excited state. The peaks labeled K, L, and
M represent conversion of electrons with principal quantum numbers of 1, 2, or 3,
respectively. (From Marmier and Sheldon, 1969, p. 332.) Copyright # Academic Press.
Reprinted by permission of Elsevier.
232 g-RAY DECAY
than with the M electrons, and so forth because the K electrons spend more time in
the nucleus than the L electrons than the M electrons, and so forth.
To characterize this decay process and its competition with g-ray emission, we
define the internal conversion coefficient, a, by the relationship
a ¼
number of internal conversion decays
number of g-ray decays
¼
lIC
lg
where a can take on values from zero to infinity. Note further that
l ¼ lIC þ lg ¼ lg(1 þ a)
One can define this ratio, the internal conversion coefficient, for electrons from the K
shell only for electrons from the M shell only, and so on, giving rise to aK, aM, and
so on. Since the total probability of decay must equal the sum of the probabilities of
decay via various paths, we have
atotal ¼ aK þ aL þ aM þ Á Á Á
The internal conversion coefficient depends primarily on the density of the atomic
electrons at the center of the nucleus, and thus it can be calculated using principles
from atomic physics. Large tables and nomographs of internal conversion coefficients
exist, such as those shown in Figure 9.6.
Rough approximate formulas for the internal conversion coefficients are
a(EL) ¼
Z3
n3
L
L þ 1
 
e2
4p10hÀ
c
 4
2mec2
E
 Lþ5=2
a(ML) ¼
Z3
n3
e2
4p10hÀ
c
 4
2mec2
E
 Lþ3=2
where Z is the atomic number of the atom in which the conversion is taking place,
n is the principal quantum number of the bound electron being ejected, and
e2
=4p10hÀ
c, the fine structure constant is 1/137. Note that the internal conversion
coefficient, a, increases approximately as Z3
, making internal conversion most
important for heavy nuclei. The last factor in the equations gives the energy and multipolarity
dependence with more internal conversion for low energies and higher
transition multipolarities. The aK/aL ratio is approximately 8 due to the n3
factor.
Example Problem Use a standard reference such as the Table of Isotopes, 8th
ed., 1996, to determine the internal conversion coefficients for each shell for the
transition from the first excited state at 0.08679 keV (2þ
) in 160
Dy to the ground
state (0þ
). Then calculate the decay rates for internal conversion and for g-ray
emission.
Solution We have already identified this transition as E2, using Appendix F,
pages 3, 7, in the Table of Isotopes, 8th ed., 1996. We have to interpolate in
9.5 INTERNAL CONVERSION 233
a graph to find:
aK (E2, Z % 65, 0:090 MeV) ¼ 1:5
aL1 (E2, Z % 65, 0:090 MeV) ¼ 0:1
aL2 (E2, Z % 65, 0:090 MeV) ¼ 5:
aL3 (E2, Z % 65, 0:090 MeV) ¼ 2:5
a ¼ aK þ aL1 þ aL2 þ aL3 ¼ 9:1
l ¼ ( ln 2)=2:02 Â 10À9
¼ 3:34 Â 108
sÀ1
¼ lg(1 þ a)
lg ¼
3:34 Â 108
sÀ1
1 þ a
¼ 3:4 Â 107
sÀ1
lIC ¼ l À lg ¼ 3 Â 108
sÀ1
E5
E4
E3
E2
E1
E4
E3
E2
E1
E1
E2
E3
E4 Z = 50
Z = 20
Z = 90
10
-5
-4
-3
-2
logaK
(a)
-1
0
1
2
2
hw, MeV
Figure 9.6 Calculated internal conversion coefficients for (a) electric transitions and
(b) magnetic transitions. (From M. A. Preston, 1962, p. 307.) Copyright # 1962 by
Addison-Wesley Publishing Company. Reprinted by permission of Pearson Education.
234 g-RAY DECAY
Note that internal conversion occurs approximately 10 times faster than g-ray emission
for this transition in this nucleus.
9.6 ANGULAR CORRELATIONS
One of the features of the derivation of the emission rate for g rays that we glossed
over is that the angular distribution of the emitted radiation from a single state must
be isotropic. The isotropy comes from the fact that the nuclei are oriented at random,
and the process sums over all the internal magnetic substates and thus includes all
10 2
hw, MeV
-5
-4
-3
-2
logaK
-1
0
1
2
3
M4
M3
M2
M1
M4
M3
M2
M4
M3
M2
M1
M1
M5
Z = 20
Z = 50
Z = 90
(b)
Figure 9.6 Continued.
9.6 ANGULAR CORRELATIONS 235
possible angular distributions. We used this fact in the derivation by using the
“reduced (or double-barred) matrix elements.” Anisotropic angular distributions
can only be observed when a preferred direction or nuclear orientation is established
prior to the emission of the photon. There are several techniques to establish such
preferred orientations that rely on observing an angular correlation with either an
external magnetic field or another particle or photon emitted in “cascade” from
the same nucleus. All of these techniques rely on unequal populations of the magnetic
substates of the emitting nuclear state. Two of these techniques are shown
schematically in Figure 9.7. Another important application of angular correlations
is to determine the multipolarity of the electromagnetic transition. We have seen
that the selection rules often provide a range of possibilities for the spin change,
and the lifetimes of the states depend on the nuclear matrix elements as well as
the multipolarity. In order to reliably identify the multipolarity, we have to
measure the angular distribution of the radiation; however, we need a reference axis.
The conceptually simplest technique to observe an angular correlation is to
measure the angular distribution of radiation from an excited nucleus relative to
an external, applied, magnetic field. The magnetic substates of nuclear excited
states that have angular momenta, I, greater than 0 will split in proportion to the
strength of the external magnetic field, Bext, and the magnetic field provides the
reference axis. This substate splitting provides the basis for NMR and MRI techniques,
of course. The difficulty with this correlation technique is that the
(Zeeman) splitting of the nuclear spin substates, DEm, given by the simple expression
DEm ¼ gIBextm0
is a very small energy. In this expression g is the gyromagnetic ratio or g factor for the
state, and m0 ¼ ehÀ
=2mpc is the nuclear magneton. We should note that this energy
splitting is much too small compared to the energy of a nuclear transition so that
Figure 9.7 Schematic examples of two techniques to prepare a nuclear state with unequal
populations of the internal magnetic substates: (a) correlating the sequential emission of
two g rays and (b) correlating the emission of a b particle with a subsequent g ray. (From
de Shalit and Feshbach, 1974, p. 693.) Copyright # 1974 John Wiley & Sons, Inc.
Reprinted by permission of John Wiley & Sons, Inc.
236 g-RAY DECAY
we could not expect to directly observe different energy transitions with different
angular distributions. Rather, we can obtain unequal populations of the substates
through the Boltzmann distribution of thermal energy when the sample is cooled
to a temperature where kBT is small compared to the energy splitting. The typical
temperature to maintain nuclear orientation in an external magnetic field is on the
order of 10 mK. Cooling small samples to such low temperatures is possible, but
it requires special techniques such as 3
He dilution refrigeration.
Example Problem Determine the temperature at which kBT is equal to the
energy level splitting for the metastable state of 123
Te at 247.6 keV in an external
magnetic field of 4.0 tesla (T).
Solution This state I ¼ 11
2 decays by (M4) IT to the ground state 1
2 with a halflife
of 119.7 d. The gyromagnetic ratio, or g factor, for this state is 0.1685:
DEm ¼ gIBextmN ¼ kBT
T ¼
gIBextmN
kB
T ¼
0:1685 11
2
À Á
(4:0T) 5:05084 Â 10À27
J=T
À Á
1:38066 Â 10À23 J=K
T ¼ 1:4 Â 10À3
K
A much more common technique for observing angular correlations relies on detecting
the direction of radiation from a process that feeds the excited state and then
observing the angular distribution relative to that direction. As indicated in
Figure 9.7 this process could be a g-ray transition from a higher lying excited
state, or it could be a b or a particle emitted by a parent nucleus. The first particle
provides the reference axis, but it must also introduce an unequal population of the
magnetic substates of the intermediate state in order for the second transition to have
an anisotropic angular distribution.
The angular distribution of the intensity of electromagnetic radiation is given by
specific analytic functions written in terms of an angle, W(u, mI), relative to the
quantization axis, Z, and the magnetic quantum number, mI. The patterns depend
on the order of the multipole, dipole, quadrupole, and so forth, but they are the
same for electric and magnetic transitions with the same order. For example, the
angular distributions for dipole radiation are
Wdipole(u, mI ¼ 0) ¼
3
8p
sin2
u
Wdipole(u, mI ¼ þ1) ¼
3
16p
1 þ cos2
u
À Á
Wdipole(u, mI ¼ À1) ¼
3
16p
1 þ cos2
u
À Á
9.6 ANGULAR CORRELATIONS 237
A schematic representation of these angular distributions is shown in Figure 9.8.
First, we should notice that these functions depend on only one angle, and thus
they are cylindrically symmetric. Therefore, we will not find any asymmetry in radiation
from systems with only two substates, that is, I ¼ 1
2 , mI ¼ + 1
2. Notice also that
Figure 9.8 Schematic diagram of how angular correlations occur. Panel (a) shows the
angular distribution of dipole radiation for Dm ¼ 0 and Dm ¼ +1. Panel (b) shows the
magnetic substates populated in a g1g2 cascade from J ¼ 0 to J ¼ 1 to J ¼ 0. When g1
defines the Z axis, then the m1 ¼ 0 state cannot be fed and one has only Dm1 ¼ +1 and
Dm2 ¼ +1, causing g2 to have an anisotropic distribution relative to g1 shown in panel
(c). [From Marmier and Sheldon, 1969.] Copyright # 1969 Academic Press. Reprinted by
permission of Elsevier.
238 g-RAY DECAY
the intensity of m ¼ 0 for dipole radiation is exactly zero along the Z axis because
the sine function becomes zero, whereas the m ¼ +1 distributions have nonzero
minima perpendicular to the Z axis.
Let us consider the specific case of the cascade of two electromagnetic dipoles
from nuclear states with Ia ¼ 0 to Ib ¼ 1 to Igs ¼ 0. The energy level diagram is
shown in Figure 9.8. The radiation pattern of the second photon will have the
(1 þ cos2
u) form relative to the direction of the first photon in this case. The fact
that the radiation will not be isotropic can be seen from a classical as well as
from a quantum mechanical standpoint. The simple definition that the Z axis is
the direction of this first photon forces the observed photon to have mI ¼ +1
because Wdipole (u, mI ¼ 0) ¼ 0 at u ¼ 0, and such a photon will not be observed
along the Z axis. The first dipole transition from Ia ¼ 0 to Ib ¼ 1 will carry away
one unit of angular momentum, and the angular momentum of the residual
nucleus must be oriented in such a way as to be equal and opposite to that of the
photon. Then, to conserve angular momentum in the second transition, the next
photon must also have mI ¼ +1 and will follow the (1 þ cos2
u) distribution relative
to the first photon and the Z axis. From a quantum mechanical standpoint the
argument depends simply on spin algebra. The only allowed magnetic substates
for the initial and final states are ma ¼ mgs ¼ 0 (because Ia ¼ Igs ¼ 0). The only
allowed value for the multipolarity of both photons, l1 ¼ l2, is one by the relation
0 À 1j j l (0 þ 1). The coupling of the angular momentum of the photon with
that of the initial state to create the intermediate state requires that only allowed
magnetic substate of Ib ¼ 1 be Mb ¼+1. Thus, both photons must have m ¼+l
and follow the (1 þ cos2
u) distribution.
Figure 9.8 Continued.
9.6 ANGULAR CORRELATIONS 239
The angular distributions for g-ray cascades have been worked out, but each case
requires substantial and sophisticated algebra that will not be presented here. The
general result is that the angular distributions can be written in terms of a sum of
Legendre polynomials that depends on the multipolarities of the photons, l1, l2
and the spin of the intervening state. It is common to analyze the observed
angular correlations in terms of a power series of cos u that is normalized with
W(u ¼ 908) ¼ 1 so
W uð Þ ¼ (1 þ a2 cos2
u þ a4 cos4
u þ a6 cos6
u þ Á Á Á þ a2L cos2L
u
where the coefficients, a2, a4, and so on are fitted to the data and can be compared to
predicted values for assumed values of Ia, Ib, Ic, l1, and l2. The number of radiation
patterns or angular distributions may seem extensive with these five variables, but
there are certain rules that simplify the situation. The highest even power of the
cosine function, 2L, is determined by the smallest value of 2Ib, 2l1, and 2l2 and is
one unit less than the smallest if the smallest of these is an odd number. For
example, when Ib ¼ 0 or 1
2, then 2L ¼ 0 and W(u) ¼ 1; when Ib ¼ 1, then 2L ¼ 2
and W(u) ¼ (1 þ a2 cos2
u). The theoretical coefficients for a few types of pure
dipole and pure quadrupole transitions are given in Table 9.3.
The third technique for establishing a reference axis for angular correlations
can be applied to nuclear reactions when the direction of a particle involved in
the reaction is detected. This direction provides a reference axis that can be
related to the angular momentum axis, but each nuclear reaction has its own pecularities
and constraints on the angular momentum vector. For example, the direction
of an a particle from a decay process that feeds an excited state can be detected as
indicated in Figure 9.7, but, as is discussed in Chapter 7, the energetics of a decay
TABLE 9.3 Angular Correlation Coefficients for Some
g–g Cascades with Pure Multipolarities
Ia(l1); Ib(l2); Ic a2 a4
0(1); 1(1); 0 1 0
1(1); 1(1); 0 À 1
3 0
1(2); 1(1); 0 À 1
3 0
2(1); 1(1); 0 1
13 0
3(2); 1(1); 0 À 3
29 0
0(2); 2(2); 0 23 4
1(2); 2(2); 0 À 1
3 0
2(2); 2(2); 0 3
7 0
2(2); 2(2); 0 À 15
13 À 16
33
3(2); 2(2); 0 À 3
29 0
4(2); 2(2); 0 1
8
1
24
Source: From Evans (1955).
240 g-RAY DECAY
are such that decay to excited states or decays with large orbital angular momenta
are hindered. Nuclear reactions can produce nuclei with large amounts of angular
momenta with characteristic distributions. The motion of the center of mass
provides a good reference that coincides with the direction of the initial beam for
the usual case of a target at rest in the lab system. The angular momentum vector
must lie in the plane perpendicular to the beam direction in compound nuclear
reactions. The angular momentum vector is further confined in two-body scattering
reactions to be normal to the plane containing the beam (or center of mass vector)
and the two particles. The effects of angular momentum on nuclear reactions are
discussed further in Chapter 10.
At this point we have established techniques to identify the multipolarity of a
transition through its angular distribution. We still have the ambiguity of the
parity of the electromagnetic wave, that is, whether it was produced by an electric
or a magnetic transition in the nucleus. The parity of the radiation corresponds to
the plane of polarization of the electromagnetic radiation. The polarization of the
wave can be determined from knowledge of the direction of the plane of the electric
vector of the photons relative to the plane containing two coincident photons. The
direction of the electron emitted in the Compton scattering process is sensitive to
the direction of the electric vector of the incoming photon and has been used to
determine the parity of electromagnetic transitions. Alternatively, the number and
type of conversion electrons emitted in the decay is also sensitive to the electric
or magnetic nature of the radiation. Measurements of the conversion coefficients
are used to establish the character of the radiation.
9.7 MO¨ SSBAUER EFFECT
We could imagine that the inverse of g-ray emission from an excited nuclear state to
the ground state might be possible if a nucleus in its ground state was bathed in sufficient
energy. A large difficulty with causing this absorption to take place is the very
large amount of energy associated with nuclear transitions (MeV), compared to the
amount of available (terrestrial) thermal energies (1026
MeV). This inverse process
can occur in two situations: (a) in nuclear reactions called Coulomb excitation that
take place when heavy ions pass very near to large target nuclei (Chapter 10), and (b)
when there is a resonant absorption of a g ray emitted by nuclear deexcitation in
another identical nucleus. The latter process is called the Mo¨ssbauer effect, and
the process requires some special conditions in order to take place. The energies
of the nuclear states are very precise so that the resonant absorption or energy matching
is very sensitive to the chemical environment of the nucleus. As we will see,
there are relatively few nuclei that are suitable for Mo¨ssbauer studies due to the
requirements of a half-life that allows a high specific activity with a reasonable
useful period, a single g-ray transition, and the absorbing nucleus must be a
stable isotope of an important/practical chemical element. The important examples
are 57
Fe, 191
Ir, and 198
Hg.
9.7 MO¨ SSBAUER EFFECT 241
The first nucleus in which the resonant absorption of photons was observed was
191
Ir. The excited states of this nucleus are fed by the electron capture decay of 191
Pt,
one of which decays by a 129.43-keV M1 transition to the ground state. Now we can
ask what will happen if we shine g rays from a radioactive source of 191
Pt onto a set
of stable 191
Ir nuclei? We could use an iridium foil because iridium only has two
stable isotopes 191 (37.3%) and 193 (72.7%). Without careful preparation, the
answer is that very few photons will be absorbed by the 191
Ir nuclei! The difficulty
comes from the fact that in order to be absorbed the g ray will have to exactly match
the energy of the transition. Remember that quantum mechanics dictates that the
absorption of the g ray will move the nucleus from its ground state to a single
and specific excited state that has an exact energy. A single nucleus cannot
absorb a random amount of energy. Several important effects shift the energy of
the emitted photon, but first we could ask how accurately do we have to match
the energy of the state in order to be absorbed? This corresponds to the natural
width of the state.
The measured half-life of the state is 89.4 ps, which corresponds to a energy
width, G, or DE, due to the Heisenberg uncertainty principle of:
G ¼
hÀ
t
¼ hÀ
l ¼ hÀ
Â
ln 2
t1=2
¼
4:6 Â 10À16
eV-s
t1=2(s)
where t is the mean life or the reciprocal of the decay constant l ¼ ln 2=t1=2. In this
case the energy width of the excited state is only the tiny amount of 5.1 Â 1026
eV,
a factor of 2 Â 10210
less than the energy of the state. Such narrow widths are a
general property of nuclear excited states that decay by g-ray emission. Thus, the
energy matching of the nuclear state and photon energy has to be incredibly exact
for significant absorption to take place.
The linewidth of an observed transition is broadened by the random thermal
motion of the nuclei that emit the photon. That is, the energies of photons emitted
along the direction of thermal motion of the atom will be slightly higher than the
average and vice versa for those emitted opposite. The value of the energy of a
photon emitted by a moving source is shifted according to the expression:
E0
g ¼ Eg 0 1 + bx
À Á
where Eg0 is the energy of the transition and bx ¼ vx/c is the familiar ratio of the
velocity along the photon direction to the speed of light. As an upper limit, we
could use the kinetic theory of gases and the Maxwell–Boltzmann velocity
distribution to estimate of the width of the velocity distribution for gaseous iron
nuclei. (The motion of atoms in liquids and the vibrations of atoms in solids are
smaller but not zero.) The Boltzmann (thermal) probability distribution for the
kinetic energy of an atom, P(KE), is always a decreasing exponential function,
P(KE) % eÀmv2
=2kBT
, and it applies to the total kinetic energy and to the kinetic
energy along one coordinate of a normal gas in a closed container. Solving the
242 g-RAY DECAY
Doppler expression for vx in terms of E0
g
vx ¼ c 1 + (E0
g=Eg0)
h i
and substituting that expression into the Boltzmann probability we find
P(E0
g) / eÀmc2
(1+E0
g=Eg0)2
=2kBT
Selecting one sign for the direction, multiplying through, and collecting constants,
we find
P(E0
g) / eÀmc2
E0
g
2
=(2E2
g0kBT)
This expression shows that the distribution of emitted g-ray energies follows a
Gaussian distribution with a variance something like
s2
% E2
g0kBT=mc2
In the present example of 191
Ir decay at room temperature, kBT ¼ 0.025 eV,
Eg0 ¼ 0.1294 MeV, and mc2
¼ 191 Â 931.5 MeV, which combine to give
s  7 Â 1022
eV, which, although small and an upper limit for gaseous atoms, is
still six orders of magnitude larger than the natural linewidth of the state. Therefore,
it is not very often that we will be able to actually observe the natural linewidth of a
g-ray emitting state. This broadening works in favor of the absorption of a photon
because it allows the thermal motion to help match the energy of the whole
system, nucleus in the atom, to the photon energy.
In addition, as we have already discussed, the emission of a photon induces a
recoil by the nucleus in order to conserve momentum. The energy of the photon
is less than the energy of the nuclear transition by the amount Tr ¼ Eg
2
/(2mc2
).
Notice that to conserve energy and momentum in the reverse process of g-ray
absorption, a nucleus initially at rest will recoil with the same value of the recoil
energy after absorbing a photon. In the present example of 191
Ir, the recoil energy
is Tr ¼ 4.7 Â 1022
eV and is a similar magnitude to the thermal Doppler shift for
a gas. We probably can expect the radioactive platinum atoms to be in a metal
lattice so their motion would correspond to lattice vibrational motion and be
somewhat less than that in a gas. The relative energy distributions expected for
the emitted and absorbed photons are shown in Figure 9.9 using the estimate
of the thermal widths. Notice that the recoil energy moves the peaks apart, and
the thermal width provides only a partial overlap. It is these photons in the
overlap region that have the proper energy to be absorbed; they must encounter a
nucleus, of course, in order to actually be absorbed.
We might imagine that we could prepare a system that physically moves
the source of the radiation toward the absorbing nuclei with sufficient speed that
the Doppler shift compensates for the energy difference. Restricting the motion
9.7 MO¨ SSBAUER EFFECT 243
to the approaching direction, we can rearrange the expression above to obtain the
velocity in terms of the Doppler shift:
DE ¼ (E0
g À Eg 0) ¼ Eg0
vx
c
 
The necessary velocity that would create a Doppler shift corresponding to twice the
recoil energy is
DE ¼ 2Tr ¼
2E2
g 0
2mc2
¼ Eg 0
vx
c
 
vx
c
¼
Eg 0
mc2
For the example of 198
Hg, bx ¼ 2.2 Â 1026
or vx ¼ 670 m/s and corresponds to a
kinetic energy of 0.92 eV. The magnitude of this difference is visible in
Figure 9.9 because the separation between the two peaks is about twice the
thermal width. Such a high velocity is difficult to attain with any macroscopic,
that is, physical source.
The Mo¨ssbauer effect relies on a very different technique for overcoming the
energy mismatch of twice the recoil energy between nuclear emission and nuclear
absorption. Notice that the recoil energies that we have calculated are small fractions
of an electron volt per atom. You might recall that chemical bonds have energies on
the order of a few electron volts per bond and are stronger in some sense than the
recoil effect. Mo¨ssbauer showed that the resonant emission/absorption of photons
could be strongly enhanced by binding the emitting atoms and the absorbing
Figure 9.9 Schematic indication of the position and widths of the emitted and absorbed
radiation corresponding to the 412-keV transition in 198
Hg. (From Krane, 1988, p. 364.)
Copyright # John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.
244 g-RAY DECAY
atoms into crystal lattices. In practice the emitter is produced by a b decay of a
parent nuclide that is a different chemical element from the absorber; thus, two separate
crystals are used. Due to the chemical bonds or the lattice energy of the crystal,
the atom that absorbs the photon is held in place, and the entire, macroscopic, lattice
“recoils” to conserve momentum. The mass of the entire lattice should be used to
calculate the recoil velocity, but this mass is on the order of Avogadro’s number
larger than that of an atom, so that there is effectively no recoil. One analogy is
to compare the difference that you would feel if you hit a single stone with a bat
compared to that you would feel if you hit the same stone if it was part of a
cement wall in a concrete building. The actual difference in the atomic case is
orders of magnitude larger. Thus, with the atoms bound into the crystal lattice,
the Doppler motion is limited to the vibrational motion of the atoms, and the linewidth
shrinks to essentially the natural width of the state. In this case the energy
of the emitted photon and the energy absorbed in the nuclear excitation overlap.
The Mo¨ssbauer experiment is then to remove the overlap between the photon energies
by moving one crystal lattice with respect to the other. The relative velocity is
on the order of cm/s, which is, of course, much smaller than that necessary to
compensate for the nuclear recoil. The resonance is then seen as a preferential
absorption as a function of relative velocity between the emitter and absorber.
Notice that the Mo¨ssbauer effect is very sensitive to the energy of the nuclear
state; changes on the order of 1026
eV are readily detected. This is the level at
which atomic orbitals can shift nuclear states through the penetration of electron
density into the nucleus. As a first approximation, we could imagine that the interaction
of the electron wave function with the nucleus will depend on the size, that is,
radius, of the nuclear wave function. The nuclear wave function for the excited state
will be (slightly) different, and thus the penetration of the electrons into the excited
nucleus will be slightly different. Thus, the transition energy will be different, albeit
by a very small amount, from the pure nuclear transition that would occur in a bare
nucleus (no electrons). When the chemical state or environment of both the absorber
and the emitter are the same, the transition will occur at a definite but different
energy, but one could not perform the measurement of the pure nuclear transition
(without electrons). Finally, when the chemical environment of the emitter and
absorber are different, then the transition will occur at a new energy. The shift of
the energy of the resonance between the identical environments and different
environments is called the chemical shift in analogy to NMR work. In practice,
the chemical shift in the Mo¨ssbauer resonance lines provides a probe for the
overall chemical environment of the absorbing nuclei.
The most extensively used nuclide for Mo¨ssbauer studies is 57
Fe due to the very
low energy of the nuclear transition. Let us consider the low-lying excited states of
57
Fe shown in Figure 9.10. The first excited state in 57
Fe lies at only 14.4125 keV,
and it decays to the ground state with a half-life of 98 ns. As shown in Figure 9.10,
the b decay of the parent nucleus, 57
Co, feeds this excited state of the daughter
nucleus so that we can image having a strong source of the low-energy g rays.
As shown in the example calculation, the energy of this transition is so low
that the recoil energy is also quite low and comparable to the thermal energy.
9.7 MO¨ SSBAUER EFFECT 245
Thus, studies can be performed with the source bound in a crystal lattice, but the
absorber can be in solution.
Example Problem Calculate the natural linewidth of the state at 14.4 keV in
57
Fe given that t1/2 ¼ 98 ns. Then calculate the velocity of the source lattice that
would correspond to twice the natural width and would lie outside the Mo¨ssbauer
resonance effect:
Solution
DE ¼ G ¼ hÀ
=t ¼ hÀ
½ln (2)=t1=2Š
DE ¼ G ¼
4:135 Â 10À15
eV-s
2p
ln (2)
98 Â 10À9 s
DE ¼ G ¼ 4:65 Â 10À9
eV
The velocity that would correspond to twice this energy can be found from the
nonrelativistic expression for the kinetic energy:
KE ¼
1
2
mv2
x ¼ 2G
vx
c
¼
ffiffiffiffiffiffiffiffi
4G
mc2
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 Â 4:65 Â 10À9 eV
57 Â 931:5 Â 106 eV
r
vx
c
¼ 5:92 Â 10À10
) vx ¼ 0:178 m=s
Figure 9.10 Energy level diagram of two members of the A ¼ 57 mass chain. 57
Co decays
to excited states of 57
Fe, which result in the M1 transition from the 3
2
À
state at 14.41 keV to
the 1
2
À
ground state. (Figure also appears in color figure section.)
246 g-RAY DECAY
PROBLEMS
1. 195
Pt has a ground-state spin and parity of 1
2
À
, with excited states at 0.029 MeV
( 3
2
À
) and 0.130 MeV (5
2
À
). Does the 5
2 level decay primarily to the 5
2
À
level or to
the 1
2
À
level? Why? What is the transition multipolarity?
2. The 1
2
À
isomeric state of 95
Nb decays to the 9
2
þ
ground state by means of an M4
transition. The half-life of the isomeric state is 90 h while the half-life of the
ground state is 35 d (atotal ¼ 4.5). Calculate the partial half-life for the g-ray
decay of the isomeric state.
3. Consider the following decay schemes for 60
Co and 60
Com
:
0.0 MeV
0.059 MeV
IC
5+
2+
60
Co
m
t1/2 = 10.5 m
(0.028%)
b-
aK=35
aK/aK/aM=14/3/1
g
a. Classify the most likely multipolarity for the g-ray decay of 60
Com
.
b. Calculate the partial decay constants for b2
, internal conversion, and g-ray
decay.
c. What is the width of 60
Com
in eV?
4. 52
Mn has an excited state at 0.377 MeV above the ground state. This excited state
decays to the ground state with t1/2 ¼ 21.1 min. The Jp values of initial excited
state and the ground state are 2þ
and 6þ
, respectively. (a) What is the lowest multipole
order that can contribute to the transition? Calculate the decay constant and
compare it to the experimental value. (b) Suppose we wanted to check whether,
in the initial state, there was any mixture of other angular momenta than 2. Set a
rough upper limit to the amplitude of a J ¼ 1 component of the initial state, using
as data only the measured half-life and transition energy. Assume parity
conservation.
5. Consider 10
B. The ground state has Jp ¼ 3þ
and the excited states in order of
increasing excitation energy are 1þ
, 0þ
, 1þ
, 2þ
, 3þ
, 22
, 2þ
. . . . (a) Explain
why 10
B is stable even though it is odd–odd. (b) The first excited state is at
0.72 MeV, and the second excited state is at 1.74 MeV. What are the energies,
PROBLEMS 247
multipolarities, and relative intensities of the g rays that are emitted in the
deexcitation of the second excited state?
6. A 64-d isomer of an even Z, and an odd nucleus with A  90 occurs at 105 keV
above the ground state. The isomeric state decays 10% by EC and 90% by IT. If
the internal conversion coefficient a ¼ 50, what is the g-ray lifetime and the most
likely multipolarity of the isomeric transition? If this is a magnetic transition and
the isomeric state has Jp ¼ 1
2
À
, what is the Jp of the ground state?
7. 51
V has a ground-state spin and parity of 7
2
À
with excited states at 0.3198 MeV
( 5
2
À
) and at 0.930 MeV ( 3
2
À
). What is the energy and multipolarity of the
principal g ray that deexcites each excited state?
8. The ground state of 61
Ni has Jp ¼ 3
2
À
. 61
Co (t1/2 ¼ 1.65 h) decays by b2
emission with Emax ¼ 1.24 MeV to a 0.067-MeV excited state of 61
Ni.
The 0.067-MeV transition has aK ¼ 0.10, aK/aL ¼ 8. The branching ratio for
the transition from 61
Co to the 61
Ni ground state is 1026
. What is Jp for the
ground state of 61
Co and the first excited state (0.067 MeV) of 61
Ni?
REFERENCES
Condon, E. U. and H. Odishaw. Handbook of Physics, 2nd ed., McGraw-Hill, New York,
1967.
de Shalit, A. and H. Feshbach. Theoretical Nuclear Physics, Vol. 1, Wiley, New York, 1974.
Evans, R. D. The Atomic Nucleus, McGraw-Hill, New York, 1955.
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1988.
Marmier, P. and E. Sheldon. Physics of Nuclei and Particles, Vol. 1, Academic, New York,
1969.
Preston, M. A. Physics of the Nucleus, Addison-Wesley, Reading, MA, 1962.
BIBLIOGRAPHY
Bohr, A. and B. Mottelson. Nuclear Structure, Vol. 1, Benjamin, New York, 1969.
Wong, S. S. M. Introductory Nuclear Physics, 2nd ed., Wiley, New York, 1998.
248 g-RAY DECAY
CHAPTER 10
NUCLEAR REACTIONS
10.1 INTRODUCTION
The study of nuclear reactions is important for a number of reasons. Progress in the
understanding of nuclear reactions has occurred at a faster pace, and generally a
higher level of sophistication has been achieved compared to similar studies of
chemical reactions. The approaches used to understand nuclear reactions are of
value to any chemist who wishes a deeper insight into chemical reactions. There
are certain nuclear reactions that play a preeminent role in the affairs of humans
and our understanding of the natural world in which we live. For example, life on
Earth would not be possible without the energy provided to us by the sun. That
energy is the energy released in the nuclear reactions that power the sun and
other stars. For better or worse, the nuclear reactions, fission and fusion, are the
basis for nuclear weapons, which have shaped much of the geopolitical dialog for
the last 50 years. Apart from the intrinsically interesting nature of these dynamic
processes, their practical importance would be enough to justify their study.
To discuss nuclear reactions effectively we must understand the notation or
jargon that is widely used to describe them. Let us begin by considering the
nuclear reaction
4
He þ 14
N À! 17
O þ 1
H
Most nuclear reactions are studied by inducing a collision between two nuclei where
one of the reacting nuclei is at rest (the target nucleus) while the other nucleus (the
249
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
projectile nucleus) is in motion. (Exceptions to this occur both in nature and in the
laboratory in studies where both the colliding nuclei are in motion relative to one
another.) But let us stick to the scenario of a moving projectile and a stationary
target nucleus. Such nuclear reactions can be described generically as:
Projectile P þ target T À! emitted particle x and residual nucleus R
For example, the first reaction discussed above might occur by bombarding 14
N with
a particles to generate an emitted particle, the proton and a residual nucleus 17
O.
A shorthand way to denote such reactions is, for the general case,
T(P, x)R
or for the specific example
14
N(a, p)17
O
In a nuclear reaction, there is conservation of the number of protons and neutrons
(and thus the number of nucleons). Thus, the total number of neutrons (protons)
on the left and right sides of the equations must be equal.
Example Problem Consider the reaction 59
Co(p, n). What is the product of the
reaction?
Solution
1
1H þ 59
27Co À! 1
0n þ Y
XZ
On the left side of the equation we have 27 þ 1 protons. On the right side we have
0 þ X protons where X is atomic number of the product. Obviously X ¼ 28 (Ni).
On the left hand side we have 59 þ 1 nucleons, and on the right side we must have
1 þ Y nucleons where Y ¼ 59. So the product is 59
Ni.
There is also conservation of energy, linear momentum, angular momentum, and
parity, which will be discussed below.
10.2 ENERGETICS OF NUCLEAR REACTIONS
Consider the T (P, x) R reaction. Neglecting electron binding energies, we have, for
the energy balance in the reaction,
mPc2
þ TP þ mTc2
¼ mRc2
þ TR þ mxc2
þ Tx
where Ti is the kinetic energy of the ith particle and mi represents the mass-energy of
the ith species. (Note that since R and x may be formed in an excited state, the values
of m may be different than the ground-state masses.)
250 NUCLEAR REACTIONS
The Q value of the reaction is defined as the difference in mass energies of the
product and reactants, that is,
Q ¼ ½mP þ mT À (mx þ mR)Šc2
¼ Tx þ TR À TP
Note that if Q is positive, the reaction is exoergic while if Q is negative, the reaction
is endoergic. Thus, the sign convention for Q is exactly the opposite of the familiar
DH in chemical reactions. A necessary but not sufficient condition for the occurrence
of a nuclear reaction is that
Q þ TP . 0
Note that Q is an important quantity for nuclear reactions. If the masses of both the
products and reactants are known (Appendix B), the Q value can be calculated using
the mass defects, D, as:
Q ¼ D(projectile) þ D(target) À SD(products)
It can be obtained by measuring the masses or kinetic energies of the reactants and
products in a nuclear reaction. However, we can show, using conservation of
momentum, that only Tx and the angle u of x with respect to the direction of
motion of P suffice to determine Q in these two-body reactions.
In the laboratory system, a typical nuclear collision can be depicted as shown in
Figure 10.1. Conserving momentum in the x direction, we can write
mPvP ¼ mxvx cos u þ mRvR cos f
Applying conservation momentum in the y direction, we have
0 ¼ Àmxvx sin u þ mRvR sin f
Figure 10.1 Schematic diagram of a nuclear reaction.
10.2 ENERGETICS OF NUCLEAR REACTIONS 251
where mi and vi are the mass and velocity of the ith species. If we remember that the
momentum p ¼ mv ¼ (2mT)1/2
, we can substitute in the above equation as
(mPTP)1=2
À (mxTx)1=2
cos u ¼ (mRTR)1=2
cos f
(mxTx)1=2
sin u ¼ (mRTR)1=2
sin f
Squaring and adding the equations, we have
mPTP À 2(mPTPmxTx)1=2
cos u þ mxTx ¼ mRTR
Previously, we had said that
Q ¼ Tx À TP À TR
Plugging in this definition of Q, the value of TR, which we have just calculated,
we get
Q ¼ Tx 1 þ
mx
mR
 
À TP 1 À
mP
mR
 
À
2
mR
mPTPmxTxð Þ1=2
cos u
This is the all-important Q equation. What does it say? It says that if we measure the
kinetic energy of the emitted particle x and the angle at which it is emitted in a reaction,
and we know the identities of the reactants and products of the reactions, we
can determine the Q value of the reaction. In short, we can measure the energy
release for any two-body reaction by measuring the properties of one of the products.
If we calculate the Q value of a reaction using a mass table, then we can turn this
equation around to calculate the energy of the emitted particle using the equation
T1=2
x ¼
mPmxTPð Þ1=2
cos u + {mPmxTP cos2
u þ mR þ mxð Þ½mRQ þ (mR À mP)TPŠ}1=2
mR þ mx
For additional insight, let us now consider the same reaction as described in the
center-of-mass (cm) coordinate system. In the cm system the total momentum of
the particles is zero, before and after the collisions. The reaction as viewed in the
laboratory, and cm system is shown in Figure 10.2.
The kinetic energy of the center of mass is
Tcm ¼
(mP þ mT) v2
cm
2
where vcm[¼ vPmP/(mP + mT)] is the speed of the center of mass. Substituting, in the
above equation, we have
Tcm ¼
1
2
(mP þ mT)
mPvP
mP þ mT
!2
¼
1
2
mPv2
P
mP
mP þ mT
 
¼ Tlab
mP
mP þ mT
 
252 NUCLEAR REACTIONS
where Tlab is the kinetic energy in the lab system before the reaction, that is,
Tlab ¼ 1
2 mPv2
P
The kinetic energy carried in by the projectile, Tlab, is not fully available to be
dissipated in the reaction. Instead, an amount Tcm must be carried away by the
center of mass. Thus, the available energy to be dissipated is Tlab 2 Tcm ; T0.
The energy available for the nuclear reaction is Q þ T0. To make the reaction go,
the sum Q þ T0 must be greater than or equal to zero. Thus, rearranging a few
terms, the condition for having the reaction occur is that
TP ! ÀQ(mP þ mT)=mT
This minimum kinetic energy that the projectile must have to make the reaction go is
called the threshold energy for the reaction.
Example Problem Consider the 14
N(a, p)17
O reaction. What is the threshold
energy for this reaction?
Solution
Q ¼ ½ma þ mN À (mP þ mO)Šc2
¼ 2:425 þ 2:863 À 7:289 À (À0:809) ¼ À1:19 MeV
Ta ¼ À(À1:19)(4 þ 14)=14 ¼ 1:53 MeV
Vf
(a) before collision
(c) before collision
(b) after collision
(d) after collision
as seen in the laboratory
as seen from the center of mass
VCM
CM
vCM
x
x
y
Y
X
X
y
Y
CM
Figure 10.2 Schematic view of a nuclear reaction in the laboratory and center-of-mass
systems. [From Weidner and Sells (1973).]
10.2 ENERGETICS OF NUCLEAR REACTIONS 253
10.3 REACTION TYPES AND MECHANISMS
Nuclear reactions, like chemical reactions, can occur via different reaction mechanisms.
Weisskopf has presented a simple conceptual model (Fig. 10.3) for illustrating
the relationships between the various nuclear reaction mechanisms.
Consider a general nuclear reaction of the type A (a, b) B, bearing in mind that
for some cases, the nuclei b and B may be identical to a and A. As the projectile a
moves near the target nucleus A, it will have a certain probability of interacting
with the nuclear force field of A, causing it to change direction but not to lose
any energy (Q ¼ 0) (Fig. 10.3). This reaction mechanism is called shape elastic
scattering. If shape elastic scattering does not occur, then the projectile may interact
with A via a two-body collision between the projectile and some nucleon of A,
raising the nucleon of A to an unfilled level (Fig. 10.3). If the struck nucleon leaves
the nucleus, a direct reaction is said to have occurred. If the struck nucleon does
not leave the nucleus, further two-body collisions may occur, and eventually the
entire kinetic energy of the projectile nucleus may be distributed between the
nucleons of the a þ A combination leading to the formation of a compound
nucleus C (see Fig. 10.3). Because of the complicated set of interactions leading
to the formation of the compound nucleus, loosely speaking, it “forgets” its
mode of formation, and its subsequent breakup only depends on the excitation
energy, angular momentum of C, and so forth and not the nature of the projectile
and target nuclei. Sometimes the compound nucleus may emit a particle of the
same kind as the projectile (or even the projectile itself) with the same energy
as the projectile had. If this happens, we say compound elastic scattering has
occurred. Also C may decay into reaction products that are unlike the projectile
or target nuclei. We shall spend much of this chapter discussing these reaction
mechanisms and some others not yet mentioned. But before doing so, let us see
what general properties of nuclear reactions we can deduce from relatively
simple arguments.
INITIAL
STAGE
INCIDENT
PARTICLE
DIRECT
REACTION
COMPOUND
ELASTIC
SCATTERING
CN
SHAPE ELASTIC
SCATTERING
INTERMEDIATE
STAGE
FINAL
STAGE
DECAY OF
COMPOUND
NUCLEUS
Figure 10.3 Conceptual view of the stages of a nuclear reaction. [After Weisskopf (1959).]
254 NUCLEAR REACTIONS
10.4 NUCLEAR REACTION CROSS SECTIONS
Consider the situation (Fig. 10.4) where a beam of projectile nuclei of intensity f0
particles/second is incident upon a thin foil of target nuclei with the result that the
beam is attenuated by reactions in the foil such that the transmitted intensity is f
particles/second. We can ask what fraction of the incident particles disappear
from the beam, that is, react, in passing through the foil. Let us assume the beam
intersects an area A on the foil. We can then assert that the fraction of beam particles
that is blocked (reacts) is the fraction of the area A that is covered by target nuclei. If
the foil contains N atoms/cm2
, then the area a that is covered by nuclei is N (atoms/
cm2
) Â a (cm2
) Â (the effective area subtended by one atom) (cm2
/atom). This
latter term, the effective area subtended by one atom, is called the cross section,
s, for the reaction under study. Then the fraction of the area A that is blocked is
a/A or N (atoms/cm2
) s (cm2
/atom). If we say the number of projectile nuclei
absorbed per unit time is Df, then we have
Df ¼ ÀfNs
As an aside, we note the units of N are atoms/cm2
or thickness, Dx (cm) Â density
n (atoms/cm3
).
Expressing the above equation as a differential equation, we have
Àdf ¼ fNs
Thus, upon rearranging, we have
df
f
¼ ÀNs ¼ Àns dx
ðftrans
finitial
df
f
¼ Àns
ðx
0
dx
ln
ftrans
finitial
¼ Ànsx
ftrans ¼ finitialeÀnsx
Figure 10.4 Schematic diagram showing the attenuation of an incident projectile beam.
10.4 NUCLEAR REACTION CROSS SECTIONS 255
Thus, we see exponential absorption of the incident projectile beam (and have thus
derived a form of the Lambert–Beers law). The number of reactions that are occurring
is the difference between the initial and transmitted flux, that is,
finitial À ftrans ¼ finitial ½1 À exp (Àns x)Š
The foregoing discussion focused on the attenuation of the incident beam and
thus refers to all reactions. In many cases,we are interested in only one of several reactions
that may be taking place. We can refer to the cross section for that particular
reaction. In addition, we may be interested not only in a specific product but
a particular product moving in a particular direction relative to the direction of the
projectile beam (see Fig. 10.5 for a sketch of a typical experimental measurement).
In this case, we can speak of a differential cross section, or the cross section per
unit solid angle, ds/dV. For a thin target in which the attenuation of the beam is not
significant, we have
dN
dV
¼ fn
ds
dV
 
dx
where dN/dV is the number of particles detected moving in a particular direction
per unit solid angle. The total cross section, s, is given as
s ¼
ð2p
0
ðp
0
ds
dV
(u) sin u du df
The description given above is appropriate for work at accelerators, where one
has a beam of particles that is smaller than the target. In this case, the beam intensity
AfterBefore
Aσ
b
θ
B
dΩ
Figure 10.5 Schematic diagram of a typical experimental setup.
256 NUCLEAR REACTIONS
f is given in particles/seconds and the target density N is given in atoms/cm2
. In a
nuclear reactor, we immerse a small target in a sea of neutrons. In this case, the
neutron flux f represents the number of neutrons passing through the target per
cm2
per sec, and N is the total number of atoms in the target. Otherwise the arithmetic
is the same. For charged particles from an accelerator, the beam intensity is
usually measured as a current. Thus, for a beam of protons with a current of
1 mA, we have
f ¼ (1 mA)(10À6
C=s=mA)
1
1:602 Â 10À19 C=proton
 
¼ 6:24 Â 1012
protons=second
For a beam of some other ion with charge q, one simply divides by the charge on
the ion to get the projectile beam intensity. Thus, for a beam of 4 mA of Ar+17
ions, we have
f ¼ (4 Â 10À6
C=s)
1
17 Â 1:602 Â 10À19 C=Ar
 
¼ 1:47 Â 1012
Ar=s
To put the intensities of beams of differing charges on a common footing, it is
common to divide the electric current by the charge state and quote charge particle
beam intensities in units of particle microamperes or particle nanoamperes where
1 particle microampere = 6.24 Â 1012
ions/s.
It is easy to calculate the number of product nuclei produced during an
irradiation, N. If we assume the product nuclei are stable, then the number of
nuclei produced is the (rate of production) Â (length of the irradiation, t). For a
thick target irradiation, we have
N ¼ f½1 À exp (Àns Dx)Št
For a thin target if we expand the function [1 2 exp(2ns Dx)], we have
N ¼ fns Dxt
But, what if the products are radioactive? Then some of them will decay during the
irradiation. In this case, we can set up the familiar differential equations:
dN
dt
¼ (rate of production) À (rate of decay)
dN
dt
¼ ns Dx f À lN
dN
ns Dx f À lN
¼ dt
10.4 NUCLEAR REACTION CROSS SECTIONS 257
Multiplying by l and rearranging
d(lN)
lN À ns Dx f
¼ Àl dt
Integrating, we have
ln (lN À ns Dx f) N
0 ¼ Àlt



t
0
Nl À ns Dx f
Àns Dx f
¼ eÀlt
A ¼ lN ¼ ns Dxf(1 À eÀlt
)
where A is the disintegration rate of product nuclei at the end of the irradiation.
The number of product nuclei, N, present at the end of the irradiation is A/l or
N ¼
ns Dx f
l
1 À eÀlt
À Á
This relationship is shown in Figure 10.6.
Note that in the limit of infinitely long irradiation, e2lt
! 0, and thus the activity
present is ns Dx f, which is termed the saturation activity. Note also that for very
short times (compared to the half-life of the product nuclei), e2lt
! 1 2 lt þ . . ..
Thus, the activity increases linearly with time. In general, we note that we
achieve one-half the saturation activity after an irradiation of one half-life, threefourths
of the saturation activity after irradiating two half-lives, seven-eighths of
the saturation activity after irradiating three half-lives, and so forth. Thus, it does
not pay to make the irradiation longer than one to two half-lives. (This effect can
be used to tune the length of the irradiation to maximize the yield of the product
of interest relative to the other reaction products.)
1.00
RelativeactivityA/Amax
Relative irradiation time t /t1/2
Saturation activity
0.75
0.50
0.25
0
0 1 2 3 4 5 6 7 8 9 10
Figure 10.6 Variation of product activity during an irradiation. [From Lieser (1997).]
258 NUCLEAR REACTIONS
Example Problem Calculate the activity of 254
No (t1/2 ¼ 55 s) produced in a
1-minute irradiation of 208
Pb by 48
Ca. Assume the 208
Pb target thickness is
0.5 mg/cm2
, the 48
Ca beam current is 0.5 particle microamperes, and the
208
Pb(48
Ca, 2n) reaction cross section is 3.0 mb.
Solution
A ¼ Nsf(1 À eÀlti
)
N ¼ (0:5 Â 10À3
g=cm2
)(6:02 Â 1023
atoms=g-at.-wt.)=208 g=g-at.-wt.
¼ 1:44 Â 1018
atoms=cm2
s ¼ 3 Â 10À30
cm2
f ¼ (0:5 Â 10À6
C=s)=1:602 Â 10À19
C=ion ¼ 3:12 Â 1012
ions=s
ti ¼ 60 s
l ¼ ( ln 2)=55 s ¼ 1:26 Â 10À2
sÀ1
A ¼ 7:2 dis=s
Let us consider what we can learn about cross sections from some general considerations.
Consider the reaction of an uncharged particle (a neutron) with a nucleus
as shown in Figure 10.7. The neutron makes a grazing collision with the nucleus.
The impact parameter b is taken to be the sum of the radii of the projectile and
target nuclei. Thus, the cross section can be written as
s % p (R þ r0
)2
¼ pr2
0(AP þ AT)2
where r0
is the radius of the projectile. Applying classical mechanics to this problem,
we can write for the orbital angular momentum, l,
l ¼ r  p ¼ pb
In quantum mechanics, l ! lhÀ
, and the momentum p is given by
p ¼
hÀ
lÀ
Figure 10.7 Schematic diagram of a grazing collision of a neutron with a nucleus.
10.4 NUCLEAR REACTION CROSS SECTIONS 259
Thus, we have
lhÀ
¼
hÀ
b
lÀ
b ¼ llÀ
This is not quite right because l is quantized but b is not. We get around this by associating
b with certain rings or zones on the target (Fig. 10.8). Figure 10.8 suggests that
for head-on collisions (l ¼ 0), the range of b is from 0 to l, whereas for l ¼ 1
collisions, the range of b is from l to 2l. Thus, the cross section is larger for larger
impact parameters, and these larger impact parameters are associated with larger
angular momenta. We can write the cross section for a specific value of l as:
sl ¼ p l þ 1ð Þ2
lÀ2
À pl2
lÀ2
sl ¼ plÀ2
(l2
þ 2l þ 1 À l2
)
sl ¼ plÀ2
(2l þ 1)
The total reaction cross section is obtained by summing over all l values as:
stotal ¼
X
l
sl ¼
Xlmax
l¼0
plÀ2
(2l þ 1) ¼ plÀ2
Xlmax
l¼0
(2l þ 1) ¼ plÀ2
(lmax þ 1)2
We can write for the maximum angular momentum, lmax,
lmax ¼
R
lÀ
lmax þ 1 ¼
R þ lÀ
lÀ
Figure 10.8 Schematic bulls-eye view of the target nucleus.
260 NUCLEAR REACTIONS
Thus, we have for the total cross section
stotal ¼ p(R þ lÀ)2
The total cross section is proportional to the size of the target nucleus and the “size” of
the projectile nucleus. Since the wavelength of the projectile, l, goes to infinity as the
projectile energy goes to zero, the cross sections for neutrons at low energies can be
very large. The above discussion is based upon semiclassical mechanics. We need to
indicate how the problem would look if we used quantum mechanics to treat it. In
quantum mechanics, we can write a similar expression for the total reaction cross
section:
stotal ¼ plÀ2
X1
l¼0
2l þ 1ð ÞTl
where the transmission coefficient Tl varies between 0 and 1. The transmission coefficient
expresses the probability that a given angular momentum transfer l will occur. At
high projectile energies, Tl ¼ 1 for l lmax and Tl ¼ 0 for l ! lmax. (This is called the
“sharp cutoff limit”.) At very low projectile energies, Tl ¼ 11/2
for l ¼ 0 and Tl ¼ 0
for l . 0, where 1 is the projectile energy. Thus, at very low energies, we have
stotal / plÀ2 ffiffiffi
1
p
/ p
hÀ 2
2m1
ffiffiffi
1
p
/
1
ffiffiffi
1
p
Such behavior of the cross sections for neutron-induced reactions is referred to as
“1/v” behavior.
Now let us consider the interaction of a charged particle with a nucleus as shown
in Figure 10.9. As the projectile approaches the target nucleus, it feels the long-range
Coulomb force and is deflected. As a consequence, the range of collisions corresponds
to a smaller range of impact parameters. If the incident projectile has an
energy 1 at an infinite separation from the target nucleus, at the distance of
closest approach R, it has a kinetic energy of 1 2 B where B, the Coulomb
barrier, is given by
B ¼ Z1Z2e2
=R
Figure 10.9 Schematic diagram of a charged-particle-induced reaction.
10.4 NUCLEAR REACTION CROSS SECTIONS 261
At the point of closest approach, the momentum p of the projectile is (2mT)1/2
.
Thus, we can write
p ¼ (2mT)1=2
¼ (2m)1=2
(1 À B)1=2
¼ (2m1)1=2
(1 À B=1)1=2
where m is the reduced mass of the system [¼A1A2/(A1 þ A2)]. Classically, we have,
for the orbital angular momentum,
l ¼ r  p
lmax ¼ R 2m1ð Þ1=2
1 À
B
1
 1=2
Quantum mechanically, we have l ! lhÀ
. So we can write
stotal ¼ plÀ2
(lmax þ 1)2
% plÀ2
l2
max ¼ plÀ2
R2 2m1
hÀ 2
1 À
B
1
 
¼ plÀ2
R2 1
lÀ2
1 À
B
1
 
stotal ¼ pR2
1 À
B
1
 
Note this last classical expression is valid only when 1 . B. The combined general
properties of cross sections for charged and uncharged particles are shown in
Figure 10.10.
Figure 10.10 Near threshold behavior of neutron and charged-particle-induced reactions.
[From Ehmann and Vance (1991).]
262 NUCLEAR REACTIONS
Example Problem Calculate the energy dependence of the total reaction cross
section for the 48
Ca þ 208
Pb reaction.
Solution
stotal ¼ pR2
1 À
B
1
 
R ¼ RPb þ RCa ¼ 1:2(2081=3
þ 481=3
) ¼ 11:47 fm
B ¼ Z1Z2e2
=R ¼ (82)(20)(1:44 MeV-fm)=11:47 fm ¼ 205:9 MeV
1 ¼ energy of the projectile in the cm system
1 (MeV) Elab (MCV) s (mb)
208 256 41.7
210 258.5 80.7
220 270.8 264.9
230 283.1 433.1
240 295.4 587.2
250 307.7 729.1
Aside on Barriers In our semiclassical treatment of the properties of chargedparticle-induced
reaction cross sections, we have equated the reaction barrier B to
the Coulomb barrier. This is, in reality, a simplification that is applicable to many
but not all charged-particle-induced reactions.
The actual force (potential energy) felt by an incoming projectile is the sum of the
nuclear, Coulomb, and centrifugal forces (Fig. 10.11). The Coulomb potential,
VC(r), is approximated as the potential between a point charge Z1e and a
homogeneous charged sphere with charge Z2e and radius RC as
VC(r) ¼ Z1Z2=r for r . RC
VC(r) ¼ (Z1Z2=RC) 3
2 À 1
2 (r2
=R2
C)
 Ã
for r , RC
The nuclear potential is frequently represented by a Woods–Saxon form
(Chapter 5) as:
Vnucl(r) ¼ V0={(1 þ exp½(r À R=a)Š}
while the centrifugal potential is taken as:
Vcent(r) ¼
hÀ 2
2m
l(l þ 1)
r2
10.4 NUCLEAR REACTION CROSS SECTIONS 263
where lhÀ
is the orbital angular momentum of the incident projectile. The total potential,
Vtot(r), is the sum VC(r) þ Vnucl(r) þ Vcent(r). These different potentials are
shown in Figure 10.11 using the 16
O þ 208
Pb reaction as an example and input
angular momenta of l ¼ 0, 10, and 100hÀ
. Note that for the highest angular
momentum, l ¼ 100hÀ
, the total potential is repulsive at all distances, that is, the
ions do not fuse.
The actual interaction barrier is the value of Vtotal(r) at the point when the colliding
nuclei touch. That is slightly different from VC(r) at r ¼ RC, the Coulomb barrier.
10.5 REACTION OBSERVABLES
What do we typically measure when we study a nuclear reaction? We might measure
sR, the total reaction cross section. This might be measured by a beam attenuation
method (Ftransmitted vs. Fincident) or by measuring all possible exit channels for a
reaction where
sR ¼
XbþB
i
si(b, B)
We might measure the cross section for producing a particular product at the end of
the reaction, s (Z, A). We might do this by measuring the radioactivity of the reaction
products. We might, as discussed previously, measure the products emerging in
a particular angular range, ds(u, f)/dV. This measurement is especially relevant for
Figure 10.11 Nuclear, Coulomb, and total potentials for the interaction of 16
O with 208
Pb
for three values of the orbital angular momentum.
264 NUCLEAR REACTIONS
experiments with charge-particle-induced reactions where the incident beam
provides a reference axis for u and f. The energy spectra of the emitted particles
can be measured as ds/dE, or we might observe the products emerging at a
particular angle and with a particular energy, d2
s/dE dV.
10.6 RUTHERFORD SCATTERING
One of the first possible outcomes of the collision of a charged particle with a
nucleus is Rutherford or Coulomb scattering. The incident charged particle feels
the long-range Coulomb force of the positively charged nucleus and is deflected
from its path (Fig. 10.12).
The Coulomb force acting between a projectile of mass m, charge Z1e, and a
target nucleus with charge Z2e is given as:
FCoul ¼
Z1Ze2
r2
where r is the distance between the projectile and target nuclei. The potential energy
(PE) in this interaction is given as:
PE ¼
Z1Z2e2
r
Consider a target nucleus that is much heavier than the projectile nucleus so that we
can neglect the recoil of the target nucleus in the interaction. The projectile will
Figure 10.12 Schematic diagram of Rutherford scattering. [From Satchler (1990).]
10.6 RUTHERFORD SCATTERING 265
follow a hyperbolic orbit, as shown in Figure 10.12 where b is the impact parameter,
TP is the kinetic energy of the projectile and d is the distance of closest approach.
At infinity, the projectile velocity is v. At r ¼ d, the projectile velocity is v0.
Conservation of energy gives
1
2
mv2
¼
1
2
mv2
0 þ
Z1Z2e2
d
Rearranging, we have
v0
v
 2
¼ 1 À
d0
d
where d0 is given as
d0 ¼
2Z1Z2e2
mv2
¼
Z1Z2e2
TP
If we now invoke the conservation of angular momentum, we can write
mvb ¼ mv0d
b2
¼
v0
v
 2
d2
¼ d(d À d0)
It is a property of a hyperbola that
d ¼ b cot (a=2)
Substituting from above, we have
tan a ¼ 2b=d0
Since u ¼ p À 2a, we can write
cot
u
2
 
¼
2b
d0
In Figure 10.13, we show the expected orbits of the projectile nuclei after undergoing
Rutherford scattering for a typical case. Note that the most probable
grazing trajectories result in projectiles being scattered to forward angles but that
some nearly head-on collisions result in large angle scattering. It was these latter
events that led Rutherford to conclude that there was a massive object at the
center of the atom.
We can make these observations more quantitative by considering the situation
where a flux of I0 particles/unit area is incident on a plane normal to the beam
266 NUCLEAR REACTIONS
direction. The flux of particles passing through a ring of width db and with impact
parameters between b and b þ db is given as:
dI ¼
flux
unit area
 
(area of ring)
dI ¼ I0(2pb db)
Substituting from above, we have
dI ¼
1
4
pI0 d2
0
cos (u=2)
sin3
(u=2)
du
If we want to calculate the number of projectile nuclei that undergo Rutherford
scattering into a solid angle dV at a plane angle u, we can write
ds
dV
¼
dI
I0
1
dV
¼
d0
4
 2
1
sin4
(u=2)
¼
Z1Z2e2
4Tcm
P
 2
1
sin4
(u=2)
if we remember that
dV ¼ 2p sin u du sr:
Note the strong dependence of the Rutherford scattering cross section upon scattering
angle. Remember that Rutherford scattering is not a nuclear reaction, as it does
not involve the nuclear force, only the Coulomb force between the charged nuclei.
Remember also that Rutherford scattering will occur to some extent in all studies of
Figure 10.13 Diagram showing some representative projectile orbits for the interaction of
130 MeV 16
O with 208
Pb. [From Satchler (1990).]
10.6 RUTHERFORD SCATTERING 267
charged-particle-induced reactions and will furnish a “background” of scattered
particles at forward angles.
Example Calculation Calculate the differential cross section for the
Rutherford scattering of 215 MeV (lab energy) 48
Ca from 208
Pb at an angle of 208.
Solution
ds
dV
¼
Z1Z2e2
4Tcm
P
 2
1
sin4
u=2ð Þ
Tcm
P ¼ 215 Â
208
256
¼ 174:7 MeV
ds
dV
¼
20 Â 82 Â 1:44
4 Â 174:7
 2
1
sin4
20=2ð Þ
¼ 12562 fm2
=sr ¼ 125:6 b=sr
10.7 ELASTIC (DIFFRACTIVE) SCATTERING
Suppose we picture the interaction of the incident projectile nucleus with the target
nucleus as it undergoes shape elastic scattering. It is convenient to think of this interaction
as that of a plane wave interacting with the nucleus as depicted in Figure 10.14.
Imagine further that all interactions take place on the nuclear surface. Assume
that only points A and B on the nucleus scatter particles and that all other points
on the surface absorb them. To get constructive interference between the incoming
and outgoing wave, we must fulfill the condition that
CB þ BD ¼ nl
where l is the wavelength of the incident particle and n is an integer. Hence peaks
should occur in the scattering cross section when
nl ¼ 2 † 2R † sin
u
2
 
Figure 10.14 Schematic diagram of the interaction of a plane wave with the nucleus. [From
Meyerhof (1967).]
268 NUCLEAR REACTIONS
In Figure 10.15, we show the angular distribution for the elastic scattering of
800 MeV protons from 208
Pb. The de Broglie wavelength of the projectile is 0.85
fm, while the nuclear radius R is about 7.6 fm [1.28(208)1/3
]. We expect peaks
(n ¼ 2, 3, 4 . . .) with a spacing between them, Du, of 3.28 while one observes a
spacing of 3.58. [This discussion of 800 MeV proton scattering is taken from
Bertsch and Kashy (1993).]
Aside on the Optical Model The optical model is a tool to understand and parameterize
studies of elastic scattering. It likens the interaction of projectile and target
nucleus with that of a beam of light interacting with a glass ball. To simulate the
occurrence of both elastic scattering and absorption (reactions) in the interaction,
the glass ball is imagined to be somewhat cloudy.
In formal terms, the nucleus is represented by a nuclear potential that has a real
and an imaginary part:
Unucl(r) ¼ V(r) þ iW(r)
where the imaginary potential W(r) describes absorption (reactions) as the depletion
of flux into nonelastic channels and the real potential V(r) describes the elastic scattering.
Frequently, the nuclear potential is taken to have the Woods–Saxon form:
Unucl(r) ¼ ÀV0fR(r) À iW0 fI(r)
10
10
10
10
1.0
10-1
10-2
4
3
2
1
ds/dW(mb/sr)
q (deg)
2 4 6 8 10 12 14 16 18 20 22 24 26
208Pb(p,p) 0.8 GeV
Figure 10.15 Angular distribution of 800 MeV protons that have been elastically scattered
from 208
Pb. [From Blanpied et al. (1978).]
10.7 ELASTIC (DIFFRACTIVE) SCATTERING 269
where
fR,I(r) ¼ 1 þ exp
r À RR, I
aR, I
 !À1
The potential is thus described in terms of six parameters, the potential depths, V0,
W0; the radii RR, RI; and the surface diffuseness aR, aI. By solving the Schro¨dinger
equation with this nuclear potential (along with the Coulomb and centrifugal potentials),
one can predict the cross section for elastic scattering, the angular distribution
for elastic scattering, and the total reaction cross section. The meaning of the
imaginary potential depth W can be understood by noting that the mean free path
of a nucleon in the nucleus, L, can be given as
L ¼
vhÀ
2W0
where v is the relative velocity. By fitting measurements of elastic scattering cross
sections and angular distributions over a wide range of projectiles, targets, and beam
energies, one might hope to gain a universal set of parameters to describe elastic
scattering (and the nuclear potential). That hope is only partially realized because
only the tail of the nuclear potential affects elastic scattering, and there are families
of parameters that fit the data equally well, as long as the potential energy functions
agree in the exterior regions of the nucleus.
10.8 DIRECT REACTIONS
As we recall from our general description of nuclear reactions, a direct reaction is
said to occur if one of the participants in the initial two-body interaction involving
the incident projectile leaves the nucleus. Generally speaking, these direct reactions
are divided into two classes, the stripping reactions in which part of the incident projectile
is “stripped away” and enters the target nucleus and the pickup reactions in
which the outgoing emitted particle is a combination of the incident projectile and
one or a few target nucleons.
Let us consider stripping reactions first and, in particular, the most commonly
encountered stripping reaction, the (d, p) reaction. Formally, the result of a (d, p)
reaction is to introduce a neutron into the target nucleus, and thus this reaction
should bear some resemblance to the simple neutron capture reaction. But
because of the generally higher angular momenta associated with the (d, p) reaction,
there can be differences between the two reactions. Consider the A (d, p) BÃ
reaction
where the recoil nucleus B is produced in an excited state BÃ
. We sketch out a simple
picture of this reaction and the momentum relations in Figure 10.16.
The momentum diagram for the reaction shown in Figure 10.16 assumes the
momentum of the incident deuteron is kdhÀ
, the momentum of the emitted proton
is kphÀ
, while knhÀ
is the momentum of the stripped neutron. From conservation of
270 NUCLEAR REACTIONS
momentum, we have
k2
n ¼ k2
d þ k2
p À 2kdkp cos u
If the neutron is captured at impact parameter R, the orbital angular momentum
transferred to the nucleus, lnÉ, is given by
lnhÀ
¼ r  p ¼ RknhÀ
ln ¼ Rkn
Since we have previously shown that kn is a function of the angle u, we can now
associate each orbital angular momentum transfer in the reaction with a given
angle u corresponding to the direction of motion of the outgoing proton. Thus, the
(d, p) reaction becomes a very powerful spectroscopic tool. By measuring the
energy of the outgoing proton, we can deduce the Q value of the reaction and
thus the energy of any excited state of the residual nucleus that is formed. From
the direction of motion of the proton, we can deduce the orbital angular momentum
transfer in the reaction, ln. If we know the ground-state spin and parity of the residual
nucleus, we can deduce information about the spin and parity of the excited states of
the residual nucleus using the rules
jJA À lnj À 1
2
À Á


 JBÃ JA þ ln þ 1
2
pApBÃ ¼ (À1)l
Other stripping reactions are reactions such as (a, t), (a, d), and so forth. Typical
pickup reactions are (p, d), (p, t), (a, 6
Li), and so forth.
d
A
B
p
Kp Kn
Kd
Figure 10.16 Sketch of a (d, p) reaction and the associated momentum triangle.
10.8 DIRECT REACTIONS 271
Example Problem Calculate the angle at which the (d, p) cross section has a
maximum for l ¼ 0, 1, 2, 3; and 4. Assume a deuteron energy of 7 MeV and a
proton energy of 13 MeV. Use R ¼ 6 fm.
Solution
kd ¼ 0:82 fmÀ1
kp ¼ 0:79 fmÀ1
kn ¼
l
R
Thus, for l ¼ 0, 1, 2, 3, and 4, kn ¼ 0, 0:17 fmÀ1
, 0:33 fmÀ1
, 0:50 fmÀ1
, and
0:67 fmÀ1
. Solving the momentum triangle,
cos q ¼
Àk2
n þ k2
d þ kp
2kdkp
u ¼ 08, 128, 248, 368, 498 for l ¼ 0, 1, 2, 3, 4
(A somewhat more correct expression would say knR ¼ l(l þ 1)½ Š1=2
.)
10.9 COMPOUND NUCLEUS REACTIONS
The compound nucleus is a relatively long-lived reaction intermediate that is the
result of a complicated set of two-body interactions in which the energy of the projectile
is distributed among all the nucleons of the composite system. How long does
the compound nucleus live? From our definition above, we can say the compound
nucleus must live for at least several times the time it would take a nucleon to traverse
the nucleus (10222
s). Thus, the time scale of compound nuclear reactions is of
the order of 10218
–10216
s. Lifetimes as long as 10214
s have been observed. These
relatively long times should be compared to the typical time scale of a direct reaction
that takes place in one transit of the nucleus of 10222
s.
Another important feature of compound nucleus reactions is that the mode of
decay of the compound nucleus is independent of its mode of formation (the Bohr
independence hypothesis or the amnesia assumption). While this statement is not
true in general, it remains a useful tool for understanding certain features of compound
nuclear reactions. For example, let us consider the classical work of
Ghoshal (1950). Ghoshal formed the compound nucleus 64
Zn in two ways, that is,
by bombarding 63
Cu with protons and by bombarding 60
Ni with a particles. He
examined the relative amounts of 62
Cu, 62
Zn, and 63
Zn found in the two bombardments
and within his experimental uncertainty of 10%, he found the amounts of the
products were the same in both bombardments. (Later experiments have shown
smaller scale deviations from the independence hypothesis.) Because of the long
time scale of the reaction and the “amnesia” of the compound nucleus about its
mode of formation, one can show that the angular distribution of the products is
symmetric about 908 (in the frame of the moving compound nucleus).
272 NUCLEAR REACTIONS
The cross section for a compound nuclear reaction can be written as the product
of two factors, the probability of forming the compound nucleus and the probability
that the compound nucleus decays in a given way. As described above, the
probability of forming the compound nucleus can be written as:
s ¼ plÀ2
X1
l¼0
2l þ 1ð ÞTl
The probability of decay of the compound nucleus (CN) into a given set of products
b can be written as:
Probability ¼
Tb
l Eb
À Á
P
lg, Eg
Tg
l Eg
À Á
2
6
4
3
7
5
where TI is the transmission coefficient for CN decay into products i. Figure 10.17
shows a schematic view of the levels of the compound nucleus. Note the increasing
number of levels as the CN excitation energy increases. Quantitatively, the number
of levels per MeV of excitation energy E increases approximately exponentially
as E1/2
.
The interesting categories of CN reactions can be defined by the ratio of the width
of a compound nucleus level, G, to the average spacing between compound nuclear
levels, D. (Recall from the Heisenberg uncertainty principle that G † t ! hÀ
, where t
is the mean life of a compound nucleus level.) The categories are (a) G=D ( 1, that
is, the case of isolated nonoverlapping levels of the compound nucleus and (b)
G=D ) 1, the case of many overlapping levels in the compound nucleus
(Fig. 10.17). Intuitively category (a) reactions are those in which the excitation
energy of the compound nucleus is low, while category (b) reactions are those in
which the excitation energy is high.
G/D >> 1
G/D << 1}
}
Figure 10.17 Schematic view of the levels of a compound nucleus.
10.9 COMPOUND NUCLEUS REACTIONS 273
Let us first consider the case of G=D ( 1. This means that at certain values of the
compound nucleus excitation energy, individual levels of the compound nucleus can
be excited (i.e., when the excitation energy exactly equals the energy of a given CN
level). When this happens, there will be a sharp rise, or resonance, in the reaction
cross section akin to the absorption of infrared radiation by sodium chloride when
the radiation frequency equals the natural crystal oscillation frequency. In this
case, the formula for the cross section (the Breit–Wigner single-level formula) for
the reaction a þ A ! C ! b þ B is
s ¼ pD2 2JC þ 1ð Þ
2JA þ 1ð Þ 2Ja þ 1ð Þ
GaAGbB
1 À 10ð Þ2
þ(G=2)2
where Ji is the spin of ith nucleus, GaA, GbB, and G are the partial widths for the formation
of C, the decay of C into b þ B, and the total width for the decay of C,
respectively. The symbols 1 and 10 refer to the energy of the projectile nucleus
and the projectile energy corresponding to the excitation of a single isolated level.
Applying this formula to the case of (n, g) reactions gives
sn,g ¼ pD2 2JC þ 1ð Þ
2JA þ 1ð Þ 2ð Þ
GnGg
1 À 10ð Þ2
þ G=2ð Þ2
An example of this behavior is shown in Figure 10.18.
Resonances are seen in low-energy neutron-induced reactions where one is populating
levels in the compound nucleus at excitation energies of the order of the
neutron binding energy where the spacing between levels is of the order of electron
volts. For neutron energies well below 10, so that (1 2 10)2
% 10
2
, then the cross
section for the (n, g) reaction goes as 1/v where v is the neutron velocity, a
general behavior described earlier.
5000
3000
1000
0.5 0.7
Energy (eV)
0.9
s(barns)
Figure 10.18 Resonance behavior in (n, g) reactions.
274 NUCLEAR REACTIONS
Let us now consider the case where G=D ) 1, that is, many overlapping levels
of the compound nucleus are populated. (We are also tacitly assuming a large
range of compound nuclear excitation energies.) The cross section for the reaction
a þ A ! C ! b þ B can be written as
sab ¼ sCPC(b)
where sC (a) is the cross section for the formation of the compound nucleus C and
PC is the probability that C will decay to form b þ B. Clearly SPC(b) ¼ 1. Now let
us consider, in detail, the probability that emitted particle b has an energy 1b. First of
all, we can write down that the maximum energy that b can have is EÃ
C 2 Sb where
EÃ
C is the excitation energy of the compound nucleus and Sb is the separation energy
of b in the residual nucleus B. But b can be emitted with a variety of energies less
than this with the result that the nucleus B will be left in an excited state. By using
the arguments of detailed balance from statistical mechanics (see Lefort, 1968 or
Friedlander, Kennedy, Miller, and Macias (FKMM)) we can write for the probability
of emitting a particle b with an energy 1b (,1max, leaving the nucleus B at an
excitation energy EÃ
B)
Wb 1bð Þd1b ¼
(2Jb þ 1)m
p2hÀ 3
1bsinv
r EÃ
B
À Á
r EÃ
C
À Á d1b
In this equation, m is the reduced mass of the system, and sinv is the cross section for
the inverse process in which the particle b is captured by the nucleus B where b has
an energy, 1b. The symbols r(EÃ
B) and r(EÃ
C) refer to the level density in the nucleus B
excited to an excitation energy EÃ
B and the level density in the compound nucleus C
excited to an excitation energy, EÃ
C. The inverse cross section can be calculated using
the same formulas used to calculate the compound nucleus formation cross section.
Using the Fermi gas model, we can calculate the level densities of the excited
nucleus as
r (EÃ
) ¼ C exp ½2(aEÃ
)1=2
Š
where the level density parameter, a, is A=12 À A=8. The nuclear temperature T is
given by the relation
EÃ
¼ aT2
À T
The ratio of emission widths for emitted particles x and y is given as:
Gx
Gy
¼
gxmx
gymy
Rx
Ry
ax
ay
exp 2 axRxð Þ1=2
À2 ayRy
À Á1=2
h i
where gi is the spin of the ith particle, ai and Ri are the level density parameter and
maximum exciation energy for the residual nucleus that results from the emission of
10.9 COMPOUND NUCLEUS REACTIONS 275
the ith particle; R is formally EÃ
2 S 2 1s where 1s is the threshold for charged
particle emission (1s for neutrons is 0).
If the emitted particles are neutrons, the emitted neutron energy spectrum has
the form
N(1) d1 ¼
1
T2
exp
À1
T
 
d1
as shown in Figure 10.19. In other words, the particles are emitted with a Maxwellian
energy distribution. The most probable energy is T while the average energy is
2T. The compound nucleus appears to “evaporate” particles like molecules leaving
the surface of a hot liquid. By measuring the energy spectrum of the particles emitted
in a compound nuclear reaction, we are using a “nuclear thermometer” in that
d ln N 1ð Þd1½ Š=1sinv
d1
¼
1
T
Charged particles may also be evaporated, except the minimum kinetic energy is not
zero as it is for neutrons. Instead, the threshold for charged particle emission 1s
(which is approximately the Coulomb barrier) determines the minimum energy of
an evaporated particle (see Fig. 10.10). The energy spectrum of evaporated
charged particles is
N(1) d1 ¼
1 À 1s
T2
exp
À1 À 1s
T
 
d1
What will be the distribution in space of the reaction products? Let us assume that
because the compound nucleus has “forgotten” its mode of formation, there should
be no preferential direction for the emission of the decay products. Thus, we might
7
6
5
4
3
2
1
0 1 2 3 4 5
EN (MeV)
EN
6 7 8 9 10
Most Probable
Energy
Average
Energy
Figure 10.19 Spectrum of evaporated neutrons.
276 NUCLEAR REACTIONS
expect that all angles of emission of the particles, u, to be equally probable. Thus, we
would expect that P(u), the probability of emitting a particle at an angle u, might be a
constant. Then, we would expect that ds/dV(u) would be given as:
ds
dV
uð Þ ¼
ð
P qð Þ
du
dV
This assumes that we are making the measurement of the emitted particles angular
distribution in the frame of the moving compound nucleus. In the laboratory
frame, there will appear to be more particles emitted in the forward direction
(with higher energies) than are emitted in the backward direction due to the
motion of the center of mass system.
The energy variation of the cross section (the excitation function) for processes
involving evaporation is fairly distinctive, as shown in Figure 10.20, where the excitation
function for the 209
Bi(a, xn) reaction is shown. Starting from the threshold 1s,
the cross section rises with increasing energy because the formation cross section for
the compound nucleus is increasing. Eventually, the excitation energy of the compound
nucleus becomes large enough that emission of two neutrons is energetically
possible. This “2n out” process will dominate over the “1n out” process, and the
cross section for the “1n out” process will decrease. Eventually, the “3n out”
process will dominate over the “2n out” process. We expect the peaks for the individual
“xn out” processes to be at Sn1 þ 2T, Sn1 þ Sn2 þ 4T, Sn1 þ Sn2 þ Sn3 þ 6T,
and so on (where we neglect any changes in T during the emission process).
Let us recapitulate what we have said about compound nuclear reactions. We
have said that they are basically nuclear reactions with a long-lived reaction
intermediate, which is formed by a complicated set of two-body interactions. We
can write down a set of equations that describes the overall compound nuclear
cross section. We have shown how this general formula simplifies for specific
1.5
0.5
30
ra (MeV)
2010 40
s(barns)
a,n
a,2n
a,3n
s
btotal(a,n
+
a,2
n
+
a,3
n)
S1 + S2Es
Figure 10.20 Excitation function for the 209
Bi(a, xn) reaction.
10.9 COMPOUND NUCLEUS REACTIONS 277
cases, the case of exciting a single level of the compound nucleus where we see
spikes or resonances in the cross section and the case of higher excitation energies
where the compound nucleus behaves like a hot liquid, evaporating particles. At all
excitation energies, the angular distribution of the reaction products is symmetric
with respect to a plane perpendicular to the incident particle direction.
10.10 PHOTONUCLEAR REACTIONS
Photonuclear reactions are a subset of nuclear reactions in which the incident projectile
is a photon and the emitted particles are either charged particles or neutrons.
Examples of such reactions are reactions like (g, p), (g, n), (g, a), and so forth. The
high-energy photons needed to induce these reactions can be furnished from the
annihilation of positrons in flight (producing monoenergetic photons) or the energetic
bremsstrahlung from slowing down high-energy electrons (producing a continuous
distribution of photon energies). A special feature of the excitation
function for photonuclear reactions is the appearance of a large bump in the cross
section at 25 MeV for reaction with a 16
O target that slowly decreases with
increasing A until it is at 15 MeV for 208
Pb (Fig. 10.21).
This bump is called the giant dipole resonance (GDR). Goldhaber and Teller
(1948) provided a model for this reaction in which the giant dipole resonance is
500
400
300
200
100
0
7 8 9
(g, n) (g, 2n)
Photon energy, MeV
79Au197
(g, 3n)
10 11 12 13 14 15 16 17 18 19 20 2221 23 24
Figure 10.21 Photonuclear cross section of 197
Au. [From Fultz et al. (1962).]
278 NUCLEAR REACTIONS
due to a huge collective vibration of all the neutrons versus all the protons. This
model suggests the energy of the GDR should vary as A21/6
, in fair agreement
with observations. In deformed nuclei, the GDR is split into two components, representing
oscillations along the major and minor nuclear axes. One further feature of
photonuclear reactions should be noted. The sum of the absorption cross section for
dipole photons (over all energies) equals some constant, that is,
ð1
0
sabs Eg
À Á
dEg /
NZ
A
% 0:058
NZ
A
MeV-barns
This is called the dipole-sum rule.
10.11 HEAVY ION REACTIONS
Heavy-ion-induced reactions are usually taken as reactions induced by projectiles
heavier than an a particle. The span of projectiles studied is large, ranging from
the light ions, C, O, and Ne, to the medium mass ions, such as S, Ar, Ca, and Kr,
to the heavy projectiles, Xe, Au, and even U. Reactions induced by heavy ions
have certain unique characteristics that distinguish them from other reactions. The
wavelength of a heavy ion at an energy of 5 MeV/nucleon or more is small compared
to the dimensions of the ion. As a result, the interactions of these ions can
be described classically. The value of the angular momentum in these collisions
is relatively large. For example, we can write
lmax ¼
R
lÀ 1 À
B
E
 1=2
For the reaction of 226 MeV 40
Ar þ 165
Ho, we find that lmax ¼ 163hÀ
. This is relatively
large compared to the angular momenta involved in nucleon-induced reactions.
Lastly, quite often the product of the atomic numbers of the projectile and
target is quite large (.1000), indicating the presence of large Coulomb forces
acting in these collisions.
The study of heavy-ion-induced reactions is a forefront area of nuclear research.
By using heavy-ion-induced reactions to make unusual nuclear species, one can
explore various aspects of nuclear structure and dynamics “at its limits.” Another
major thrust is to study the dynamics and thermodynamics of the colliding nuclei.
In Figure 10.22, we show a cartoon of the various impact parameters and trajectories
one might see in a heavy ion reaction. The most distant collisions lead to
elastic scattering and Coulomb excitation. Coulomb excitation is the transfer of
energy to the target nucleus via the long-range Coulomb interaction that excites
the low-lying levels of the target nucleus. Grazing collisions lead to inelastic scattering
and the onset of nucleon exchange. Head-on or near-head-on collisions
lead to fusion of the reacting nuclei, which can lead to the formation of a compound
10.11 HEAVY ION REACTIONS 279
nucleus or a “quasi-fusion” reaction in which there is substantial mass and energy
exchange between the projectile and target nuclei without the “true amnesia”
characteristic of compound nucleus formation. For impact parameters between the
grazing and head-on collisions, one observes a new type of nuclear reaction mechanism,
deep inelastic scattering. In deep inelastic scattering, the colliding nuclei
touch, partially amalgamate, exchange substantial amounts of energy and mass,
rotate as a partially fused complex, and then reseparate under the influence of
their mutual Coulomb repulsion before forming a compound nucleus.
The same range of reaction mechanisms can be depicted in terms of the angular
momentum transfer associated with each of the mechanisms (Fig. 10.23). The most
peripheral collisions lead to elastic scattering and thus have the highest values of the
angular momentum transfer, l. The grazing collisions lead to inelastic scattering and
nucleon exchange reactions, which are lumped together as “quasi-elastic” reactions.
Solid-contact collisions lead to deep inelastic collisions, corresponding to intermediate
values of l. The most head-on collisions correspond to compound nucleus
formation and thus the lowest values of the angular momentum transfer, l. Slightly
more peripheral collisions lead to the fusionlike or quasi-fusion reactions.
10.11.1 Coulomb Excitation
The potential energy due to the Coulomb interaction between a heavy ion and a
nucleus can be written as:
EC ¼ (Z1Z2e2
=R)  1:2(Z1Z2=A1=3
) MeV
Because of the strong, long-range electric field between projectile and target
nuclei, it is possible for the incident heavy ion to excite the target nucleus
Peripheral collisions
Grazing collisions
Distant collisions
Fusion
Elastic scattering
direct reactions
Incomplete fusion and
deep inelastic collisions
Elastic (Rutherford) scattering
Coulomb excitations
Figure 10.22 Classification scheme of collisions based upon impact parameter. [From
Hodgson et al. (1997).]
280 NUCLEAR REACTIONS
electromagnetically. This is called Coulomb excitation, or Coulex. In particular,
rotational bands in deformed target nuclei may be easily excited by the absorption
of dipole photons. This technique is useful for studying the structure of such nuclei.
Since the cross sections for these reactions are very large (involving long-range
interactions with the nucleus), they are especially suitable for use when studying
the structure of exotic nuclei with radioactive beams where the intensities are
low. At relativistic bombarding energies, the strong electric field of the incident
ion may be used to disintegrate the target nucleus (electromagnetic dissociation).
10.11.2 Elastic Scattering
In Figure 10.24, we compare elastic scattering for the collision of light nuclei with
that observed in collisions involving much heavier nuclei. Collisions between the
light nuclei show the characteristic Fraunhofer diffraction pattern discussed
earlier in connection with the scattering of nucleons. The large Coulomb force
associated with the heavier nucleus acts as a diverging lens, causing the diffraction
pattern to be that of Fresnel diffraction. For the case of Fresnel diffraction, special
emphasis is given to the point in the angular distribution of the scattered particle
where the cross section is one-fourth that of the Rutherford scattering cross
section. This “quarter-point angle” is taken to be the classical grazing angle. Note
that the elastic scattering cross section equals the Rutherford scattering cross
section at angles significantly less than the quarter-point angle. Since the Rutherford
scattering cross section is readily calculable, experimentalists measure the number
of elastically scattered particles at angles less than the quarter-point angle to
deduce/monitor the beam intensity in heavy-ion-induced reaction studies.
Figure 10.23 Schematic illustration of the l dependence of the partial cross section for
compound nucleus (CN), fusionlike (FL), deep inelastic (D), quasi-elastic (QE), Coulomb
excitation (CE), and elastic (EL) processes. [From Schroeder and Huizenga (1984, p. 242).]
10.11 HEAVY ION REACTIONS 281
10.11.3 Fusion Reactions
In Figure 10.25, we show another representation of the difference between the
various reaction mechanisms in terms of the energy needed to induce the reactions.
The energy needed to bring the ions in contact and thus interact, is the interaction
barrier, V(Rint). Formally, Bass (1980) has shown the reaction cross section can
be expressed in terms of a one-dimensional interaction barrier as:
sR ¼ pR2
int 1 À
V Rintð Þ
Ecm
!
where the interaction radius is given as:
Rint ¼ R1 þ R2 þ 3:2 fm
where the radius of the ith nucleus is
RI ¼ 1:12A1=3
i À 0:94AÀ1=3
I fm
and the interaction barrier is given as:
V(Rint) ¼ 1:44
Z1Z2
Rint
À b
R1R2
R1 þ R2
where b  1 MeV=fm. (This expression is similar to that developed in Section 10.4
but represents a better description of the cross section.) The energy necessary to
cause the ions to interpenetrate further to cause quasi-fusion is called the extra
500 2.0
1.0
0.5
0.2
10° 40°35°30°25°20°15°
0.1
100
10
1
0.1
20° 30°
qC.M. qC.M.
40°
s(q)¥mb/sr
s/sR
Experiment
EC.M. = 72 MeV
EC.M. = 158.0 MeV
(a) (b)
Figure 10.24 Angular distribution for 12
C þ 16
O elastic scattering, showing Fraunhofer
diffraction and the elastic scattering of 16
O with 208
Pb, which shows Fresnel diffraction.
[From Valentin (1981).]
282 NUCLEAR REACTIONS
push energy. The energy necessary to cause the ions to truly fuse and forget their
mode of formation is referred to as the extra–extra push energy.
The probability of fusion is a sensitive function of the product of the atomic
numbers of the colliding ions. The abrupt decline of the fusion cross section as
the Coulomb force between the ions increases is due to the emergence of the
deep inelastic reaction mechanism. This decline and other features of the fusion
cross section can be explained in terms of the potential between the colliding
ions. This potential consists of three contributions, the Coulomb potential, the
nuclear potential, and the centrifugal potential. The variation of this potential as a
function of the angular momentum l and radial separation is shown as Figure 10.26.
Note that at small values of the angular momentum, there is a pocket in the potential.
Fusion occurs when the ions get trapped in this pocket. If they do not get
trapped, they do not fuse. With high values of the Coulomb potential, there are
few or no pockets in the potential for any value of l, thus no fusion occurs. For a
given projectile energy and Coulomb potential, there is a value of the angular
momentum above which there are no pockets in the potential (the critical value of
the angular momentum) and thus no fusion occurs.
As shown in Figure 10.25, there is an l-dependent barrier to fusion that is the sum
of the nuclear, Coulomb, and centrifugal potentials. This barrier is also a sensitive
function of the relative deformation and orientation of the colliding ions. In
Figure 10.27, we show the excitation function for fusion of 16
O with various isotopes
of Sm that span a wide range of deformations.
Figure 10.25 Schematic illustration of the three critical energies and the four types of heavy
ion nuclear reactions. [From S. Bjornholm and W. J. Swiatecki, Nucl. Phys. A391, 471 (1982).]
10.11 HEAVY ION REACTIONS 283
150
100
50
0
0 10 20
R
30
10
L=9020
40
50
60
70
80
VN+VC+VL
E
Figure 10.26 Sum of the nuclear, Coulomb, and centrifugal potential for 18
O þ 120
Sn as a
function of radial distance for various values of the orbital angular momentum l.
103
102
101
100
10−1
1
a[v](mb)
Eicb (MeV)
Fusion of 16O + ASm
60 65
154
a b2
152
150
148
0.20
0.18
0.14
0.00
70 75
Figure 10.27 Fusion cross sections for 16
O þ A
Sm.
284 NUCLEAR REACTIONS
One observes a significantly lower threshold and enhanced cross section for the
case where the 16
O ion interacts with deformed 154
Sm compared to near-spherical
148
Sm. This enhancement is the result of the lowering of the fusion barrier for the
collision with the deformed nucleus due to the fact that the ions can contact at a
larger value of R resulting in a lower Coulomb component of the potential. Let us
now consider what happens after the formation of a compound nucleus in a
heavy-ion fusion reaction. In Figure 10.28, we show the predictions for the decay
of the compound nuclei formed in the reaction of 147 MeV 40
Ar with 124
Sn to
8
4
0
(mb)s
0 20 40 60 80
Input Angular
Momentum
40 124 164-xAr( Sn, xn) Er
E = 161 MevBeam
E*= 53.8 Mev
50
40
30
20
10
0
ExcitationEnergy(MeV)
0 20 40 60 80
n
3n
4n
5n
YrastLine
8
4
0
Intensity
0 20 40 60 80
4n
5n
3n
l (h)
30
20
10
0
Intensity
0102030
4n
5n
3n
Figure 10.28 Predicted decay of the 164
Er compound nuclei formed in the reaction of 40
Ar
with 124
Sn. [From Stokstad (1985).]
10.11 HEAVY ION REACTIONS 285
form 164
Er at an excitation energy of 53.8 MeV. The angular momentum distribution
in the compound nucleus shows population of states with l ¼ 0 À 60hÀ
. The excitation
energy is such that energetically the preferred reaction channel involves the
evaporation of four neutrons from the compound nucleus. As the compound
nucleus evaporates neutrons, the angular momentum does not change dramatically
since each neutron removes a relatively small amount of angular momentum.
Eventually, the yrast line restricts the population of states in the E-J plane. The
yrast line is the locus of the lowest lying state of a given angular momentum
for a given J value. Below the yrast line for a given J, there are not states of the
nucleus. (The word yrast is from the Old Norse for the “dizziest.”) When the
system reaches the yrast line, it decays by a cascade of g rays. Heavy-ion reactions
are thus a tool to excite levels of the highest spin in nuclei, allowing the study of
nuclear structure at high angular momentum.
10.11.4 Deep Inelastic Scattering
Now let us turn our attention to the case of deep inelastic scattering. In the early
1970s, as part of a quest to form superheavy elements by the fusion of Ar and Kr
ions with heavy target nuclei, a new nuclear reaction mechanism, deep inelastic scattering
was discovered. For example, in the reaction of 84
Kr with 209
Bi (Fig. 10.29)
instead of observing the fission of the completely fused nuclei (to form nuclei in the
region denoted by the triangle), one observed projectile and targetlike nuclei and a
new and unexpected group of fragments with masses near that of the target and
projectile but with kinetic energies that were much lower than those expected
from elastic or quasi-elastic scattering.
These nuclei appeared to be nuclei that had undergone an inelastic scattering that
had resulted in the loss of a large amount of the incident projectile kinetic energy.
Further measurement revealed this to be a general phenomenon in reactions
where the product of the atomic numbers of the colliding ions was large
(.2000). As described earlier, the ions come together, interpenetrate partially,
exchange some mass, and charge with dissipation of a large amount of kinetic
energy in a diffusion process, and then reseparate under the influence of their
mutual Coulomb repulsion. The initial projectile energy is damped into the
excitation energy of the projectile and targetlike fragments. As a consequence, the
larger the kinetic energy loss, the broader the distribution of the final products
becomes partly due to the evaporation of nucleons after the nuclei reseparate.
10.11.5 Incomplete Fusion
In the course of the fusion of the projectile and target nuclei, it is possible that one of
the partners will emit a single nucleon or a nucleonic cluster prior to the formation of
a completely fused system. Such processes are referred to as preequilibrium emission
(in the case of nucleon emission) or incomplete fusion (in the case of cluster
emission). As the projectile energy increases, these emission processes become
more important and they generally dominate over fusion at projectile energies
286 NUCLEAR REACTIONS
above 20 MeV/nucleon. As a consequence of these processes, the resulting product
nucleus has a momentum that is reduced relative to complete fusion events.
Measurement of the momentum transfer in the collision serves as a measure of
the occurrence of these phenomena. In the spectra of emitted particles, a highenergy
tail on the normal evaporation spectrum is another signature of preequilibrium
emission.
10.11.6 Reactions Induced by Radioactive Projectiles
There are a few hundred stable nuclei but several thousand nuclei that are radioactive
and have experimentally useful lifetimes. Since 1990, one of the fastest
growing areas of research in nuclear science has been the study of nuclear reactions
induced by radioactive projectiles. Using either ISOL (Isotope Separator On-Line)
or PF (Projectile Fragmentation) techniques, several hundred new radioactive
nuclear beams have become available (see Chapter 14).
The principal attraction in these studies is the ability to form reaction products or
reaction intermediates with unusual N/Z ratios. By starting with nuclei that are either
very proton rich or very neutron rich, new regions of nuclei can be reached and their
properties studied. At higher energies, the unusual isospin of the intermediate species
allows one to determine the effect of isospin on the properties of highly excited
Figure 10.29 Measurement of the product energy and mass distributions in the reaction of
84
Kr with 209
Bi. [From M. Lefort et al., Nucl. Phys. A216, 166 (1973).]
10.11 HEAVY ION REACTIONS 287
nuclear matter. Occasionally, the radioactive beams themselves have unusual
structure, for example, 11
Li, and their properties and reactions are of interest.
10.12 HIGH-ENERGY NUCLEAR REACTIONS
A nuclear reaction is said to be a low-energy reaction if the projectile energy is
10 MeV/nucleon that is, near the Coulomb barrier. A nuclear reaction is termed
a high-energy reaction if the projectile energy is much greater than the Coulomb
barrier and approaching the rest mass of the nucleon, for example, 400 MeV/
nucleon. (Not surprisingly the reactions induced by 20–250 MeV/nucleon projectiles
are called intermediate-energy reactions.)
What distinguishes low- and high-energy reactions? In low-energy nuclear collisions,
the nucleons of the projectile interact with the average or mean nuclear
force field associated with the entire target nucleus. In a high-energy reaction, the
nucleons of the projectile interact with the nucleons of the target nucleus individually,
as nucleon–nucleon collisions. To see why this might occur, let us compute
the de Broglie wavelength of a 10-MeV proton and a 1000-MeV proton. We get
l10 MeV ¼ 9.0 fm and l1000 MeV ¼ 0.73 fm. The average spacing between nucleons
in a nucleus is 1.2 fm. Thus, we conclude that at low energies the projectile
nucleons can interact with several nucleons at once, whereas at high energies
collisions occur between pairs of nucleons.
10.12.1 Spallation/Fragmentation
What type of reactions do we observe at high energies? Because we are dealing with
processes dominated by nucleon–nucleon collisions, we do not expect any significant
amount of compound nucleus formation. Instead most reactions should be
direct reactions taking place on a short time scale. In Figure 10.30, we show a
typical distribution of the masses of the residual nuclei from the interaction of
GeV protons with a heavy nucleus, such as 209
Bi.
At the highest energy, one observes a continuous distribution of product masses
ranging from the target mass to very low values of A. Three regions can be identified
in the yield distributions. One region is centered around Atarget/2 (A ¼ 50–140) and
consists of the products of the fission of a targetlike nucleus. For larger A values
[Afragment ! 2
3 Atarget] the products are thought to arise from a direct reaction
process termed spallation. The incident proton knocks out several nucleons in a
series of two-body collisions, leaving behind a highly excited heavy nucleus. This
nucleus decays by the evaporation of charged particles and neutrons, forming a continuous
distribution of products ranging downward in A from the target mass
number. The term spallation was given to this phenomenon by one of us (GTS)
after consultation with a professor of English who assured him that the verb “to
spall” was a very appropriate term for this phenomenon. For the lowest mass
numbers [Afragment
1
3 Atarget] one observes another group of fragments that are
termed to be intermediate mass fragments (IMFs). These fragments are thought to
288 NUCLEAR REACTIONS
arise from the very highly excited remnants of the most head-on collisions by
either sequential particle emission or a nuclear disintegration process called
multifragmentation.
The course of a reaction at high energies is different than one occurring at lower
energies. As mentioned earlier, collisions occur between pairs of nucleons rather
than having one nucleon collide with several nucleons simultaneously. The cross
section for nucleon–nucleon scattering varies inversely with projectile energy. At
the highest energies, this cross section may become so small that some nucleons
will pass through the nucleus without undergoing any collisions, that is, the
nucleus appears to be transparent.
In this regard, a useful quantitative measure of the number of collisions a nucleon
undergoes in traversing the nucleus is the mean free path, L. Formally, we have
L ¼ 1=rs
where s is the average nucleon–nucleon scattering cross section (30 millibarns)
and r is the nuclear density (1038
nucleons/cm3
). Thus, the mean free path is
3 Â 10213
cm. In each collision, the kinetic energy imparted to the struck
nucleon is 25 MeV, and thus the struck nucleon may collide with other nucleons,
generating a cascade of struck particles (see Fig. 10.31). If the energy of the incident
103
0.1
102
10
1
Crosssections(mb)
0 0.25 0.50
3000 Mev
480 Mev
40 Mev
0.75
Relative mass number
1.00
Figure 10.30 Mass distribution for p þ 209
Bi. [From Miller and Hudis (1959).]
10.12 HIGH-ENERGY NUCLEAR REACTIONS 289
nucleon exceeds 300 MeV, then it is possible to generate p mesons in these
collisions, which, in turn, can interact with other nucleons. A typical time scale
for the cascade is 10222
s. The result of this intranuclear cascade is an excited
nucleus, which may decay by preequilibrium emission of particles, evaporation of
nucleons, sequential emission of IMFs, or disintegration into multiple fragments.
In the mid-1970s, at the Bevalac in Berkeley, heavy ion reactions at very high
energies (0.250–2.1 GeV/nucleon) were studied. At these high projectile energies,
a number of observations were interpreted in terms of a simple geometric model
referred to as the abrasion-ablation or fireball model. (Fig. 10.32). In this model
the incoming projectile sheared off a sector of the target (corresponding to the
overlap region of the projectile and target nucleus—the “abrasion” step). The nonoverlapping
regions of the target and projectile nuclei were assumed to be left
essentially undisturbed and unheated, the so-called “spectators” to the collision.
The hot overlap region (the “participants”) formed a “fireball” that decayed with
the release of nucleons and fragments. The wounded target nucleus was expected
to have a region of extra surface area exposed by the projectile cut through it.
Associated with this extra surface area is an excitation energy corresponding to
the surface area term of the semiempirical mass equation of about 1 MeV per
excess fm2
of surface area. As the nucleus relaxes, this excess surface energy
becomes available as excitation energy and results in the normal emission of
nucleons and fragments (the “ablation” step).
The use of this simple model for high-energy nucleus–nucleus collisions has
resulted in a general categorization of energetic nucleus–nucleus collisions as
either “peripheral” or “central.” In peripheral reactions, one has large impact
parameters and small momentum transfer. Such reactions, which produce surviving
large spectators, are referred to as fragmentation reactions. Such reactions are of
interest in the production of new radioactive nuclei and radioactive beams.
Figure 10.31 Schematic view of nuclear cascade. [From Lieser (1997).]
290 NUCLEAR REACTIONS
10.12.2 Multifragmentation
In central collisions one has smaller impact parameters and larger energy and
momentum transfer. In central nucleus–nucleus collisions at intermediate energies
(20–200 MeV/nucleon), large values of the nuclear excitation energy
(.1000 MeV) and temperature (.10 MeV) may be achieved for short periods of
time (10222
s). Nuclei at these high excitation energies can decay by the emission
of complex or intermediate mass fragments. (An IMF is defined as a reaction
product whose mass is greater than 4 and less than that of a fission fragment.)
Multifragmentation occurs when several IMFs are produced in a reaction. This
can be the result of sequential binary processes, “statistical” decay into many fragments
(described by passage through a transition state or the establishment of
statistical equilibrium among fragments in a critical volume), or a dynamical
process in which the system evolves into regions of volume and surface instabilities
leading to multifragment production.
To investigate these phenomena, it is necessary to measure as many of the
emitted fragments and particles from a reaction. As a result, various multidetector
arrays have been constructed and used. Quite often these arrays consist of several
hundred individual detectors to detect the emitted IMFs, light charged particles,
neutrons, target fragments, and the like. As a consequence of the high granularity
of these detectors, the analysis of the experimental data is time consuming and difficult.
Nonetheless, several interesting developments have occurred in recent years.
ABRASION
(a)
(b)
“DIRTY CUT” ABLASION
Figure 10.32 Abrasion–ablation model of relativistic nuclear collisions.
10.12 HIGH-ENERGY NUCLEAR REACTIONS 291
One theory to describe multifragmentation postulates the formation of a hot
nuclear vapor during the reaction, which subsequently condenses into droplets of
liquid (IMFs) somewhere near the critical temperature. First postulated to occur
in the interaction of GeV protons with Xe, recent experiments with heavy ions
have resulted in deduced temperatures and excitation energies (Fig. 10.33) that
resemble calculations for a liquid–gas phase transition. This somewhat speculative
“caloric curve” shows an initial rise in temperature with excitation energy typical
of heating a liquid, followed by a flat region (the phase transition), followed by a
region corresponding to heating a vapor.
Finally, there has been an extended debate and discussion of the relative role of
statistical and dynamical factors in multifragmentation. The debate has focused on
the observation that the data from several reactions could be plotted such that the
probability of emitting multiple fragments, p, could be expressed in a form,
p exp(2B/T), suggestive of the dependence of the fragment emission probabilities
upon a single fragment emission barrier, B, a feature suggesting the importance of
statistical factors. Others have criticized this observation. The criticisms have
focused on the details of the correlation and evidence for dynamic effects.
10.12.3 Quark–Gluon Plasma
The primary thrust of studies of central collisions of massive nuclei at ultrarelativistic
energies (.5 GeV/nucleon) is to create and observe a new form of matter,
the quark–gluon plasma (QGP). The modern theory of the strong interaction,
quantum chromodynamics, predicts that while quarks and gluons will be confined
within a nucleonic “bag” under normal conditions, deconfinement will occur at
sufficiently high energies and densities. This phase transition (from normal
nuclear matter to the QGP) is predicted to occur at energy densities of
Figure 10.33 Caloric curve as calculated by a multifragmentation model and as measured.
292 NUCLEAR REACTIONS
1–3 GeV/fm3
, which can be achieved in collisions at cm energies of 17 GeV/
nucleon.
The experimental signatures of a phase transition include: (a) suppression of
production of the heavy vector mesons J/C and C0
and the upsilon states, (b) the
creation of a large number of ss quark–antiquark pairs, and (c) the momentum
spectra, abundance, and direction of emission of di-lepton pairs. The first phase
experiments in this field have been carried out. Energy densities of 2 GeV/fm3
were created. Strong J/C suppression has been observed relative to p-A collisions
along with an increase in strangeness production.
PROBLEMS
1. Consider the reaction of 16
O with 64
Ni at a cm energy of 48 MeV. What is the
lab kinetic energy of the 16
O? What is the Coulomb barrier for the reaction?
What is the total reaction cross section? What is the maximum angular momentum
brought in by the 16
O projectile?
2. One reaction proposed for the synthesis of element 110 is the reaction of 59
Co
with 209
Bi at a laboratory energy of 295 MeV. Calculate the expected total
reaction cross section.
3. Define or describe the following terms or phenomena: direct reaction, compound
nucleus, and stripping reaction.
4. A piece of Au that is 1 mm thick is bombarded for 15 h by a slow neutron beam
of intensity 106
/s. How many disintegrations per second of 198
Au are present in
the sample 24 h after the end of the bombardment? s (n, g) ¼ 98.8 barns,
t1/2 (198
Au) ¼ 2.7 d.
5. What was the rate of production, in atoms per second, of 128
I during a constant
1-h cyclotron (neutron) irradiation of an iodine sample if the sample was found
to contain 2.00 mCi of 128
I activity at 15 min after end of the irradiation?
6. What is the excitation energy of the 116
Sb compound nuclei formed by the
bombardment of 103
Rh with 50 MeV 13
C ions?
7. Neutrons evaporated from a compound nucleus have an average kinetic energy
of 2T, where T is the nuclear temperature of the residual nucleus. What is the
optimum bombarding energy for the production of 66
Ga via the 65
Cu(a, 3n)
reaction if the average nuclear temperature is 1.6 MeV?
8. A 100-mg/cm2
-thick natural Zr target is bombarded with a beam of 11-MeV
protons for one hour (beam current ¼ 25 mA). The 95
Nbm
from the reaction
96
Zr(p, 2n) was isolated chemically (100% yield), and the k x-rays resulting
from the internal conversion decay of 95
Nbm
were counted. In a 2-h count beginning
20 h after the end of bombardment, 1000 counts were observed in the Nb
Ka x-ray peak. Given the 95
Nb decay scheme shown below and the data given
PROBLEMS 293
below, calculate the cross section for the 95
Zr(p, 2n) 95
Nbm
reaction.
Fluorescence yield ¼ 0.7, efficiency of detection of the K x-ray is 1023
,
ak ¼ 2.21.
294 NUCLEAR REACTIONS
9. What is the number of 60
Co atoms produced in a 10-mg sample of cobalt metal
exposed for 2 min to a thermal neutron flux of 2 Â 1013
n/cm2
-s in a reactor?
The cross section for producing 10.5-min 60
Com
is 16 barns, while the cross
section for producing 5.3 y 60
Co is 20 barns. What is the disintegration rate
of the cobalt sample 4 h after the end of the irradiation?
10. Consider the 48
Ca þ 248
Cm reaction where the lab energy of the 48
Ca is
300 MeV. What is the excitation energy of the putative compound nucleus
296
116? What is the expected total reaction cross section?
11. Consider the reaction of 30 MeV/nucleon 129
Xe with 238
U. What is the energy
of the elastically scattered 129
Xe detected at 108 in the lab system?
12. Consider the 40
Ca(d, p) reaction. What would be the most probable angle to
detect the protons leading to the first excited state 3
2
À
of 41
Ca? What would
be their energy? Assume the energy of the incident deuteron beam was
21.0 MeV.
13. Consider you want to make 18
F for use in PET studies. What would be the
maximum specific activity (dpm/g F) of the 18
F made by irradiating 1.0 g of
KF in a flux of 1010
fast neutrons/cm2
-s. You may assume the 18
F(n, 2n)
cross section is 300 millibarns. Consider you want to produce the 18
F carrier
free (i.e., with no stable fluorine present). Devise a synthetic scheme for
producing the carrier-free 18
F. Defend your choice of reactions.
14. Consider the nuclide 99
Tcm
that is the daughter of 99
Mo. In the United States
most diagnostic procedures involving radioactivity involve 99
Tcm
. Explain
how you would produce 99
Mo (the 66.0-h parent of 6.0-h 99
Tcm
). Consider
two choices, production of 99
Mo as a fission product or via the 98
Mo(n, g)
reaction.
15. Calculate the activity of 254
No (t1/2 ¼ 55 s) present 5 min after a 10-min
irradiation of a 0.001-in.-thick 208
Pb foil by 48
Ca projectiles (f ¼ 6.28 Â 1012
particles/second). Assume s(48
Ca, 2n) is 3 Â 10230
cm2
.
16. Consider the reaction 12
C(a, n) where the laboratory energy of the incident projectile
is 14.6 MeV. What is the excitation energy of the compound nucleus?
The reaction cross section is 25 millibars. Assuming a carbon target thickness
of 0.10 mg/cm2
and a beam current of 25 nA, compute the 15
O activity after
a 4-min irradiation.
17. The cross section for the 60
Ni(a, pn) reaction is 0.9 barn for 32-MeV a particles.
Calculate the number of disintegrations per minute of 62
Cu at 15 min after
a 15-min bombardment of a 50-mg/cm2
foil of 60
Ni with 10 mA of 32 MeV
a particles.
18. Consider the reaction 29
Si(18
O, p2n) which populates the metastable and ground
states of 44
Sc. Using the decay scheme shown below, and the fact that at end of
bombardment (EOB) one observed 1000 photons/second of energy 271.2 keV
and 1000 photons/second of energy 1157.0 keV, calculate the ratio of the cross
PROBLEMS 295
section for the production of 44
Scm
, sm, to the cross section for the production of
44
Sc, sg. Neglect any decay of 44
Scm
to 44
Sc during the irradiation and assume
the length of the irradiation was 6 h.
Ca
44
0+
0+
B+, EC
2+
2+
99.89%
0
1157.0 keV
Sc
44
Sc
44m
86.6%
271.2 keV
0 t1/2 = 3.93 k
t1/2 = 2.44 d6+
REFERENCES
Bass, R. Nuclear Reactions with Heavy Ions, Springer, Berlin, 1980.
Bertsch, G. F. and E. Kashy. Am. J. Phys. 61, 858 (1993).
Blanpied, G. S. et al. Phys. Rev. C 18, 1436 (1978).
Ehmann, W. D. and D. E. Vance. Radiochemistry and Nuclear Methods of Analysis, Wiley,
New York, 1991.
Fultz, C. et al. Phys. Rev. 127, 1273 (1962).
Ghoshal, X. Phys. Rev. 80, 939 (1980).
Goldhaber, M. and E. Teller. Phys. Rev. 74, 1046 (1948).
Lefort, M. Nuclear Chemistry, Van Nostrand, Princeton, 1968.
Lieser, K. H. Nuclear and Radiochemistry, VCH, Wertheim, 1997.
Meyerhof, W. E. Elements of Nuclear Physics, McGraw-Hill, New York, 1967.
Miller, J. M. and J. Hudis. Ann. Rev. Nucl. Sci. 9, 159 (1959).
Satchler, G. R. Introduction of Nuclear Reactions, 2nd ed., Oxford, New York, 1990.
Schroeder, W. U. and J. R. Huizenga. In D. A. Bromley, Ed., Treatise on Heavy Ion Science,
Vol. 2, Plenum, New York, 1984.
Stokstad, R. In D. A. Bromley, Ed., Treatise on Heavy Ion Science, Vol. 3, Plenum,
New York, 1985.
296 NUCLEAR REACTIONS
Valentin, L. Subatomic Physics: Nuclei and Particles, Vol. II, North-Holland, Amsterdam,
1981.
Weidner, R. T. and R. L. Sells. Elementary Modern Physics, Boston, Allyn and Bacon,
Boston, 1973.
Weisskopf, V. F. Rev. Mod. Phys. 29, 174 (1959).
BIBLIOGRAPHY
General Electric, Nuclides and Isotopes, 6th ed., KAPL, 2002.
Hodgson, P. E., E. Gladioli, and E. Gladioli-Erba. Introductory Nuclear Physics, Clarendon,
Oxford, 1997.
Jelley, N. A. Fundamentals of Nuclear Physics, Cambridge, Cambridge, 1990.
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1988.
Lilley, J. S. Nuclear Physics, Wiley, Chichester, 2001.
Wong, S. S. M. Nuclear Physics, 2nd ed., Wiley, New York, 1998.
BIBLIOGRAPHY 297
CHAPTER 11
FISSION
11.1 INTRODUCTION
Fission has a unique importance among nuclear reactions. Apart from the nuclear
reactions that drive the sun, no other nuclear reaction has had such a profound
impact on the affairs of humans. The discovery of fission, and the developments
that proceeded from it, have altered the world forever and have impinged on the
consciousness of every literate human being. The exploitation of nuclear energy,
which followed the discovery of fission, particularly in weapons of mass destruction,
has been of profound importance to humankind.
Chemists have played an important role in the study of fission. Fission was
discovered by the chemists Otto Hahn and Fritz Strassman in 1938 (Hahn and
Strassman, 1938). By painstakingly difficult chemical separations, they were able
to show that the neutron irradiation of uranium led, not to many new elements as
had been thought, but to products such as barium, lanthanum, and the like. The
uranium nucleus had been split! That conclusion caused Hahn and Strassman
(1938) much concern as they wrote, “As ‘nuclear chemists’ working very close to
the field of physics, we cannot bring ourselves yet to take such a drastic step (to
conclude that uranium had fissioned), which goes against all previous experience
in nuclear physics.” (Am. J. Phys. 32, 15 (1964)). Nuclear chemists have continued
their role in studying fission, first using chemical techniques and, more recently,
using physical techniques.
299
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
Knowledge of fission and its consequences is important for the nuclear power
industry and the related fields of nuclear waste management and environmental
cleanup. From the point of view of basic research, fission is interesting in its own
right as a large-scale collective motion of the nucleus, as an important exit
channel for many nuclear reactions, and as a source of neutron-rich nuclei for
nuclear structure studies and use as radioactive beams.
The reader should be cautioned that understanding the fission process represents a
very difficult problem. Some of the best minds in chemistry and physics have
worked on the problem since the discovery of fission. Yet, while we understand
many aspects of the fission process, there is no overall theoretical framework that
gives a satisfactory account of the basic observations.
In Figure 11.1, we show a schematic view of the fission process. A nucleus with
some equilibrium deformation absorbs energy, becoming excited, and deforms to a
configuration known as the transition state or saddle point configuration. As it
deforms, the nuclear Coulomb energy decreases (as the average distance between
the nuclear protons increases) while the nuclear surface energy increases (as
the nuclear surface area increases). At the saddle point, the rate of change of the
Coulomb energy is equal to the rate of change of the nuclear surface energy. The
formation and decay of this transition state nucleus is the rate-determining step in
the fission process and corresponds to the passage over an activation energy
barrier to the reaction. If the nucleus deforms beyond this point, it is irretrievably
committed to fission. When this happens, then in a very short time the neck
between the nascent fragments disappears and the nucleus divides into two fragments
at the scission point. At the scission point, one has two highly charged,
deformed fragments in contact with each other. The large Coulomb repulsion
between the two fragments accelerates them to 90% of their final kinetic energy
within 10220
s. As these accelerated primary fragments move away from one
another, they contract to more spherical shapes, converting their potential energy
of deformation into internal excitation energy, that is, they get “hot.” This excitation
energy is removed by the emission of the “prompt” neutrons from the fully accelerated
fragments and then, in competition with the last neutrons to be emitted, the
nucleus emits g rays. Finally, on a longer time scale the neutron-rich fragments
emit b2
particles. Occasionally, one of these b decays populates a high-lying
excited state of a daughter that is unstable with respect to neutron emission, and
the resulting emitted neutrons are called the “delayed” neutrons. Note that this schematic
view conflicts with some presentations of fission in elementary textbooks. For
example, if the neutrons are emitted primarily from the fully accelerated fragments,
their spatial distribution is along the direction of motion of the fragments. They do
not emerge randomly from the nucleus as many artists’ conceptions of fission depict.
Also note that the energy release in fission is primarily in the form of the kinetic
energies of the fragments not in the neutrons, photons, or other emitted particles.
This energy is the “mass-energy” released in fission due to the increased stability
of the fission fragments.
Because of the large amount of information available about fission, it is beyond
the scope of this chapter to present a complete treatment of fission research. We shall
300 FISSION
attempt to emphasize the fundamental aspects of the subject. The reader is referred
to one of the excellent monographs or reviews of fission (Vandenbosch and
Huizenga, 1973; Wagemans, 1991; Oganessian and Lazarev, 1985; Hoffman
et al., 1996) for further information.
Target nucleus
in equilibrium
deformation
(Z0,A0)
Transition state nucleus
with saddle deformation
and (E − Ef) MeV of excitation.
Lifetime depends on
E − Ef and is about 10−5 sec
for thermal neutron capture
Scission configuration, two
nuclear potentials. fragments
highly deformed with the
degree of deformation depending
upon fragment stiffness;
kinetic energies of fragments
small (?); occasional small
particle emitted although neutron
yield may be relatively large.
(Z1,A1) (Z2,A2)
Accelerated primary fragments
separated by a relatively large
distance
10-21 sec £ t £ 10-20 sec (?)
Nuclear Reaction Excited nucleus with
equilibrium deformation
of target and E MeV of
excitation energy
(Z1,A1) (Z2,A2)
(Z3,A3)
and
(Z4,A4)
(Z1,A3) (Z2,A4)
Stable
end
products
Primary fission products. secondary
deexcited fission fragments in their
ground state. These nuclei are
far removed from beta stability and
are radioactive.
Neutron emission in time
period of the order of
10-15 – 10-18 sec
Accelerating primary fragments
under coulomb interaction;
approximately 10-20 sec for
fragments to reach 90% of
their final K.E.
(Z1,A1) (Z2,A2)
(Z0,A0)
such as particle
capture or a direct
reaction, e.g.
(d,p) or (a,a')
(Z1,A3) (Z2,A4)
g
g
g-ray emission in time
period of about 10-11 sec
g
bRadioactive
decay; very slow process;
t > 10-3 sec; occasionally beta
decay populates a neutron-unstable
level to give delay neutrons;
radiochemists observe the relatively
longer-lived members in
this radioactive decay process.
g
bFigure
11.1 Schematic view of the fission process. (From J. E. Gindler and J. R. Huizenga,
“Nuclear Fission” in Nuclear Chemistry, Vol. II, L. Yaffe, Ed. Copyright # 1968 Academic
Press, Reprinted by permission of Elsevier.)
11.1 INTRODUCTION 301
11.2 PROBABILITY OF FISSION
11.2.1 Liquid Drop Model
Figure 11.1 suggests fission proceeds in two steps, the ascent to the saddle point and
the passage through the scission point. We shall present our discussion of fission
from this point of view. We shall assert that like chemical reactions, the reaction
probability is determined by the passage through the transition state. We shall
also assert, more controversially, that the distribution of fission product energies,
masses, and so forth is determined at or near the scission point.
Let us begin with a discussion of the probability of fission. For the first approximation
to the estimation of the fission barrier, we shall use the liquid drop model
(Chapter 2). We can parameterize the small nonequilibrium deformations, that is,
elongations, of the nuclear surface as
R(q) ¼ R0½1 þ a2P2( cos q)Š (11:1)
where a2 is the quadrupole distortion parameter [¼ ( 5
4 p)1=2
b2] and P2 is the secondorder
Legendre polynomial. For small distortions, the surface, Es, and Coulomb, EC,
energies are given by
Es ¼ E0
s 1 þ
2
5
a2
2
 
EC ¼ E0
C 1 À
1
5
a2
2
  (11:2)
where Es
0
and EC
0
are the surface and Coulomb energies of the undistorted spherical
drops. When the changes in the Coulomb and surface energies (DEC ¼ EC
0
2 EC,
DEs ¼ Es 2 Es
0
) are equal, the nucleus becomes spontaneously unstable with
respect to fission. At that point we find that
E0
c
2E0
s
¼ 1 (11:3)
Thus, it is natural to express the fissionability of nuclei in terms of a parameter x, that
is, this energy ratio, and is called the fissionability parameter. Thus
x ¼
E0
C
2E0
s
¼
1
2
Coulomb energy of a charged sphere
surface energy of the sphere
 
(11:4)
We can approximate the Coulomb and surface energies of a uniformly charged
sphere by the following expressions:
E0
C ¼
3
5
Z 2
e2
R0A1=3
¼ aC
Z2
A1=3
 
(11:5)
302 FISSION
where aC ¼ 3e2
/5R0 and
E0
s ¼ 4pR2
0SA2=3
¼ asA2=3
(11:6)
where S is the surface tension per unit area and as ¼ 4pR0
2
S. Then the equation for x
becomes
x ¼
aC
2as
 
Z2
A
 
¼
ðZ2
=AÞ
(Z2=A)critical
(11:7)
where the ratio of the constants (aC/2as)21
is referred to as (Z2
/A)critical. The fissility
of a given nucleus can be categorized relative to (Z2
/A)critical. More sophisticated
treatments of the fissionability of nuclei show that (Z2
/A)critical varies slightly
from nucleus to nucleus (due to the isospin asymmetry) and is given by
Z2
A
 
critical
¼ 50:883 1 À 1:7826
N À Z
A
 2
" #
(11:8)
The parameters Z2
/A and x provide measures of the relative fissionability of
nuclei. The greater the value of these parameters, the more “fissionable” the
nuclei are. Very fissionable nuclei, such as 239
Pu, have Z2
/A values of 36.97
while less fissionable nuclei such as 209
Bi have Z2
/A values of 32.96. Recall that
the Z2
/A factor is simply proportional to the ratio of the disruptive Coulomb
energy (/Z2
=A1=3
) to the cohesive surface (nuclear) energy (/A2=3
).
Note that the parameter (Z2
/A)critical is the ratio of two empirical constants related
to the strength of the Coulomb and surface (nuclear) forces. If we take the view that
the limit to the size of the periodic table is given by the point at which the heaviest
nuclei spontaneously fission
E0
C
2E0
s
¼ 1 (11:9)
We can rearrange these equations to find the value of the atomic number Z at which
this occurs. Thus, Zlimit is given by
Z2
limit ¼ 2(as=aC)Alimit (11:10)
If we remember that the neutron/proton ratio in heavy nuclei is about 1.5, then Zlimit
will be about 5(as/aC). Thus, the upper bound to the periodic table is given as a ratio
of two constants relating to the strength of the nuclear and Coulomb forces. The ratio
as/aC is about 20–25, and thus we expect about 100–125 chemical elements.
For all stable nuclei, x must be less than 1. In that case, the total deformation
energy of nuclei undergoing fission will increase by an amount, ( 1
5 )a2
2(2E0
s À E0
C),
11.2 PROBABILITY OF FISSION 303
as the nucleus deforms toward fission. This increase in potential energy can be
thought of as an activation energy barrier for the reaction. Eventually if the deformation
proceeds far enough, the decrease in Coulomb energy will overwhelm the
increase in surface energy and the deformation energy will decrease. (In this case,
the simple deformation energy formulas used so far in our discussion become inaccurate,
and more complicated formulas must be used.) One can appreciate the difficulty
of these calculations by a simple example. The liquid drop fission barrier for
238
U is 4.8 MeV. Equating this to ( 1
5 )a2
2(2E0
s À E0
C) and calculating values of
983 MeV for EC
0
and 695 MeV for Es
0
for 238
U, one can calculate a value of the
deforma1tion parameter a2 of 0.243, which gives us changes in the surface and
Coulomb energies (DEs
0
and DEC
0
) of 16.4 and 11.6 MeV, respectively. Thus, one
sees that the resulting fission barrier heights are calculated as the small differences
between two large numbers that are difficult to determine. Modern calculations of
the potential energy of deformation for the liquid drop model involve many deformation
coordinates (not just the a2 used above) and represent major computational tasks.
11.2.2 Shell Corrections
Figure 11.2 shows some of the basic features of fission barriers. In Figure 11.2, the
fission barriers as estimated from the liquid drop model for a range of actinide nuclei
are shown. The fission barrier height decreases, and the maximum (saddle point)
LIQUID
DROP
SHELL
FISSION
BARRIER
Th
Pu
Cf
DEFORMATION
POTENTIALENERGY
Th
Pu
Cf
Figure 11.2 Qualitative features of the fission barriers for actinide nuclei. (From H. C. Britt,
“Fission Properties of the Actinides” in Actinides in Perspective, N. Edelstein, Ed. Copyright
# 1982 Pergamon Press, Ltd. Reprinted by permission of H. C. Britt.)
304 FISSION
moves to smaller deformations as Z2
/A increases. In the lighter nuclei the saddle
point and scission point configurations are more similar, that is, have a similar deformation,
than in the heavier nuclei.
As we learned in Chapter 2, it is necessary to include shell effects in the liquid
drop model if we want to get reasonable values for nuclear masses. Similarly, we
must devise a way to include these same shell effects into the liquid drop model
description of the effect of deforming nuclei. Strutinsky (1967) proposed such a
method to calculate these “shell corrections” (and also corrections for nuclear
pairing) to the liquid drop model. In this method, the total energy of the nucleus
is taken as the sum of a liquid drop model (LDM) energy, ELDM, and the shell
(dS) and pairing (dP) corrections to this energy,
E ¼ ELDM þ
X
p,n
(dS þ dP) (11:11)
The shell corrections, just like the liquid drop (LD) energy, are functions of the
nuclear deformation. The shell corrections tend to lower the ground-state masses of
spherical nuclei with magic or near-magic numbers of neutrons and protons. They
also tend to lower the ground-state mass of midshell nuclei at some finite deformation
(b2 % 0:3), thus accounting for the deformed nature of the actinides. A
large minimum in the shell correction energies occurs when the ratios of the
major/minor nuclear axes are in the ratio of small whole numbers, as 3 : 2 or 2 : 1
(corresponding to level bunchings in the single-particle levels). The qualitative
result of combining these deformation-dependent shell corrections with the liquid
drop barriers is shown in Figure 11.2. The stable ground-state shape is predicted
to have some finite deformation (b2  0.2) rather than zero deformation (a
sphere), and a secondary minimum in the barrier appears at b2  0.6 (axes ratio
of 2 : 1). In the heaviest nuclei (Z ! 106), where the liquid drop fission barriers
are very small or nonexistent, the fission barrier heights are enhanced relative to
the liquid drop model due primarily to a lowering of the ground-state mass by
shell corrections. Without these shell effects, the heaviest nuclei could not be
observed as they would decay by spontaneous fission on a time scale much
shorter than we can observe (t1/2 , ms).
Notice that this combination of macroscopic (LD) and microscopic (shell) effects
predicts that for nuclei in the uranium–plutonium region, a double-humped fission
barrier with equal barrier heights and a deep secondary minimum will occur. For
heavier nuclei, like californium, the first barrier is predicted to be much larger
than the second barrier, and passage over this first barrier is the rate determining
step. In effect, these heavy nuclei (Z ! 100) behave as though they have a high
thin single barrier to fission. For the lighter nuclei (radium, thorium), the predicted
barrier shape is triple-humped in many cases. The reader should be aware that the
situation is even more complicated than this cursory description would indicate as
considerations of the nuclear shapes make it clear that these fission barriers are
multidimensional in character with a complicated dependence on asymmetric and
11.2 PROBABILITY OF FISSION 305
symmetric deformations. In general, there is ample experimental and theoretical evidence
that the lowest energy path in the fission process corresponds to having the
nucleus, initially in an axially symmetric and mass (reflection) symmetric shape,
pass over the first maximum in the fission barrier with an axially asymmetric but
mass symmetric shape and then to pass over the second maximum in the barrier
with an axially symmetric but mass (reflection) asymmetric shape. Because of the
complicated multidimensional character of the fission process, there are no simple
formulas for the fission barrier heights. However, the reader can find (Vandenbosch
and Huizenga, 1973; Wagemans, 1991) extensive tabulations of experimental
characterizations of the fission barrier heights for various nuclei.
Nuclei can be trapped in the secondary minimum of the fission barrier. Such
trapped nuclei will experience a significant hindrance of their g-ray decay back to
the ground state (because of the large shape change involved) and an enhancement
of their decay by spontaneous fission (due to the “thinner” barrier they would have to
penetrate.) Such nuclei are called spontaneously fissioning isomers, and they were
first observed in 1962 and are discussed below. They are members of a general
class of nuclei, called superdeformed nuclei, that have shapes with axes ratios of
2 : 1. These nuclei are all trapped in a pocket in the potential energy surface due
to a shell effect at this deformation.
11.2.3 Spontaneous Fission
In 1940 Petrzhak and Flerov discovered that 238
U could decay by spontaneously
fissioning into two large fragments (with a probability that was 5 Â 1027
of that
of undergoing a decay). Over 100 examples of this decay mode have been found
since then. Spontaneous fission is a rare decay mode in the light actinides and
increases in importance with increasing atomic number until it is a stability-limiting
mode for nuclei with Z ! 98. The spontaneous fission half-lives change by a factor
of 1029
in going from the longest lived uranium nuclei to the short-lived isotopes of
fermium.
It is clear from these basic facts and our picture of fission that spontaneous fission
is a barrier penetration phenomenon similar to a or proton decay. The nucleus
“tunnels” from its ground state through the fission barrier to the scission point.
Therefore, we would expect the spontaneous fission (SF) half-life to have the form
tSF
1=2 ¼
ln 2
fP
(11:12)
where f is the frequency of assaults on the fission barrier in the first minimum
(1020
/s) and P is the barrier penetrability. As in a or proton decay, the penetrability
factor is the most important term. The calculation of the barrier penetrability
is complicated by the double or triple humped shape of the multidimensional
barrier. A simple model for the barrier (near its top) is that of an inverted harmonic
oscillator potential (a parabola) (Fig. 11.3). The Hill–Wheeler formula describes the
306 FISSION
transmission coefficient for penetration of such a barrier as:
P ¼ (1 þ exp½2p(Bf )=hÀ
vŠ)À1
(11:13)
where Bf is the fission barrier height and hÀ
v is the barrier curvature (spacing
between the levels in the corresponding “normal” harmonic oscillator potential).
Large values of hÀ
v imply tall, thin barriers with high penetrabilities; low values
of hÀ
v imply short, thick barriers with low penetrabilities. Combining equations
gives
tSF
1=2 ¼ 2:77 Â 10À21
exp½2pBf =hÀ
vŠ (11:14)
As an exercise, let us compare the spontaneous fission half-lives of two nuclei with
barrier heights of 5 and 6 MeV, respectively, and barrier curvatures of 0.5 MeV. One
quickly calculates that the spontaneous fission half-lives of these two nuclei differ by
a factor of 3 Â 105
. The barrier heights and curvatures in this example are relevant
for the actinides and illustrate the difficulty that a 1-MeV uncertainty in the fission
barrier height corresponds to a factor of 105
in the spontaneous fission half-life.
In our previous discussion, we showed that the fission barrier heights depend on
Z2
/A and thus so should the spontaneous fission half-lives. In Figure 11.4, we show
the dependence of the known spontaneous fission half-lives on x, the fissionability
parameter. There is an overall decrease in spontaneous fission half-life with increasing
x, but clearly the spontaneous fission half-life does not depend only on Z2
/A.
One also observes that the odd A nuclei have abnormally long half-lives relative
to the even–even nuclei. Also the spontaneous fission half-lives of the heaviest
nuclei (Z ! 104) are roughly similar with values of milliseconds.
Let us see if we can explain these observations. Swiatecki has shown that there is
a correlation between the deviations of the spontaneous fission half-lives from the
r
Dr
Ef
½hw
Figure 11.3 Simple parabolic fission barrier. (From R. Vandenbosch and J. R. Huizenga,
Nuclear Fission. Copyright # 1973 Academic Press. Reprinted by permission of Elsevier.)
11.2 PROBABILITY OF FISSION 307
smooth trend with Z2
/A and the deviations of the ground-state masses from those
expected from the liquid drop model. (These deviations are exactly the shell and
pairing corrections discussed above.) Following the prescription developed by
Swiatecki, we can plot log t1/2(exp) þ 5dm (where dm represents the deviation of
the ground-state mass from the liquid drop model) vs. x (Fig. 11.5). The correlation
becomes much better, indicating we have captured the essence of the phenomenon.
We should note (Fig. 11.5) that the half-lives of the odd A nuclei are still significantly
longer than those of the neighboring even–even nuclei even though we have
corrected for the effect of the ground-state masses. We can parameterize this difference
by calculating a hindrance factor similar to that used in a-decay systematics. In
the present case, the hindrance factor is defined as the log of the ratio of the observed
half-life for the odd A nucleus to that of the neighboring even–even nuclei. For the
odd A nuclei, typical hindrance factors of 5 are observed, that is, the odd A half-lives
are 105
times longer than those of their even–even neighbors (Hoffman et al.,
1996).
11.2.4 Spontaneously Fissioning Isomers
Since the discovery of the first spontaneously fissioning isomer, a number of other
examples have been found. The positions of these nuclei in the chart of nuclides are
0.78
-12
-8
-4
Log10SpontaneousFissiont½(yr)
4
8
12
16
0
0.80 0.82 0.84
Fm
Cf
Cm
Pu
U
x
Figure 11.4 Spontaneous fission half-lives of even–even (*) and even–odd (W) nuclides
as a function of fissionability parameter, x. (From R. Vandenbosch and J. R. Huizenga,
Nuclear Fission. Copyright # 1973 Academic Press. Reprinted by permission of Elsevier.)
308 FISSION
shown in Figure 11.6. These isomers range from thorium to berkelium, forming an
island with a point of maximum stability around 242
Am. g-Ray decay back to the
ground state limits the number of isomers with lower Z and N than those in this
island, whereas spontaneous fission decay limits the number of cases with high Z
and N. The half-lives range from 1029
to 1023
s, whereas the ground-state halflives
are 1025
–1030
times longer. The typical excitation energy of these isomers
Pu
U
.77
-20
-16
-12
-8
-4
0
logt½(yr)+58m
4
8
12
16
.78 .79 .80 .81 .82 .83 .84 .85
x
Fm
Cf
Cm
( )
( )
( )
Figure 11.5 Spontaneous fission half-lives, corrected according to the method of Swiatecki,
vs. fissionability parameter x. (From R. Vandenbosch and J. R. Huizenga, Nuclear Fission.
Copyright # 1973 Academic Press. Reprinted by permission of Elsevier.)
Fission Isomers
144
Fm 100
Es 99
Cf 98
Bk 97
Cm 96
Am 95
Pu 94
Np 93
U 92
Pa 91
Th 90
140
148
152
Figure 11.6 Position of the known spontaneously fissioning isomers in the nuclide chart.
(Figure also appears in color figure section.)
11.2 PROBABILITY OF FISSION 309
is 2–3 MeV. Spectroscopic studies of the transitions between the states in the
second minimum has shown that the moments of inertia associated with the
rotational bands are those expected for an object with an axes ratio of 2 : 1—a
result confirmed in quadrupole moment studies.
11.2.5 Transition Nucleus
In analogy to chemical reactions, we might expect the probability of fission as
expressed in the fission width, Gf (¼hÀ
=t) to be given as
Gf ¼ A exp (ÀBf =T) (11:15)
where Bf is the fission barrier height. It turns out that this is an oversimplification, but
it has certain pedagogical uses. For example, in an early study describing fission,
Bohr and Wheeler were able to use this idea to show that a rare odd A isotope of
uranium, 235
U, was responsible for the fission of uranium by thermal neutrons,
not the more abundant even–even isotope 238
U.
Example Problem Thermal neutrons whose kinetic energy is 0.025 eV cause
235
U to fission but not 238
U. Why?
Solution Let us calculate the energy released when a neutron is captured by
235
U and 238
U. (This will be equivalent to the binding energy of the
last neutron in 236
U and 239
U.) For 235
U þ n, we have Erelease ¼ (M235 þ Mn 2
M236)c2
¼ 40.913 þ 8.071 2 42.440 ¼ 6.5 MeV. For 238
U þ n, we have
Erelease ¼ (M238 þ Mn 2 M239)c2
¼ 47.305 þ 8.071 2 50.570 ¼ 4.8 MeV. The
fission barrier in 235,238
U is 5.7 MeV. Thus, in 235
U þ n, we exceed it with
“zero-energy” neutrons while with 238
U þ n, we will need 1-MeV neutrons
to cause fission. In fact, this example suggests 235
U would fission even if we bombarded
it with “negative kinetic energy” neutrons. Where would we find such
neutrons? Consider the reaction 235
U(d, p). This reaction, it turns out, is equivalent
to adding a neutron to the nucleus and can even correspond, in some cases, to
the addition of “negative kinetic energy” neutrons to the nucleus, allowing
studies of near-barrier phenomena in the odd A actinides.
The ability to cause odd A actinide nuclei to undergo fission when bombarded
with thermal neutrons is of great practical importance. Because of the large cross
sections associated with thermal neutrons due to their long wavelength, the fission
cross sections for these odd A nuclei are very large. For the “big three” nuclei,
233
U, 235
U, and 239
Pu, these cross sections have the values of 530, 586, and 752
barns, respectively. These actinides are the fuel for nuclear reactors and nuclear
weapons utilizing fission by thermal neutrons.
We should note that once again, the probability of fission is more complicated
than the simple relation given above would indicate. In a paper written shortly
310 FISSION
after the discovery of fission, Bohr and Wheeler showed that fission has to compete
with other modes of nuclear deexcitation. They showed that Gf should be written as
Gf ¼
Nf
Nf þ Nn þ Ng þ Nch:p
(11:16)
where Ni is a measure of the number of ways (open channels) to accomplish each
possible deexcitation process (¼2pG/D). When evaluating Nf , one must evaluate
rf , the density of levels in the transition state nucleus. Nn is the principal term in
this equation for heavy nuclei (why?) and is taken as the number of final states of
the nucleus (after emitting a neutron) times the neutron kinetic energy. Bohr and
Wheeler’s predictions of the probability of fission in 238
U as a function of excitation
energy are shown in Figure 11.7.
In nuclear reactors one has neutrons with energies ranging from thermal
(0.025 eV) to several MeV. There are a series of sharp peaks in the total cross
section for neutrons with energies between 0.2 and 3000 eV that are called “resonances.”
These resonances correspond to exciting a specific isolated level in the compound
nucleus that can decay by fission. The situation is particularly interesting for
the neutron irradiation of even–even nuclei, such as 240
Pu at subthreshold energies
0 2 4
excitation energy (MeV)
BOHR and WHEELER 1939
Bn
10-6
10-5
10-4
10-3
10-2
10-1
G(eV)
Gn
Gf
Gγ
102
103
1
10
6 8
0
0.2
0.4
U238 + n
0.6
s
t
(b)
1 2
En (MeV)
3
10
Figure 11.7 Schematic diagram of neutron, fission, and g-ray widths of a typical nucleus
with a neutron binding energy slightly less than 6 MeV. The inset shows the predicted
fission excitation function for a nucleus with Bf 2 Bn ¼ 0.75 MeV together with more
recent data.
11.2 PROBABILITY OF FISSION 311
(Fig. 11.8). The resonances associated with fission appear to cluster in bunches. Not
all resonances in the compound nucleus lead to fission. We can understand this situation
with the help of Figure 11.9. The normal resonances correspond to excitation
of levels in the compound nucleus, which are levels in the first minimum in
Figure 11.9. When one of these metastable levels exactly corresponds to a level
in the second minimum, then there will be an enhanced tunneling through the
fission barrier and an enhanced fission cross section.
When higher energy (E . 1 MeV) neutrons interact with nuclei such as 238
U
where the fission barrier height is greater than the neutron separation energy, a stairstep
pattern in the excitation function occurs (Fig. 11.10). The first rise and plateau is
due to the occurrence of the (n, f ) reaction, the second rise and plateau is due to the
(n, nf ) reaction (“second-chance fission”), and the third rise and plateau is due to the
(n, 2nf ) reaction (“third-chance fission”), and so forth. For nuclei where Bf , Bn,
the same pattern occurs but lies on top of rapidly decreasing cross sections (with
increasing energy) at low energies due to 1/v absorption of neutrons.
How do we estimate the factors determining the fission probability when the excitation
energy of the fissioning system is 10 MeV or more? (How do we calculate the
various widths?) At these excitation energies, we have reached the point where the
statistical model of nuclear reactions can be used. The relevant terms are only Gf and
Gn. The available experimental data on Gn/Gf at excitation energies of 5–25 MeV is
shown in Figure 11.11. One notes the general trend in Gn/Gf with increasing Z and A
(consistent with the qualitative dependence on Z2
/A for fission). For this limited
500
-2
0
sf(10−24cm2)
2
4
6
8
10
12
14
1000 1500 2000 2500 3000
NEUTRON ENERGY (eV)
240
Pu SUBTHRESHOLD FISSION CROSS-SECTION
500
0
200
400
600
800
1000
stot(10−24cm2)
1000 1500 2000 2500 3000
NEUTRON ENERGY (eV)
240
Pu TOTAL CROSS-SECTION
Figure 11.8 Neutron total and subbarrier fission cross sections of 240
Pu as a function of
neutron energy between 0.5 and 3 keV. (From H. Weigmann, “Neutron-Induced Fission
Cross Sections” in C. Wagemans, The Nuclear Fission Process. Copyright # 1991 CRC
Press. Reprinted by permission of CRC Press.)
312 FISSION
range of energies, the ratio Gn/Gf can be parameterized as:
Gn
Gf
¼
2TA2=3
10
exp
Bf À Bn
T
 
(11:17)
where Bf , Bn, the nuclear temperature T [ ¼ (8EÃ
/A)1/2
] refer to the fissioning
system. A more rigorous expression that can be used over a wider range of
DeformationeI eII
II
EB
EA
V(PotentialEnergy)
Spontaneous fission
Isomer fission
Subbarrier fission
Normal fission
Mass distribution
Figure 11.9 Schematic representation of the double humped fission barrier. Intrinsic
excitations in the first and second minimum are shown along with the path of fission from
isomeric states and ground-state spontaneous fission.
0
1.0
sf(barns)
2.0
238
U
5 10 15 20 25
Energy of neutrons (MeV)
Figure 11.10 Fission excitation function for n þ 238
U for Eneutron between 1 and 22 MeV.
11.2 PROBABILITY OF FISSION 313
excitation energies is
Gn
Gf
¼
gmr2
0
hÀ 2
4A2=3
af (EÃ
À Bn)
an½2a1=2
f (EÃ À Bf )1=2
À 1Š
exp½2a1=2
n (EÃ
À Bf )1=2
À 2a1=2
f (EÃ
À Bf )1=2
Š
(11:18)
where an is the level density parameter of the residual nucleus after emission of a
neutron and af is that of the deformed transition state nucleus. Note that Gn/Gf is
related to the difference (Bf 2 Bn) as shown in Figure 11.12.
Example Problem Consider the bombardment of 238
U with 42-MeV a particles.
What fraction of the initial nuclei undergoes first-chance fission?
Solution The excitation energy of the compound nucleus EÃ
will be
EÃ
¼ 42(238=242) þ QCN
QCN ¼ (M238 þ Ma À M242)c2
QCN ¼ 47:305 þ 2:425 À 54:712 ¼ À5:0
EÃ
¼ 36:3 MeV
228
0.1
1.0
Γn
10
230 232 234 236 238 240 242
MASS NUMBER
244 246 248 250 252 254
Γf
Z = 90
Z = 91
Z = 92
Z = 93 Z = 94
Z = 95
Z = 96
Z = 97
Z = 98
Z = 99
Z = 100
x
Figure 11.11 Values of Gn/Gf as a function of A of the fissioning system. (From
R. Vandenbosch and J. R. Huizenga, Nuclear Fission. Copyright # 1973 Academic Press.
Reprinted by permission of Elsevier.)
314 FISSION
In 242
Pu, Bn ¼ 6.3 MeV, Bf ¼ 5.3 MeV. From Figure 11.12, Gn/Gf ¼ 3, that is,
25% of the initial nuclei fission at the first chance.
For reactions induced by heavy ions or high-energy charged particles, these
expressions should be corrected for the effect of angular momentum. For
example, there will be excitation energy tied up in rotation, which is unavailable
for fission (Vandenbosch and Huizenga, 1973) and the fission barriers are lower
for rotating nuclei. For reactions involving less fissionable nuclei (x , 0.7),
especially at higher energies, one frequently sees that the primary reaction
products first decay by sequential emission of neutrons or charged particles,
and, then, as Z2
/A increases, fission occurs at the last stages of the evaporation
chains.
0
0.1
1
10
100
Gn/Gf
1000
10,000
5 10 15
Ef – Bn (MeV)
20 25 30
Ef – Bn = –3
Ef – Bn = –1
Ef – Bn = 0
Ef – Bn = 0.5
Ef – Bn = 1
Ef – Bn = 3
Ef – Bn = 6
35
Figure 11.12 Excitation energy dependence of Gn/Gf for different values of (Bf 2 Bn).
(From R. Vandenbosch and J. R. Huizenga, Nuclear Fission. Copyright # 1973 Academic
Press. Reprinted by permission of Elsevier.)
11.2 PROBABILITY OF FISSION 315
11.3 FISSION PRODUCT DISTRIBUTIONS
Up to this point, we have focused on describing the factors that control the probability
of fission to occur. Now we will focus our attention on the distributions of
the products in mass, energy, charge, and so forth. In doing so, we will mostly be
discussing “scission point” or “postfission” phenomena. Our treatment of these
phenomena is, of necessity, somewhat superficial, and the reader is referred to the
excellent monograph of Vandenbosch and Huizenga (1973) for a more authoritative
account.
11.3.1 Total Kinetic Energy (TKE) Release in Fission
To a first approximation, one can assume that the kinetic energies of the fission fragments
are the result of the Coulomb repulsion of the fragments after scission. A
handy pocket formula that gives the total kinetic energy is
TKE ¼
Z1Z2e2
1:8(A1=3
1 þ A1=3
2 )
MeV (11:19)
where Z1, A1, Z2, A2, refer to the atomic and mass numbers of the two fragments. The
factor of 1.8 (instead of the usual value of r0 of 1.2) results from the fact that the
fragments at scission are unusually deformed. More detailed empirical prescriptions
for the TKE are available (Viola et al., 1985), but the above formula seems to work
quite well over a range of excitation energies and fissioning nuclei. The most significant
deviations from these formulas appear in the very heavy actinides, 258,259
Fm
and 260
Md, where the observed kinetic energies are evidence (Hoffman et al.,
1996) for an unusually compact scission configuration.
11.3.2 Fission Product Mass Distributions
One of the first big surprises in early studies of fission was the fission product mass
distribution. Investigations of the thermal neutron-induced fission of uranium and
plutonium nuclides (and later the spontaneous fission of 252
Cf) showed the most
probable division of mass was asymmetric (MH/ML ¼ 1.3–1.5). The liquid drop
model would predict that the greatest energy release and, therefore, the most probable
mass split, would be a symmetric one, that is, MH/ML ¼ 1.0. This situation is
shown in Figure 11.13 where the mass distributions for the thermal neutroninduced
fission of the “big three nuclides,” 233
U, 235
U, and 239
Pu, are shown.
Symmetric fission is suppressed by at least two orders of magnitude relative to
asymmetric fission. Note the peak-to-valley ratio of the distributions.
The key to understanding this situation can be seen in Figures 11.13 and 11.14.
In these figures, we show that as the mass of the fissioning system increases, the
position of the heavy peak in the fission mass distribution remains constant while
the position of the light peak increases with increasing fissioning system mass.
316 FISSION
This observation, along with the observation that the lower edge of the heavy fragment
peak is anchored at A ¼ 132 has suggested that the preference for asymmetric
fission is due to the special stability of having one fragment with Z ¼ 50, N ¼ 82, a
doubly magic spherical nucleus.
Further evidence for this influence of “magic” (shell model) configurations on the
fission mass distributions is found in the fragment mass distributions for spontaneous
fission (Fig. 11.15) and low-energy-induced fission of the “preactinides”
(Fig. 11.16). One observes, in the case of spontaneous fission, a sharp transition
between asymmetric fission and symmetric fission as one goes from 257
Fm to
258
Fm. The addition of a single neutron to the nucleus causes a large change in
the fission product mass distribution. Similarly, a shift of two protons in going
from 225
Ac to 227
Pa causes the mass distribution to shift from purely symmetric
to dominantly asymmetric. These changes occur at neutron and proton numbers
60
10-4
10-3
10-2
Fissionyield(percent)
10-1
100
101
80 100 120
233
U
235
U
239Pu
Mass number, A
140 160 180
Figure 11.13 Smoothed fragment mass distributions for the thermal neutron-induced
fission of 233
U, 235
U, and 239
Pu. [From Seaborg and Loveland (1990).]
11.3 FISSION PRODUCT DISTRIBUTIONS 317
that are not the so-called magic numbers for spherical nuclei. The key is that the fissioning
system and the fragments themselves are initially quite deformed. Thus, the
relevant magic numbers, that is, configurations of special stability are those expected
for deformed nuclei and as shown in Chapter 6, the actual configurations change
with deformation. A detailed theory of fission scission point properties based on
these ideas developed by Wilkins, Steinberg, and Chasman (1976) has been quite
successful in describing the observed trends.
Qualitatively, if these explanations of the fission mass distributions for lowenergy-induced
fission are correct, one might expect as the excitation energy of
the fissioning system is raised, the influence of the ground-state shell structure of
the nascent fragments decreases, and the fission mass distributions shows a
greater amount of symmetric fission. That is exactly what happens and at high
energy all nuclei fission symmetrically (Fig. 11.17).
11.3.3 Fission Product Charge Distributions
If one were to plot the yield of fission fragments as a function of their atomic
numbers (as in Fig. 11.16), the result would look very much like the fission mass
225
86
88
90
92
94
96
98
100
MEANMASSOFTHEFISSIONPRODUCTGROUP
102
104
106
108
110
112
136
138
140
142
144 HEAVY GROUP
LIGHT GROUP
230 235
MASS OF THE FISSIONING NUCLEUS
240 245
SPONTANEOUS FISSION
THERMAL-NEUTRON-INDUCED FISSION
REACTOR-NEUTRON-INDUCED FISSION
250 255 260
Figure 11.14 Average masses of the light and heavy fragments as a function of the mass of
the fissioning system. [From K. Flynn, et al., Phys. Rev. C5, 1728 (1972). Copyright (1972) by
the American Physical Society.]
318 FISSION
Figure 11.15 Schematic representation of the mass yield distributions for the spontaneous
fission of the trans-berkelium nuclides. (From D. C. Hoffman, et al., “Spontaneous Fission” in
Nuclear Decay Modes, D. N. Poenaru, Ed. Copyright # 1996 IOP Press. Reprinted by
permission of IOP Press.)
132 133 134 135 136 137
Neutron number
138 139 140 141 142
Ac
Th
Pa
U
89
90
Protonnumber
91
92
Figure 11.16 Atomic number (Z) distributions for the low-energy fission of several actinide
nuclei are shown in each panel. (From K.-H. Schmidt et al., in Heavy Ion Physics, Y. T.
Oganessiam and R. Kalpakchieva, Eds. Copyright # 1998 World Scientific. Reprinted by
permission of World Scientific.)
11.3 FISSION PRODUCT DISTRIBUTIONS 319
distribution. Nuclear matter is not very polarizable and, to first order, the protons
will divide like the neutrons. The primary fission fragments thus have neutron/
proton ratios like that of the fissioning system and thus lie on the neutron-rich
side of b stability. Enhanced yields for even Z nuclides relative to odd Z nuclides
are observed (Fig. 11.18) due to the stabilization from proton pairing.
The yield of any given nuclide in fission is called its independent yield. It can be
shown that the independent yield of isobars in fission has a Gaussian form:
P(Z) ¼
1
ffiffiffiffiffiffi
cp
p exp À
(Z À Zp)2
c
 
(11:20)
where c has an average value of 0.80 + 0.14 for low-energy fission and Zp is the
most probable primary fragment atomic number (noninteger) for that isobar.
[Large tables of Zp exist for common fissioning systems (Wahl, 1988).] One
Figure 11.17 Fission mass distributions for 232
Th(p, f).
320 FISSION
consequence of this small value of c is that, for a given A, only a few isobars will
have significant yields. Two effects tend to favor narrow charge distributions: (a)
the high energetic cost of unfavorable charge splits and (b) the existence of
ground-state correlations between neutrons and protons in the fragments.
In discussions of fission, one frequently hears the terms cumulative yield and
independent yield. The independent yield of a nuclide is just what it appears, the
yield of that nucleus as a primary fission product. Because the fission products
are all b2
emitters, they decay toward the bottom of the valley of b stability,
populating several different members of an isobaric series, as, for example, with
A ¼ 140 fragments:
140
Xe À!
bÀ
140
Cs À!
bÀ
140
Ba À!
bÀ
140
La À!
bÀ
140
Ce
The yield of each member of the isobaric series integrates, by virtue of the intervening
b decay, the yields of its precursors. Such yields are referred to as cumulative
yields. For example, the cumulative yield of the mass 140 chain in the thermal
neutron-induced fission of 235
U is 6.25%.
Figure 11.18 Yields of products from the thermal neutron-induced fission of 235
U. (From
A. C. Wahl, “Nuclear Charge Distribution in Fission,” in New Directions in Physics,
N. Metropolis, D. M. Kerr, and G. C. Rota, Eds. Copyright # 1987 by Academic Press,
Inc. Reprinted by permission of Elsevier.)
11.3 FISSION PRODUCT DISTRIBUTIONS 321
Example Problem In the above example, what is the independent yield of
140
Ba for the thermal neutron-induced fission of 235
U and what is its cumulative
yield?
Solution The fractional independent yield is given as:
P(Z) ¼
1
ffiffiffiffiffiffi
cp
p exp À
(Z À Zp)2
c
 
For the mass 140 chain, Zp ¼ 54:55 (Wahl, 1988). Note that this tabulated value
of Zp/A (¼54.55/140) is very close to that of the fissioning system, 92/236,
that is, the N/Z ratio of the fragments is approximately that of the fissioning
system. This idea is called the UCD (unchanged charge distribution) prescription.
Therefore
P(56) ¼
1
ffiffiffiffiffiffiffiffiffiffi
0:8p
p exp À
(56 À 54:55)2
0:8
 
¼ 4:56 Â 10À2
Therefore
Yield of 140
Ba ¼ 6:25 Â 0:0456 ¼ 0:28%
The fractional cumulative yield (FCY) of 140
Ba is defined as:
FCY ¼
1
s
ffiffiffiffiffiffi
2p
p
ðZþ1=2
À1
exp À
(n À Zp)2
2s2
 
dn
where the familiar Gaussian parameter s is related to c by Sheppard’s relation
c ¼ 2(s2
þ 1
12 ). Evaluating this integral gives FCY ¼ 0.9978.
11.4 EXCITATION ENERGY OF THE FISSION FRAGMENTS
The excitation energy of the fission fragments is equal to the difference between the
total energy release, Q, and the total kinetic energy of the fragments. The excitation
energy should be calculated for each mass split. Here we will do an average accounting
to see where the energy goes. For the thermal neutron-induced fission of 235
U,
this corresponds to 200 MeV 2 172 MeV or about 28 MeV (14% of the
total energy release), averaged over all mass splits. The average number of
emitted prompt neutrons is 2.4, and each neutron has a kinetic energy of
2 MeV while the emitting fragments have average neutron binding energies
of 5.5 MeV. Thus about 18 MeV [¼2.4 Â (2 þ 5.5)] of the fragment excitation
energy is carried away by the prompt neutrons. Prompt photon emission carries
away 7.5 MeV, which leaves about 2.5 MeV, in this crude accounting, to be
emitted in the form of b particles, neutrinos, delayed neutrons, and so forth.
As noted earlier, the prompt neutrons are emitted from the fully accelerated fragments
after scission. The number of these neutrons, nT , (¼ 2.4 in the case of 235
U) as
322 FISSION
a function of the mass of the fissioning system is shown in Figure 11.19. The general
increase in nT with mass of the fissioning system is due to the increase in fragment
excitation energy. For very heavy systems (Z  114), nT is predicted to be 7,
allowing the critical mass for a self-sustaining fission reaction to be quite small.
Example Problem How did we get the estimate for nT for the fission of 298
114?
Assume this nucleus fissions symmetrically.
Solution The total energy released can be estimated using a modern mass
formula (Liran and Zeldes, 1976) as 311 MeV. The total kinetic energy of the
fragments can be calculated as
TKE ¼
(57)(57)(1:440)
(1:8)(1491=3)(2)
¼ 245 MeV
This leaves a total fragment excitation energy of 311 2 245 ¼ 66 MeV. Since the
g rays and b2
, are emitted only to take away the energy not emitted as neutrons,
we shall assume this energy is the same as in 235
U(nth, f) or 10 MeV. This
230
1
2
Th
U
Pu
Am
Cm
Bk
Cf
Fm
No
3
4
235 240
– Spontaneous Fission
– Thermal Neutron Fission, Corrected
245
A
250 255 260
nT
+
+
++
+ + +
+
+
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+
Figure 11.19 Average total number of neutrons, nT, as a function of the mass number of the
fissioning system for spontaneous and thermal neutron-induced fission. The values for thermal
neutron-induced fission have been corrected to zero excitation energy (spontaneous fission)
assuming dn/dt ¼ 0.12 MeV21
.
11.4 EXCITATION ENERGY OF THE FISSION FRAGMENTS 323
leaves a “neutron excitation energy” of 66 2 10 ¼ 56 MeV. The neutron binding
energy in a typical fragment is 6 MeV (Liran and Zeldes, 1976). Thus, we
would calculate nT ¼ 56/(6 þ 2) ¼ 7.
The average neutron kinetic energy is 2 MeV. In the frame of the moving fragment,
the distribution of fragment energies is Maxwellian, P(E) ¼ En exp(2En/T).
Transforming this spectrum into the laboratory frame gives a spectrum of the Watt
form, that is,
P(En) ¼ eÀEn=T
sinh(4EnEf =T2
)1=2
(11:21)
where En and Ef are the laboratory system energies of the neutron and fission
fragment (in MeV/nucleon) and T is the nuclear temperature.
Another important aspect of neutron emission is the variation of the number of
emitted neutrons as a function of the fragment mass, n(A) (Fig. 11.20). The striking
features of these data are the nearly universal dependence of n(A) upon A, independent
of fissioning system for these actinide nuclei (which again suggests the role of
fragment shell structure upon this property) and the sawtooth dependence of n(A).
Note the correlation of low values of n(A) for those fragments whose structure is
that of a magic nucleus, that is, a nucleus of special stability. These fragments are
expected to have low excitation energies due to shell effects (Wilkins et al., 1976).
Prompt g-ray emission competes with or follows the last stages of prompt
neutron emission. These photons are emitted in times from 10215
–1027
s.
Typical g-ray multiplicities of 7–10 photons/fission are observed. These photons,
as indicated earlier, carry away 7.5 MeV. This g-ray yield is considerably larger
than one would predict if g-ray emission followed neutron emission instead of competing
with it. Because of the significant angular momentum of the fission fragments
(7–10 hÀ
) even in spontaneous fission, photon emission can compete with neutron
emission. The emitted g rays are mostly dipole radiation with some significant
admixture of quadrupole radiation, due to “stretched” E2 transitions (Jf ¼ Ji 2 2).
100 110 120
INITIAL FRAGMENT MASS
130 140 150 160 17070
0
1
2
NEUTRONSperFRAGMENT
3
4
5
6
80
Z=28 N=50 U233
U233
U235
U235Pu239
Pu239
Cf252
Cf252
Z=50
N=82
90
nL nH
Figure 11.20 Dependence of n(A) upon fragment mass number A for some actinide nuclei.
324 FISSION
Because of the large number of possible neutron-rich fragments produced in fission,
the study of the g rays emitted by the fragments can lead to useful information about
the nuclear structure of these exotic, short-lived nuclei far from stability.
As indicated in our discussion of prompt g-ray emission, the fission fragments
have a significant angular momentum. There are two origins for this angular
momentum: (a) the existence of off-axis torques given the fragments during
the scission process and (b) the excitation of bending and wriggling modes in
the nascent fragments at the saddle point, which persist to scission and are
amplified by (a).
11.5 DYNAMICAL PROPERTIES OF THE FISSION FRAGMENTS
One of the properties of fission fragments that is often exploited is the angular distribution.
Fission is generally considered to be a “slow” process in which the fissioning
nucleus is in statistical equilibrium. The angular distribution of the fission
fragments will, therefore, be symmetric with respect to a plane perpendicular to
the direction of motion of the fissioning system, that is, the fragment angular
distributions will be symmetric about 908 in the frame of the fissioning system.
A typical fission fragment angular distribution for a heavy-ion-induced fission
reaction is shown in Figure 11.21. As one can see, the fragments are emitted
DRAWN
SYMMETRIC
WITH = 90°θ
0 20 40 60 80 90100 120 140 160 180
5.0
4.0
3.0
2.0
1.0
0
1
SIN θ
θ CM
Rel. dσ
dω
Figure 11.21 Fission fragment angular distribution for the carbon-ion-induced fission of
gold. [From G. E. Gordon et al., Phys. Rev. 120, 1341 (1960). Copyright (1960) by the
American Physical Society.]
11.5 DYNAMICAL PROPERTIES OF THE FISSION FRAGMENTS 325
preferentially forwards and backwards with respect to the direction of motion of the
fissioning system. In this case involving a reaction in which the fissioning system has
a significant angular momentum (36hÀ
), the distribution closely resembles the
function 1/sin(u).
To understand these distributions, one needs to consider the fission transition
nucleus. Figure 11.22 shows a coordinate system for describing this nucleus in
terms of its quantum numbers J, the total angular momentum M, the projection of
J upon a space-fixed axis, usually taken to be the direction of motion of the fissioning
system, and K, the projection of J upon the nuclear symmetry axis.
In most collisions, the angular momentum vectors, J, will be concentrated in a
plane perpendicular to the beam direction (M ¼ 0). In this case, we can easily see
a relation between J, K, and the fission fragment angular distribution. Consider
the case where J ¼ K, the nuclear symmetry axis is perpendicular to the beam,
and the fragments emerge sidewise to the beam. Similarly, if K ¼ 0, the symmetry
axis of the nucleus is oriented perpendicular to J, that is, along the beam direction,
and the fragments are emitted preferentially forward and backward. It is this extreme
that leads to the 1/sin(u) form. [If J is perpendicular to the beam, and the vectors
describing the possible directions of the nuclear symmetry axis are uniformly distributed
over the surface of a sphere, then the probability of having a symmetry
axis at angle u with respect to the beam will go as 1/sin(u).]
For the general case, Wheeler (1963) showed that the probability of emitting a
fragment at an angle u from a transition nucleus characterized by J, K, and M is
PJ
M,K(u) ¼ (2 J þ 1)
2pR2
sin u du
4pR2
 
dJ
M,k(u)



2
(11:22)
The first term represents a statistical weighting factor, the second term a solid angle
factor for the probability of getting the angle u, and the third term is a symmetric top
wave function.
K
J
R
Z
M
q
Figure 11.22 Angular momentum coordinate system for a deformed nucleus.
326 FISSION
In low-energy fission or photofission, one can populate individual states of the
fission transition nucleus and one can observe fragment angular distributions that
change from forward to sidewise peaked and back again as a function of the J, K,
M of the transition nucleus (Vandenbosch and Huizenga, 1973). At higher energies
(EÃ
. 10 MeV), one describes the states of the transition nucleus using a statistical
model. One assumes there is a Gaussian distribution of K values for the transition
nucleus
r(K) / exp (ÀK2
=K2
0 ) K , J
¼ 0 K . J
(11:23)
Then K0
2
, the root-mean-square projection of J upon the nuclear symmetry axis, can
be calculated as:
K2
0 ¼
IeffT
hÀ 2
(11:24)
where Ieff is the effective moment of inertia of the transition nucleus with temperature
T. The fission fragment angular distributions W(u) can be written as:
W(u) /
X1
J¼0
(2J þ 1)TJ
XJ
K¼ÀJ
(2J þ 1) dJ
M¼0,K(u)



2
exp (ÀK=2K2
0 )
PJ
K¼ÀJ exp (ÀK2=2K2
0 )
(11:25)
where TJ is the transmission coefficient for forming the fissioning nucleus with
total angular momentum J. Under the assumption that M ¼ 0, we get the handy
“pocket” formula
W(u) /
X1
J¼0
(2J þ 1)2
TJ exp½À(J þ 0:5)2
sin2
u=4K2
0 Š
ÂJ0½i(J þ 0:5)2
sin2
u=4K2
0 Š
erf½(J þ 0:5)=(2K2
0 )1=2
Š
(11:26)
where J0 is the zero-order Bessel function with imaginary argument and
erf[(J þ 0.5)/(2K0
2
)1/2
] is the error function defined by
erf(x) ¼
2
p1=2
ðx
0
exp (Àt2
) dt (11:27)
If one can estimate K0
2
, then the fission angular distributions can be used to measure J
or vice versa.
One other aspect of the spatial distribution of the fission fragments that has
proven to be a useful tool in studying nuclear reactions is the angular correlation
between the two fission fragments. When a fission event occurs, the two fragments
emerge with an angle of 1808 between them (to conserve angular momentum). If the
11.5 DYNAMICAL PROPERTIES OF THE FISSION FRAGMENTS 327
fissioning nucleus is in motion, then the initial momentum of the fissioning system
must be shared between the two fragments to give the final (laboratory system)
fragment momenta. Complete fusion events can thus be differentiated from incomplete
fusion events by observing the mean angle between coincident fission
fragments.
Example Problem Consider the case of 240-MeV 32
S interacting with 181
Ta,
which fissions. What would be the laboratory correlation angle between the
fragments if the full linear momentum of the projectile was transferred to the
fissioning system?
Solution
32
S þ 181
Ta À! 213
Ac
The momentum of the beam nucleus (CN) is given by
P ¼ (2mT)1=2
¼ (2 Â 32 Â 240)1=2
¼ 123:9 (MeV-amu)1=2
For the symmetric fission of 213
Ac, we expect
TKE ¼
(89=2)2
e2
1:8(213=2)1=3
(2)
¼ 167 MeV
The momentum of each fragment is then
P ¼ (2mT)1=2
¼ 2
213
2
 
167
2
  1=2
¼ 133:4 (MeV-amu)1=2
The half correlation angle is
u ¼ tanÀ1
(133:4=123:9=2)) ¼ 658
The correlation angle would be 2u ¼ 1308.
A dynamical scission point phenomenon is the violent snapping of the neck
between the nascent fragments, which can result in the emission of particles into
the region between the fragments. The phenomenon is rare, occurring in about 1
in 300 to 1 in 1000 of the fission events for a particles and in lesser frequency
for other charged particles. (Neutrons can be emitted by this same mechanism in
a few percent of all fission events.) The charged particles, being born in the
region between the fragments, are strongly focused by the Coulomb field of the fragments
and emerge at 908 with respect to the direction of motion of the separating
fragments, with energies (15 MeV for a particles) characteristic of the Coulomb
fields of the separating fragments.
328 FISSION
PROBLEMS
1. Why is 240
Pu not fissionable by thermal neutrons, but 239
Pu is?
2. What is the expected total kinetic energy release in the fission of 272
110 assuming
fission occurs symmetrically?
3. What is the meaning of the terms prompt and delayed with respect to the fission
neutrons?
4. Sketch the fission excitation function for the reaction of 232
Th with neutrons. The
fission barrier is 6.5 MeV, and the binding energies of the last neutron in 232
Th
and 233
Th are 6.90 and 4.93 MeV, respectively.
5. What are the values of the fissionability parameter x for 209
Bi, 226
Ra, 232
Th,
242
Pu, and 252
Cf?
6. What is the fraction of fission neutrons with energies greater than 2 MeV (in the
laboratory frame)?
7. What is the independent yield of 99
Mo in the thermal neutron-induced fission of
239
Pu? Zp is 39.921 and the yield of the A ¼ 99 chain is 6.15%.
8. What is the value of Gn/Gf for a 210
Po nucleus produced in the bombardment of
209
Bi with 10.5-MeV protons. Bf ¼ 20.4 MeV.
REFERENCES
Britt, H. C. In N. M. Edelstein, Ed., Actinides in Perspective, Pergamon, Oxford, 1982, p. 245.
Flynn, K. F., E. P. Horwitz, C. A. A. Bloomquist, R. F. Barnes, R. K. Sjoblom, P. R. Fields,
and L. E. Glendenin. Phys. Rev. C5, 1725 (1972).
Gindler, J. and J. R. Huizenga. Nuclear Fission, In Vol. II, L. Yaffe, Ed., Nuclear Chemistry,
Academic Press, New York, 1964, p. 1.
Hahn, O. and F. Strassman. Naturwiss. 26, 756 (1938); see translation by H. Graetzer,
Am. J. Phys. 32, 10 (1964).
Hoffman, D. C., T. M. Hamilton, and M. R. Lane. Spontaneous Fission, In D. N. Poenaru, Ed.,
Nuclear Decay Modes, IOP, Bristol, 1996, pp. 393–432.
Liran, S. and N. Zeldes. At. Data Nucl. Data Tables 17, 431 (1976).
Oganessian, Y. T. and Y. A. Lazarev. Heavy Ions and Nuclear Fission, In D. A.
Bromley, Ed., Treatise on Heavy-Ion Nuclear Science, Vol. 4, Plenum, New York,
1985, pp. 1–254.
Schmidt, K.-H., et al. In Y. T. Oganessian and R. Kalpakchieva, Eds., Heavy Ion Physics,
World, Singapore, 1998, p. 667.
Seaborg, G. T. and W. Loveland. The Elements Beyond Uranium, Wiley, New York, 1990.
Strutinsky, V. M. Nucl. Phys. A 95, 420 (1967).
Vandenbosch, R. and J. R. Huizenga. Nuclear Fission, Academic, New York, 1973.
Viola, V. E., Jr., K. Kwiatkowski, and M. Walker. Phys. Rev. C31, 1550 (1985).
REFERENCES 329
Wagemans, C. The Nuclear Fission Process, CRC Press, Boca Raton, FL, 1991.
Wahl, A. C. At. Data Nuclear Data Tables 39, 1 (1988).
Wheeler, J. A. In J. B. Marion and J. L. Fowler, Eds., Fast Neutron Physics, Wiley, New York,
1963.
Wilkins, B. D., E. P. Steinberg, and R. R. Chasman. Phys. Rev. C14, 1832 (1976).
330 FISSION
CHAPTER 12
NUCLEAR REACTIONS IN NATURE:
NUCLEAR ASTROPHYSICS
12.1 INTRODUCTION
An important mystery that is still unfolding today is how did the chemical elements
that we have here on Earth come into existence? We know that the readily available
chemical elements are restricted in number to less than 90 and that they are essentially
immutable by chemical reactions. The large-scale nuclear reactions that are
taking place are those induced by (external) cosmic rays and radioactive decay;
induced nuclear reactions such as fission take place on a tiny scale. Thus, the vast
bulk of chemical elements that we have today are those that were present when
the Earth was formed. The elements have undergone an enormous range of geochemical,
geological, and biochemical processes, but all such processes retain the
integrity of the atom. Thus, the origin of the elements is certainly extraterrestrial,
but questions remain as to where and how they were formed?
These questions lie in the field of nuclear astrophysics, an area concerned with the
connection of fundamental information on the properties of nuclei and their reactions
to the perceived properties of astrological objects and processes that occur
in space. The universe is composed of a large variety of massive objects distributed
in an enormous volume. Most of the volume is very empty (,1 Â 10218
kg/m3
) and
very cold (3 K). On the other hand, the massive objects, stars, and such are very
dense (sun’s core 2 Â 105
kg/m3
) and very hot (sun’s core 16 Â 106
K). These
temperatures and densities are such that the light elements are ionized and have
331
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
high enough thermal velocities to occasionally induce a nuclear reaction. Thus, the
general understanding of the synthesis of the heavier elements is that they were
created by a variety of nuclear processes in massive stellar systems. These
massive objects exert large gravitational forces, and so one might expect the new
materials to remain in the stars. The stellar processing systems must explode at
some point in order to disperse the heavy elements. When we look at the details
of the distribution of isotopes here on Earth, we will find that some number of explosive
cycles must have taken place before the Earth was formed.
In this chapter we will first consider the underlying information on the elemental
abundances and some of the implications of the isotopic abundances. Then, we will
consider the nuclear processes that took place to produce the primordial elements
and those that processed the primordial light elements into those that we have
here on Earth.
12.2 ELEMENTAL AND ISOTOPIC ABUNDANCES
Many students of chemistry have given little thought to the relative abundances of
the chemical elements. Everyone realizes that some elements and their compounds
are more common than others. The oxygen in water, for example, must be plentiful
compared to mercury or gold. But what if we compare elements that are closer in the
periodic table, for example, what is the amount of lead (Z ¼ 82) compared to
mercury (Z ¼ 80) or what is the amount iron (Z ¼ 26) compared to copper
(Z ¼ 27)? Oddly enough, the answer one gets depends on what is sampled. The relative
abundances of the first 40 elements are shown in Figure 12.1 as a percentage by
mass of Earth’s crust and as a percentage by mass of our solar system. Notice that the
scale is logarithmic and the data spans almost 11 orders of magnitude. Earth is predominantly
oxygen, silicon, aluminum, iron, and calcium, which comprise more
than 90% of Earth’s crust. On the other hand, the mass of the solar system is dominated
by the mass of the sun so that the solar system is mostly hydrogen, with some
helium, and everything else is present at the trace level. The differences between the
solar system abundances and those on Earth are due to geophysical and geochemical
processing of the solar material. Therefore, in this text we will concentrate on
understanding the solar system abundances that reflect nuclear processes. The abundances
of the isotopes and the elements are the basic factual information that we
have to test theories of nucleosynthesis. We have data from Earth, the moon,
and meteorites, from spectroscopic measurements of the sun, and recently from
spectroscopic measurements of distant stars. Many studies have characterized and
then attempted to explain the similarities and differences from what we observe
in the solar system.
The solar abundances of all of the chemical elements are shown in Figure 12.2.
These abundances are derived primarily from knowledge of the elemental abundances
in CI carbonaceous chrondritic meteorites and stellar spectra. Note that
99% of the mass is in the form of hydrogen and helium. Notice that there is a
general logarithmic decline in the elemental abundance with atomic number with
332 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
Figure 12.1 Abundances of the first 40 elements as a percentage by mass of Earth’s crust
(filled squares) and in the solar system (open circles). Data from the CRC Handbook of
Chemistry and Physics, 75th ed., 1994. (Figure also appears in color figure section.)
Figure 12.2 Abundances of the elements as a percentage by mass of the solar system. Data
from the CRC Handbook of Chemistry and Physics, 75th ed., 1994.
12.2 ELEMENTAL AND ISOTOPIC ABUNDANCES 333
the exceptions of a large dip at beryllium (Z ¼ 4) and of peaks at carbon and oxygen
(Z ¼ 6–8), iron (Z  26) and the platinum (Z ¼ 78) to lead (Z ¼ 82) region. Also
notice that there is a strong odd–even staggering and that all the even Z elements
with Z . 6 are more abundant than their odd atomic number neighbors. We have
already encountered an explanation for this effect, that is, recall from earlier discussions
on nuclear stability that there are many more stable nuclei for elements with an
even number of protons than there are for elements with an odd number of protons
simply because there are very few stable odd–odd nuclei. Thus, the simple number
of stable product nuclei, whatever the production mechanism, will have an effect on
the observed populations because nearly all radioactive decay will have taken place
since the astrophysical production leaving (only) the stable remains. There are
exceptions, of course, and contemporary research emphasizes observing recently
produced radioactive nuclei in the cosmos.
Given what we know about nuclear structure, it is reasonable to consider the
isotopic distribution rather than the elemental distribution. An example of the isotopic
abundances of the top-row elements is shown in Figure 12.3. Once again the
very strong staggering is seen and the depression of masses between 5 and 10 is
more apparent. This mass region has gaps (no stable nuclei with A ¼ 5 or 8), and
the remaining nuclei are all relatively fragile and have small binding energies.
For the lightest nuclei, the nuclei whose mass numbers are a multiple of 4 have
the highest abundances. Again, simple nuclear stability considerations affect
Figure 12.3 Mass fractions of the top-row elements in the sun from Anders and Grevesse
(1989). Elements with an odd atomic number are shown by square symbols, even by
circles. The isotopes of a given chemical element are connected by line segments.
334 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
the amount of beryllium we find relative to the amount of carbon or oxygen, but
the many orders of magnitude difference in the abundance of elements such as
beryllium and carbon must be due to some effects from the production
mechanisms.
The sun is a typical star (see below), and one uses the solar abundances to represent
the elemental abundances in the universe (the “cosmic” abundances). It will turn out
that several nucleosynthetic processes are necessary to explain the details of the
observed abundances. In Figure 12.4, we jump ahead of our discussion to show a
rough association between the elemental abundances and the nucleosynthetic processes
that created them. Figure 12.4 is based upon a pioneering study by Burbidge,
Burbidge, Fowler and Hoyle (B2
FH) (1957) and an independent analysis of Cameron
(1957). These works have served as a framework for the discussion of nucleosynthesis
since their publication in the 1950s, and we will follow a similar route in our
discussion.
Figure 12.4 Atomic abundances of the elements in the solar system and the major
nucleosynthetic processes responsible for the observed abundances. [From E. M. Burbidge,
et al., Rev. Mod. Phys. 29, 547 (1957). Copyright (1957) by the American Physical Society.]
12.2 ELEMENTAL AND ISOTOPIC ABUNDANCES 335
12.3 PRIMORDIAL NUCLEOSYNTHESIS
The universe is between 10 and 20 billion years old, with the best estimate of its age
being 14 + 1 Â 109
y old. The universe is thought to have begun with a cataclysmic
explosion called the Big Bang. Since the Big Bang, the universe has been expanding
with a decrease of temperature and density.
One important piece of evidence to support the idea of the Big Bang is the 2.7-K
microwave radiation background in the universe. This blackbody radiation was discovered
by Penzias and Wilson in 1965 and represents the thermal remnants of the
electromagnetic radiation that existed shortly after the Big Bang. Weinberg (1977)
tells how Penzias and Wilson found a microwave noise at 7.35 cm that was independent
of direction in their radio antenna at the Bell Telephone Laboratories in New
Jersey. After ruling out a number of sources for this noise, they noted a pair of
pigeons had been roosting in the antenna. The pigeons were caught, shipped to a
new site, reappeared, were caught again, and were “discouraged by more decisive
means.” The pigeons, it was noted, had coated the antenna with a “white dielectric
material.” After removal of this material, the microwave background was still there.
It was soon realized that this 7.35-cm radiation corresponded to an equivalent temperature
of the noise of about 3.5 K, which was eventually recognized as the remnants
of the Big Bang. (Subsequent measurements have characterized this radiation as
having a temperature of 2.7 K with a photon density of 400 photons/cm3
in the
universe.)
A pictorial representation of some of the important events in the “thermal”
history of the universe is shown in Figure 12.5. The description of the evolution
of the universe begins at 10243
s after the Big Bang, the so-called Planck time.
The universe at that time had a temperature of 1032
K (kBT 1019
GeV) and a
volume that was 10231
of its current volume. [To convert temperature in K to
Figure 12.5 Outline of the events in the universe since the Big Bang. (From C. E. Rolfs and
W. S. Rodney, Cauldrons in the Cosmos, Chicago University Press, Chicago, 1988.)
336 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
energy (kBT) in electron volts, note that kBT (eV) ¼ 8.6 Â 1025
T (K)] Matter
existed in a state unknown to us, a plasma of quarks and gluons. All particles
were present and in statistical equilibrium, where each particle had a production
rate equal to the rate at which it was destroyed. As the universe expanded, it
cooled and some species fell out of statistical equilibrium. At a time of 1026
s
(T  1013
K), the photons from the blackbody radiation could not sustain the production
of the massive particles and the hadronic matter condensed into a gas of
nucleons and mesons. At this point, the universe consisted of nucleons, mesons, neutrinos
(and antineutrinos), photons, electrons (and positrons). The ratio of baryons to
photons was 1029
.
At a time of 1022
s (T  1011
K), the density of the universe dropped to
4 Â 106
kg/m3
. [In this photon-dominated era, the temperature T (K) was given
by the relation
T (K) ¼
1:5 Â 1010
ffiffiffiffiffiffiffiffiffi
t (s)
p
where t is the age in seconds.] At this time, the neutrons and protons interconvert by
the weak interactions
ne þ p () eþ
þ n
ne þ n () p þ eÀ
[One neglects the free decay of the neutron to the proton because the half-life for that
decay (10.6 m) is too long to be relevant.] The neutron–proton ratio, n : p, was determined
by a Boltzmann factor, that is,
n : p ¼ exp (ÀDmc2
=kT)
where Dmc2
is the n-p mass difference (1.29 MeV). At T ¼ 1012
K, n/p  1, at
T ¼ 1011
K, n/p  0.86, and so forth. At T ¼ 1011
K, no complex nuclei were
formed because the temperature was too high to allow deuterons to form. When
the temperature fell to T ¼ 1010
K (t  1 s), the creation of eþ
/e2
pairs (pair
production) ceased because kT , 1.02 MeV and the neutron/proton ratio was
17/83. At a time of 225 s, this ratio was 13/87, the temperature was T  109
K,
then density was 2 Â 104
kg/m3
, and the first nucleosynthetic reactions occurred.
These primordial nucleosynthesis reactions began with the production of deuterium
by the simple radiative capture process:
n þ p À! d þ g
Notice that the deuteron can be destroyed by the absorption of a high-energy photon
in the reverse process. At this time, the deuteron survived long enough to allow the
subsequent reactions
p þ d À! 3
He þ g
12.3 PRIMORDIAL NUCLEOSYNTHESIS 337
and
n þ d À! 3
H þ g
3
H and 3
He are more strongly bound allowing further reactions that produce the very
strongly bound a particles:
3
H þ p À! 4
He þ g
3
He þ n À! 4
He þ g
3
H þ d À! 4
He þ n
d þ d À! 4
He þ g
Further reactions to produce the A ¼ 5 nuclei do not occur because there are no
stable nuclei with A ¼ 5 (or A ¼ 8). A small amount of 7
Li is produced in the
reactions:
4
He þ 3
H À! 7
Li þ g
3
He þ 3
He À! 7
Be þ g eÀ
À! 7
Li þ g
but the 7
Li is also very weakly bound and is rapidly destroyed. Thus, the synthesis of
larger nuclei was blocked. After about 30 m, nucleosynthesis ceased. The temperature
was 3 Â 108
K and the density was 30 kg/m3
. (For reference, please note
that water vapor at 1 atm has a density of 1 kg/m3
and liquid water has a
density of 103
kg/m3
). Nuclear matter was 76% by mass protons, 24% a particles
with traces of deuterium, 3
He and 7
Li. The g : n : p ratio was 109
: 13 : 87. The relative
ratio of p : 4
He : d : 3
He : 7
Li is a sensitive function of the baryon density of the
universe (Fig. 12.6), a fact that is used to constrain models of the Big Bang. (The
cross sections for the reactions that convert one product to another are generally
known, and detailed network calculations of the reaction rates can be performed
as a function of temperature and density. The resulting abundances can be compared
to estimates from observations of stellar matter.) Chemistry began about 106
y later,
when the temperature has fallen to 2000 K and the electrons and protons could
combine to form atoms. Further nucleosynthesis continues to occur in the interiors
of stars.
12.4 STELLAR EVOLUTION
As discussed above, nucleosynthesis occurred in two steps, the primordial nucleosynthesis
that occurred in the Big Bang forming only the lightest nuclei and later
processes, beginning 106
y after the Big Bang and continuing to the present, of
nucleosynthesis in the stars. Big Bang nucleosynthesis produced hydrogen,
338 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
helium, and traces of 7
Li, whereas the rest of the elements are the result of stellar
nucleosynthesis. For example, recent observations of stellar spectral lines
showing the presence of 2 Â 105
y 99
Tc indicate ongoing stellar nucleosynthesis.
To understand the nuclear reactions that make the stars shine and generate the
bulk of the elements, one needs to understand how stars work. That is the focus
of this section.
After the Big Bang explosion, the material of the universe was dispersed. Inhomogenities
developed, which under the influence of gravity, condensed to form
the galaxies. Within these galaxies, clouds of hydrogen and helium gas can
collapse under the influence of gravity. At first, the internal heat of this collapse
is radiated away. As the gas becomes denser, however, the opacity increases, and
the gravitational energy associated with the collapse is stored in the interior rather
0.25
0.24
0.23
0.22
10–4
10
–5
10
–9
10
–10
1 2 5
Baryon density (10
–31
g cm
–3
)
Li
He
D
7
3
0.01 0.02 0.05
Fraction of critical density
4
HeMassfractionNumberrelativetoH
Figure 12.6 Variation of the relative abundances of the Big Bang nuclei and the 4
He mass
fraction versus the baryon density. The boxes indicate the measured data and an estimate of
the uncertainty. The curves indicate the dependence of the yield on the baryon density in the
Big Bang; the vertical bar indicates the region of concordence. (Figure also appears in color
figure section.)
12.4 STELLAR EVOLUTION 339
than being radiated into space. Eventually a radiative equilibrium is established with
the development of a protostar. The protostar continues to shrink under the influence
of gravity with continued heating of the stellar interior. When the interior temperature
reaches 107
K, thermonuclear reactions between the hydrogen nuclei can
begin because some of the protons have sufficient kinetic energies to overcome
the Coulomb repulsion between the nuclei.
The first generation of stars that formed in this way is called (for historical
reasons) population III stars. They consisted of hydrogen and helium, were
massive, had relatively short lifetimes, and are now extinct. The debris from these
stars has been dispersed and was incorporated into later generation of stars.
The second generation of stars, called population II stars, was comprised of
hydrogen, helium, and about 1% of the heavier elements such as carbon and
oxygen. Finally, there was a third generation of stars, like our sun, called population
I stars. These stars consist of hydrogen, helium, and 2–5% of the heavier elements.
Our sun is a typical population I star. It has a mass of 2.0 Â 1030
kg, a radius of
7.0 Â 106
m (an average density of 1.41 Â 103
kg/m3
), a surface temperature of
6000 K, and a luminosity of 3.83 Â 1026
W. Our sun is 4.5 Â 109
y old.
The Danish astronomer Ejnar Hertzsprung and the American astronomer Henry
Norris Russell independently observed a very well defined correlation between the
luminosity and surface temperature of stars. That correlation is shown in Figure 12.7
and is called a Hertzsprung–Russell, or H–R, diagram. Most stars, such as our sun,
fall in a narrow band on this diagram called the main sequence. Stars in this main
sequence have luminosities L that are approximately proportional to T5:5
surface, or in
terms of their mass, M, L / M3.5
. How long a star stays on the main sequence
will depend on its mass, which, in turn, is related to the reaction rates in its interior.
Figure 12.7 Schematic representation of a Hertzsprung–Russell diagram. (From C. E.
Rolfs and W. S. Rodney, Cauldrons in the Cosmos, Chicago University Press, Chicago, 1988.)
340 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
In the upper-right portion of the H–R diagram, one sees a group of stars, the red
giants or super giants, with large radii that are relatively cool (3000–4000 K). Stars
on the main sequence move to this region when the nuclear energy liberated in the
nuclear reactions occurring in the star is not enough to sustain main sequence luminosity
values.
Our sun is expected to spend 7 Â 109
more years on the main sequence before
becoming a red giant. In a shorter time of 1.1–1.5 Â 109
y, the sun will increase
slowly in luminosity by 10%, probably leading to a cessation of life on Earth.
(In short, terrestrial life has used up 3
4 of its allotted time since it began
3.5 Â 109
y ago.)
In the lower left of the H–R diagram, one sees a group of small dense, bright stars
(T . 104
K) called white dwarfs. The white dwarfs represent one evolutionary
outcome for the red giants with masses between 0.1 and 1.4 solar masses. Red giants
are helium-burning stars and as the helium is burnt, the stars become unstable, ejecting
theirenvelopes,creatingaplanetarynebula,andmovingacrossthemainsequenceonthe
H–R diagram to be white dwarfs. (See Fig. 12.7 for a schematic view of this evolution.)
For massive red giants (M . 8 solar masses), one finds they undergo a more spectacular
death cycle, with contractions, increases in temperature leading to helium
burning, carbon–oxygen burning, silicon burning, and the like with the production
of the elements near iron, followed by an explosive end (Fig. 12.8).
Figure 12.8 Schematic diagram of the evolution of (a) a star with a mass near that of the sun
and (b) a more massive star. (From C. E. Rolfs and W. S. Rodney, Cauldrons in the Cosmos,
Chicago University Press, Chicago, 1988.)
12.4 STELLAR EVOLUTION 341
The explosive end for these stars can lead to the formation of novas and supernovas.
The name nova means new and connotates a star that undergoes a sudden
increase in brightness, followed by a fading characteristic of an explosion. In
this process, the outer part of the star containing only 1023
of the stellar mass
is ejected with the release of 1045
ergs. Supernovas are spectacular stellar
explosions in which the stellar brightness increases by a factor of 106
–109
, releasing
1051
ergs on a time scale of seconds. We have observed about 10 nova/year
but only 2–3 supernova per century. Supernovas are classified as type I (low
hydrogen, high “heavy” elements, such as oxygen to iron), and type II (primarily
hydrogen, with lesser amounts of the heavy elements). Some supernovas lead to
the formation of neutron stars, which are giant nuclei of essentially pure neutronic
matter.
12.5 THERMONUCLEAR REACTION RATES
Before discussing the nuclear reactions involved in stellar nucleosynthesis, we need
to discuss the rates of these reactions, which take place in a “thermal soup” as
opposed to reactions studied in the laboratory. The rates of these reactions will
tell us what reactions are most important. When we speak of thermonuclear
reactions, we mean nuclear reactions in which the energy of the colliding nuclei
is the thermal energy of the particles in a hot gas. Both reacting nuclei are moving,
and thus it is their relative velocity (center-of-mass energy) that is important. In
ordinary nuclear reactions in the laboratory, we write for the rate of the reaction R:
R ¼ Nsf
where the reaction rate R is in reactions/second, s is the reaction cross section (in
cm2
), f the incident particle flux in particles/second and N is the number of target
atoms/cm2
. For astrophysical reactions, we write
R ¼ NxNy
ð1
0
s(v)v dv ¼ NxNyksvl
where v is the relative velocity between nuclei x and y, each present in a concentration
of Ni particles/cm3
, the quantity ksvl is the temperature-averaged reaction
rate per particle pair. To be sure that double counting of collisions between identical
particles does not occur, it is conventional to express the above equation as:
R ¼
NxNyksvl
1 þ dxy
where dxy is the Kronecker delta (which is 0 when x = y and 1 when x ¼ y). Note
the mean lifetime of nuclei x is then 1/(Nxksvl).
342 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
In a hot gas the velocity distribution of each component will be given by a
Maxwell–Boltzmann distribution:
P(v) ¼
m
2pkT
 3=2
exp À
mv2
2kT
 
where m is the particle mass, k is Boltzmann’s constant, and T is the gas temperature.
Integrating over all velocities for the reacting particles, x and y, gives
ksvl ¼
8
pm
 1=2
1
(kT)3=2
ð1
0
s(E)E exp À
E
kT
 
dE
where m is the reduced mass ½mxmy=(mx þ my)Š. The rates, R, of stellar nuclear
reactions are directly proportional to ksvl, which depends on the temperature T.
For slow neutron-induced reactions that do not involve resonances, we know
(Chapter 10) that sn(E) / 1/vn so that ksvl is a constant. For charged particle reactions,
one must overcome the repulsive Coulomb force between the positively
charged nuclei. For the simplest reaction, p þ p, the Coulomb barrier is 550 keV.
But, in a typical star such as the sun, kT is 1.3 keV, that is, the nuclear reactions
that occur are subbarrier, and the resulting reactions are the result of barrier penetration.
(At a proton–proton center-of-mass energy of 1 keV, the barrier penetration
probability is 2 Â 10210
). At these extreme subbarrier energies, the
barrier penetration factor can be approximated as:
P ¼ exp À
2pZ1Z2e2
hÀ
v
 
¼ exp À31:29Z1Z2
m
E
 1=2
 
where E is in keV and m in amu. This tunneling probability is referred to as the
Gamow factor. The cross section (Chapter 10) is also proportional to plÀ2
/ 1=E.
Thus, the cross section for nonresonant charged-particle-induced reactions can be
written as:
s(E) ¼
1
E
exp À31:29Z1Z2
m
E
 1=2
 
S(E)
where the function S(E), the so-called astrophysical S factor, contains all the constants
and terms related to the nuclei involved. Substituting this expression into
the equation for ksvl, we have
ksvl ¼
8
pm
 1=2
1
(kT)3=2
ð1
0
S(E) exp À
E
kT
À
b
E1=2
 
dE
where b is 0.989Z1Z2m1/2
(MeV)1/2
. This equation represents the overlap between
the Maxwell–Boltzmann distribution, which is peaked at low energies, and the
12.5 THERMONUCLEAR REACTION RATES 343
Gamow factor, which increases with increasing energy. The product of these two
terms produces a peak in the overlap region of these two functions called the
Gamow peak (Fig. 12.9). This peak occurs at an energy E0 ¼ (bkT/2)2/3
.
For reactions involving isolated single resonances or broad resonances, it is possible
to derive additional formulas for s(E) [R þ R] in the Breit–Wigner form, that is,
s(E) ¼ plÀ2 2Jr þ 1
(2Jx þ 1)(2Jy þ 1)
 
GinGout
(E À Er)2
þ
G2
tot
4
where Jx, Jy, Jr are the spins of the interacting particles and the resonance and
Gin, Gout, Gtot are the partial widths of the entrance and exit channels and the total
width, respectively.
12.6 STELLAR NUCLEOSYNTHESIS
12.6.1 Introduction
After Big Bang nucleosynthesis, we have a universe that is 75% hydrogen and
25% helium with a trace of 7
Li. Stellar nucleosynthesis continues the synthesis
of the chemical elements. Beginning 106
y after the Big Bang, as described in
Section 12.4, the sequence of gravitational collapse and an increasing temperature
cause the onset of nuclear fusion reactions, releasing energy that stops the collapse.
Starting from hydrogen and helium, fusion reactions produce the nuclei up to the
maximum in the nuclear binding energy curve at A  60. The limiting temperature
of these reactions is about 5 Â 109
K, where kT  0.4 MeV. A rough outline of the
nuclear reactions involved is given in Table 12.1.
The products from these reactions are distributed into the galaxies by slow emission
from the red giants and by the catastrophic explosions of novas and supernovas.
This dispersed material condenses in the population II and later the population I stars
Figure 12.9 Rate of nonresonant stellar nuclear reactions as a function of temperature.
(From S. S. M. Wong, Introductory Nuclear Physics. Copyright # 1988 by John Wiley &
Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.)
344 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
where additional nuclear reactions (see below) create the odd A nuclei and sources of
free neutrons. These neutrons allow us to get slow neutron capture reactions
(s process) synthesizing many of the nuclei with A . 60. High-temperature photonuclear
reactions and rapid neutron capture reactions in supernovas complete the
bulk of the nucleosynthesis reactions.
12.6.2 Hydrogen Burning
The first stage of stellar nucleosynthesis, which is still occurring in stars such as our
sun, is hydrogen burning. In hydrogen burning, protons are converted to 4
He nuclei.
Since there are no free neutrons present, the reactions differ from those of Big Bang
nucleosynthesis. The first reaction that occurs is
p þ p À! d þ eþ
þ ne Q ¼ 0:42 MeV
where most of the released energy is shared between the two leptons. In our sun,
T  15 Â 106
K (kT  1 keV). The proton–proton (pp) reaction is a weak interaction
process and has, therefore, a very small cross section, 10247
cm2
, at these proton
energies. The resulting reaction rate is 5 Â 10218
reactions/second/proton.
There is an improbable (0.4%) variant of this reaction, called the pep reaction
that also leads to deuteron production. It is
p þ eÀ
þ p À! d þ ne Q ¼ 1:42 MeV
This rare reaction is a source of energetic neutrinos from the sun.
The next reaction in the sequence is
d þ p À! 3
He þ g Q ¼ 5:49 MeV
leading to the synthesis of 3
He. The rate of this strong interaction is 1016
times
greater than the weak p þ p reaction. The product 3
He can undergo two possible
reactions. In 86% of the cases, the reaction is
3
He þ 3
He À! 4
He þ 2p Q ¼ 12:96 MeV
TABLE 12.1 Nuclear Reactions Involved in Stellar
Nucleosynthesis
Fuel T (K) kT (MeV) Products
1
H 5 Â 107
0.002 4
He
4
He 2 Â 108
0.02 12
C, 16
O, 20
Ne
12
C 8 Â 108
0.07 16
O, 20
Ne, 24
Mg
16
O 2 Â 109
0.2 20
Ne, 28
Si, 32
S
20
Ne 1.5 Â 109
0.13 16
O, 24
Mg
28
Si 3.5 Â 109
0.3 A , 60
12.6 STELLAR NUCLEOSYNTHESIS 345
This reaction, when combined with the two previous reactions (p þ p and d þ p)
corresponds to an overall reaction of
4p À! 4
He þ 2eþ
þ 2ne A ¼ 26:7 MeV
This sequence of reactions is called the ppI chain and is responsible for 91% of the
sun’s energy. A schematic view of this reaction is shown in Figure 12.10.
In 14% of the cases, the 3
He product undergoes the side reaction with an
a particle:
3
He þ 4
He À! 7
Be þ ne
The 7
Be undergoes subsequent electron capture decay:
eÀ
þ 7
Be À! 7
Li þ ne Q ¼ 0:86 MeV
Note that this EC decay process does not involve capture of the orbital electron of
the 7
Be since it is fully ionized in a star but rather involves capture of a free continuum
electron. As a consequence, the half-life of this decay is 120 d rather than
the terrestrial half-life of 77 d. The resulting 7
Li undergoes proton capture as:
p þ 7
Li À! 2 4
He
This sequence of reactions [p þ p, d þ p, 3
He þ 4
He, 7
Be EC, 7
Li(p,a)] constitutes
the ppII process, which accounts for 7% of the sun’s energy.
Figure 12.10 Schematic view of the ppI chain. (From C. E. Rolfs and W. S. Rodney,
Cauldrons in the Cosmos, Chicago University Press, Chicago, 1988.)
346 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
A small fraction of the 7
Be from the 3
He þ 4
He reaction can undergo proton
capture, leading to
7
Be þ p À! 8
B þ g
8
B À! 8
BeÃ
þ eþ
þ ne
8
BeÃ
À! 2 4
He
This sequence [p þ p, p þ d, 3
He þ 4
He, 7
Be(p,g), 8
B ! 8
BeÃ
! 2 4
He] constitutes
the ppIII chain (which provides about 0.015% of the sun’s energy). In each of the pp
processes, some energy is carried away by the emitted neutrinos. Quantitatively, in
the ppI process, the loss is 2%, in the ppII process 4%, and 28.3% in the ppIII
process. These pp chains are shown together in Figure 12.11.
In population II and population I stars, heavy elements such as carbon, nitrogen,
and oxygen (CNO) are present, leading to the occurrence of another set of nuclear
reactions whose net effect in the conversion 4p À! 4
He þ 2eþ
þ 2ne. The “heavy”
nuclei act as catalysts for this reaction. A typical cycle is
12
C þ p À! 13
N þ g
13
N À! 13
C þ eþ
þ ne
13
C þ p À! 14
N þ g
14
N þ p À! 15
O þ g
15
O À! 15
N þ eþ
þ ne
15
N þ p À! 12
C þ 4
He
Figure 12.11 Three chains of nuclear reactions that constitute hydrogen burning and convert
protons into 4
He. The rate-limiting step in all reactions is the first reaction to create the deuterium.
12.6 STELLAR NUCLEOSYNTHESIS 347
This group of reactions is referred to as the CNO cycle and is favored at higher temperatures
where the Coulomb barrier for these reactions can be more easily overcome.
In our sun, 98% of the energy comes from the pp chain and only 2% from the CNO
cycle. Several side chains of this reaction cycle are possible, as illustrated in
Figure 12.12.
12.6.3 Helium Burning
Eventually, the hydrogen fuel of the star will be exhausted and further gravitational
collapse will occur. This will give rise to a temperature increase up to
1–2 Â 108
K (with a density of 108
kg/m3
). In this red giant, helium burning
will commence.
One might think the first reaction might be
4
He þ 4
He À! 8
Be Q ¼ À0:0191 MeV
but 8
Be is unstable (t1/2 ¼ 6.7 Â 10217
s), and thus that process is hindered by the
short lifetime and low transient population of the Be nuclei. Instead one gets the
so-called 3a process:
3 4
He À! 12
C Q ¼ 7:37 MeV
Three-body reactions are rare, but the reaction proceeds through a resonance in 12
C
at 7.65 MeV corresponding to the second excited state of 12
C (Jp ¼ 0þ
). This
excited state has a more favorable configuration than the 12
C ground state for allowing
the collision to occur. (In a triumph for nuclear astrophysics, the existence of this
state was postulated by astrophysicists to explain nucleosynthetic rates before it was
found in the laboratory.)
Figure 12.12 CNO cycle with side chains. (From S. S. M. Wong, Introductory Nuclear
Physics. Copyright # 1988 by John Wiley & Sons, Inc. Reprinted by permission of John
Wiley & Sons, Inc.)
348 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
After a significant amount of 12
C is formed, one gets the a-capture reactions:
4
He þ 12
C À! 16
O þ g Q ¼ 7:16 MeV
4
He þ 16
O À! 20
Ne þ g Q ¼ 4:73 MeV
A brief interlude of neon burning then occurs with reactions such as:
20
Ne þ g À! 4
He þ 16
O
4
He þ 20
Ne À! 24
Mg þ g
The relative rates of these and related processes are shown in Figure 12.13.
12.6.4 Synthesis of Nuclei with A < 60
Eventually, the helium of the star will be exhausted, leading to further gravitational
collapse with a temperature increase to 6 Â 108
–2 Â 109
K (kT  100–200 keV).
At this point the fusion reactions of the “a-cluster” nuclei are possible. For example,
Figure 12.13 Mean lifetimes for various nucleosynthesis reactions involving a particles as
a function of temperature. The mean lifetime is inversely related to the reaction rate. [From
E. M. Burbidge et al., Rev. Mod. Phys., 29, 547 (1957). Copyright (1957) by the American
Physical Society.]
12.6 STELLAR NUCLEOSYNTHESIS 349
carbon and oxygen burning occurs in charged particle reactions such as:
12
C þ 12
C À! 20
Ne þ 4
He
12
C þ 12
C À! 23
Na þ p
12
C þ 12
C À! 23
Mg þ n
12
C þ 12
C À! 24
Mg þ g
16
O þ 16
O À! 24
Mg þ 2 4
He
16
O þ 16
O À! 28
Si þ 4
He
16
O þ 16
O À! 31
P þ p
16
O þ 16
O À! 31
S þ n
16
O þ 16
O À! 32
S þ g
with the production of 28
Si and 32
S being the most important branches of the oxygenburning
reactions. Further rises in temperature up to 5 Â 109
K result in a series of
silicon-burning reactions involving an equilibrium between photodisintegration and
radiative capture processes such as:
28
Si þ g À! 24
Mg þ 4
He
4
He þ 28
Si À! 32
S þ g
The nuclei up to A  60 are produced in these equilibrium processes. In such equilibrium
processes, the final yields of various nuclei are directly related to their nuclear
stability with the more stable nuclei having higher yields. So one observes greater
yields of even–even nuclei than odd A nuclei (due to the pairing term in the mass
formula) and even N isotopes are more abundant than odd A isotopes of an element.
The relative time scales of the various reactions leading to nuclei with A , 60 are
shown in Table 12.2. Note these time scales are inversely proportional to the
reaction rates.
TABLE 12.2 Time Scales of Nucleosynthetic Reactions
in a 1 Solar Mass Star
Reaction Time
H burning 6 Â 109
y
He burning 0.5 Â 106
y
C burning 200 y
Ne burning 1 y
O burning Few months
Si burning Days
350 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
12.6.5 Synthesis of Nuclei with A > 60
The binding energy per nucleon curve peaks at A  60 and decreases as A increases
beyond 60. This indicates that fusion reactions using charged particles are not energetically
favored to make these nuclei. However, another possible nuclear reaction is
neutron capture, that is, (n, g). These reactions have no Coulomb barriers to inhibit
them, and the rates are then governed by the Maxwell–Boltzmann distribution of
velocities in a hot gas and the availability of free neutrons. If s(n, g)  1/v, then
the reaction rate Nnksvl is largely governed by Nn, the neutron density. We can
identify two types of neutron capture processes for nucleosynthesis. The first of
these is slow neutron capture, the so-called s process, where the time scale of the
neutron capture process, treaction ) tb, where tb is the b2
-decay lifetime. In this
process, each neutron capture proceeds in competition with b2
decay. For
example, consider the stable nucleus 56
Fe. If it captures a neutron, the following
reactions can occur:
56
Fe þ n À! 57
Fe (stable) þ g
57
Fe þ n À! 58
Fe (stable) þ g
58
Fe þ n À! 59
Fe (t1=2 ¼ 44:5 d) þ g
Then, 44.5 d 59
Fe will undergo b2
decay before another neutron is captured, that is,
59
Fe À! 59
Co (stable) þ bÀ
þ ne
and further captures will start with 59
Co. The mean times of neutron capture
reactions treaction ¼ ln 2=rate ¼ ln 2=Nnksvl. If Nn  1011
/m3
, s ¼ 0.1 barn,
En  50 keV, then t  105
y. Then, one will see neutron capture by all stable
nuclei and many of the long-lived nuclei. A typical s-process path of nucleosynthesis
for the nuclei with Z ¼ 45–60 is shown in Figure 12.14. The production of nuclei
follows a zig-zag path through the chart of nuclides, with increases in mass when
a neutron is captured and increases in atomic number when b decay precedes the
next neutron capture.
The s process terminates at 209
Bi because of the cyclic sequence:
209
Bi(n, g)210
Bi ÀÀÀ!
bÀ
210
Po ÀÀÀ!
a 206
Pb ÀÀÀ!
3(n,g) 209
Pb ÀÀÀ!
bÀ
209
Bi
The source of the neutrons for the s process is (a, n) reactions on n-rich nuclei
such as 13
C or 21
Ne, with the latter being most important. In population II and I
stars, one can get side reactions such as:
20
Ne( p, g)21
Na
21
Na À! 21
Ne þ eþ
þ ne
that produce the target nuclei for the (a, n) reactions.
12.6 STELLAR NUCLEOSYNTHESIS 351
For the slow neutron capture process, there is an equilibrium between the production
and loss of adjacent nuclei. Stable nuclei are only destroyed by neutron
capture. For such nuclei, we can write for the rate of change of a nucleus with
mass number A:
dNA
dt
¼ sAÀ1NAÀ1 À sANA
where sI and Ni are the capture cross sections and number of nuclei (abundance) for
nucleus i, respectively. At equilibrium,
dNA
dt
¼ 0
Thus
sAÀ1NAÀ1 ¼ sANA
This relationship between the abundances of neighboring stable nuclei is a signature
for the s process.
If the time scale of neutron capture reactions is very much less than b2
-decay
lifetimes, then rapid neutron capture or the r process occurs. For r-process nucleosynthesis,
one needs large neutron densities, 1028
/m3
, which lead to capture times
of the order of fractions of a second. The astrophysical environment where such processes
can occur is now thought to be in supernovas. In the r process, a large number
of sequential captures will occur until the process is terminated by neutron emission
or, in the case of the heavy elements, fission or b-delayed fission. The lighter “seed”
nuclei capture neutrons until they reach the point where b2
-decay lifetimes have
Figure 12.14 Section of the chart of nuclides showing the s-process path. (From C. E. Rolfs
and W. S. Rodney, Cauldrons in the Cosmos, Chicago University Press, Chicago, 1988.)
352 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
decreased and b2
decay will compete with neutron capture. The r process is responsible
for the synthesis of all nuclei with A . 209 and many lower mass nuclei. In a
plot of abundances vs. mass number A (Fig. 12.4) one sees two peaks in the abundance
distributions near the magic neutron numbers (N ¼ 50, 82, 126). The lower A
peak is due to the r process, which reaches the magic number of neutrons at a lower Z
value than the s process. The peaks occur because of the relative stability of N ¼ 50,
82, 126 nuclei against neutron capture. A typical r-process path is shown in
Figure 12.15. Notice that the path climbs up in atomic number along the neutron
magic numbers. The nuclei in each zig-zag region are the places of maxima in
the isotopic yields after decay.
Another important process leading to the synthesis of some proton-rich nuclei
with 70 , A , 200 is the so-called p process. The p process consists of a series
of photonuclear reactions (g, p), (g, a), (g, n) on “seed” nuclei from the s or r processes
that produce these nuclei. (Originally, it was believed that proton capture processes
during supernovas were responsible for these nuclei, but it was found that the
proton densities are too small to explain the observed abundances.) The temperature
during a supernova explosion is 3 Â 109
K, producing blackbody radiation that
can cause photonuclear reactions. The p-process contribution to the abundances
of most elements is very small, but there are some nuclei (190
Pt, 168
Yb) that seem
to have been exclusively made by this process. The relative importance of s, r,
and p processes in nucleosynthesis in a given region is shown in Figure 12.16.
A process that is sometimes related to the p process is the rp process, the rapid
proton capture process. This process makes proton-rich nuclei with Z ¼ 7 2 26.
This process involves a set of (p, g) and bþ
decays that populate the p-rich
nuclei. The process starts as a “breakout” from the CNO cycle, that is, a side
Figure 12.15 Neutron capture paths are shown for the s and r processes. (From C. E. Rolfs
and W. S. Rodney, Cauldrons in the Cosmos, Chicago University Press, Chicago, 1988.)
12.6 STELLAR NUCLEOSYNTHESIS 353
chain of the CNO cycle that produces p-rich nuclei such as 21
Na and 19
Ne. These
“seed” nuclei can form the basis for further proton captures, leading to the nucleosynthetic
path shown in Figure 12.17. The rp process creates a small number of
nuclei with A , 100. The process follows a path analogous to the r process but
on the proton-rich side of stability. At present, the source of the protons for this
process are certain binary stars. Note this process while close to the line of b
stability, approaches the proton drip line as the nuclei become heavier.
12.7 SOLAR NEUTRINO PROBLEM
12.7.1 Introduction
Many of the nuclear reactions that provide the energy of the stars also result in the
emission of neutrinos. Because of the small absorption cross sections for neutrinos
interacting with matter (sabs  10244
cm2
), these neutrinos are not generally
absorbed in the sun and other stars. (This loss of neutrinos corresponds to a loss
of 2% of the energy of our sun.) Because of this, the neutrinos are a window
into the stellar interior. The small absorption cross sections also make neutrinos
difficult to detect, with almost all neutrinos passing through planet Earth without
interacting.
76
74
72
70
Hf 176 Hf 177 Hf 178
Lu 175 Lu 177
6.7 d(s)
Yb 174 Yb 175 Yb 176
4.2 d (r)
Lu 176
2.6×10
10
y
Ta 180
>1.6×10
13
y
(p)
Hf 179 Hf 180 Hf 181
42 d
115 d
Ta 182Ta 181
W 182 W 183 W 184 W 185 W 186
75 d (r)
(r)91 h
Re 185 Re 186 Re 187
5×10
10
y
(s) (s)
Os 186 Os 187 Os 188 Os 189 Os 190Os 184
0.018
W 180
0.14
(p)
(p)
Ta 180
>1.6×10
13
y
Ta 180
>1.6×10
13
y
p–process nuclei
MECHANISMS OF HEAVY ELEMENT SYNTHESIS
(s–process capture path)
r–process contributions
104 106 108 110 112 114
Neutron Number
ProtonNumber
Figure 12.16 Typical portion of the heavy element chart of the nuclides showing the
relative importance of s, r, and p processes in nucleosynthesis. [From J. W. Truran,
“Nucleosynthesis” in Ann. Rev. Nucl. Part. Sci., 34, 53 (1984). Copyright # 1984 by
Annual Reviews, Inc. Reprinted by permission of Annual Reviews, Inc.]
354 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
Recently, a good deal of attention has been given to the “solar neutrino problem”
and its solution. The 2002 Nobel Prize in physics was awarded to Ray Davis and
Masatoshi Koshita for their pioneering work on this problem. Of special interest
is the important role of nuclear and radiochemistry in this work as Davis is a
nuclear chemist. The definition and solution of this problem is thought to be one
of the major scientific advances of recent years.
12.7.2 Expected Solar Neutrino Sources, Energies, and Fluxes
The sun is a major source of neutrinos reaching the surface of Earth due to its close
proximity. The sun emits 1.8 Â 1038
neutrinos/second, which, after an 8-min
transport time, reach the surface of Earth at a rate of 6.4 Â 1010
neutrinos/s/cm2
.
The predictions of the standard solar model for the neutrino fluxes at the surface
of Earth due to various nuclear reactions are shown in Table 12.3.
The predicted energy distributions are shown in Figure 12.18. Each nuclear
reaction has a characteristic neutrino energy distribution.
The source labeled “pp” in the Table 12.3 and Figure 12.18 refers to the reaction
p þ p À! d þ eþ
þ ne
and is the most important reaction, producing one neutrino for each 4
He nucleus
made. The “pep” source is the reaction
p þ p þ eÀ
À! d þ ne
Figure 12.17 Position of the rp process relative to the line of b stability.
12.7 SOLAR NEUTRINO PROBLEM 355
which produces monoenergetic neutrinos, while the source “hep” refers to the
reaction
p þ 3
He À! 4
He þ eþ
þ ne
This latter reaction produces the highest energy neutrinos with a maximum energy
of 18.77 MeV due to the high reaction Q value. The intensity of this source is about
TABLE 12.3 Predicted Solar Neutrino Fluxes (Bahcall
and Pena-Garay)
Source Flux (particles/s/cm2
)
pp 5.94 Â 1010
pep 1.40 Â 108
hep 7.88 Â 103
7
Be 4.86 Â 107
8
B 5.82 Â 106
13
N 5.71 Â 108
15
O 5.03 Â 108
17
F 5.91 Â 106
0.1 0.3 1 3 10
1012
1011
1010
109
108
107
106
105
104
103
102
101
Neutrino Energy (MeV)
NeutrinoFlux
±12%
±12%
±1%
±23%
±16%
±2%
hep
pp
7
Be 7
Be
8B
pep
Bahcall–Pinsonneault 2004
SuperK, SNO
Chlorine
Gallium
Figure 12.18 Log–log plot of predicted neutrino fluxes from various solar nuclear
reactions. The energy regions to which the neutrino detectors are sensitive are shown at the
top. [From Bahcall ( from Bahcall website).]
356 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
107
less than the pp source. The “7
Be” source refers to the pp chain electron capture
decay reaction
eÀ
þ 7
Be À! 7
Li þ ne
This reaction produces two groups of neutrinos, one in which the ground state of 7
Li
is populated (90% abundance) and one in which the 0.477-MeV excited state is
populated (10% abundance). The source “8
B” refers to the positron decay
8
B À! 8
BeÃ
þ eþ
þ ne
in which the first excited state of 8
Be (at 3.040 MeV) is populated. The weak
sources “13
N,” “15
O,” and “17
F” refer to bþ
decays that occur in the CNO cycle,
that is,
13
N À! 13
C þ bþ
þ ne
15
O À! 15
N þ bþ
þ ne
17
F À! 17
O þ bþ
þ ne
12.7.3 Detection of Neutrinos
As indicated above, the detection of these weakly interacting neutrinos is difficult
because of the low absorption cross sections. Two classes of detectors have been
used to overcome this obstacle, radiochemical detectors and Cerenkov detectors.
Radiochemical detectors rely on detecting the products of neutrino-induced
nuclear reactions, whereas the Cerenkov detectors observe the scattering of neutrinos.
The most famous radiochemical detector is that constructed by Davis and
co-workers in the Homestake Gold Mine in South Dakota. In a cavern about
1500 m below the surface of Earth, a massive detector, consisting of 100,000 gal
of a cleaning fluid, C2Cl4, was mounted. The cleaning fluid weighed 610 tons and
corresponded to the volume of 10 railway tanker cars. The nuclear reaction occurring
in the detector was
ne þ 37
Cl À! 37
Ar þ eÀ
The 37
Ar product nucleus decays by electron capture with a 35-d half-life. After the
cleaning fluid has been exposed to solar neutrinos for a period of time, the individual
37
Ar product nuclei are flushed from the detector with a stream of He gas and put
into a proportional counter where the 2.8-keV Auger electrons from the EC decay
are detected. The detection reaction has a threshold of 0.813 MeV, making it sensitive
to the 8
B, hep, pep, and 7
Be (ground-state decay) neutrinos with the 8
B being the
most important. Typically 3 atoms of 37
Ar are produced per week and must be isolated
from the 1030
atoms of cleaning fluid in the tank, a radiochemical tour de force.
12.7 SOLAR NEUTRINO PROBLEM 357
The detector was placed deep underground to shield against cosmic ray background
reactions.
The Davis or Cl detector was the detector used to define the “solar neutrino
problem,” and another type of radiochemical detector, the SAGE/GALLEX detectors,
was used to further define the problem. These detectors, GALLEX in Italy and
SAGE in Russia, are based on the reaction
ne þ 71
Ga À! 71
Ge þ eÀ
These detectors have a threshold of 0.232 MeV and can be used to directly detect the
dominant pp neutrinos from the sun. The gallium is present as a solution of GaCl3.
The 71
Ge is collected by sweeping the detector solution with N2 and converting the
Ge to GeH4 before counting. These detectors utilize 30–100 tons of gallium and
contain a significant fraction of the world’s yearly gallium production.
The Cerenkov detectors involve the scattering of neutrinos by charged particles,
causing the charged particles to emit Cerenkov radiation that can be detected by
scintillation detectors. The first of these detectors were placed in a mine at
Kamioka, Japan. The largest of the detectors at Kamioka is called Super Kamiokande
and consists of 50,000 tons of high-purity water. The detection reaction in
this case is a scattering reaction
n þ eÀ
À! n þ eÀ
and the detection threshold is about 8 MeV, allowing one to observe the 8
B
neutrinos.
The Sudbury Neutrino Observatory (SNO) detector, located at Sudbury, Ontario,
Canada, consists of 1000 tons of heavy water (D2O) mounted 2 km below the
surface in the Sudbury Nickel Mine. In addition to neutrino–electron scattering,
this detector can also utilize nuclear reactions involving deuterium:
ne þ d À! 2p þ eÀ
n þ d À! n þ p þ n
where, in the latter reaction, the reaction occurs for all types of neutrinos, ne, nm, and
nt. The former reaction is sensitive to electron neutrinos only. These different types
of reactions can be exploited to look for neutrino oscillations (see below). In the
latter reaction, the emitted neutron is detected by an (n,g) reaction in which the g
ray is detected by scintillation detectors. (The heavy water of the detector is
surrounded by 7000 tons of ordinary water to shield against neutrons from radioactivity
in the rock walls of the mine.) This detector also poses radiochemical
challenges as the water purity must be such that there are ,10 U or Th atoms per
1015
water molecules.
358 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
12.7.4 Solar Neutrino Problem
The solar neutrino “problem” was identified by the first results of Davis et al. using
the Cl detector at the Homestake Mine. Davis et al. observed only about one-third of
the expected solar neutrino flux as predicted by standard models of the sun, which
assume 98.5% of the energy is from the pp chain and 1.5% of the energy is from the
CNO cycle. The final result of the Cl detector experiment is that the observed solar
neutrino flux is 2.1 + 0.3 SNU compared to the predicted 7.9 + 2.4 SNU, where the
solar neutrino unit (SNU) is defined as 10236
neutrino captures/second/target atom.
The GALLEX and SAGE detectors subsequently reported solar neutrino fluxes of
77 + 10 SNU and 69 + 13 SNU, which are to be compared to the standard solar
model prediction of 127 SNU for the neutrinos detected by these reactions.
Such large discrepancies indicated that clearly either the models of the sun were
wrong or something fundamental was wrong in our ideas of the nuclear physics
involved.
12.7.5 Solution of the Problem—Neutrino Oscillations
The solution to the solar neutrino problem is that something is wrong with our ideas
of the fundamental structure of matter, the so-called standard model. This difficulty
takes the form of “neutrino oscillations” as the solution to the solar neutrino
Figure 12.19 Current status of the comparison between standard solar model predictions
and experimental measurements. [From Bahcall (from Bahcall website http://www.
sns.ias.edu/jnb).] (Figure also appears in color figure section.)
12.7 SOLAR NEUTRINO PROBLEM 359
problem. (The standard model predicts that the three types of neutrinos are massless
and that, once created, they retain their identity for all time.) The basic idea is that as
neutrinos come from the sun, they “oscillate,” that is, they change from being
electron neutrinos to muon neutrinos and back, and so on. This oscillation is possible
if the neutrinos have mass and there is a mass difference between the electron and
muon neutrinos and that they have mass. These neutrino oscillations are enhanced
by neutrino–electron interactions in the sun. It is believed that the masstneutrino .
massmneutrino . masselectron neutrino. The current upper limits on these masses are
m(ne) , 2:2 eV
m(nm) , 170 keV
m(nt) , 15:5 MeV
The direct observational evidence for the occurrence of neutrino oscillations came
from observations with the Cerenkov detectors. The SNO detector found one-third the
expected number of electron neutrinos coming from the sun in agreement with previous
work with the radiochemical detectors. The Super Kamiokande detector,
which is primarily sensitive to electron neutrinos, but has some sensitivity to other
neutrino types found about one-half the neutrino flux predicted by the standard
1
10–1
10–2
10–3
10–4
10 100 1 GeV 10 GeV
Flux
He
C
O
B
Li
N
Be
Figure 12.20 Energy spectrum of the GCR. (From J. Audouze and S. Vauclair, An
Introduction to Nuclear Astrophysics, Reidel, Dordrecht, 1980.)
360 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
solar models. If all neutrino types behaved similarly, the SNO and Super Kamiokande
detectors should have detected the same fraction of neutrinos. Further experiments
with the SNO detector operating in a mode to simultaneously detect all types of
neutrinos found neutrino fluxes in agreement with the solar models. This situation
is summarized in Figure 12.19.
12.8 SYNTHESIS OF Li, Be, AND B
Big Bang nucleosynthesis is responsible for the synthesis of hydrogen and helium
and some of the 7
Li. (Stellar nucleosynthesis in main sequence stars transforms
about 7% of the hydrogen into 4
He.) However, neither stellar nucleosynthesis or
Big Bang nucleosynthesis can produce the observed abundances of Li, Be, and
B. Consequently, the abundances of Li, Be, and B are suppressed by a factor of
106
relative to the abundances of the neighboring elements (Fig. 12.2).
4
10-6
10
-4
Be
Li
B
F
Sc
Ar
Fe
Ne
O
H
He
C
N
10
-2
10
2
104
10
6
10
0
RelativeAbundance
8 12 16
Atomic Number
20 24 28
Solar System
Cosmic Rays
Figure 12.21 Relativeelementalabundances inthesolarsystemandcosmicrays.(FromC.E.
Rolfs and N. S. Rodney, Cauldrons in the Cosmos, Chicago University Press, Chicago, 1988.)
12.8 SYNTHESIS OF Li, Be, AND B 361
The extremelylow abundance is theresult of two factors, therelative fragilityof the
isotopes of Li, Be, and B and the high binding energy of 4
He, which makes the isotopes
ofLi,Be,andB unstablewithrespect todecay/reactionsthatleadto4
He.Forexample,
the nuclei 6
Li, 7
Li, 9
Be, 11
B, and 10
B are destroyed by stellar proton irradiations at
temperatures of 2.0, 2.5, 3.5, 5.0, and 5.3 Â 106
K, respectively. Thus, these nuclei
cannot survive the stellar environment. (Only the rapid cooling following the Big
Bang allows the survival of the products of primordial nucleosynthesis.)
Li, Be, and B are believed to be produced in spallation reactions in which the
interstellar 12
C and 16
O interact with protons in the galactic cosmic rays (GCR).
These reactions are high-energy reactions with thresholds of 10–20 MeV. The
energy spectrum of the GCR is shown in Figure 12.20.
Typical cross sections for these spallation reactions are 1–100 mb for
Ep . 0.1 GeV. The time scale of the irradiation is 1010
y. The product nuclei
are not subject to high temperatures after synthesis and can survive. A further
testimonial to this mechanism is the relative abundances of the elements in the
GCR relative to the solar abundances (Fig. 12.21), which shows the enhanced
yields of Li, Be, and B in the GCR. This pattern is similar to the yield distributions
of the fragments from the reactions of high-energy projectiles.
PROBLEMS
1. Assume an absorption cross section of 10244
cm2
for solar neutrinos interacting
with matter. Calculate the probability of a neutrino interacting as it passes
through Earth.
2. What is the most probable kinetic energy of a proton in the interior of the sun
(T ¼ 1.5 Â 107
K). What fraction of these protons has an energy greater than
0.5 MeV?
3. If we want to study the reaction of 4
He with 16
O under stellar conditions, what
laboratory energy would we use for the 4
He?
4. If Earth was a neutron star, what would be its radius and density?
5. If the interior temperature of the sun is 1.5 Â 107
K, what is the peak energy of
the p þ 14
N À! 15
O þ g reaction.
6. Which nucleosynthetic processes are responsible for the following nuclei: 7
Li,
12
C, 20
Ne, 56
Fe, 84
Sr, 96
Zr, 114
Sn, 124
Sn, 209
Bi, 238
U.
7. Outline how you would construct a radiochemical neutrino detector based upon
115
In.
8. Estimate the Coulomb barrier height for the following pairs of nuclei: (a) p þ p,
(b) 16
O þ 16
O, and (c) 28
Si þ 28
Si.
9. Calculate the rate of fusion reactions in the sun. Be sure to correct for the energy
loss due to neutrino emission.
362 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
10. Assuming the sun will continue to shine at its present rate, calculate how long
the sun will shine.
11. From the data given on the Davis detector, and the assumption that the 37
Ar production
rate is 0.5 atoms/day, calculate the neutrino capture rate in SNU.
Assume the effective cross section for the 8
B neutrinos is 10242
cm2
.
12. Calculate the evolution of the n : p ratio in the primordial universe from the
information given as the temporal dependence of the temperature.
13. Make an estimate of the neutron-to-proton ratio in the center of the sun if the
only source of neutrons is thermal equilibrium of the weak interactions.
14. Using the information on the r-process and the s-process paths in Figures 12.14
and 12.15, make estimates of the average atomic numbers of the nuclei in the
peaks for N ¼ Z in the mass abundance curves. Do the masses of these nuclei
correspond to the peaks in Figure 12.4?
REFERENCES
Anders, E. and N. Grevasse. Geochim. Cosmochem. Acta 53, 197 (1989).
Burbidge, E. M., G. R. Burbidge, W. A. Fowler, and F. Hoyle. Rev. Mod. Phys. 29, 547
(1957).
Cameron, A. G. W. Atomic Energy of Canada, CRL-41.
Rolfs, C. E. and W. S. Rodney. Cauldrons in the Cosmos, Chicago University Press, Chicago,
1988.
Truran, J. W. Ann. Rev. Nucl. Part. Sci. 34, 53 (1984).
Weinberg, S. The First Three Minutes, Basic, New York, 1977.
Wong, S. S. M. Introductory Nuclear Physics, 2nd ed., Wiley, New York, 1998.
BIBLIOGRAPHY
Audouze, J. and S. Vauclair. An Introduction to Nuclear Astrophysics, Reidel, Dordrecht,
1980.
Bahcall J. N. (http://www.sns.ias.edu/jnb); a comprehensive account of the solar neutrino
problem and its solution is found at the website.
Bahcall, J. N. and C. Pena-Garay. New J. Phys. 6, 63 (2004).
Barnes, C. A., D. D. Clayton, and D. N. Schramm, Eds. Essays in Nuclear Astrophysics,
Cambridge University Press, Cambridge, 1982.
Choppin, G. R., J. O. Liljenzin, and J. Rydberg. Radiochemistry and Nuclear Chemistry,
Butterworth-Heinemann, Woburn, 2002.
Cottingham, W. N. and D. A. Greenwood. An Introduction to Nuclear Physics, Cambridge
University Press, Cambridge, 1986.
BIBLIOGRAPHY 363
Clayton, D. D. Principles of Stellar Evolution and Nucleosynthesis, Chicago University Press,
Chicago, 1983.
Dunlap, R. An Introduction to the Physics of Nuclei and Particles, Thomson, Toronto, 2004.
Ehmann, W. D. and D. E. Vance. Radiochemistry and Nuclear Methods of Analysis, Wiley,
New York, 1991.
Filippone, B. W. Ann. Rev. Nucl. Part. Sci. 36, 717 (1986).
Friedlander, G., J. W. Kennedy, E. S. Macias, and J. M. Miller. Nuclear and Radiochemistry,
3rd ed., Wiley, New York, 1981.
Heyde, K. Basic Ideas and Concepts in Nuclear Physics, IOP, Bristol, 1999.
Hodgson, P. E., E. Gadioli, and E. Gadioli-Erba. Introductory Nuclear Physics, Clarendon,
Oxford, 1997.
Kappeler, F., F.-K. Thielemann, and M. Wiescher. Ann. Rev. Nucl. Part. Sci. 48, 175 (1998).
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1988.
Nuclear Physics: The Core of Matter, The Fuel of Stars, NAS, Washington, 1999.
Reeves, H. Rev. Mod. Phys. 66, 193 (1994).
Viola, V. E. LiBeB Nucleosynthesis and Clues to the Chemical Evolution of the Universe,
in O. Manuel, Ed., Origin of Elements in the Solar System: Implications of Post-1957
Observations, Kluwer, New York, 2000.
Williams, W. S. C. Nuclear and Particle Physics, Clarendon, Oxford, 1991.
364 NUCLEAR REACTIONS IN NATURE: NUCLEAR ASTROPHYSICS
CHAPTER 13
ANALYTICAL APPLICATIONS OF
NUCLEAR REACTIONS
As mentioned previously (Chapter 4), one of the compelling reasons to use nuclear
analytical methods is their high sensitivity. The radiation from the decay or excitation
of a single nucleus can be readily detected. Even when one has to have the intervening
step of a nuclear reaction to produce or excite the decaying species, one still has
the ability to detect very small quantities of material. This chapter deals with those
nuclear analytical methods [activation analysis, particle-induced X-ray emission
(PIXE), Rutherford backscattering (RBS)] in which a nuclear reaction is the necessary
first step in the analysis procedure. The techniques to be discussed are known for
their sensitivity, the ability to do nondestructive analysis of a large number of
samples, sometimes quickly, and the ability to analyze the surfaces of materials.
All these techniques are elemental analysis techniques and do not, in general, give
information about the chemical form of the element, any attached ligands, and the
like. This lack of speciation information is a drawback of these methods.
13.1 ACTIVATION ANALYSIS
13.1.1 Basic Description of Method
Activation analysis is an analytical technique that allows one to determine the
amount of a given element X contained in some material Y. The basic steps in
the activation technique are as follows:
365
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
1. Irradiate Y with a source of ionizing radiation so that X will change into XÃ
, a
radioactive isotope of X.
2. Using chemical or instrumental techniques, “isolate” X and XÃ
from all other
elements in Y (not necessarily quantitatively) and measure the activity of XÃ
.
Chemical “isolation” of the activity of interest is performed simply by separating
it chemically from all other activities. Instrumental “isolation” of the
activity of interest involves the detection of radiation that can uniquely identify
the nuclide in question.
3. Calculate the amount of X present.
These basic steps are shown schematically for neutron activation analysis in
Figure 13.1.
How does one calculate the amount of X present, knowing the activity of X produced
in the irradiation? It can be shown that
AXÃ ¼ NXsf(1 À eÀlXÃ ti
)eÀlXÃ td
(13:1)
where AXÃ is the activity of XÃ
present at a time td after the end of the bombardment,
NX is the number of X nuclei present initially, s is the nuclear reaction cross section,
f is the flux of activating particles, ti is the length of the irradiation, and lXÃ is the
decay constant of XÃ
. From this equation one could calculate NX from AXÃ, knowing
all the other variables. (The above equation for AXÃ is valid for “thin targets,” that is,
samples that absorb ,5% of the flux of activating particles.)
This method of analysis is called absolute activation analysis and is done rarely.
The reasons for this are the need for detailed knowledge of the flux and energy of the
bombarding particles in the sample, the compounding of the uncertainties of our
knowledge of cross sections, decay branching ratios, and the like in the final
results. A simpler technique is to irradiate and count a known amount of pure X
under the same conditions used for the mixture of X inY. Then
Mass X in Y ¼ (known mass X)
activity of XÃ
in Y
activity of XÃ
in pure X
 
(13:2)
This is known as the comparator technique and is the most widely used method
of activation analysis. It depends on irradiating and counting standards of known
amounts of pure material using the same conditions as the samples being analyzed.
13.1.2 Advantages and Disadvantages of Activation Analysis
Since we know that A ¼ 1lN where A is the measured radioactivity, l is the decay
constant, N is the number of radioactive nuclei present, and 1 is a constant representing
the detection efficiency, we know that just a few radioactive nuclei need to be
present to give measurable activities. Use of activation analysis can lead to measurement
of elemental abundances of the order of 1026
–10212
g. The actual detection
366 ANALYTICAL APPLICATIONS OF NUCLEAR REACTIONS
sensitivities for activation analysis of various elements, as practiced by a commercial
activation analysis service, are shown in Figure 13.2. One can detect microgram
levels of over two-thirds of the elements using activation analysis.
Although the high sensitivity of activation analysis is perhaps its most striking
advantage, there are a number of other favorable aspects as well. Activation analysis
is basically a multielemental technique. Many elements in the sample will become
radioactive during the irradiation; and if each of these elements can be “isolated”
chemically or instrumentally, their abundances may be determined simultaneously.
Activation analysis can be a nondestructive method of analysis. Numerous tests have
shown that with careful experimental manipulation, activation analysis is an
accurate (1% accuracy) and precise (5% precision) method of measuring
elemental concentrations.
Figure 13.1 Schematic representation of activation analysis. [From Corliss (1963).]
13.1 ACTIVATION ANALYSIS 367
Activation analysis is not without its drawbacks, however. Among them are the
need to use expensive equipment and irradiation facilities, the inability to determine
the chemical state of the elements in question, the need to work with significant
levels of radioactivity, with their attendant radiation safety and legal problems,
the long times needed to complete some analyses, and complex analysis sometimes
needed to unscramble the g-ray spectra in a given experiment.
The ultimate test of the utility of activation analysis as an analytical technique is
whether there are competitive technologies that have the advantages of activation
analysis with fewer drawbacks. One candidate for this designation is inductively
coupled plasma–mass spectroscopy (ICP–MS).
The detection limits in ICP–MS are certainly equal to those achieved by activation
analysis. In addition, ICP–MS apparatus is frequently connected to ordinary
chemical separation apparatus, such as liquid chromatography (LC), thus allowing a
sensitive determination of both the amount and chemical species present for both
metals and nonmetals.
In recent years, there has been increasing use of ICP–MS techniques to replace
those of activation analysis, although there still are a large number of applications of
activation analysis each year, especially in the geological sciences.
13.1.3 Practical Considerations in Activation Analysis
To better understand the practical details of how activation analysis may be applied
to a given problem in elemental analysis, let us consider the various aspects of a
Figure 13.2 Table of activation analysis sensitivities as offered by General Atomic
Company, San Diego, California.
368 ANALYTICAL APPLICATIONS OF NUCLEAR REACTIONS
typical activation analysis problem. To make our discussion more concrete, let us
consider a specific problem, the measurement of the aluminum content of rocks
and meteorites (Loveland et al., 1969). The choice of this problem as an example
was dictated by its pedagogic simplicity and the fact that conventional chemical
analyses of aluminum in rocks are known to be inaccurate for low aluminum
concentrations and, in general, not very precise.
The first step in an activation analysis procedure is sample preparation. The
unknown and known samples (sometimes referred to as the unknown and standard
samples) should have the same size, composition, and homogeneity insofar as possible
to ensure that any attenuation of the incoming radiation, or the sample radiation
before counting, or any count rate-dependent effects, are exactly the same. In practice,
this step is accomplished by making sure that the unknown sample and known
sample have the same physical volume, are irradiated in a homogenous flux, and are
counted under exactly the same conditions (geometry, detector, etc.). Preirradiation
treatment of the sample should be kept to a minimum so as to lessen the possibility
of sample contamination. The standards are either aqueous solutions of the elements
in question or multielemental standard reference materials whose composition is
certified by a national or international agency (IAEA, U.S. National Institute of
Standards and Technology, etc.).
The second step in an activation analysis concerns the choice of nuclear reaction
to change X into XÃ
, plus the irradiation facility in which the reaction will be carried
out. In addition, the length of irradiation and decay prior to counting must be chosen
so the produced XÃ
activity is enhanced relative to all other activities produced.
Most activation analysis is done with thermal neutrons produced in nuclear reactors
for the following reasons:
1. Many elements have high cross sections for the absorption of thermal neutrons
in (n, g) reactions.
2. Copious fluxes of thermal neutrons (f  1012
n/cm2
/s) are available in
nuclear reactors.
3. Neutrons penetrate matter easily, and therefore there are few problems related
to attenuation of the neutron flux in the sample.
4. The major elements, carbon, nitrogen, and oxygen, are scarcely activated by
thermal neutrons, making detection of other elements easier.
Although most activation analysis is done with reactor thermal neutrons, several
other nuclear reactions and irradiation facilities can be used. Spontaneous fission of
252
Cf furnishes 3.8 neutrons per fission, and fluxes of up to 109
n/cm2
/s are available
from 252
Cf isotopic neutron sources. Cockroft–Walton accelerators can be
used to accelerate deuterons to energies of 150 keV, and then, using the 3
H(d, n)
reaction, 14 MeV neutrons can be produced ( fast neutrons). Typical neutron
generators of this type give fluxes of 109
n/cm2
/s of 14-MeV neutrons. These
fast neutrons are useful for activating the light elements, such as silicon, nitrogen,
fluorine, and oxygen, via (n, p) or (n, a) reactions, leading to sensitivities of
50–200 ppm and, thus, is complementary to slow neutron activation analysis.
13.1 ACTIVATION ANALYSIS 369
Charged particle or photon-induced reactions can also be used for activation. The
typical charged particles used are protons, deuterons, 3
He, and a particles. Charged
particle activation analysis (CPAA) is frequently complementary to neutron activation
analysis (NAA). NAA has poor sensitivity for the lighter elements,
whereas CPAA has good sensitivity. Because of the limited penetrating power of
charged particles in matter, CPAA either requires a thin sample or is used for
surface analysis. This attenuation of the primary radiation by the sample puts
especially stringent requirements on sample preparation.
Activation by photons (PAA) usually takes place via the (g, n) reaction, although
other reactions such as (g, p), (g, a), and the like are possible. Of special interest is
the determination of lead by PAA with a detection limit of 0.5 mg. [Lead is very
hard to detect using NAA (Fig. 13.2).] Photon sources are usually electron accelerators,
which produce high-energy photons through the bremsstrahlung process when
the electrons strike a heavy-metal target.
For the sample problem of determining the Al content of rocks, the activating
nuclear reaction was chosen to be 27
Al(n, g)28
Al, with the irradiation source
being a nuclear reactor. The 28
Al decays with a 2.2-min half-life and emits a b2
particle and a high-energy (1.78 MeV) g ray.
Even if you have chosen to irradiate a sample with thermal neutrons from a
nuclear reactor, you may be surprised to learn that several other neutron energies
may be present and cause reactions. For the popular TRIGA design of reactor,
only 25% of the neutrons at a typical irradiation position are “thermal” neutrons
(0 , En , 0.05 eV). The rest of the neutrons have higher energies, with neutrons
with 0.05 eV, En , 0.1 MeV being called epithermal neutrons and neutrons with
0.1 , En , 15 MeV being called fast neutrons. The capture cross sections for
epithermal neutrons frequently involve resonance capture (Chapters 10 and 11)
and can involve very large cross sections (.104
barns). Usually one uses epithermal
neutrons as the activating particle when one wants to avoid interfering activities in
the sample due to thermal neutron capture. For example, suppose a sample has a
large content of sodium. Sodium is easily activated via the 23
Na (n, g) reaction
giving rise to copious quantities of 15-h 24
Na in the sample, which may interfere
with the detection and measurement of other activities. How do we get rid of this
sodium? We can surround our sample with a metallic cadmium cover (0.1 cm
thick). Cadmium has a very large capture cross section for neutrons in the energy
region below 1.0 eV and effectively “cuts off” or removes these neutrons.
The resulting neutron flux in the sample consists of the higher energy (epithermal)
neutrons. Frequently, one measures a “Cd ratio” for activation of a specific element
to get some idea of how much of the produced activity is due to epithermal
activation. This Cd ratio, R, is defined as:
R ¼
(activity)no Cd cover
(activity)with Cd cover
(13:3)
Typical values of R range from 2 to 1000 depending on the reactor irradiation
position. Epithermal activation is advantageous for Ag, As, Au, Ba, Cs, Ga, In,
Mo, Pt, Rb, Sb, Se, Sr, Tb, Th, Tm, U, W, Zn, and Zr among others.
370 ANALYTICAL APPLICATIONS OF NUCLEAR REACTIONS
Once a nuclear reaction and an irradiation facility have been selected, the possibility
of interfering reactions must be carefully considered. This term means that
quite often, although X will change to XÃ
during the irradiation, some other
element Z may also change to XÃ
during the irradiation. Thus, the activity of XÃ
is
proportional to the abundances of Z and X in the sample, not just X. This effect is
referred to as an interfering reaction or interference, and a correction must be made
for it. In the case of the aluminum analysis, there is a very important interference—
namely the occurrence of the 28
Si(n, p)28
Al reaction whereby silicon in the rock is
converted into 28
Al by reactions involving fast neutrons present in any reactor
(along with the desired thermal neutrons). Thus, the measured 28
Al activity will be
due to the activation of 27
Al and 28
Si. By irradiating a known amount of silicon
and counting it, and from the well-known Si abundances of rocks, a correction for
the 28
Al produced by the 28
Si(n, p)28
Al reaction can be calculated. Other possible
interferences are the fission of any uranium in the sample, or the occurrence of
two nuclides that emit g rays that have similar energies that cannot be resolved.
The final decision concerning irradiation conditions involves the determination
of the flux and irradiation duration. A rough rule is that the longer one irradiates
the sample and the longer one lets the sample decay before counting, the greater
the activity of the long-lived species relative to the short-lived species. One must
keep in mind the saturation properties of irradiations are such that it rarely pays
to irradiate any material for a time corresponding to more than two half-lives of
the desired activity. (In the Al analysis, a sample irradiation time of 1.0 min and a
neutron flux of 5 Â 1010
n/cm2
/s were used.)
Frequently, multiple irradiations of a sample are made. The first irradiation is
short (minutes) to determine the short-lived radioisotopes (of Ag, Al, Ba, Br, Ca,
Cl, Co, Cu, Dy, F, I, In, K, Mg, Mn, Na, Se, Sb, Si, Sr, Ti, U, and V), and the subsequent
irradiations (hours) are to determine the intermediate (As, Au, Br, Cd, Ga,
Ge, Hg, Ho, K, La, Mo, Na, Pd, Sb, Sm, U, W, and Zn) or long-lived (Ag, Ce, Cr, Cs,
Co, Eu, Fe, Hf, Hg, Lu, Nd, Ni, Rb, Sb, Sc, Se, Sn, Sr, Ta, Tb, Th, Tm, Yb, Zn, and
Zr) radionuclides. In the long irradiations, it is common to let the sample “decay” for
several days to get rid of the 15-h 24
Na.
The next major step in any activation analysis procedure is the selection of a
method of isolating the activity of interest, XÃ
, to measure it. Two methods of
isolating XÃ
are commonly used—instrumental activation analysis (IAA) and radiochemical
activation analysis (RAA). In instrumental activation analysis, the characteristic
energies of the g rays emitted by the radionuclides in the activated sample are
used to identify them, and the corresponding photopeak areas give a measure of the
activities. Instrumental activation analysis is nondestructive, allowing further use of
the sample. Furthermore, it permits the use of short-lived activities to identify
various elements that might not be possible if a lengthy chemical separation
would precede the counting. Also, instrumental activation analysis (IAA) lends
itself to automation and reduces the time spent per sample in the analysis. The
use of Ge semiconductor detectors with excellent energy resolution has made
IAA the preferred method of activation analysis.
Although most investigators prefer to use IAA, in some situations radiochemistry
must be done prior to counting the sample, to isolate the activity of interest.
13.1 ACTIVATION ANALYSIS 371
An example of the need for radiochemistry is the determination of trace elements in
biological materials, such as blood, which have a very high sodium content. Large
quantities of 24
Na are produced via the 23
Na(n, g)24
Na reaction, and they tend to
“mask” the trace element activities in the blood by creating a large Compton background
in the region where the photopeaks of other trace-element activities are
found (see the discussion in Chapter 18 on g-ray detectors). One solution to this
problem is to separate the sodium chemically from the irradiated blood (using ion
exchange with hydrated antimony pentoxide) and then to instrumentally analyze
the purified blood. This example does illustrate a feature of modern radiochemical
activation analysis—that of not completely separating the element of interest but of
making a group separation of a relatively small number of activities and further
resolving these activities by g-ray spectroscopy.
All of our discussions up to now have focused on detecting the g rays from the
decaying activation products. There is another approach that has been used in
some cases. This approach is called prompt g-ray activation analysis (PGAA) in
which one detects the prompt g radiation emitted during the activating reaction.
For neutron activation via the (n, g) reaction, one detects the g rays emitted
during the neutron capture. Such analyses must be carried out with beams of
activating particles (such as neutrons) and usually involves detecting high-energy
(.5 MeV) g rays. Because of these constraints, this rapid analysis method is restricted
usually to the determination of the major elemental constituents of the sample.
13.1.4 Applications of Activation Analysis
The applications of activation analysis are almost innumerable. In the physical
sciences, activation analysis is used in trace-element analysis of semiconductor
materials, metals, meteorites, lunar samples, and terrestrial rocks. In most cases,
the multielemental analysis feature of activation analysis is used to measure the concentrations
of several trace-elements simultaneously. From these detailed studies of
trace-element abundance patterns, one has been able to deduce information about
the thermal and chemical history of the Earth, moon, Mars, and meteorites, as
well as the source or age of an object.
The use of activation analysis in criminal investigations ( forensic activation
analysis) is also well established. The basic idea here is to match the traceelement
distributions found in bullets, paint, oil, and so on found at the scene of a
crime with the trace-element distributions in objects found with criminal suspects.
Such identification is rapid and nondestructive (allowing the actual evidence to be
presented in court). Moreover, the probability of its correctness can be ascertained
quantitatively. Other prominent examples of the use of forensic activation analysis
involve confirmation of the notion that Napoleon was poisoned (by finding significant
amounts of arsenic in hair from his head) and the finding that the activation
analysis of the wipe samples taken from a suspect’s hand can reveal not only if
he or she has fired a gun recently but also the type of gun and ammunition used.
Applications of activation analysis in the environmental sciences are routine.
Determinations of the trace element content of urban atmospheres, lakes, streams,
372 ANALYTICAL APPLICATIONS OF NUCLEAR REACTIONS
and similar areas have been used to trace the flow of pollutants in various ecosystems.
In addition, a few of the trace elements whose abundances have been measured by
activation analysis have turned out to be biologically significant by themselves.
The classic example is mercury and the significant mercury concentration in fish
and other foodstuffs revealed by activation analysis. A particular combination of activation
analysis and radiotracer methods has found important applications in the
environmental sciences. This combination involves the use of stable isotopes
instead of radioactive isotopes as tracers in various systems, with activation analysis
of the samples collected after tracer dispersal being used to measure the tracer concentrations.
Such a technique avoids the need to introduce radioactive materials into a
system (such as the environment with its subsequent health and legal complications)
and yet retains the selectivity and sensitivity of radiation measurements. The stable
isotopes are called stable activable tracers. Kruger (1971) has described their use.
In summary, activation analysis is a multielemental, nondestructive, very accurate
method of analysis. The best-case sensitivities are picograms per gram with
an irregular variation from element to element. It is best suited for the analysis of
solid samples and can be “tuned” using changes in irradiation conditions, particles,
and the like, and postirradiation sample treatment. Disadvantages are the long analysis
times, the need for access to an irradiation facility (usually a reactor), the need to
handle radioactivity, the labor-intensive nature of sample counting, and the inability
to get speciation information.
13.2 PARTICLE-INDUCED X-RAY EMISSION
Particle-induced X-ray emission (PIXE) is an analytical technique based upon
observing fluorescent X-rays. As such, it really is not a nuclear technique since it
involves an atomic process, X-ray emission. But the atomic electron shell vacancies
that are filled when the X-ray is emitted are created using particle-accelerator beams
and one uses typical semiconductor radiation detectors, Si (Li) detectors, to detect
the X-rays.
The essential features of a PIXE setup are shown schematically in Figure 13.3. A
beam of charged particles from an accelerator, typically 2- to 4-MeV protons,
impinges on a thin sample in a vacuum chamber. The protons collide with the electrons
in the material and some eject inner shell electrons from the atoms in the
sample. A Faraday cup is used to collect the charge deposited by the incident
protons, and this is integrated electronically to give the beam current. The sample
is typically a thin, uniform deposit of the material to be analyzed on a thin
backing material. The characteristic X-rays from the sample are detected with a
Si (Li) detector. A typical spectrum is shown in Figure 13.4. The spectrum consists
of discrete X-ray peaks superimposed on a continuous background of bremsstrahlung.
One can see the Ka and Kb lines of the lighter elements (from the filling of
the K shell vacancies) and the L lines of the heaviest elements. The peaks corresponding
to a given element are integrated to give peak areas and the amounts of
that element obtained either from a knowledge of the absolute ionization cross
13.2 PARTICLE-INDUCED X-RAY EMISSION 373
Figure 13.3 Schematic diagram of a PIXE setup. [Reproduced from Ehmann and Vance
(1991).]
Figure 13.4 PIXE spectrum of a rainwater sample. [From Johansson and Johansson
(1976).]
374 ANALYTICAL APPLICATIONS OF NUCLEAR REACTIONS
sections (1–104
barns), fluorescence yields (0.1–0.9), beam current, and geometry
or by comparison to the results obtained from a thin elemental standard. The term
fluorescence yield refers to the fraction of the electron vacancies filled by X-ray
emission vs. the ejection of Auger electrons.
Typical detection limits for various elements in a biological sample are shown in
Figure 13.5. Typically, PIXE has sensitivity at the parts per million level for many
elements. About 25% of the applications of PIXE are in biology and medicine. The
light-element matrices lead to smaller continuous backgrounds, and many trace and
toxic elements are easily detected by PIXE. (There are no “holes” in detection limits
as there are in activation analysis as all the elements emit some X-rays.) Considerable
attention has been and must be devoted to the preparation of thin, representative
samples. Note that PIXE is only sensitive to the elemental composition of the sample
and not to the isotopic composition.
One of the most successful applications of PIXE has been in the analysis of air
pollution particulate matter. Atmospheric particulate matter is typically collected
by impaction on a filter paper, which provides an ideal thin sample for PIXE analysis.
Another aspect of PIXE that is very important for the analysis of aerosol samples
is the ability to analyze a large number of samples in a short time. PIXE analyses
typically take less than a minute, and the entire irradiation, counting, sample
changing, and analysis procedure can be automated.
Figure 13.5 Detection limits in a PIXE analysis of a biological sample. [From Ishii and
Morita (1990).]
13.2 PARTICLE-INDUCED X-RAY EMISSION 375
An important variant on PIXE is micro-PIXE. By using a proton beam whose
spatial dimension is 0.5 mm (rather than the usual 10 mm), one can determine
the trace-element content of a small portion of the sample, giving one a
“trace-element microscope.” This application is important in probing samples of
medical interest. A related technique is used in the electron microprobe where the
ionization is caused by electron impact.
13.3 RUTHERFORD BACKSCATTERING (RBS)
One of the earliest experiments in nuclear physics was Rutherford’s demonstration
of large-angle scattering of a particles by gold nuclei. This experiment established
the existence of a small nucleus within the atom (Chapter 10). The force acting
in this process, called Rutherford scattering, is the repulsive Coulomb force
between the positively charged nuclei. A schematic diagram of the phenomena is
shown in Figure 13.6.
Rutherford scattering is an elastic event, that is, no excitation of either the projectile
or target nuclei occurs. However, due to conservation of energy and momentum
in the interaction, the kinetic energy of the backscattered ion is less than that of
the incident ion. The relation between these energies is the kinematic factor, K,
which is given by the expression
K ¼
E1
E0
¼
(M2
2 À M2
1 sin2
u)1=2
þ M1 cos u
M1 þ M2
 2
(13:4)
where M1 and M2 are the masses of the incident and target atoms, respectively, and u
is the angle between the direction of the incident and scattered ions. Note the relative
shift in energy in the collision depends only on the masses of the ions and the angle
of the detector. If one measures the scattering angle and the energy shift, one can
calculate the mass (identity) of the scattering atom. The largest change in energy
Figure 13.6 Schematic diagram of Rutherford backscattering. [From Rauhala (1994).]
376 ANALYTICAL APPLICATIONS OF NUCLEAR REACTIONS
occurs for u ¼ 1808 where
K ¼
M2 À M1
M2 þ M1
 2
(13:5)
A geometry that allows detection of the scattered a particles at very large angles is
usually selected.
The probability or cross section for Rutherford scattering (Chapter 10) is given
(Segre, 1977) as:
ds
dV
¼
Z1Z2e2
4E
 2
4
sin4
u
½cos u þ (1 À x2
sin2
u)1=2
Š2
(1 À x2 sin2
u)1=2
(13:6)
where x ¼ M1/M2, e2
is the square of the electronic charge, and E is the energy of
the incident ion. Note the probability of scattering goes as (Z1Z2)2
and as 1/E2
. If
this were all that went into Rutherford backscattering, we would expect a spectrum
of backscattered particles that consisted of a peak for each element in the sample
with a relative height (area) /Z2
. The elemental abundances could be calculated
using the relation
N ¼
D
F
ð
ds
dV
dV
(13:7)
where N is the number of target atoms, D is the number of detected events, and F is
the incident ion flux. This is the situation if one has a very thin film as the target
material or if one scatters particles from the surface of a thick sample.
In reality, the situation is usually more complicated because the incident ions lose
energy as they penetrate into the sample, thus continuously changing the probability
of scattering and the energies of the scattered particles. The resulting spectrum for
scattering from a single element at varying depths is shown in Figure 13.7, where the
Figure 13.7 Energy depth scale in Rutherford backscattering. [From Rauhala (1994).]
13.3 RUTHERFORD BACKSCATTERING (RBS) 377
incident ion energy is E0, the energy of ions scattered from the surface is KE0, and
the energy of ions scattered from a depth x is E1. In this situation, the energy loss in
traversing (into and back out of) a foil of thickness Nx is
DE ¼ KE0 À E ¼ ½1ŠBSNx (13:8)
½1ŠBS ¼
K
cos u1
1in þ
1
cos u2
1out (13:9)
where 1in and 1out are the energy-dependent stopping cross sections (Ziegler) on the
inward and outward paths of the ion.
Rutherford backscattering is an important method for determining the composition
and structure of surfaces and thin films. In Figure 13.8, we show the results
of an RBS measurement with 2.0-MeV 4
He incident on an Si surface with a Co
impurity that was diffused into the bulk material. One can clearly detect the Co
and its depth profile.
Another important application of this technique has been to determine the
elemental composition of the lunar and Martian surfaces. Turkevich et al. (1969)
constructed a rugged device to measure the backscattering of a particles from the
lunar surface, which flew on three Surveyor missions in 1967–68 and yielded
the first complete and accurate analysis of the lunar surface. The a particles came
from a radioactive source (242
Cm) that was part of the instrument package. The
results of these experiments, which showed an unexpected and comparatively
high abundance of Ti, were confirmed by laboratory analysis of lunar samples
gathered in the Apollo missions. Since then, this technique has been used to study
Martian rocks and soil.
Figure 13.8 Rutherford backscattering for 2.0 MeV 4
He ions incident on a Si (Co) sample.
Dots represent the experimental data while the solid line is a simulated spectrum. Scattering
angle Q ¼ 1708, with u1 ¼ u2 ¼ 58. [From Saarilahti and Rauhala (1992).]
378 ANALYTICAL APPLICATIONS OF NUCLEAR REACTIONS
PROBLEMS
1. For each of the following analyses, indicate what role, if any, activation analysis
could or should play. Be sure to clearly state the reasons for your choice.
(a) Determination of the oxygen content of steel, (b) verification of the authenticity
of ancient paintings, (c) determination of the radionuclides present in
fallout from nuclear weapons testing, (d) determination of the extent to
which radionuclides leaking from nuclear waste storage facilities contaminate
the water of nearby streams, and (e) determination of lithium impurities
in thin films of GaAs.
2. (a) Calculate the activity (in microcuries) of 49
Ca produced when 2.7 g of CaO
are irradiated in a flux of 3 Â 1012
n/cm2
-s for 10 min. (b) Repeat this calculation
for the situation when the bombarding particle is 21 MeV deuterons, and
the deuteron beam current is 10 mA. Assume the (d, p) cross section is 50 mbarns.
3. Using the Chart of the Nuclides as a guide, estimate the sensitivity (minimum
quantity that can be detected) of neutron activation analysis for europium
using a thermal neutron flux of 3 Â 1012
n/cm2
-s. Assume no irradiation may
last more than 1 h and the minimum detectable activity is 10 dpm.
4. For the following analyses, indicate whether radiochemical neutron activation
analysis would be preferred to instrumental neutron activation analysis. If radiochemistry
is indicated, briefly sketch the separation procedures to be used: (a)
the determination of ppm levels of Mo in flathead minnows, (b) the determination
of the trace-element content of agricultural field-burning particulate
matter, (c) the use of stable activable tracers to determine flow patterns in an
ocean estuary, and (d) the determination of Dy in pine needles.
5. Consider you want to trace the deposition of particulate matter using the stable
activable tracer In. The dilution factor between the point of release and the point
of sampling is 106
. Assume the samples that are collected are activated in a
thermal neutron flux of 3 Â 1012
n/cm2
-s for 10 min. Further assume a 1%
efficiency for detecting the emitted photons. Determine the minimum amount
of In that must be released to ensure the uncertainty in the measured sample
concentrations is 5%.
6. Consider the following results obtained by neutron activation analysis of lake
water samples for their Mn content. Assume the sample volumes are 1 L.
Sample EOB Activity (cp5m)
1 1204
2 1275
3 940
4 1350
10 mg Mn standard 5000
What is the Mn content of the lake water and its uncertainty?
PROBLEMS 379
7. Two thin 1-mg samples of dysprosium are irradiated and counted in a similar
manner, except for the use of a Cd cover foil on one sample. A Cd ratio of 7
is measured, with the bare foil saturation activity of 1 Â 104
dpm. Calculate
the thermal neutron flux at the irradiation position in the reactor.
8. Devise an activation analysis scheme for determining the concentration of nitrogen
in a sample of plant material. Assume the analysis must be nondestructive
and rapid. Suggest an appropriate reaction, irradiation, and counting conditions,
and indicate possible interferences in your analysis.
9. Compute the “advantage factor” for using a reactor pulse to produce 20-s 46
Scm
compared to the activity produced by steady-state irradiation. Assume the
reactor is of the TRIGA type and produces a 15-ms 3000-MW pulse with a
peak instantaneous flux of 21 Â 1015
n/cm2
-s. Assume steady-state operation
is at 1 MW.
10. Imagine you wish to detect ppm levels of Al in a matrix containing iron,
calcium, and silicon. Assume you have access to a modern nuclear reactor.
Describe an activation analysis procedure to do this analysis. Be sure to describe
the irradiation conditions, any pre- or postirradiation chemistry, and the
counting strategy. Indicate how you would deal with any interferences in the
analysis.
REFERENCES
Corliss, W. R. Neutron Activation Analysis, USAEC, 1963.
Ehmann, W. D. and D. E. Vance. Radiochemistry and Nuclear Methods of Analysis, Wiley,
New York, 1991.
Ishii, K. and S. Morita. Int. J. PIXE 1, 1 (1990).
Johansson, S. A. E. and T. B. Johansson. Nucl. Instr. Meth. 137, 473 (1976).
Kruger, P. Principles of Activation Analysis, Wiley, New York, 1971.
Loveland, W., R. A. Schmitt, and D. E. Fisher. Geochim. Cosmochim. Acta 33, 375 (1969).
Rauhala, E. In Z. B. Alfassi, Ed., Chemical Analysis by Nuclear Methods, Wiley,
Chichester, 1994.
Saarilahti, J. and E. Rauhala. Nucl. Instr. Meth. Phys. Res. B B64, 734 (1992).
Segre, E. Nuclei and Particles, 2nd ed., Benjamin, Reading, 1977.
Turkevich, A., E. F. Franzgrote, and J. H. Patterson. Science 165, 277 (1969).
BIBLIOGRAPHY
Alfassi, Z. B., Ed. Chemical Analysis by Nuclear Methods, Wiley, Chichester, 1994. A series
of essays on various aspects of nuclear analytical chemistry. Most of them are quite good.
Brune, D., B. Forkman, and B. Persson. Nuclear Analytical Chemistry, Chartwell-Bratt,
London, 1984.
380 ANALYTICAL APPLICATIONS OF NUCLEAR REACTIONS
de Soete, D., R. Gijbels, and J. Hoste. Neutron Activation Analysis, Wiley, New York, 1974.
An encyclopedic work.
Hughes, D. J. Pile Neutron Research, Addison-Wesley, Cambridge, 1953. The bible (old
testament) of reactor neutron physics.
Johansson, S. A. E. and J. L. Campbell. PIXE: A Novel Technique for Elemental Analysis,
Wiley, Chicester, 1988.
Johansson, S. A. E., J. L. Campbell, and K.-G. Malmqvist. PIXE, Wiley, New York, 1995.
Ziegler, J. F., P. J. Scanlon, W. A. Lanford, and J. L. Duggan, Eds. Ion Beam Analysis,
North-Holland, Amsterdam, 1990.
BIBLIOGRAPHY 381
CHAPTER 14
REACTORS AND ACCELERATORS
Radioactive decay is the only nuclear reaction that commonly takes place on Earth.
The reasons that other nuclear reactions do not normally occur on Earth are simple.
Nuclear reactions that are induced by protons or heavier charged particles all have
large activation barriers and require energetic charged particles that are only present
in space and the highest regions of the atmosphere. On the other hand, nuclear reactions
induced by neutrons do not have an activation barrier, but neutrons are
unstable, decaying by b2
decay into protons with a half-life of approximately
10 min. Thus, neutrons cannot be stored very long and have to be produced in
other nuclear reactions before they can be used in subsequent nuclear reactions.
Protons and all nuclei are positively charged and strongly repel one another
through the Coulomb force. Colliding nuclei must have kinetic energies that are
far in excess of normal thermal energies to reach distances that are short enough
for the nuclear force to be effective (1 fm). We must accelerate one of the particles
until it has sufficient kinetic energy to get over the Coulomb barrier for the nuclei to
react.
The fact that neutrons can be absorbed by nuclei without overcoming a threshold
(l ¼ 0 or s-wave reactions) makes neutrons extremely effective nuclear “reactants.”
Neutron-induced reactions are the energy source for present-day commercial
nuclear power (fission reactors) while charged-particle-induced reactions remain
under study as power sources (fusion reactors). In this chapter we will consider
the general features of nuclear fission reactors, following by the general features
383
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
of charged particle accelerators, magnetic spectrometers, and the production of
beams of radioactive nuclei.
Example Problem A very simple and potentially useful fusion reaction combines
two deuterium nuclei forming 3
He and a neutron.
2
H þ 2
H À! 3
He þ 1
n þ Q
Estimate the Coulomb barrier for this reaction and the temperature of deuterium gas
that would give such an average energy.
Solution The Coulomb barrier, discussed in Chapter 10, is
VCoulomb ¼
z1z2e2
r
¼
(1)(1)1:439 MeV-fm
1:93 þ 1:93 fm
¼ 0:373 MeV
Setting the kinetic energy equal to the Coulomb barrier as would be appropriate
when all the particles are moving (in a gas), and using the thermal energy of an
ideal gas:
VCoulomb ¼ KE ¼ 1
2 mv2
¼ 3
2 kT
T ¼
2VCoulomb
3k
¼
2(0:373 MeV)1:602 Â 10À13
J=MeV
3(1:38 Â 10À23 J=K)
T ¼ 3 Â 109
K
This extraordinary temperature is characteristic of the interiors of stars and not of
terrestrial objects.
14.1 NUCLEAR REACTORS
14.1.1 Neutron-Induced Reactions
Nuclear fission reactors (“nuclear power reactors”) are devices that use controlled
neutron-induced fission to generate energy. While a complete description of the
design of these devices is beyond the scope of this book, there are certain basic principles
related to nuclear reactors that are worth studying and that can be described
and understood with a moderate effort.
Let us begin by reminding ourselves about the energy dependence of the cross
section for neutron-induced reactions. In Figure 14.1, we show the (n, f) cross
section for 235
U and 238
U as a function of neutron energy. In examining Figure 14.1,
we see that the highest cross section for fission of 235
U occurs at very low energies,
so-called thermal energies where En , 1 eV. As discussed in Chapter 10, the cross
section varies approximately as 1/v for these neutron energies. Thermal neutrons
are neutrons that have come into thermal equilibrium with the surroundings.
384 REACTORS AND ACCELERATORS
Example Problem What is the velocity and de Broglie wavelength of a
thermal neutron?
Solution The Maxwell–Boltzmann velocity distribution for the random motion
of a thermally equilibrated neutron gas is
n(v) ¼ 4pv2 m
2pkT
h i3=2
eÀmv2
=2kT
where we have normalized the function so that
ð1
0
n(v) dv ¼ 1
The most probable velocity is
vmp ¼ (2kT=m)1=2
If T ¼ 208C, then
vmp ¼
2(1:38 Â 10À23
J=K)293K
1:675 Â 10À27kg
 1=2
vmp ¼ 2200 m=s
Figure 14.1 Neutron-induced fission cross section for 235
U and 238
U as function of the
neutron energy, En. (From D. T. Hughes and R. B. Schwartz, Neutron Cross Sections, 2nd
ed., Brookhaven National Laboratory Report 325, 1958.)
14.1 NUCLEAR REACTORS 385
This velocity, 2200 m/s, is taken as the characteristic velocity of thermal neutrons,
and cross sections for neutrons of velocity 2200 m/s (En ¼ 1/
2mv2
¼ 0.0253 eV) are referred to as “thermal” cross sections.
The wavelength is
ldeBroglie ¼
h
p
¼
6:626 Â 10À34
J=s
(1:675 Â 10À27 kg)(2200 m=s)
ldeBroglie ¼ 1:80 Â 10À10
m
Notice the de Broglie wavelength of thermal neutrons is much larger than the size
of a typical nucleus (r  1–10 Â 10215
m) and similar to the size of a typical
atom. Reaction cross sections for thermal neutrons exceed the geometrical area
of the nucleus.
Two other important features of Figure 14.1 deserve further comment. The first of
these features is the large difference between the excitation functions for (n, f) reactions
with 235
U and 238
U. We can understand this difference by noting the Q values
for neutron capture by these nuclides are
235
U þ n À! 236
U þ 6:54 MeV
238
U þ n À! 239
U þ 4:80 MeV
Note the Q value for the n þ 235
U reaction is 1.7 MeV larger. The reaction with the
lighter isotope converts an even–odd nucleus into an even–even nucleus, where as
the other reaction creates an even–odd product. Roughly, we would then expect the
Q values for these reactions to differ by twice the neutron pairing energy. Since the
fission barriers for 235
U and 238
U are about the same (Bf  6.2 MeV), capture of neutrons
of any energy will cause 235
U to fission while it takes 1.4 MeV neutrons to
cause 238
U to fission. The “thermally fissionable” nuclei are thus all even–odd
nuclei where the energy release in neutron capture is greater than the fission
barrier. The most important of these nuclei from a practical point of view are the
“big three,” 233
U, 235
U, and 239
Pu.
The other feature of Figure 14.1 worthy of comment is the different regions of
neutron energies and the associated cross sections. Neutrons with energies ,1 eV
exhibit 1/v cross-section behavior, and this region is referred to as the “thermal”
region. Epithermal neutrons have energies from 1 to 100 eV, and their reactions
are characterized by large resonances in the cross section. In the neutron energy
region from 100 eV to 1 MeV, the energy levels of the excited states in the nuclei
overlap and there are no discrete resonances. Neutrons with energies greater than
1 MeV are referred to as “fast” neutrons and they can cause 238
U to fission.
As discussed in Chapter 10, there are other reaction mechanisms besides fission
when neutrons interact with heavy nuclei. They include: (a) elastic scattering where
Q ¼ 0 and kinetic energy is conserved. However, the target nucleus recoils in each
386 REACTORS AND ACCELERATORS
event and the elastically scattered neutron loses some energy. (For a collision of a
neutron of energy E with a nucleus containing A nucleons, its energy after the
collision will be ðA2
þ 1Þ=(A þ 1)2
E. Note the maximum neutron energy loss
will occur when A ¼ 1 when the neutron energy will be halved in each collision.
(b) Inelastic scattering where the neutron gives up some of its energy leaving the
struck nucleus in an excited state. The threshold energy for this process in the cm
system will be the energy of the first excited state of the struck nucleus. For 235
U
and 238
U, these energies are 14 and 44 keV, respectively. (c) Radiative capture,
that is, the (n, g) reaction in which part of the energy released by the capture of
the neutron is carried away by the emitted photon. The total cross section, stotal,
is the sum of the cross sections for these processes, that is,
stotal ¼ selas þ sinel þ sn,g þ sf
The distance neutrons travel between interactions, the mean free path l is given as:
l ¼ 1=stotalr
where r is the number density of nuclei. For uranium metal, r ¼ 4.8 Â 1028
/m3
,
and, if we assume stotal ¼ 7 barns, then l ¼ 0.03 m. If the average neutron
energy is 2 MeV, the time between interactions will be 1028
s. This mean free
path will also constrain the size of a self-sustaining assembly of fissionable material.
14.1.2 Neutron-Induced Fission
Let us review some aspects of fission discussed in Chapter 11. Consider the case of
the thermal neutron-induced fission of 235
U, that is,
1
0n þ 235
92 U À! 236
92 U À! A
ZX þ 236ÀA
92ÀZ Y
The two fission fragments X and Y will have a total kinetic energy of 168 MeV due
to their mutual Coulomb repulsion at scission with the lighter fragment carrying
away the larger energy due to conservation of momentum. The most probable
mass split will be an asymmetric one with Aheavy/Alight  1.3 – 1.4. Following
scission, the deformed fragments will contract to a more spherical shape, heating
up in the process. They will get rid of this excess energy by the emission of neutrons,
emitting 2.5 neutrons per fission event. These neutrons will have a Watt spectrum,
that is, a broad peak centered below 1 MeV with an energy distribution of the form:
N(E) ¼ 0:453eÀ1:036E
sinh ½(2:29E)0:5
Š
The total energy carried away by these neutrons is 5 MeV. In competition with the
last stages of neutron emission and when the excitation energies of the fission fragments
are less than the neutron binding energy, the fragments will deexcite by g-ray
14.1 NUCLEAR REACTORS 387
emission (the “prompt” g rays) with the energy carried away by these g rays being
8 MeV. Following prompt g-ray emission, one will be left with neutron-rich fragments
that will decay by b2
and g-ray emission toward stability. About 8 and
7 MeV will be emitted in the form of b2
particles and g rays, respectively. About
12 MeV will appear in the form of electron antineutrinos emitted in b2
decay;
but these neutrinos will escape any practical assembly. During this b2
and g
decay, in a small number of decays, the residual nucleus following b2
decay is
excited to an energy greater than the neutron binding energy. Such nuclei will
decay by emission of neutrons (the so-called delayed neutrons) on a time scale of
seconds/minutes rather than the time scale of prompt neutron emission
(10215
s). While the energy carried away by these delayed neutrons is insignificant
in the fission energy balance, these neutrons are very important in controlling the
chain reaction in reactors
For the thermal neutron-induced fission of 235
U, the total recoverable energy
release is 195 MeV/fission while it is 202 MeV for the fission of 239
Pu. These
energy releases can be transformed into practical units by noting that
200 MeV ffi 3.2 Â 10211
joules. One gram of 235
U contains about 3 Â 1021
atoms,
corresponding to an energy release of 3.2 Â 10211
 3  1021
¼ 1 MW/d. (The
burning of one ton (106
g) of coal releases about 0.36 MW/d, so that 1 g 235
U has
an energy content of about 3 Â 106
more energy than 1 g of coal.)
14.1.3 Neutron Inventory
A reactor designer must pay special attention to the inventory of neutrons in the
reactor. Each fission event in the thermal neutron-induced fission of 235
U produces
2.5 fast neutrons. Upon examining the data of Figure 14.1, we conclude that the
energy of these neutrons should be reduced to thermal energies to induce further fissions.
Thus, we place lumps of the uranium fuel in a moderator that slows down the
fission neutrons to thermal energies. Previously, we have demonstrated that neutrons
are most rapidly slowed by elastic collisions with light nuclei. So the ideal moderator
will contain light nuclei whose neutron capture cross sections are low. Graphite
is an appropriate material as is 2
H2O or Be. Ordinary hydrogen has an (n, g) cross
section of 0.33 barn for thermal neutrons, making it unsuitable as a moderator
unless the fuel is enriched in 235
U. The number of collisions necessary to thermalize
the fast neutrons from fission is 14.5 for 1
H, 92 for 12
C, and 1730 for 235
U.
Of the fast neutrons produced in fission, some of them will be “moderated” to
thermal energies and will induce other fission reactions while others will be
“lost.” The ratio of the number of neutrons in the next generation to that in the previous
generation is called the multiplication factor k. If the value of k is less than 1,
then the reactor is subcritical and the fission process is not self-sustaining. If the
value of k is greater than 1, then the number of fissions will accelerate with time
and the reactor is supercritical. The goal of reactor operation is to maintain the
system in a critical state with k exactly equal to 1. The extreme upper limit
for the multiplication factor would correspond to the mean number of neutrons
per fission (2.5 for 235
U(n, f)) if each neutron produces a secondary fission.
388 REACTORS AND ACCELERATORS
This scenario is impossible to attain and, in fact, the neutron inventory must be
carefully monitored in order to maintain a critical reactor.
Given that the number of neutrons emitted per fission event v ¼ 2.5 for the fission
of 235
U, one would think that designing a system with k ¼ 1 would be easy;
however, there are many ways in which neutrons can be lost. First of all, the core
of the reactor that contains the fuel must be finite in size. Therefore, there will be
a limit or edge of the core from which some neutrons can escape. Neutrons can
be “reflected” back into the core by a layer of material such as graphite (lowabsorption
cross section and higher mass) that surrounds the core, but the reflection
is not complete.
A second unavoidable source of neutron loss occurs in the fuel itself. Consider for
the moment a hypothetical reactor core made of uranium metal, There are two
neutron absorption reactions with uranium nuclei, 235
U(n, g)236
U, sth ¼ 98.3 barns
and 238
U(n, g)239
U, sth ¼ 2.7 barns that compete with the 235
U(n, f), (sth ¼ 583
barns) reaction. As an aside, one should remember that these cross sections for radiative
capture and fission, like all nuclear reactions, are energy dependent. For the
present discussion we will concentrate on thermal-energy (En ¼ 0.0253 eV) cross
sections. Let us define a parameter h as the average number of fission neutrons
per thermal neutron absorbed in the fuel. For a pure 235
U sample,
h
n
¼
sf
sa
¼
1
1 þ a
where a ¼ sg/sf and sa ¼ sg þ sf. For a material such as uranium metal that contains
both 235
U and 238
U
h
n
¼
xsf(235)
xsa(235) þ (1 À x)sa(238)
where x is the atom fraction (mole fraction) 235
U. For 235
U in natural uranium,
h ¼ 1.3. If the 235
U content of the fuel is greater than the natural abundance,
h will be larger.
The multiplication factor for an infinite sized reactor core is given by the fourfactor
formula:
K ¼ h fp1
where p is the fraction of fission neutrons that are thermalized (without being captured),
f is the fraction of thermal neutrons that are captured in the fuel, and 1 is the
so-called fast fission enhancement factor that expresses the fact that some fission
events are due to fast neutrons. (For a typical thermal reactor, h ¼ 1.65,
p ¼ 0.87, f ¼ 0.71, and 1 ¼ 1.02. Because of a leakage of 4% of the neutrons in a
finite reactor, k ¼ 1.00.)
For safe operation of the reactor, k must be exactly unity. That is difficult to
achieve in practice. In fact, if the mean time between generations of neutrons is t,
14.1 NUCLEAR REACTORS 389
and the multiplication factor is k, and N is the number of neutrons at time t, then
there will be kN neutrons at t þ t, k2
N neutrons at t þ 2t, and so forth. This can
be expressed as
dN ¼ (kN À N)
dt
t
or
N(t) ¼ N0e(kÀ1)t=t
Suppose k ¼ 1.01 and t ¼ 1023
s, then N(1 s) ¼ 22,000 Â N0, a dangerous
increase. The neutron inventory is regulated by inserting control rods of a neutron
absorbing material such as Cd in the fissioning assembly. But such regulation
cannot take place on a 1-s time scale. Fortunately, about 0.65% of the fission neutrons
are “delayed neutrons” emitted on a time scale of seconds/minutes. The resulting
average time constant for the “prompt þ delayed” neutrons t is 0.1 s, instead
of 1023
s, which allows control of the reactor.
As mentioned above, reactor control is achieved using control rods containing
113
Cd (sn,g ¼ 20,000 barns) or 10
B (sn,a ¼ 3800 barns). Another important
aspect of reactor control is the fact that certain fission products have very high
neutron capture cross sections and thus depress the neutron inventory. Foremost
among these nuclei, known as poisons, are 135
Xe (sn,g ¼ 2.65 Â 106
barns)
and 149
Sm (sn,g ¼ 4.1 Â 104
barns). These nuclei lower both f and k. In an
ordinary reactor, the amount of these nuclei is regulated by their decay and their
destruction by neutron capture, although their presence does affect the neutron
inventory.
14.1.4 Light Water Reactors
A large number of light-water-cooled nuclear reactors have been constructed around
the world. All of these reactors rely on the thermal fission of enriched uranium and
on normal or “light” water for neutron moderation and heat transfer. These machines
fall into two major categories of research reactors and power reactors. We have
described the principles that underlie the construction and operation of these
machines but the implementation is different. All nuclear reactors can be categorized
by the thermal generating power of the core, usually given in megawatts.
Power reactors are also categorized by the electrical generating power of the
plant in giga- or megawatts. Note that the ratio of the electrical power to the
thermal power of a power plant is the efficiency of the conversion process (a
number always less than 1, see below).
Generally speaking, the research reactors are small, on the order of 1 MW of
thermal power, and are optimized to provide intense neutron fluxes for the
irradiation of samples. These reactors are fueled with a few kilograms of enriched
uranium (20–90% 235
U) in fuel rods that are clad with a zirconium alloy or with
aluminum. The entire core assembly of a 1-MW research reactor is on the order
390 REACTORS AND ACCELERATORS
of 1 m3
and can produce an internal neutron flux on the order of 1013
/cm2
-s. The two
largest research reactors reach internal fluxes of 1 Â 1015
/cm2
-s (ILL, Grenoble,
France) and 3 Â 1015
/cm2
-s (HFIR, Oak Ridge, Tennessee). The small reactors
are usually submerged in a pool of water and are cooled by convection. The
water layer is designed to be deep enough to provide sufficient radiation shielding
so that a person can look through the water and observe the operating reactor.
The heat generated by the core of a research reactor is dissipated in the large pool
of water.
Nuclear power reactors are generally much larger, on the order of 2 GW of
thermal power, and are designed to produce electricity by the adiabatic expansion
of steam in a turbine. There are two competing designs in western countries for
nuclear power reactors that differ in the primary cooling loop. In one case the
water is allowed to boil (boiling water reactor, BWR) and in the other design superheated
water is held in the liquid phase under pressure (pressurized water reactor,
PWR) (Fig. 14.2). We will briefly consider the two types of power reactors.
Boiling water reactors are characterized by having only two coolant loops. The
water in the primary coolant loop circulates through the reactor core and boils at
approximately 1 atm pressure and is heated to approximately 3008C. The steam is
passed to a turbine system to generate electricity, is condensed, and is cycled
back to the core. A second coolant loop is used to maintain a constant output
temperature at the exit of the turbines; this loop removes the so-called waste
heat at the end of the thermodynamic cycle. Such cooling loops are commonly
included in machines that use adiabatic expansion to do work; for example,
radiators are connected to gasoline engines in cars. The waste-heat loop in a
nuclear power plant is usually an external, open loop. The waste heat is released
in the atmosphere in large evaporative-cooling towers or released into rivers,
lakes, or the ocean. The primary coolant is also the neutron moderator and is
subject to intense irradiation in the core. It will contain radioactivities from
impurities extracted from the walls, and so forth, and as a result the turbines will
become contaminated. Thus, the important feature of the BWR design with the
primary coolant circulating through the turbines necessitates placing them inside
the containment shielding.
In the other design, PWRs have two closed loops of water circulating in the plant
plus a third, external loop to remove the waste heat. Water is pumped through the
reactor core in the primary coolant loop to moderate the neutrons and to remove
the heat from the core as in the BWR. However, the reactor vessel is pressurized
so that the water does not boil. Steam is necessary to run the turbines, so the
primary loop transfers the heat to a secondary loop. The water in the secondary
loop is allowed to boil, producing steam that is isolated from both the core and
the outside. The water in the primary loop usually contains boron (as boric acid
H3BO3  0.025 M) to control the reactivity of the reactor. The steam in the secondary
loop is allowed to expand and cool through a set of turbines as in the BWR; the
cold steam condenses and is returned to the primary heat exchanger. A third loop
of water is used to maintain the low-temperature end of the expansion near room
temperature and remove the waste heat.
14.1 NUCLEAR REACTORS 391
The PWR is more expensive to build because the reactor vessel must be stronger
to withstand the higher water pressure, and there is a secondary coolant loop with
pumps and so on. The BWR, while less expensive to build, is more complicated
to service since the turbines are part of the primary coolant loop. The details of
the core design are different as well. Approximately twice as many PWRs have
been constructed as BWRs.
Figure 14.2 Schematic diagram of boiling water (top) and pressurized-water reactors
(bottom). (From Krane, 1988.)
392 REACTORS AND ACCELERATORS
A limit on the efficiency of the electrical energy conduction can be obtained by
applying the second law of thermodynamics to the secondary loop. The maximum
thermal efficiency, 1th, is given in terms of the input and output heats:
1th ¼ (qin À qout)=qin
Note that the output heat is the waste heat. In the limit that the machine operates in a
Carnot cycle, which can be characterized by constant temperatures at the input and
output, then the maximum efficiency is
1carnot ¼ (Tin À Tout)=Tin
The output temperature is given by the ambient temperature of the waste-heat loop
and can be taken to be 308C for purposes of estimation. The input temperature of the
steam is limited by physical constraints on the reactor primary cooling loop to be
about 3008C. Therefore, the maximum Carnot efficiency is approximately
1carnot ¼ (573 K–303 K)/573 K ¼ 0.47, whereas the actual efficiency is typically
1elec ¼ 0.35 when measured as electrical power outside the plant to total thermal
power in the core. For comparison, a coal-powered plant might have values of
1carnot ¼ 0.65, 1elec ¼ 0.5 due to higher steam temperatures
Side Neutron Reactions in Water One of the interesting side reactions that occurs in
water-moderated nuclear reactors is the (n, p) reaction on 16
O, which occurs with a
cross section of 0.017 mbarns. The 16
N product rapidly decays back to 16
O with a
half-life of 7.13 s; thus,thenetreaction canbe calleda catalysis of theneutronb decay.
16
O þ 1
n À! 16
N þ 1
H þ Qm
rxn
16
N À! 16
O þ bÀ
þ n þ Qb
NET 1
n À! 1
H þ bÀ
þ n þ Q
Make an estimate of the equilibrium activity per liter of cooling water due to 16
N in a
reactor that has an internal flux of 1013
neutrons/cm 2
-s.
Recall that the equilibrium activity, also called the saturation activity, occurs
when the rate of production is equal to the rate of decay and requires that the
sample be irradiated for more than three half-lives, or 22 s for 16
N. Also,
1 mbarn is 1 Â 10227
cm2
, thus,
Activity ¼ rate of production ¼ rate of decay ¼ N0sf
A ¼ N0 1013 1
cm2s
 
0:017 Â 10À27
cm2
À Á
A ¼
1 kg
L
NA Â 0:997616
O atoms=moleH2O
0:0180 kg=moleH2O
 
0:017 Â 10À14
=s
A ¼ 3:34 Â 1025
=L Â 0:017 Â 10À14
=s ¼ 5:68 Â 109
Bq=L
14.1 NUCLEAR REACTORS 393
At this point it is appropriate to mention the two most significant accidents that
have occurred at nuclear power plants. In both cases the seriousness of the accidents
were dramatically increased by human error. In both cases the difficulties were
caused by chemical reactions and not by nuclear fission.
An accident occurred at the Three-Mile Island PWR in Pennsylvania in 1979 in
which the water stopped flowing due to a mechanical failure in the primary coolant
loop. Subsequent actions by the operators caused the water level in the core to drop,
uncovering the upper part of the fuel rods. The nuclear fission process rapidly ceased
due to the loss of the water moderator, but the fuel continued to generate heat due to
the decay of fission products from prior operation. This residual decay heat is a
general feature of all nuclear reactors. Parts of fuel rods melted, which indicates
that the local temperature reached 30008C. As part of the accident, contaminated
water from the primary coolant loop was released inside the containment building
and soaked into the concrete. The noble gas fission products and a fraction of the
iodine fission products were released to the environment. The difficulty of melted
fuel notwithstanding, the extreme heating of the zirconium alloy that is used to
clad the fuel opened the door to an exothermic chemical reaction with steam that
produces hydrogen:
Zr(s) þ 2H2O À! 4ZrO2(s) þ 2H2(g)
An important concern during the accident was the potential chemical explosion of
this hydrogen gas with oxygen inside the containment building. The cleanup
process necessary inside the building continued for many years, and the perception
that nuclear power is somehow very dangerous has not subsided after more than 25
years.
A much more serious accident occurred at the Chernobyl power station near Kiev
in 1986 that was entirely the result of human error. This reactor relied on a large
amount of graphite to moderate the neutrons with water-filled tubes to remove the
heat and generate steam. This general reactor design, which also needs large
amounts of uranium, was used in the United States to produce plutonium during
the Cold War era but was not used for power generation. The accident in Chernobyl
occurred when the operators manually removed the control rods from the reactor
during a “test.” The chain reaction accelerated due to the core design and the
system became very hot. The cooling water was suddenly vaporized and caused
the core to explode. The nuclear fission stopped due to the loss of the moderator,
but the graphite was ignited and continued to burn for some time, spewing radioactivity
into the air. Approximately 10% of the graphite and large fractions of the
radioactive fission products were volatilized, the fraction depending on their chemical
nature, all of which was spread across western Europe by the wind. The burning
facility was too dangerous to approach, and the fire was extinguished by dropping
sand, clay, lead, and boron onto the fire from helicopters. What was left of the
reactor was buried in concrete and a massive cleanup was necessary. The inherent
difficulties in the design of this reactor continue to exist in numerous other
reactor facilities in the former Soviet Union.
394 REACTORS AND ACCELERATORS
14.1.5 The Oklo Phenomenon
We should not leave our discussion of nuclear reactors without mentioning “the
Oklo phenomenon.” In 1972, French scientists analyzing uranium ore from the
Oklo uranium mine in Gabon found ore that was depleted in 235
U. Further investigation
showed the presence of high abundances of certain Nd isotopes, which are
formed as fission products. The relative isotopic abundances of these isotopes
were very different from natural abundance patterns. The conclusion was that a
natural uranium chain reaction had occurred 1.8 billion years ago.
At that time, the isotopic abundance of 235
U would have been different than
today, due to the differing half-lives of 235
U and 238
U. At t ¼ 1.9 Â 109
y ago,
the isotopic abundance of 235
U was 3%, a number characteristic of the fuels of
nuclear reactors. Water apparently entered the ore deposit, acted as a moderator,
and initiated chain reactions. These chain reactions appear to have lasted for
106
y, ebbing and flowing as the water boiled away and returned. The power
level was 10 kW. Some attention has been paid to the fact that these fission
product deposits remained stable for more than 109
y, possibly supporting the
notion of geologic storage of nuclear waste.
14.2 NEUTRON SOURCES
Occasionally, one may need to use a radionuclide neutron source. For example,
in geological applications, one may need to have a portable neutron source.
Radionuclide neutron sources are generally based on either the (a, n) reaction or
spontaneous fission. Older (a, n) sources utilized the 5.3-MeV a aprticles from
138-d 210
Po, but modern sources utilize 238
Pu, 241
Am, or 242
Cm. The a particles
emitted from these nuclei interact with Be to produce neutrons via the (a, n) reaction.
The resulting neutron spectrum is broad because the alpha particles can react
with the Be at any point in the stopping process, usually ranging from a few MeV
to greater than 10 MeV. The yield of these sources and the accompanying g
radiation is shown in Table 14.1.
TABLE 14.1 Properties of Radionuclide Neutron Sources
Radionuclide t1/2
Neutron Yield
(n/Ci)
g-Radiation
(mr/h at 1 m/Ci)
210
Po 138 d 2.5 Â 106
—
238
Pu 87.8 y 2.2 Â 106
,1
241
Am 433 y 2.2 Â 106
,2.5
242
Cm 163 d 2.5 Â 106
2.5
252
Cf 2.65 y 4.3 Â 109
300
Source: From Brune, 1984, Kruger, 1971.
14.2 NEUTRON SOURCES 395
252
Cf acts as the basis of a radionuclide neutron source because 3.2% of its decays
are by spontaneous fission, yielding 3.76 neutrons per fission. The neutron emission
rate/Ci of material is quite high and these sources have found widespread use.
14.3 NEUTRON GENERATORS
Commercial neutron generators are compact charged particle accelerators designed
to produce a beam of neutrons by an appropriate nuclear reaction. The most commonly
used nuclear reactions are
2
H(d, n) Q ¼ 3:25 MeV
3
H(d, n) Q ¼ 17:6 MeV
7
Li(p, n) Q ¼ À1:646 MeV
9
Be(d, n) Q ¼ 3:79 MeV
The most common sources are based on the 3
H(d, n) reaction. Deuterons are accelerated
to 150 keV with currents 2.5 mA and strike a tritium target. They produce
2 Â 1011
of 14-MeV neutrons/s under these conditions. The neutrons produced are
widely used in fast neutron activation analysis for the determination of light elements.
The tritium targets are typically metals such as Ti, which have been loaded with
titanium tritide. The accelerators are usually small Cockcroft–Walton machines or
small sealed-tube devices where the ion source and accelerator structure are combined
to produce a less expensive device with neutron yields 108
/s.
The 7
Li(p, n) reaction is used commonly to produce approximately monoenergetic
fast neutrons. The protons are accelerated to an energy of a few MeV by a
small van de Graaff accelerator and strike a cooled rotating Li target. Thick target
neutron yields are . 109
n/s-mA. The energy of the neutrons can be obtained
from the Q value equation (Chapter 10), which can be expressed (for 08 neutrons) as
8Tn À 6Tp À 2(TnTp)1=2
¼ À11:522
where Tn and Tp are the kinetic energies of the neutron and proton in MeV.
The ultimate in accelerator-based neutron sources are the spallation neutron
sources. Neutrons are generated by the spallation reaction, which occurs when
high-energy (1 GeV) protons interact with heavy nuclei, such as mercury, releasing
20–30 neutrons/reacting proton. The proton beams (and the resulting neutrons)
are pulsed allowing the use of time-of-flight techniques to measure the energies of
the neutrons. Expressed as a thermal neutron flux, yields of 1017
n/cm2
-s are
possible, exceeding the neutron yields of reactors by orders of magnitude. These
high-intensity neutron fluxes can be used for neutron scattering experiments in
materials science and biology.
396 REACTORS AND ACCELERATORS
14.4 ACCELERATORS
As we have already indicated, charged particles must be accelerated to kinetic energies
on the order of millions of electron volts (MeV) in order to overcome the
Coulomb repulsion of another nucleus and induce a nuclear reaction. The
Coulomb potential grows with the inverse of the separation between the two ions:
Vcoul ¼ q1q2e2
=r
The Coulomb barrier is defined as the value of this potential energy for the two bare
nuclei at a separation that corresponds to the sum of their radii, or
VB ¼ z1z2e2
=(r1 þ r2)
The kinetic energy necessary for a moving projectile with mass number Al to react
with a stationary target with mass number A2 is KEthreshold ¼ VB(A1 þ A2)/A2 being
larger due to conservation of momentum.
High kinetic energies can be obtained by producing an ion with charge q on a
high-voltage platform held at a static potential E and simply allowing the ion to
move (fall) toward ground potential. The kinetic energy gain will be equal to the
loss of potential energy, thus, KE ¼ qE. The earliest accelerators were exactly of
this type, but physical breakdown of the insulating materials limit the maximum
electrostatic potential. Modern accelerators use electrode structures with alternating
electric fields to accelerate charged ions. We will consider the general features
of accelerators for heavy charged particles. (The acceleration of electrons is a
special case due to the relative ease of production, the very large charge-to-mass
(m/q) ratio, and the fact that energetic electrons are relativistic and travel with
essentially the same velocity, v % c.) We will start with consideration of ion
sources and then consider the various machines for accelerating charged particles
roughly in order of increasing final energy.
14.4.1 Ion Sources
All accelerators operate by the manipulation of charged ions in vacuum. Such particles
do not exist naturally and must be produced in ion sources. Positive ions of all
chemical elements can be produced, in principle, by ionization of atoms already in
the vapor phase. The difficulty of producing an ion depends dramatically on the
chemical species. A few elements have a exothermic electron affinity and can be
produced as singly charged negative ions. As a result only a few accelerators
utilize negative ions. Here we will consider three classes of ion sources that can
produce positively charged ions.
The simplest ion sources create positive ions by bombardment of the residual gas
inside a tube by electrons emitted from a hot filament (Fig. 14.3). The electrons can
be accelerated to a few hundred volts, and the electron impact on atoms and small
14.4 ACCELERATORS 397
molecules can easily create singly charged positive ions. Filament ion sources work
well for producing hydrogen and helium ion beams and other gaseous elements
that can be bled into the ionization region. These sources can have long lifetimes
as the electrodes are not directly damaged by the ionization process, and the ions
are produced from the feed material.
The energy of the electron beam can be increased in sources that are based on the
features of a Penning ion gauge (PIG) that was developed to measure low pressures.
A high-voltage electron arc is created between two electrodes and is confined by an
external magnetic field. The arc vaporizes the electrode material and ionizes any
residual gas in its path. A PIG source can produce electrons on the order of a few
thousand volts and can remove substantial numbers of electrons from the top-row
elements. These sources have erratic and relatively short lifetimes (10 h or less).
The electrodes are worn out as they are vaporized by the arc, which also tends to
produce metal coating on insulators leading to short circuits.
Very highly charged ions are produced by modern ion sources that rely on magnetically
confined plasmas such as the electron-cyclotron resonance ion source
(ECRIS or ECR) and the electron-beam ion source (EBIS). An ECRIS uses the
superposition of axially symmetric magnetic fields with an electric field from
end-cap electrodes to trap electrons in a magnetic bottle. The electrons are forced
to oscillate with radiofrequency (RF) radiation that corresponds to the oscillation
in the magnetic field (see discussion below about cyclotrons). The electrons move
in the plasma for a long time and collide with the residual gas, creating positive
ions and more electrons. The positive ions drift toward the extraction electrode
Figure 14.3 Schematic diagram of a hot filament ion source. (From Choppin, Rydberg, and
Liljenzin, 2002.)
398 REACTORS AND ACCELERATORS
and do not absorb the RF power that is tuned to the m/q ratio of the electron. The
plasma can remove even the innermost electrons from second-row elements.
All of these ion sources emit beams of positive ions at relatively low velocities,
the ions drift or are pulled out from the ionization region with relatively small electrostatic
potentials (U  20 kV). These beams of charged particles can be focused
and transported in vacuum to the main accelerating machines.
14.4.2 Electrostatic Machines
An ion source that is held at a large and stable positive electrostatic potential, V, will
accelerate positive ions to a kinetic energy of KE ¼ qV. The maximum potential is
limited by the ability to sustain the high voltage without breaking down the intervening
dielectric material (sparking). Consideration of the formula for the electric field
E at the surface of a sphere with a radius r, carrying a total charge Q, surrounded by a
medium with a dielectric constant 1, which is E ¼ Q/1r2
, leads to several common
features of electrostatic accelerators. The high-voltage terminal should be as large a
sphere as possible without any sharp points (the ends of sharp points have small
radii) and the terminal should be surrounded by a material with a large dielectric
constant. A large, carefully prepared terminal can be held at a maximum voltage
of % þ750 kV in dry air. The kinetic energies of ions from such systems are
(only) sufficient to induce nuclear reactions among the lightest elements and are
often used to generate neutrons via the d þ t !a þ n reaction.
The breakdown voltages of various gases were studied as a function of pressure in
the 19th century by Paschen, and he showed, not surprisingly, that pure gases have
higher breakdown potentials than air. Thus, higher electrostatic potentials can be
maintained by insulating the platform with an inert gas such as N2 or SF6 and by
increasing the gas pressure. An important distinguishing feature of electrostatic
accelerators is that the beam is emitted “continuously” from the ion source and is
literally a direct current (DC) beam of particles This feature can be good or bad
depending on the application, but a DC beam can be chopped, switched, or
pulsed to produce an alternating beam (AC) beam.
The underlying principle for production of high voltages is that the leakage
current should be as small as possible, microamps or less, and thus the dissipated
power remains low. Notice that a current of 1 mA from a 1 MV platform will
have a power drain of 1 watt. The techniques developed for the production of the
very high voltages necessary for electrostatic accelerators fall into two categories:
direct and mechanical production. Direct production relies on electronic circuits,
whereas mechanical production relies on the mechanical transportation of charge
to the platform. A voltage multiplication circuit for the direct production of very
high voltages was developed by Cockcroft and Walton. A schematic drawing of
the circuit is shown in Figure 14.4.
An alternating current is applied to the transformer at the bottom of the circuit,
which is rectified and multiplied by the stack to produce the high voltage. The resulting
voltage depends on the number of elements in the stack and on the input voltage.
The high voltage is not precisely constant in that it has a small variation or ripple in
14.4 ACCELERATORS 399
proportion to the input frequency. The Cockcroft–Walton design continues to be
used in small machines to generate neutrons via the d þ t reaction and to provide
the bias voltages for ion sources for large accelerator complexes.
An important feature of all electrostatic machines is that the beam is accelerated
down to ground potential by a series of electrodes at intermediate potentials. The
electrodes provide a weak focusing effect that causes the beam particles to move
toward the center of the tube. The focusing is due to the cylindrically symmetric
shape of the electric field combined with the fact that the particle spends more
time in the focusing region (first half of the gap where the field lines move
toward the center) than in the defocusing region (second half of the gap) due to
acceleration. Note that the Cockcroft–Walton circuit can also provide these potentials
at the intermediate stages in the stacked circuit.
Mechanical production of high voltages for ion acceleration is the basis of the
class of machines called Van de Graaff accelerators. These machines rely on the
principle from electrostatics that if a charge is placed inside a hollow conducting
sphere then the charge will migrate to the outer surface regardless of the amount
of charge already on the sphere. Thus, a Van de Graaff uses the mechanical transportation
of positive charges on an insulting belt from ground potential up to a
contact on the inside of a large hollow electrode, as indicated in Figure 14.5.
The mechanical generation of a few thousand volts is relatively easy and many,
many variations have been developed. You may be familiar with classroom demonstrations
that use a small Van de Graaff machine to charge up a metal sphere (radius
10 cm) and make a person’s hair stand on end. Larger generators that can produce
moderately high potentials, a few MV, have been constructed on a large scale and
have been used in spectacular displays of artificial lighting. The electrostatic potential
can be written in terms of the capacitance with respect to ground of the terminal,
Figure 14.4 Schematic diagram of a Cockcroft–Walton accelerator system on the left and
the electronic circuit used to provide the high voltage. (From Segre, 1977.)
400 REACTORS AND ACCELERATORS
V ¼ Q/C and is limited by the designed breakdown at intermediate points along the
insulating support column that provide fixed potentials for focusing the ion beam or
by leakage in the surrounding gas. In practice, a Van de Graaff system reaches an
equilibrium in which the added charge just compensates for charge leaking from
the terminal. Rapid discharges or sparks need to be avoided because they can
damage the components. An important feature of Van de Graaff high-voltage generators
is that the terminal voltage can be extremely stable and ripple free.
To be used as an accelerator, a source of positive ions must be mounted inside the
high-voltage terminal, and an insulating vacuum tube is needed to allow the ions to
be accelerated to ground potential. The positive ions will be accelerated toward
ground potential as in the Cockcroft–Walton machines. The terminal, accelerating
column, and charging system is usually placed inside a pressurized chamber that is
filled with an insulating gas such as pure N2 or SF6 at several atmospheres. The constraints
that come from placing the ion source inside the terminal generally limit the
Figure 14.5 A highly schematic view of the important components in a Van de Graaff
accelerator. Positive ions created by a corona discharge near ground potential are swept by
a moving belt to a similar corona contact attached to the inside of the high-voltage
terminal. The positive ions then evenly distribute themselves on the surface of the terminal
(Krane, 1988).
14.4 ACCELERATORS 401
charge on the ions to be one or two plus and thus directly limit the energy of the
accelerated beam. Even though technically challenging, many so-called singleended
Van de Graaffs were produced in the 1960s and early 1970s and used in
detailed studies of low-energy nuclear reactions. Note that the beam itself provides
a drain on the terminal voltage, and its intensity is therefore limited to be on the
order of microamperes or less. A number of advances in the technology of the
construction of the belt system and of the vacuum tube and electrode structure
have been made so that a modern accelerator terminal can routinely sustain 25
MV. For example, the high-voltage generator for the accelerator at Oak Ridge
National Laboratory is located inside a 100-ft high, 33-ft diameter pressure vessel
that is filled with SF6 at a pressure of approximately 75 psig (Ptotal  6 atm).
An important improvement of the single-ended Van de Graaff accelerators came
through the replacement of the positive ion source inside the high-voltage terminal
with an external negative ion source (Fig 14.6).
The negative ions are accelerated toward the high potential in one vacuum
column, and they strike a very thin foil or a layer of gas placed at the center of
the terminal. Electrons are readily stripped from the energetic negative ions, and
the positive ions are then accelerated away from the high-voltage terminal in a
second accelerating column. The final kinetic energy of the ions is the combination
of the initial acceleration of the negative ion plus the final acceleration of the (multiply
charged) positive ion. The charge state of the positive ion is usually much
larger than one, so that the final beam energy is more than twice the terminal
voltage. These devices are called tandem Van de Graaff accelerators or simply
tandems. Even though there are serious limitations on the number of chemical
elements that can be used to produce negative ions, the significantly higher
energy beams from tandems and the relocation of the ion source away from the terminal
have outweighed the limitations. Tandem accelerators have been used extensively
to study low-energy nuclear reactions, particularly direct reactions induced
Figure 14.6 Two-stage tandem Van de Graaff accelerator. [From R. J. Van de Graaff, Nucl.
Instr. Meth. 8, 195 (1960).]
402 REACTORS AND ACCELERATORS
by the lightest ions and fusion reactions induced by elements in the top row of the
periodic table. The kinetic energy of the beam can be very precisely controlled and is
very stable; however, the total energy is still limited by the terminal voltage. Attaining
significantly higher kinetic energies requires a booster accelerator based on the
electric fields from alternating sources of current.
14.4.3 Linear Accelerators
The production of very high-energy beams that are necessary for the production and
study of new and exotic subnucleonic particles would require acceleration from a
high-voltage platform at potentials that are unattainable in a steady state.
However, one can imagine that a group of particles can be accelerated in small
steps along a series of electrodes if the potential on the electrodes is synchronized
with the motion of the particles (Fig. 14.7)
For example, positively charged particles will be repelled by positive electrodes
and attracted to negatively charged electrodes, gaining kinetic energy as they cross
the gap. (From a simple standpoint each pair of electrodes acts like an instantaneous
electrostatic terminal and ground.) The particles will slowly gain energy as they synchronously
cross each successive gap.
The synchronization of the arrival of the particles at the electrode gaps with an
accelerating electric field can be done with an alternating electric field, as a given
electrode should have the polarity that will first attract a set of particles toward it
and then later repel the same particles after they have passed through the electrode.
The potential difference would follow a sine function: V ¼ V0 sin(vt), where V0 is
the peak voltage and v is the frequency. The key ingredient in such a simple linear
accelerator is to have the electrodes be hollow tubes and have the beam particles
pass through the center of each tube. The beam will be accelerated across the gap
between the electrodes by the electric field, but the beam will drift while in the
field-free region inside the tubular electrodes (hence the electrodes are called drift
tubes). The alternating power supply can change the polarity while the beam
pulse is inside a drift tube. Notice that the amount of time that the pulse of beam
particles spends inside the drift tubes must be held constant to provide a uniform
acceleration. The particles should reach the next gap at a time T ¼ 2p/v. Thus,
the length of each drift tube must increase along the path of the beam in proportion
to the velocity, L ¼ viT/2 where vi is the velocity inside the ith drift tube. The
physical dimensions of the drift tubes will have a finite acceptance range in velocity,
and so linear accelerators (linacs) have to be designed for specific velocity ranges.
In practice, linacs are used as energy boosters that accept beams, usually from an
electrostatic injector, started with significant initial velocities.
The phasing of the arrival of the particles with respect to the RF has two effects,
one good and one bad. At first glance, one might think that the arrival of the particles
should coincide with the maximum accelerating voltage. However, this point in time
coincides with the top of the sine wave, and particles that arrive slightly earlier
(faster particles) or slightly later (slower particles) will receive lower accelerations,
and the bunch will spread out in velocity, arriving at the next electrode at different
14.4 ACCELERATORS 403
times. If the center of the pulse arrives at the electrode gap while the voltage is
increasing somewhat linearly, then the faster particles will receive a lower acceleration
than that applied to the average particles and the slower particles will receive a
larger acceleration. These small differences in accelerations will tend to minimize
the velocity distribution of the beam pulse. This phase stability is a good feature
of linacs that work with nonrelativistic particles. Tuning the arrival of the beam
pulse to coincide with the increasing portion of the accelerating field has the bad
Figure 14.7 (a) Basic design of a linear accelerator. (b) Electric field in the gap between
two drift tables. (c) Phase stability in a linear accelerator. (From Krane, 1988.)
404 REACTORS AND ACCELERATORS
feature that the field across the gap sensed by a moving beam particle is asymmetric.
A symmetric field provides the weak focusing in the accelerating columns of electrostatic
machines, mentioned above. When the field is asymmetric, then the particle
will not be focused toward the center of the tube. Phase stability is more important
than the weak focusing effect, and linacs include additional components inside the
drift tubes to focus the beam.
A classical linac (the Wideroe design) with fixed drift tubes connected to an
external oscillator has a rather limited velocity acceptance and therefore would be
used in specific applications. A much more flexible linac design (the Alvarez
design) relies on creating either a standing electromagnetic wave in a resonant
cavity or a traveling wave in a waveguide. Most booster accelerators used to accelerate
heavy ions (nuclides more massive than helium) utilize resonant cavities to
provide the accelerating voltages. Various shapes have been used to create the accelerating
gaps and the drift tube regions. Early designs used copper surfaces to reflect
the power, but significant power was still dissipated in the walls. Recent designs
have used superconducting niobium or lead surfaces that have a much lower dissipation.
Recall that the velocity of all relativistic particles is essentially constant.
Thus, accelerating structures for the highest energy particles generally rely on standing
wave cavities with constant drift tube lengths, L ¼ cT/2. Electrons become relativistic
at comparatively low energies (recall that me ¼ 0.511 MeV), and so electron
accelerators have simpler designs than heavy-ion accelerators.
The number of drift tubes in a linac has to be relatively large because the acceleration
per gap is modest. A typical value of the effective accelerating electric field
in a superconducting cavity is 1–2 MV/q per meter of cavity. Booster accelerators
for heavy-ion beams can be 50 m long, and the electron linac at Stanford (SLAC) is
2 miles long. Linacs have the obvious difficulty that the drift tubes have to increase
in length as the velocity of the particle grows. For example, the drift tube length
for a relativistic particle is inversely proportional to the AC frequency:
L ¼ c(2p/2v) ¼ cp/v. A typical value for the frequency in such an accelerator
is 300 MHz, so that L  3 m. The accelerator has to lie in a straight line, and thus
space, alignment, and construction costs are important concerns. Notice that a
given pulse of particles will only pass through the accelerating structures one
time. The beam from a linac will arrive in pulses that follow the time structure of
the oscillations applied to the accelerating gaps. The frequency is usually in the
megahertz region (radio wave region of the spectrum) so that the pulses are
usually separated by tens of nanoseconds. The time structure on this scale is
usually called the beam microstructure. From a practical standpoint, a beam with
such a small time separation between pulses appears to be continuous. The beam
from a linac with standing-wave cavities is often pulsed on the millisecond time
scale in order to allow time to dissipate heat in the walls, such pulsing makes up
the macrostructure of the beam and has a large effect on experimental measurements.
The fraction of time that the macrostructure of a linac is “on” is called the
duty factor. The operating principles of linear accelerators were established by
1930, but the compact design of cyclotrons, which use a magnetic field to “reuse”
the accelerating electrodes, overtook the linac development.
14.4 ACCELERATORS 405
14.4.4 Cyclotrons, Synchrotrons, and Rings
It is well known that when a charged particle moves through a magnetic field it will
experience a force that acts perpendicular to the direction of motion. Thus, a moving
charged particle can be made to move in a circular orbit by placing it in a suitably
large and uniform magnetic field. Essentially all of the accelerators that have been
developed to produce very high-energy beams use a magnetic field to cause the particles
to circle through accelerating structures. The first device to rely on a magnetic
field was the cyclotron invented in 1929 by E. O. Lawrence.
The original cyclotron had a pancake-shaped vacuum chamber that was placed
between the north and south poles of an electromagnet (Fig. 14.8). Two large electrodes
were placed inside the vacuum chamber to provide one acceleration gap and
two drift regions. Each electrode was a hollow cavity in the shape of the letter “D”
and was fitted, back-to-back, inside the vacuum chamber with the acceleration gap
between the straight sides of the two D’s. The ions to be accelerated, in the first case
H2
þ
, were created at the center of the circle and were accelerated across the gap and
enter the drift space. Thus, moving, the particles experience the Lorentz force, move
on a circular path, and return to the accelerating gap! As the particles drift around,
the phase of the accelerating voltage is switched to the opposite polarity as in a linac.
When the particles arrive at the gap, they are accelerated again, gain energy, and
move into the drift region.
Figure 14.8 Schematic view of components of a cyclotron. A pulse of beam particles starts
in the center of the machine and is accelerated across the gap, circles through the drift space
inside the D-electrode, and is accelerated again when it returns to the gap. Eventually the
beam reaches the edge of the machine and can be extracted along a tangent. [From Krane,
1988.]
406 REACTORS AND ACCELERATORS
For many years, filament and PIG ion sources were placed in the center of the
cyclotrons. However, the development of more complicated and powerful ion
sources such as the ECR sources required more space than was available at the
center of the machines. Present-day cyclotrons have external ion sources and a
low-energy beam from an electrostatic injector is threaded into the center via an
axial channel and inflector or via a radial channel in separated-sector machines
(described below).
The developers of the cyclotron recognized that the frequency of the circular
motion of the particle is constant. Consider the Lorentz force acting on a moving
charged particle, Florentz ¼ B Â qv, where B is the (vector) magnetic field, q is the
charge, and v is the (vector) velocity. The cross product follows the “right-hand
rule” so that the force is perpendicular to the motion. The radius of the motion, r,
can be found by setting the Lorentz force equal to the mass times the centripetal
acceleration:
FLorentz ¼ Bqv ¼ mv2
=r
Solving for the radius,
r ¼
mv
Bq
The time, tcyc, that it will take a particle to complete one orbit is the circumference
divided by the velocity:
tcyc ¼
2pr
v
¼
2pmv
Bqv
¼
2pm
Bq
and is independent of the velocity of the ion for a given value of the magnetic field.
This constant time for a charged particle to orbit in a magnetic field is usually stated
as a frequency called the cyclotron resonance frequency of that particle. As an aside,
the circular or cyclotron motion forms the basis of mass measurements in ioncyclotron-resonance
(ICR) mass spectrometers. As long as the particles are
nonrelativistic, all the beam particles in a cyclotron will drift through the D’s in
the same amount of time and arrive at the accelerating gaps in phase, that is, the
orbits are isochronous. Notice also that we can use the concept of phase stability
from linear accelerators to maintain the bunch structure of the initial beam pulse.
The orbital radii will increase as the velocity of the particle increases at each gap
so that the particles will appear to spiral out from the center of the cyclotron. The
maximum velocity will occur when the particles reach the maximum physical
radius, rmax, of the vacuum chamber and D’s. Solving for the maximum kinetic
energy, Tmax, assuming a nonrelativistic beam:
Tmax ¼
1
2
mv2
max ¼
Bqrmax
À Á2
2m
¼
Brmax
À Á2
2
q2
m
 
14.4 ACCELERATORS 407
Notice that the maximum kinetic energy depends on the two machine parameters B
and rmax times a ratio of the square of the charge to the mass of the beam particles.
The first term in the expression for the maximum beam energy, (Brmax)2
/2, is often
called the K value of the cyclotron and is given in units of MeV. The oscillation
frequency of the D’s can be tuned over a limited range (e.g., 15–30 MHz) in
present-day cyclotrons to provide beams with different values of (q/m) at various
energies. Small cyclotrons used to produce specific isotopes for radiopharmaceuticals
have K  30 MeV and can provide 30-MeV protons (q2
/m ¼ 1). The highest
energy cyclotron in the world has K ¼ 1200 MeV and provides heavy ions up to
E/A ¼ 200 MeV (q2
/m  m/4) and is limited by the vertical focusing and the
relativistic mass increase of the beam.
The beam has to make many revolutions in a cyclotron in order to be accelerated
up to the full energy. For example, a typical accelerating gap might have a potential
difference of 100 kV so that a proton beam with total energy of 30 MeV that crosses
the gap twice per revolution still requires on the order of nturns  Etotal/(qV2) ¼
30/(1 Â 0.100 Â 2) ¼ 150 turns. It is important that vertical focusing be applied
to the beam so that it remains, at least on average, on the central plane of the cyclotron.
There are two main techniques for vertical focusing in cyclotrons. A weak
focusing occurs in the simple case in which the magnetic field of the cyclotron
runs between flat, uniform, and finite faces (pole faces) of an electromagnet. The
magnetic field between flat pole faces will only be exactly perpendicular in the
center and will increasingly bow out as one moves toward the edge. The curved
shape of the magnetic field will provide a weak restoring force for particles that
leave the median plane. A stronger focusing effect can be produced by dividing
the flat pole face into sections that are higher (hills) and lower (valleys). As indicated
in Figure 14.9, the average magnetic field should remain the same as that obtained
with a flat pole face, but the local magnetic field is higher between the hills and
lower between the valleys. The magnetic field will bow out from the hill region
High field
Highfield
Low field
Low field
(a) (b)
Figure 14.9 Top view of the sectors, or the hills and valleys, in a sector-focused cyclotrons.
In (a) the concept of strong vertical focusing at the transition regions, edges, of the sectors is
shown for straight sectors. A larger focusing effect is obtained by spiraling the sectors as
shown in (b). Note that the particles travel counterclockwise in this figure.
408 REACTORS AND ACCELERATORS
into the valley region along the path of the beam particles and provide a restoring
force toward the median plane each time a particle crosses the transition. The vertical
focusing can be increased by using a spiraled pole sector rather than a straight
sector. Such cyclotrons are usually called sector-focused cyclotrons and are
extensively used to provide moderate and high-energy beams of heavy ions.
Another technique to produce strong focusing is to make the hill regions from separate
wedge-shaped electromagnets and to leave the valley regions open. There has
to be a common vacuum chamber for the particles to circulate and to house the D’s.
The complexity of running four large individual magnets is compensated by the very
large variation in the magnetic field between the hills and valleys (called the flutter)
leading to a large vertical focusing effect, and it is easier to install and service all the
auxiliary equipment needed to inject, accelerate, and extract the beam. Such
machines are called separated-sector cyclotrons.
An important limitation on the maximum beam energy available from cyclotrons
comes from the relativistic increase of the mass of a particle with velocity. We have
seen elsewhere that m ¼ m0/[1 2 (v/c)2
]1/2
; thus, m ¼ 1.02 m0 for v ¼ 0.2c. Notice
that the ratio B/m occurs in all of the cyclotron equations given above. Thus, there
are two classes of isochronous cyclotrons, low-energy machines in which the mass
increase is small enough to be ignored, for example, v/c , 0.2, and medium-energy
machines in which the increasing mass is compensated by increasing the strength of
the magnetic field with radius. The field is usually increased by adding extra, concentric
coils to the magnet pole pieces that are called trim coils. The field could also
be increased by decreasing the gap between the pole faces. The field lines in a cyclotron
with a magnetic field that increases with radius will bow in toward the middle
(opposite to that described above) a feature that will produce a weak vertical
defocusing effect! Thus, fixed frequency cyclotrons cannot produce extremely
high-energy beams.
A large number of cyclotrons have been constructed over the years. For the most
part, they have been spiral sector-focused machines with trim coils. The K values
range from 30 to 150 MeV for electromagnets with normal conducting (or resistive)
coils. A number of sector-focused cyclotrons have been constructed with superconducting
coils based on designs developed at Michigan State University that have K
values of 500, and the largest machine has K ¼ 1200. Several large separated-sector,
normally conducting cyclotrons were constructed at the end of the 20th century at
GANIL in France (K ¼ 440) and at RIKEN in Japan (K ¼ 540). A project is underway
at RIKEN to design and build a separated-sector machine using superconducting
coils. The beams from cyclotrons will have a microstructure similar to that from
a linac and appear to be continuous in most applications. A cyclotron beam will not
have a macrostructure unless one is applied to the beam from the ion source for
specific experimental reasons. The acceleration of the beam from a cyclotron can
be rapidly stopped by simply shifting the relative phases of the D’s.
A number of attempts were made to develop cyclotrons that could accelerate
protons up to energies on the order of a GeV (the proton rest mass). For example,
the resonant frequency of the cyclotron could be decreased in proportion to the
mass increase. Such a frequency-modulated (FM) cyclotron (synchrocyclotron)
14.4 ACCELERATORS 409
could accelerate a single pulse of particles up to high-energy but would have to be
reset to start the next pulse and thus would have a low-duty factor. In addition, the
size of the magnet becomes extremely large and costly. The largest cyclotron
magnet ever constructed was a 184-in. diameter machine at LBL in Berkeley,
originally designed to provide 100-MeV protons, but completed in 1946 as a
synchrocyclotron.
The successful acceleration of protons and heavier nuclei to relativistic energies
was realized through the compensation of the increasing mass of the particle with an
increasing magnetic field. The early machines were fixed-frequency cyclotrons in
which the acceleration process was synchronized with a changing magnetic field
produced by a very large electromagnet. The mechanical design was changed
later to be just a ring of individual magnets. Simple geometry indicates that the
set of individual magnets necessary to construct a ring requires much, much less
iron that that of a large cyclotron magnet. The synchrotron design has proven to
be extremely robust and is used in all of the machines built to produce the highest
energy nuclear beams.
Synchrotrons use the concept that the particles are to be confined to move in a
circular orbit with essentially a constant radius, that is, a ring, regardless of the
energy of the particles. Thus, synchrotrons are pulsed machines that operate on a
cycle in which a modest energy beam is injected into the ring, the beam is accelerated,
the high-energy beam is dumped out, and the ring is returned to the injection
state. A low magnetic field is necessary to confine the low-energy particles at injection.
After a sufficiently large number of particles have been fed into the ring, one or
more accelerating structures (originally drift tubes, now resonant cavities) are turned
on and the beam begins to gain energy. The energy gain per turn is usually low
(100 kV) and during the acceleration process the magnetic field is ramped up
toward the maximum value that the magnets can provide. If the synchrotron has
to accept nonrelativistic particles, then the revolution frequency of the particles
will increase as the velocity increases (as the radius of the orbit is constant) so
that the frequency of the accelerating structure has to increase as well. Thus, both
the magnetic field and the accelerator have to be synchronized with the energy of
the particles. The particles can be extracted from the ring providing a single macrocycle
beam pulse. The magnetic field is then returned to the initial, low value. The
highest energy synchrotrons accept particles that are already relativistic (from prior
acceleration in booster synchrotrons) and the revolution frequency remains constant.
The time necessary for a single macrocycle is on the order of seconds and is dictated
by the maximum rate of change of the magnetic field. Modern rapid-cycling synchrotrons
run at 1 or 2 Hz, while the original machines from the 1960s typically
ran at 1–5 Hz.
The principle of phase stability is used in synchrotrons to maintain a narrow
energy distribution of the beam bunches during acceleration. The problems associated
with vertical focusing in cyclotrons will be present in synchrotrons. The original
machines relied on weak focusing in magnets with flat pole faces. A variation of
the strong focusing obtained with hills and valleys in sector-focused cyclotrons can
be obtained in synchrotrons. Rather than hills and valleys, though, the ring is divided
410 REACTORS AND ACCELERATORS
into sectors in which the gaps between the pole pieces of the magnets are wedge
shaped. The thinner side of the wedge is alternately on the inside or the outside
of the ring. As in the cyclotron, the average field is set for an isochronous orbit.
When the beam circles around the ring, it encounters a vertical focusing region
(thin edge inside) and a vertical defocusing region (thin edge outside), which produces
a net vertical restoring force. Dipole magnets that have pole pieces that are
shaped (tilted) to provide focusing are called combined-function magnets as they
are meant to perform two tasks, and synchrotrons that use this version of strong
focusing are called alternating-gradient machines. Modern synchrotrons do not
use combined-function magnets but rather use dipole magnets to bend the beam
and quadrupole doublet magnets (discussed below) to focus the beam in straight sections
between the dipoles. Such independent function magnets are easier to construct
and allow more flexible tuning.
Synchrotrons are used to accelerate protons and heavy nuclei to the highest
energies, presently 0.95 TeV protons in the TEVATRON at Fermilab and 100
GeV/nucleon heavy ions (including 197
Au nuclei at almost 20 TeV) in RHIC at
Brookhaven National Lab (BNL). The Large Hadron Collider (LHC) under construction
at CERN will collide 7 TeV protons and collide heavy ions with a total collision
energy in excess of 1250 TeV. The maximum energy of the beam is
proportional to Brmax as in a cyclotron, and the value for the ring is usually given
in tesla-meters, or T-m. Even after extensive development, the maximum field
strengths in large electromagnets is on the order of a few tesla, however, the
radius of the ring of magnets is only limited by money. For example, the booster
or intermediate energy synchrotron for the RHIC system is approximately 20 m
radius and provides 1 GeV protons. The main ring of the AGS (alternating gradient
synchrotron) at BNL is a 100-T-m system consisting of 240 combined function
magnets in a radius of approximately 85 m. The main rings of the RHIC system
are 839.5 T-m and contain 1740 separated function superconducting magnets in a
3.834-km circumference.
A similar but bigger and very versatile combination of big accelerators has been
established at CERN in Europe. The complex includes several high-energy injectors
for protons, electrons, and heavy ions, a booster synchrotron, the PS, that can accelerate
pulses of these ions (e.g., 26-GeV protons), a high-energy synchrotron, the
SPS, that can also accelerate the ions (e.g., 400-GeV protons, 170-GeV/nucleon
heavy ions in a 1.1-km radius ring), and the largest ring system, LEP, for electrons
and positrons (90-GeV electrons in 4.3-km radius). The LEP machine is being converted
into the large hadron collider, LHC, that will be able to accelerate and collide
all of these particles (e.g., 7-TeV protons in the LEP tunnel).
There are two important features of nuclear collisions that we have not considered
in our discussion of particle accelerators. First is conservation of momentum
in the collision dictates that a large fraction of the energy that goes into a collision in
the laboratory between a moving particle and a resting target nucleus will go into
kinematic motion of the products and will not be available for excitation of the products
or for new particle production. Perhaps more surprising is the second feature
that after all the effort to produce a beam of high-energy particles, the great majority
14.4 ACCELERATORS 411
of the beam particles will pass through the target material, interacting with the
electrons and slowing down, and not collide with another nucleus. Both of these
problems can be resolved if we create counterrotating beams of particles that are
circulating at constant orbits in synchrotron rings held at their maximum magnetic
fields. The beams can be forced to cross one another at specific points or interaction
regions. Circular systems of magnets have also been developed without accelerating
structures called storage rings. The net momentum of particles that collide head-on
is zero in the laboratory frame so that all of the energy is available in principle for
excitation of the products. It is much more cost effective to build a storage ring for a
synchrotron system and gain a factor of 2 in energy than to double the radius of the
synchrotron. If the counterrotating particles miss during one crossing, as is the most
likely event, then they simply continue on their orbit and literally come back around
for another try. These features, particularly the higher available energy, are part of
all of the modern high-energy accelerator complexes. All of the modern high-energy
synchrotrons mentioned above include storage rings and rely on colliding counterrotating
beams, for example, the Tevatron collides protons and antiprotons,
LEP-collided electrons and positrons, and RHIC collides heavy-ion beams from
separate rings.
14.5 CHARGED PARTICLE BEAM TRANSPORT AND ANALYSIS
The goal of accelerating particles is to induce reactions with target nuclei. Most
nuclear targets are pure elemental foils and the earliest experiments were performed
by placing a metal foil in the path of the beam at the end of the acceleration process.
For example, a foil could be placed at the largest radius of a cyclotron or at the end of
a linear accelerator. As we have just discussed, these reactions can be made to occur
between counterrotating beams in storage rings. It is very difficult to perform
experiments directly in the accelerator for a number of reasons including the high
radiation environment caused by beam loss during acceleration and physical constraints
on the available space. Thus, beam transport techniques were developed
to bring fully accelerated beams to remote and shielded vaults. These beam handling
techniques are directly analogous to optical techniques using glass prisms and lenses
to transport beams of photons.
In optical systems the light rays are diffracted when they make a transition
between two media with different indices of refraction. Prisms use converging flat
surfaces to chromatically disperse the light, lenses use curved surfaces (spherical
lenses are most common) to focus or defocus the light rays. Charged particle
beams are similarly effected by magnetic and electric fields. A beam of charged particles
will be deflected as it travels through a uniform field created between two surfaces
and can be focused or defocused as it travels through the radially increasing
field created by two, four, six, or more surfaces. The fields are usually labeled as
dipole, quadrupole, hexapole, and so on. The multipole fields can be created by electrostatic
plates or electromagnet poles. The forces acting on the moving particles are
different in electrostatic and magnetic systems. Magnetic systems have the most
412 REACTORS AND ACCELERATORS
widespread applications primarily for technical reasons as superconducting wire
technology provides a means to supply extremely large magnetic fields in
compact devices, whereas the maximum attainable electric fields are small by
comparison.
We have already seen that a moving charged particle will experience a force perpendicular
to its direction of motion when it is moving in a magnetic field. This
Lorentz force causes the particle to curve with a radius, r ¼ Bq/mv that depends
on the charge-to-mass ratio (q/m) of the ion. Thus, a simple magnetic dipole can
be used to change the direction of a beam of particles. For example, consider the
path of a beam that enters the magnetic field in the region between the poles of a
wedge-shaped magnet, as indicated in Figure 14.10. Such magnets are commonly
called magnetic sectors or sector magnets and are characterized by their magnetic
field, B, their bend radius, r, and their bend angle, u. The beam will follow the
path from O to E along an arc with the radius r and turn through an angle u. The
path of the beam need not be perpendicular to the straight edges of the sector.
These angles are labeled a1 and a2 in Figure 14.10. If all the particles are on
exactly parallel trajectories when they enter the magnetic field, they will all turn
through the angle u and emerge in a parallel bunch. If on the other hand, the particles
enter the magnetic field in a diverging bunch, then the sector will focus the beam in
the horizontal and vertical directions with focal lengths given by:
fhoriz ¼
r
sin uð Þ 1 À tan a1ð Þ tan a2ð Þ½ Š À cos uð Þ tan a1ð Þ þ tan a2ð Þ½ Š
and
fvert ¼
r
tan a1ð Þ þ tan a2ð Þ½ Š À u tan a1ð Þ tan a2ð Þ½ Š
A
a1 a2
O Magnet E
B
q
Figure 14.10 Schematic view of the bending of a charged particle beam in a wedge-shaped
dipole magnet with straight edges. The beam will be bent through an angle u; the angles al and
a2 are used to describe the angles of incidence relative to the straight edges and are important
in determining the weak focusing due to the fringing field. [From Harvey, 1964.]
14.5 CHARGED PARTICLE BEAM TRANSPORT AND ANALYSIS 413
Notice that the focal lengths are different. If the beam of particles enters and exits
along a normal to each face, a1 ¼ a2 ¼ 0, the horizontal focal length becomes
fhoriz ¼ r/sin(u) while all vertical focusing is removed, fvert ¼ 1. Thus, if a perfectly
round beam is deflected by a dipole magnet, it will often lose its symmetry
due to the different focal lengths of the dipole magnet. Subsequent ion-optical focusing
elements have to be set to compensate for this difference to produce a round
beam at the target position. It is important to remember that the bending radius,
r, is proportional to the charge-to-momentum ratio of the beam (q/mv) so that the
field strengths of all of the magnets will have to be set for each beam.
Magnetic focusing of a charged particle beam can be produced by a colinear
solenoidal field or by a pair of magnetic quadrupoles. In both cases the magnets
create a fringing field or fields that cause diverging particles to be returned to the
optical axis. If we define the z axis along the path of the beam, a single solenoid
will focus in both the x and y directions. A magnetic quadrupole can be constructed
by arranging four equal strength poles on a circle at 908 from one another with
polarities that alternate between north and south. An example of a quadrupole
magnet is shown in Figure 14.11.
If we examine the magnetic field from such a quadrupolar device, we will find
that the field along the central axis is exactly zero and increases linearly to a
maximum value at each pole face. (The pole tips should be hyperbolic surfaces to
conform to the shape of the magnetic field, although cylindrical pole tips are
often used for ease of manufacture.) Quadrupole magnets are thus characterized
by the gradient of the magnetic field, dB/dr, where r is a radial coordinate and
N
copper coil
iron yoke
N
4 in.
2.9
in.
S
S
y
xB
Figure 14.11 Schematic design of a quadrupole magnet. The arrangement of the poles will
provide x focusing in the usual right-handed coordinate system with the z axis emerging from
the page. [From O. Chumberlin, Ann. Rev. Nucl. Sci. 10, 161 (1960).]
414 REACTORS AND ACCELERATORS
their length, l, along the beam direction. A particle that moves along the central axis
stays in a region of no magnetic field, while particles that are off-axis will encounter
an increasing magnetic field that acts like the fringing field at the edge of a dipole
magnet. Notice, however, that the fringing fields will focus off-axis particles in
one direction (x in Fig. 14.11) and defocus them in the orthogonal direction (y in
Fig. 14.11). If we define the constant k2
¼ (q/mv)dB/dr, then the focal length of
one quadrupole with a length l (called a singlet) is
fquad ¼
1
k sin klð Þ
At first glance the fact that a singlet is both focusing and defocusing at the same time
might imply that uniform focusing of a beam is not possible. However, it is known
from light optics that the combination of a converging lens with a diverging lens has
a net focusing effect as is indicated in Figure 14.12. Therefore, focusing magnet
packages are most often made by combining an x-focusing quadrupole with a
y-focusing quadrupole of equal lengths into a “doublet.” Three quadrupoles
(y, x, y) with lengths (l, 2l, l ) are sometimes combined into a “triplets,” which
provide more flexibility in manipulating the beam shape.
Modern accelerator complexes rely on a large combination of magnetic dipoles
and quadrupoles to transport fully accelerated beams large distances without significant
loss. Solenoidal magnets are only used occasionally when point-to-point focii
are needed as quadrupoles are more efficient and allow the two coordinates to be
tuned independently. Higher order magnetic multipoles, sextupoles, and octupoles
are used to correct aberations in high-resolution applications such as spectrometers
and fragment separators.
Example Problem A dipole magnet deflects charged particle beams through an
angle of 22.58 with a radius of 2.0 m. For ease of construction the magnet has
rectangular pole pieces 0.5 Â 1.5 m long. The beam enters normally at the
L
f2 f1
Z
y
x
d
Z
Focusing Defocusing
Defocusing Focusing
L
Figure 14.12 Net focusing effect on a diverging charged particle beam from the
combination of a converging and diverging pair of magnetic quadrupole lenses. The key
feature of the doublet is that the particles move along paths that are closer to the optical
axis in the defocusing element of the pair. (From Harvey, 1964.)
14.5 CHARGED PARTICLE BEAM TRANSPORT AND ANALYSIS 415
center of one of the 0.5-m faces and exits at an angle from the opposite 0.5-m
edge. What are the focal lengths of this magnet?
Solution We need to evaluate the angles a1 and a2 in order to solve this
problem. From the problem definition we know that a1 ¼ 0. A little geometry
will show that in this case the exit angle is equal to the bend angle:
fhoriz ¼
r
sin (u) 1 À tan (a1) tan (a2)½ Š À cos (u) tan (a1) þ tan (a2)½ Š
¼
2:0 m
sin (22:5) 1 À tan (0) tan (22:5)½ Š À cos (22:5) tan (0) þ tan (22:5)½ Š
¼
2:0 m
sin (22:5)(1 À 0) þ cos (22:5) tan (22:5)½ Š
¼
2:0 m
0:383 þ 0:924(0:414)
¼ 2:613 m
For the vertical focal length we have
fvert ¼
h
tan (a1) þ tan (a2)½ Š À u tan (a1) tan (a2)½ Š
¼
2:0 m
tan (0) þ tan (22:5)½ Š À 22:5(2p=180) tan (0) tan (22:5)½ Š
¼
2:0 m
tan (22:5) À 0
¼
2:0 m
0:414
¼ 4:828 m
Such different focal lengths, a weaker focus in vertical compared to horizontal, as
in this situation are a common occurrence in beam transport systems.
The transport and control of charged particle beams with electrostatic elements
has the nice feature that the equations of motion do not depend on the mass of
the particle. The force felt by a charged particle in an electric field is simply
Fe ¼ qE where q is the electric charge and E is the electric field strength. When a
positively charged particle enters the region between two parallel plates with a separation
d at a voltage V, the particle will feel the electric force pulling it towards the
electrode at the lower potential, Fe ¼ qV/d, and will move on a circular orbit with a
radius, r ¼ mv2
(d/qV). Such a device can thus change the direction of the incident
beam. (The beam will undergo a weak focusing from the fringing field at the
entrance and exit of the device as discussed above.) Strong focusing of a beam of
charged particles can be produced by an Einzel lens (cylindrical focusing from a
set of three ring electrodes) and by a quadrupolar arrangement of electrodes.
416 REACTORS AND ACCELERATORS
An electrostatic mirror can be produced by an electrode at a potential energy that is
greater than the kinetic energy divided by the charge of the particle. The bending and
focusing power of electrostatic systems are limited by the maximum electric fields
that can be applied across the electrodes. Extensive electrostatic systems have been
constructed for the transport of low-energy beams, KE 50 keV, for example,
beams extracted from ion sources are usually transported with electrostatic
elements.
14.6 RADIOACTIVE ION BEAMS
Unstable nuclei with modestly short half-lives have been produced and separated as
very low energy beams for some time. Recently, techniques have been developed to
provide much more energetic beams of nuclei with half-lives as short as a few milliseconds
with sufficient energy to induce secondary nuclear reactions. The production
techniques usually rely on the creation of exotic nuclei in high-energy
reactions followed by the collection and separation of a specific exotic nucleus.
The physical techniques differ in that the products are either the residues of target
(nearly at rest in the laboratory) or of projectile nuclei (moving with nearly the
beam velocity), but in both reactions a large nucleus is fragmented into its components.
The difference in the initial velocity of the product has large consequences
for the physical separation techniques and reacceleration but no consequences for
the reaction mechanism. Thus, the same residues can and have been produced for
study in each rest frame. Very energetic proton beams (Elab  m0c2
) from synchrotrons
were used extensively in the 1960s, and the process of target fragmentation
was rapidly exploited for the production of exotic nuclei. The residual nuclei left
in the target after a beam pulse were thermalized and then ionized for separation.
This technique is usually called the ISOL technique (Fig. 14.13). This was an
acronym for isotope separator online. Today it may be more precise to associate
ISOL with ion source online to more clearly distinguish these devices from inflight
devices used to separate projectile fragments. The beam interacts with a
target, usually a refractory metal, that is heated to several thousand degrees. The
target itself can be quite thick, even thick enough to stop the beam, but it should
be thin to allow rapid release of the reaction products. This apparent paradox can
be solved by using stacks of thin metal foils. The reaction products come to
thermal equilibrium in the target matrix and, depending on their chemical nature,
can diffuse out of the matrix. Many techniques have been used to ionize the hot
atoms including surface ionization, electron beam or plasma ionization, and resonant
laser ionization. Notice that the reaction mechanism creates a broad range
of products, most of which diffuse out of the target at some rate. The ionization
process is a second chemical process that creates singly charged ions. After being
ionized, the reaction products can be readily extracted from the target system and
accelerated. The chemical selectivity provides a means to select the most exotic
nuclei that are only weakly produced in the primary reaction. The target ionsource
combination is placed on an electrostatic platform to provide very low
14.6 RADIOACTIVE ION BEAMS 417
energy ions ( 50 keV, total) for decay studies. A large number of facilities were
operated over the years, the most successful being the ISOLDE facility at CERN
that uses pulses of 1-GeV protons from the CERN PS synchrotron to irradiate
targets such as Nb, Ta, and U. The ISOLDE facility has a large electrostatic
beam-handling system with many experimental stations.
A number of projects are underway around the world to accelerate exotic beams
to energies sufficient to induce secondary nuclear reactions. These first-generation
facilities all rely on using existing accelerators and in some cases existing experimental
equipment. A facility at Louvain-la-Neuve in Belgium is producing radioactive
beams of light nuclei created in direct reactions induced by a 30-MeV
proton beam from a cyclotron developed to produce radioisotopes for medicine.
Neutral gaseous products, atoms, and small molecules are pumped from the target
into an ECR ion source to be ionized and then the ions are transferred into a
K110 cyclotron. The HRIBF project at Oak Ridge relies on using one of the first
sector-focused cyclotrons (K ¼ 100) to provide radioactive ions with an intense
proton beam. In this case the K100 cyclotron produces nuclei by simple reactions
such as 70
Ge(p, 2n)69
As in an ion source that produces 1þ
ions. The positive ions
ISOL
Heavy lon
accelarator Fragment separator
Projectile Fragmentation
Radioactive lon beam
Thin production
target
Production
accelerator
Transfer tube
Production beam
Thick, hot target
Radioactive lon beam
Ion source
Isotope / Isobar
separator
Postaccelerator
Experiment
Experiment
Figure 14.13 Schematic view of the PF and ISOL techniques for generating radioactive
beams. (Figure also appears in color figure section.)
418 REACTORS AND ACCELERATORS
have to be converted into negative ions before injection into a 25-MV tandem. The
tandem thus places some limits on the chemical elements that can be utilized. The
ISAC facility in Vancouver, B.C. uses a 500 MeV proton beam from the TRIUMF
separated sector cyclotron to irradiate heated targets. The emitted ions are then
accelerated in a superconducting linear accelerator. The hallmark of all of these
facilities is that they have an accelerator that provides an intense beam to produce
the activities (the driver) and another accelerator for the secondary beam.
Table 14.2 presents a list of ISOL facilities that is meant to indicate the variety of
approaches that are being used.
The advent in the 1970s of synchrotrons that were capable of delivering beams of
all elements with very high energies allowed the production of the same exotic
nuclei observed as target residues with sufficient kinetic energies to allow rapid
physical separation and identification. This technique is usually called the “projectile
fragmentation technique” but more correctly might be referred to as in-flight
separation since other reaction mechanisms besides projectile fragmentation can
be used to produce the nuclei (Fig. 14.13). The beams of exotic nuclei are not
stopped in this technique, and even very short-lived nuclei can be studied and
used to induce secondary reactions. In the mid-1970s it was shown that up to 1%
of a primary 12
C beam could be converted into 11
C ions and separated for implantation
into biomedical samples. Nuclear physics experiments to produce exotic
nuclei and calibrate space-flight instruments using similar techniques based on
magnetic rigidity were also pioneered with beam-line elements at LBL (Berkeley,
California) and then dramatically extended by using degraders (the LISE spectrometer
at GANIL (Caen, France)). This technique has been further extended in
four second-generation devices distributed around the world and third-generation
devices at the NSCL (Michigan State) and at RIKEN (Japan).
Fast-ion beams of exotic nuclei are separated from the primary beam and from
the other reaction products by a combination of separated function magnetic
bending dipoles and focusing quadrupole doublets acting on the distribution of
ions emerging from the target at high velocities (b  0.5c). Achromatic magnetic
systems are used, where achromatic means that the position and angle of ions at
the end of the device (called the focal plane) does not depend on the ion’s momentum.
Such achromatic magnetic systems are generally most useful for efficient
TABLE 14.2 Examples of ISOL Fragmentation Facilities
Device Accelerator System Reactiona
Country
ARENAS Cyclotron/ECR/Cyclotron (p, n), (p, 2p), etc. Belgium
HRIBF Cyclotron/1þ//12/TANDEM (p, xn) United States
ISAC Synchrotron/ECR/LINAC TF Canada
REX-ISOLDE Large Synchrotron/1þ//
EBIS/LINAC
TF Europe (CERN)
SPIRAL HI-Cyclotrons/ECR/Cyclotron PF France
a
Class of reactions: low-energy fusion and direct reactions (p, xn); target fragmentation (TF); or projectile
fragmentation (PF).
14.6 RADIOACTIVE ION BEAMS 419
separation at the highest energies because they can collect a large fraction of all the
produced fragments and focus them to a small spot. Achromatic systems have the
additional advantage that the final spot size is kept small even when the momentum
acceptance is large. The key elements in these devices are an initial bend for
momentum-to-charge ratio selection, an energy loss degrader for atomic number
separation also called a “wedge,” and a second bend for momentum-to-charge
ratio selection of a specific ion.
An aperture or a slit is used at the intermediate position to limit the momentum
acceptance of the device. Since the fragmentation mechanism produces all the nuclei
with nearly the same velocity, the initial magnetic rigidity (Br ¼ mv/q) and thus
momentum-to-charge ratio selection is equivalent to separation according to the
mass-to-charge ratio of the products. Even so, projectile fragmentation reactions
produce many different ions that have the same mass-to-charge ratio, for example,
the fragmentation of an 18
O beam can formally produce five ions with m/q ¼ 3:
3
H, 6
He, 9
Li, 12
Be, and 15
B. An energy degrader is inserted into the beam at the
intermediate dispersive image in order to remove the ions that have the same
initial mass-to-charge ratio as the fragment of interest but different atomic
numbers. Recall that all ions will lose some kinetic energy in the degrader, and the
relative amount will depend on their atomic number. The ions, therefore, will exit
the foil with different magnetic rigidities. The contaminants can then be spatially
dispersed at the focal plane by an additional bend. This Z-dependent separation is
proportional to the degrader thickness and to the ratio of the magnetic rigidity of
the second half of the system to that of the first half. A schematic diagram of the
A1900 separator operating at the NSCL is shown in Figure 14.14.
There are projectile fragmentation separators presently operating in France,
Germany, Japan, Russia, and the United States. Other similar devices are in the planning
stages or are under construction in several laboratories. A comparison of the
various parameters that describe these fragment separators is given in Table 14.3.
The LISE separator has been operated for more than 15 years and has provided
Figure 14.14 Schematic diagram of the A1900, a new projectile fragmentation separator
recently constructed at Michigan State University.
420 REACTORS AND ACCELERATORS
beams for a variety of experiments. The RIPS device has the largest solid angle and
momentum acceptance of second-generation devices.
The more recently constructed COMBAS device at the JINR at Dubna, Russia
has a significantly larger acceptance and is based on using combined function
magnets. The A1200 (now retired) and the A1900(MSU) (shown schematically in
Fig. 14.14), RCNP(Osaka), and FRS(GSl) separators are positioned at the beginning
of the beam distribution system to allow delivery of radioactive beams to any experimental
area.
Besides the obvious dependence of the RNB intensity on the intensity of the
primary beam, the secondary beam rate is also directly related to the relative separator
acceptance. The GSI device has been designed for very high kinetic energies,
where the fragmentation angular cone and relative energy spread are relatively
small. This allows the physical acceptance of the device to be smaller but yet
gives “full acceptance” of individual products. Large solid angle and large momentum
acceptance are especially important if the device is to be used to separate light
ions at 50–200 MeV/nucleon, the energy region in which most separators are operating.
Note that the larger the physical acceptance of the separator, the lower the
primary beam energy at which 100% collection efficiency is attained.
14.7 NUCLEAR WEAPONS
While a full discussion of nuclear weapons is beyond the scope of this book, some
comments about the operating principles of such devices and their connection to
reactors and accelerators are desirable.
The techniques used to produce a “nuclear explosion” (i.e., an essentially instantaneous,
self-perpetuating nuclear chain reaction) are very complex. A nuclear
explosion must utilize a high-energy neutron spectrum (fast neutrons, that is,
neutrons with energies .1 MeV). This results basically from the fact that, for an
explosion to take place, the nuclear chain reaction must be very rapid—of the
order of microseconds. Each generation in the chain reaction must occur within
TABLE 14.3 Comparison of Fragment Separators
Device V (msr)
Dp/p
(%)
Br
(T 2 m)
Resolving
Powera
Length
(m) Facility
A1200 0.8–4.3 3.0 5.4 700–1500 22 NSCL (U.S.)
A1900 8.0 4.5 6.0 2900 35 NSCL (U.S.)
COMBAS 6.4 20 4.5 4360 14.5 JINR (Russia)
LISE3 1.0 5.0 3.2 800 18. GANIL (France)
FRS 0.7–2.5 2.0 9–18 240–1500 73 GSI (Germany)
RIPS 5.0 6.0 5.76 1500 21 RIKEN (Japan)
RCNP 1.1 8.0 3.2 2000 14 RCNP (Japan)
a
Mass-to-charge resolution, see the text.
14.7 NUCLEAR WEAPONS 421
about 0.01 ms (a “shake”) or less. The energy release takes place over many
generations, although 99.9% of the energy release occurs within the last seven
generations, that is, in a time of the order of 0.1 ms. The rapid time scale of this
reaction requires the use of fast neutrons. The process by which a neutron is
degraded in energy is time consuming and largely eliminates the possibility of an
explosion. This also explains why power reactors that operate with a slow or
thermal neutron spectrum cannot undergo a nuclear explosion, even if the worst
accident is imagined. In the case of reactors that operate with higher energy neutrons,
a nuclear explosion is also precluded because of the geometrical arrangement
of the fissionable material and the rearrangement of this material if an accident
occurs.
The explosive ingredients of fission weapons are limited, in practice, to 239
Pu and
235
U because these are the only nuclides that are reasonably long-lived, capable of
being produced in significant quantities, and also capable of undergoing fission with
neutrons of all energies, from essentially zero (thermal) to the higher energies of the
secondary neutrons emitted in fission. Other nuclides, for example, 238
U or 232
Th,
can undergo fission with some of these higher energy neutrons but not with those
of lower energy. It is not possible to produce a self-sustaining chain reaction with
these nuclides since an insufficient fraction of the neutrons produced in the fission
reaction has an appropriate energy to induce, and hence perpetuate, the fission
reaction.
Fission weapons currently use 239
Pu or highly enriched 235
U (usually greater than
90%), although, in principle, enrichments as low as 10% are usable. Fission weapons
utilizing 239
Pu have higher yield-to-weight ratios and can be made with smaller sizes
and weights. One problem in plutonium-based weapons is the presence of 240
Pu,
whose high spontaneous fission rate can present problems with the preinitiation of
the weapon. Preinitiation of the weapon is defined as the initiation of the nuclear
chain reaction before the desired degree of supercriticality (see below) is achieved.
The neutrons emitted during the spontaneous fission of 240
Pu can cause such a preinitiation,
which will decrease the yield of the weapon and increase the uncertainty
in that yield. To prevent this preinitiation, weapons-grade plutonium contains less
than 7% 240
Pu while ordinary reactor-grade plutonium may contain more than
19% 240
Pu. The 240
Pu content of plutonium can be regulated by controlling the
time 238
U is left in the reactor for generating 239
Pu. Many U.S. fission weapons
contain both 239
Pu and 235
U as a trade-off between the higher efficiency of using
239
Pu and the greater availability of 235
U. (About 43 metric tons of Pu are in U.S.
nuclear weapons.) If the conditions are such that the neutrons are lost at a faster
rate than they are formed by fission, the chain reaction is not self-sustaining. The
escape of neutrons occurs at the exterior of the 239
Pu (or 235
U) mass undergoing
fission, and thus the rate of loss by escape will be determined by the surface area.
On the other hand, the fission process, which results in the formation of more neutrons,
takes place throughout the whole of the material; the rate of growth of neutron
population is therefore dependent upon the mass. If the quantity of 239
Pu (or 235
U) is
small, that is, if the ratio of the surface area to the volume is large, the proportion
of neutrons lost by escape to those producing fissions will be so great that the
422 REACTORS AND ACCELERATORS
propagation of a nuclear fission chain, and hence the production of an explosion, will
not be possible. But as the size of the piece of 239
Pu (or 235
U) is increased and the
relative loss of neutrons is thereby decreased, a point is reached at which the chain
reaction can become self-sustaining. This is referred to as the “critical mass” of the
fissionable material.
The critical mass of a bare sphere of normal density 235
U metal has been reported to
be 52 kg while the same number reported for certain phases of plutonium metal is
about 10 kg. However, the critical mass may be lowered in a number of ways. Use
of a reflector can lower the critical mass by a factor of 2–3. Compression of the
material to increase its density will also lower the value of the critical mass, with
the critical mass being approximately proportional to the inverse square of the
density. Most nuclear weapons employ only a fraction of the critical mass (at
normal density). Because of the presence of stray neutrons in the atmosphere or the
possibility of their being generated in various ways, a quantity of 239
Pu (or 235
U)
exceeding the critical mass would be likely to melt or possibly explode. It is necessary,
therefore, that before detonation a nuclear weapon should contain no single piece
of fissionable material that is as large as the critical mass for the given condition. In
order to produce an explosion, the material must then be made supercritical, that is,
made to exceed the critical mass, in a time so short as to completely preclude a subexplosive
change in the configuration, such as by melting.
Two general methods have been described for bringing about a nuclear
explosion, that is to say, for quickly converting a subcritical system into a supercritical
one. In the first procedure, two or more pieces of fissionable material, each less
than a critical mass, are brought together very rapidly in the presence of neutrons to
form one piece that exceeds the critical mass. This may be achieved in some kind of
gun-barrel device in which a high explosive is used to blow one subcritical piece of
fissionable material from the breech end of the gun into another subcritical piece
firmly held in the muzzle end. The first nuclear weapons had a mass of 235
U in
the form of a sphere with a plug removed from its center. The plug was then fired
into the center of the sphere creating a supercritical assembly. This technique is
largely of historical interest.
The second method makes use of the fact that when a subcritical quantity of an
appropriate isotope, that is, 239
Pu (or 235
U), is strongly compressed, it can become
critical or supercritical. The reason for this is that compressing the fissionable
material, that is, increasing its density increases the rate of production of neutrons
by fission relative to the rate of loss by escape. The surface area (or neutron
escape area) is decreased, while the mass (upon which the rate of propagation of
fission depends) remains constant. A self-sustaining chain reaction may then
become possible with the same mass that was subcritical in the uncompressed state.
In a fission weapon, the compression may be achieved by encompassing the
subcritical material with a shell of chemical high explosives, which is imploded
by means of a number of external detonators, so that a uniform inwardly directed
“implosion” wave is produced. The implosion wave creates overpressures of
millions of pounds per square inch in the core of the weapon, increasing the
density by a factor of 2. A simple estimate may be made to show that the resulting
14.7 NUCLEAR WEAPONS 423
assembly should have a size of 10 cm, the mean free path of a fast neutron in 235
U or
239
Pu. The implosion technique is used in modern nuclear weapons.
In both methods, high-density, heavy metals are used to surround the fissionable
material, thereby reducing or preventing the escape of neutrons from the reacting
assembly. To contain the fissionable material and ensure that a large enough fraction
of the nuclei undergo fission before the expansion of the exploding material causes
subcriticality, the fissile material is surrounded by a heavy-metal case that acts as a
tamper (and a neutron reflector).
In a thermonuclear or hydrogen bomb, a significant fraction of the energy release
occurs by nuclear fusion rather than nuclear fission. The hydrogen isotopes, 2
H
(deuterium, D) and 3
H (tritium, T), can be made to fuse, as:
2
H þ 3
H À! 4
He þ n þ 17 MeV
To initiate such a D–T fusion reaction requires temperatures of 10–100 million
degrees. Relatively large amounts of deuterium/tritium and/or lithium deuteride
can be heated to such temperatures by a fission explosion where the temperature
may be 108
K. (Tritium is generated in situ by the neutron bombardment of 6
Li
during the fusion reaction by the reaction 6
Li þ n ! 3
H þ 4
He þ n þ 17 MeV,
thus making the overall fusion reaction 6
Li þ 2
H ! 2 4
He þ 21.78 MeV).
The energy release can be enhanced further by using the high-energy neutrons
released in the fusion reactions to induce fission in the abundant isotope, 238
U.
Thus, we have fission–fusion and fission–fusion–fission weapons, which can give
rise to explosions of much greater energy than those from simple fission weapons.
In a typical modern multistage thermonuclear weapon, the radiation from a
fission explosion is used to transfer energy and compress a physically separate component
containing the fusion material. The fissile material is referred to as the
primary stage while the fusion material is called the secondary stage. A third
stage can be added in which the fast neutrons from the fusion reaction are used to
initiate the fission of 238
U. In modern multistage thermonuclear weapons, comparable
energy release is said to come from fission and fusion reactions.
A published schematic diagram of the operation of a modern multistage
thermonuclear weapon is shown in Figure 14.15. The fission stage is similar
to the implosion weapon used over Nagasaki but is only 12 in. in diameter.
The chemical explosives are arranged in a soccer ball configuration with
20 hexagons and 12 pentagons forming a sphere. Detonator wires are attached to
each face.
In this example, the fusion reaction must take place before the expanding fireball
of the exploding fission trigger destroys the fusion materials (i.e., in a time scale of
less than 100 shakes). This is accomplished through the use of x and g radiation to
transmit the energy of the fission reaction. The x and g radiation travels about 100
times faster than the exploding debris from the fission reaction to the fusion
assembly. As shown in Figure 14.15, the thermonuclear weapon in this example
is a 3 to 4-ft-long cylinder with an 18-in. diameter with the fission stage located
424 REACTORS AND ACCELERATORS
near one end and the fusion stage near the other. The x or g radiation is directed to a
tamper of polystyrene foam, which surrounds the fusion assembly. The radiation
energy is absorbed by the polystyrene foam, which is transformed into a highly energized
plasma, which compresses the fusion fuel assembly.
The “neutron bomb” or “enhanced radiation” weapon is a thermonuclear weapon
in which the energy release in the form of heat and blast is minimized, and the lethal
effects of the high-energy neutrons generated in fusion are maximized. This is
reported to be done by the elimination of the 238
U components of the weapon.
The suggested net effect of this is that the instantaneously incapacitating radius
(dose of 8000 rad) of a neutron bomb is about the same as a fission weapon with
10 times the yield. The instantaneously incapacitating radius for a one-kiloton
(kT) neutron bomb is thus about 690 m.
Nuclear weapon yields are measured in units of kilotons of TNT (1 kT of
TNT ¼ 1012
calories ¼ the explosive energy release from 60 g of fissile material).
The energy release is mostly in the form of pressure and heat with a smaller
amount (15%) released in the form of radiation. The first nuclear explosive
device, which was detonated at Alamogordo, New Mexico, had a yield of about 20
kT as did the Fat Man bomb dropped over Nagasaki, Japan (both fueled by 239
Pu).
The Little Boy bomb dropped over Hiroshima, Japan, had a yield of 12–15 kT
(fueled by 235
U). The efficiency of the plutonium-based devices was about 17%,
while the uranium-based device had an efficiency of about 1.3%. For a 20-kT
weapon, the radiation dose at 500 m from the center is estimated to be 70 Gy and
drops to 4 Gy at 1.1 km. The smallest nuclear weapons have been reported to
have weight that is about 0.5% of the Fat Man bomb (10,800 1b) and a total
size of 25–30 in. in length and 10–12 in. in diameter, with explosive yields
about 0.25 kT. Modern thermonuclear weapons with yields above 100 kT have
yield/weight ratios of 1–3 kT/kg, which is far from the theoretical maximum of
80 kT/kg.
Figure 14.15 A schematic diagram of a hydrogen bomb. Adapted from Carey Sublette,
Nuclear Weapon Archive.
14.7 NUCLEAR WEAPONS 425
PROBLEMS
1. A Cockcroft–Walton accelerator produces 400-keV protons. What is the
maximum energy of the neutrons that can be produced with this accelerator
using the d þ T reaction?
2. Given a reactor that contains 11 kg of 235
U and operates at a power level of
1 MWe, what is the antineutrino flux 15 m from the core?
3. Verify the statement that the reactor poison 135
Xe reaches a maximum 10 h
after shutdown of a high flux (.1014
n/cm2
s) reactor.
4. Given the reactor of Problem 2, how long can it run before it uses up 10% of
its fuel?
5. If natural uranium in dissolved in D2O at a concentration of 0.4 g/g D2O,
calculate k1 and the radius of a critical mass.
6. Given a reactor where the time between production and absorption of neutrons
is 1 ms and the power level is 1 MWe, calculate the number of free neutrons in
the reactor during operation.
7. Calculate the number of collisions needed to reduce a neutron’s energy from
1 MeV to 0.025 eV in H2O, D2O, and C. Calculate the neutron mean free
path in each case.
8. Given a 1-g source of 252
Cf, calculate the neutron flux 1 m from the source and
the heat produced in the source.
9. For a 1000-MWe nuclear reactor fueled with a fuel containing 5% 235
U,
calculate the uranium use in a year of full-time operation.
10. Describe and compare electron cooling and stochastic cooling.
11. Using the Web as an information source, compare the current Michigan
State University superconducting cyclotrons (K500, K1200) and the Uppsala
University GWI synchrocyclotron with a “classical” cyclotron.
12. Describe the duty cycle of a typical Van de Graaff accelerator, a linac, and a
synchrotron.
13. For an accelerator with a radius equal to that of Earth, and a magnetic field of
40 kG, calculate the maximum energy of the protons that could be accelerated.
14. A cyclotron accelerates a particles to an energy of 42 MeV. What is the K of the
cyclotron? What is the maximum energy deuteron beam that it can produce?
15. A cyclotron has a diameter of 60 in. with a frequency of 10.75 MHz. Calculate
the maximum energy of the proton beam and maximum field strength needed.
16. Explain the meaning of the term “phase stability” in regard to a linac and as used
with a synchrocyclotron.
426 REACTORS AND ACCELERATORS
17. To study the structure of the proton, what energy particles do you need?
18. Consider the following situations: (a) a 20-GeV proton collides with a target
electron, (b) a 20-GeV electron collides with a target proton, and (c) a
10-GeV proton collides with a 10-GeV electron. In each case, what is the cm
energy available to create new particles?
REFERENCES
Burcham, W. E. and M. Jobes. Nuclear and Particle Physics, Longman, Burnt Mill, 1995.
Harvey, B. G. Introduction to Nuclear Physics and Chemistry, 2nd ed., Prentice-Hall,
Englewood Cliffs, NJ, 1964.
Krane, K. S. Introductory Nuclear Physics, Wiley, New York, 1988.
Segre, E. Nuclei and Particles, 2nd ed., Benjamin, Reading, MA, 1977.
BIBLIOGRAPHY
Brune, D., B. Forkman, and B. Persson. Nuclear Analytical Chemistry, Chartwell-Bratt,
Sweden, 1984.
Choppin, G., J. O. Liljenzin, and J. Rydberg. Radiochemistry and Nuclear Chemistry, 3rd ed.,
Butterworth-Heinemann, Oxford, UK, 2001.
Cochran, T. B., W. M. Arkin, and M. M. Hoenig. Nuclear Weapons Databook, Vol I,
Ballinger, Cambridge, 1984.
Das, A. and T. Ferbel. Introduction to Nuclear and Particle Physics, Wiley, New York, 1994.
Draganic, I. G., Z. D. Draganic, and J. P. Adloff. Radiation and Radioactivity on Earth and
Beyond, CRC Press, Boca Raton, FL, 1993.
Enge, H. A. Introduction to Nuclear Physics, Addison-Wesley, Reading, 1966.
Frauenfelder, H., and E. M. Henley. Subatomic Physics, 2nd ed., Prentice-Hall, Englewood
Cliffs, NJ, 1991.
Friedlander, G., J. W. Kennedy, E. S. Macias, and J. M. Miller. Nuclear and Radiochemistry,
3rd ed., Wiley, New York, 1981.
Kruger, P. Principles of Activation Analysis, Wiley, New York, 1971.
LeFort, M. Nuclear Chemistry, van Nostrand, Princeton, NJ, 1968.
Perkins, D. H. Introduction to High Energy Physics, 4th ed., Cambridge University Press,
Cambridge, 2000.
Seaborg, G. T. and W. Loveland. The Elements Beyond Uranium, Wiley, New York, 1990.
BIBLIOGRAPHY 427
CHAPTER 15
THE TRANSURANIUM ELEMENTS
15.1 INTRODUCTION
The chemical elements are the building blocks of nature. All substances are combinations
of these elements. There are (as of 2005) 113 known chemical elements with
the heaviest naturally occurring element being uranium (Z ¼ 92). The 22 heaviest
chemical elements, the transuranium elements, are manmade. The story of their synthesis,
their properties, their impact on chemistry and physics, and their importance
to society is fascinating. This story is of particular importance to nuclear chemistry
because most of our knowledge of these elements and their properties comes from
the work of nuclear chemists, and such work continues to be a major area of nuclear
chemical research. One of us (GTS) has been intimately involved in the discovery
and characterization of these transuranium elements.
In this chapter, we will discuss how to make these elements, their chemical properties,
and their presence in the environment. The current list of transuranium
elements is shown in Table 15.1 with a modern view of their place in the periodic
table being shown in Figure 15.1.
15.2 LIMITS OF STABILITY
There are about 260 transuranium nuclei known as of 2005. All these nuclei
are unstable, with half-lives ranging from 1029
s to 1017
s (Fig. 15.2a). The
longest-lived nuclei are those with lower values of Z. As Z increases, the lifetimes
429
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
become shorter, with the lifetimes of the heaviest elements being tiny fractions of a
second. For elements with Z ! 107, there are few, if any, nuclei whose lifetimes lend
themselves to chemical studies.
All transuranium nuclei are unstable with respect to a decay, meaning that Qa is
positive for all these nuclei. In addition, nuclei with neutron/proton ratios differing
from that of nuclei along the valley of b stability will also decay by either b2
decay
or bþ
/EC decay. For most heavy nuclei, EC decay dominates over bþ
decay, and consequently
the neutron-deficient heavy nuclei decay by EC decay. As the atomic number
of these nuclei increases, the importance of decay by spontaneous fission (SF)
increases. In Figure 15.2b, we show the dominant decay mode for each of the transuranium
nuclei. (A subtle bias occurs in Fig. 15.2b in that spontaneous fission is, in
general, not an acceptable way to characterize a nucleus, due to the lack of a definitive
way of establishing the Z, A of the fissioning system. Consequently, the heaviest known
nuclei are a emitters.) As we discussed in Chapter 11, the upper bound of the periodic
table is given by spontaneous fission. At some value of (Z, A) the spontaneous fission
half-life becomes so short as to prevent observation (t1/2 , 1029
2 1026
s). Many
transuranium nuclei decay by a combination of EC, a decay, and SF with the branching
ratios for each mode depending on the (Z, A) of the nucleus.
TABLE 15.1 Transuranium Elementsa
Atomic Number Element Symbol
93 Neptunium Np
94 Plutonium Pu
95 Americium Am
96 Curium Cm
97 Berkelium Bk
98 Californium Cf
99 Einsteinium Es
100 Fermium Fm
101 Mendelevium Md
102 Nobelium No
103 Lawrencium Lr
104 Rutherfordium Rf
105 Dubnium Db
106 Seaborgium Sg
107 Bohrium Bh
108 Hassium Hs
109 Meitnerium Mt
110 Darmstadtium Ds
111 Roentgenium Rg
112
113
114
115
116
117
118
a
Elements 112–118 have not been named (as of 2005).
430 THE TRANSURANIUM ELEMENTS
What about the breadth of the distribution of heavy nuclei? What are the limits on
N/Z? As in the lighter nuclei, the limits are set by the proton dripline (Sp ¼ 0) and
the neutron dripline (Sn ¼ 0). For a typical heavy nucleus, nobelium, the proton dripline
is at N  132, while the neutron dripline is at N  236 (Mo¨ller et al., 1997).
The range of known nobelium isotopes goes from N ¼ 147 to N ¼ 160. Thus, it is
unlikely that one will be limited by the neutron dripline for heavy nuclei, while the
proton dripline may be reachable with some effort. In Figure 15.3, we show the calculated
(Mo¨ller et al., 1997) changes in the half-lives of the heavy nuclei, as they
become more neutron rich. Comparison of Figures 15.2a and 15.3 shows that the
predicted half-lives increase by orders of magnitude as the neutron number increases
modestly from those currently observed. This effect motivated recent work to make
more neutron-rich heavy nuclei to study their chemistry and atomic physics.
The underlying science behind these trends is that increasing N/Z decreases Z2
/A
(reducing SF decay), decreases Qa (reducing the a-decay probability), and this
consequently leads to nuclei that decay by b2
decay, a slower process.
15.3 ELEMENT SYNTHESIS
The synthesis of a new element involves more than just colliding two nuclei whose
atomic numbers are such that they sum to those of a previously unobserved nuclide.
Figure 15.1 Modern periodic table showing the transuranium elements. (Figure also
appears in color figure section.)
15.3 ELEMENT SYNTHESIS 431
Heavy nuclei are, in general, quite fissionable. If they are made with significant excitation,
they will decay by fission, leaving no easily identifiable heavy residue of their
formation. So one must balance carefully the factors governing the “production” of a
new nucleus with those factors governing its “survival”. The “production factors”
determine the yield of the primary reaction products while the “survival factors”
determine how the primary product nuclei deexcite by particle emission, allowing
them to survive or those nuclei that deexcite by fission, destroying them. Among
Figure 15.2 (a)Half-livesoftheknowntransuraniumnucleiplottedasafunctionofZandN.(b)
Dominant decay modes of the nuclei shown in (a). (Figure also appears in color figure section.)
432 THE TRANSURANIUM ELEMENTS
the production factors are items such as the “starting material,” the target nuclei,
which must be available in sufficient quantity and suitable form. We must have
enough transmuting projectile nuclei also. Note that the N/Z ratio of light projectile
nuclei is typically lower than that of heavy nuclei and places a limit on the fusion
process. The transmutation reaction must occur with adequate probability to
ensure a good yield of the product nucleus in a form suitable for further study.
Equally important is that the product nuclei be produced with excitation energy
and angular momentum distributions such that the product nuclei will deexcite by
particle or photon emission rather than the disastrous fission process. The competition
between particle emission and fission as deexcitation paths depends on excitation
energy, angular momentum, and the intrinsic stability of the product nucleus,
which is related to the atomic and mass numbers of the product.
Nuclear synthesis is similar in some ways to inorganic or organic chemical syntheses
with the synthetic chemist or physicist having to understand the reactions
involved and the structure and stability of the intermediate species. While, in
principle, the outcome of any synthesis reaction is calculable; in practice such calculations
are, for the most part, very difficult. Instead, the cleverness of the scientists
involved, their manipulative skills, and the instrumentation available for their use
often determine the success of many synthetic efforts.
The synthesis reactions used to “discover” the transuranium elements are given in
Table 15.2. All these reactions are complete fusion reactions in which the reacting
nuclei fuse, equilibrate, and deexcite in a manner independent of their mode of
formation. Other production reactions involving a partial capture of the projectile
nucleus are also possible.
Figure 15.3 Predicted (Mo¨ller et al., 1997) half-lives of the transuranium nuclei with
Z 112. (Figure also appears in color figure section.)
15.3 ELEMENT SYNTHESIS 433
The cross section for production of a heavy evaporation residue, sEVR, by a
complete fusion reaction can be written as:
sEVR ¼ (fusion probability)(survival probability)
TABLE 15.2 Summary of Transuranium Element Synthesis
Atomic Number Name and Symbol Synthesis Reaction Half-Life
238
U þ n À! 239
U þ g
93 Neptunium (Np) 239
U þ À!
bÀ
239
Np 2.35 d
238
U þ 2
H À! 238
Np þ 2n
94 Plutonium (Pu) 238
Np À!
bÀ
238
Pu 86.4 y
239
Pu þ n À! 240
Pu þ g
240
Pu þ n À! 240
Pu þ g
95 Americium (Am) 241
Pu À!
bÀ
241
Am 433 y
96 Curium (Cm) 239
Pu þ 4
He À! 242
Cm þ n 162.5 d
97 Berkelium (Bk) 241
Am þ 4
He À! 243
Bk þ 2n 4.5 h
98 Californium (Cf) 242
Cm þ 4
He À! 245
Cf þ n 44 min
99 Einsteinium (Es) “Mike” thermonuclear
explosion (leading to 253
Es)
20 d
100 Fermium (Fm) “Mike” thermonuclear
explosion (leading to 255
Fm)
20 h
101 Mendelevium (Md) 253
Es þ 4
He À! 256
Md þ n 76 min
102 Nobelium (No) 244
Cm þ 12
C À! 252
No þ 4n 2.3 s
103 Lawrencium (Lr) 250,251,252
Cf þ 11
B À! 258
Lr þ 3–5n
250,251,252
Cf þ 10
B À! 258
Lr þ 2–4n
4.3 s
104 Rutherfordium (Rf) 249
Cf þ 12
C À! 257
Rf þ 4n
þ 13
C À! 259
Rf þ 3n
3.4 s
3.8 s
105 Dubnium (Db) 249
Cf þ 15
N À! 260
Db þ 4n 1.5 s
106 Seaborgium (Sg) 249
Cf þ 18
O À! 263
Sg þ 4n 0.9 s
107 Bohrium (Bh) 209
Bi þ 54
Cr À! 262
Bh þ n 102 ms
108 Hassium (Hs) 208
Pb þ 58
Fe À! 265
Hs þ n 1.8 ms
109 Meitnerium (Mt) 209
Bi þ 58
Fe À! 266
Mt þ n 3.4 ms
110 Darmstadtium (Ds) 209
Bi þ 59
Co À! 267
Ds þ n
208
Pb þ 62
Ni À! 269
Ds þ n
208
Pb þ 64
Ni À! 271
Ds þ n
244
Pu þ 34
S À! 273
Ds þ 5n
4 ms
170 ms
56 ms
118 ms
111 Roentgenium (Rg) 209
Bi þ 64
Ni À! 272
Rg þ n 1.5 ms
112 — 208
Pb þ 70
Zn À! 277
112 þ n 240 ms
113 — 209
Bi þ 70
Zn À! 278
113 + n 340 ms
434 THE TRANSURANIUM ELEMENTS
where the fusion probability refers to the probability of forming a completely fused
system in the reaction, and the survival probability refers to the probability that the
excited complete fusion product will deexcite by particle emission rather than
fission, which destroys the nucleus. Recent synthesis reactions for heavy nuclei
are divided into “cold” or “hot” fusion. Cold fusion reactions involve a heavier projectile
(Ar–Kr) interacting with a Pb or Bi nucleus, where the excitation energy of
the completely fused system is low (13 MeV), giving high survival probabilities.
Unfortunately, the fusion probability in such systems is low. Hot fusion reactions
involve the use of lighter projectiles (11
B–48
Ca) interacting with actinide nuclei,
giving a high fusion probability, but a high excitation energy (EÃ
 30–50 MeV)
with a resulting low survival probability.
The reactions in Table 15.2 can be divided into four classes: the neutron-induced
reactions (Z ¼ 93, 95, 99, 100), the light-charged particle-induced reactions
(Z ¼ 94, 96–98, 101), the hot fusion reactions (Z ¼ 102–106), and the cold
fusion reactions (Z ¼ 107–113). In the neutron-induced reactions used to make
the transuranium nuclei, the capture of a neutron does not create a new element,
but the subsequent b2
decays do similar to the astrophysical s-process. Lightcharged
particle reactions with exotic actinide target nuclei allow one to increase
the atomic number of the product one or two units from the target nucleus. To
make the heaviest elements, one needs to add several protons to the target
nucleus by a reaction with a heavy ion. Such hot fusion reactions with actinide
target nuclei lead to highly excited intermediate species that decay mostly by
fission but occasionally by emitting neutrons, thus producing new heavy nuclei.
However, as the atomic number of the product nuclei increases, so does the probability
of fission leading to very poor survival probabilities for the putative new
species. The nuclear physicist Yuri Oganessian pointed out that a way around this
problem was to fuse heavier projectile nuclei with nuclei in the lead–bismuth
region. Because of the special stability of the lead–bismuth nuclei, the resulting
fused species would be formed “cold” and could, with some reasonable probability,
decay by only emitting a single neutron.
In Figure 15.4, we show current measurements (filled squares) of the cross sections
for cold fusion reactions as a function of the atomic number Z of the completely
fused system. (The cold fusion point at Z ¼ 118 is an upper limit.) Also shown
(as open circles) are the cross sections for hot fusion reactions. Clearly, future efforts
will have to focus on experiments at the 0.1- to 1.0-pb cross-section level or lower.
Current technology for cold fusion reaction studies would require 12 days to
observe one event at a cross-section level of 1 pb. Similarly, a cross section of
1 pb in a hot fusion reaction would require 6–19 days to observe one event.
From examining the data in Figure 15.4, it would also appear that hot fusion
reactions might be the reactions of choice in pursuing future research in this area.
Aside on Element Synthesis Calculations The reactions used to synthesize heavy
nuclei are, quite often, very improbable reactions, representing minor branches to
the main reaction. Their probability of occurrence with respect to the main synthesis
15.3 ELEMENT SYNTHESIS 435
reaction is frequently less than 1026
. As such, it is intrinsically difficult to accurately
describe these reactions from a theoretical point of view. Instead, workers in this
field have frequently resorted to semiempirical prescriptions to guide their efforts.
To give one a feel for the magnitude of the quantities involved, we outline below
a very simple schematic method for estimating heavy-element production cross
sections. It is intended to show the relevant factors and should not be taken too
seriously, except to indicate the order of magnitude of a particular formation
cross section.
The physicist Peter Armbruster has made an empirical systematic description of
the probability of fusion of two heavy nuclei at energies near the reaction barrier.
These systematics are shown in Figure 15.5. To use this graph, one picks values
Figure 15.4 Plot of the observed cross sections for the production of heavy elements by
cold and hot fusion reactions.
Figure 15.5 Plot of the contours of log10 sfus (where sfus is the s-wave fusion cross section
at the interaction barrier).
436 THE TRANSURANIUM ELEMENTS
of the atomic number of projectile and target nuclei and reads off the expected value
for the cross section for producing a completely fused species. The excitation energy
of the completely fused species can then be read from Figure 15.6, which is based
upon the nuclear masses of Peter Mo¨ller, J. Rayford Nix, and Karl-Ludvig Kratz.
Taking as a rough rule of thumb, for each 10 MeV of excitation energy, the survival
probability of the fused system drops by a factor of 102
, one can then compute the
cross section for producing a given species.
For example, the successful synthesis of 265
Hs(265
108) involved the reaction
58
Fe þ 208
Pb À! 265
Hs þ n
From Figure 15.5, one predicts the fusion cross section to be 10232
cm2
, while
Figure 15.6 would suggest an excitation energy of 20 MeV. Thus, one would
roughly estimate the overall cross section for producing 265
Hs to be
(10À32
)(10À2
)2
% 10À36
cm2
(The measured cross section was 2 Â 10235
cm2
.)
Aside on the Detection of Heavy Atoms The detection of atoms of a new element
has always focused on measuring the atomic number of the new species and showing
that it is different from all known values of Z. Unambiguous methods for establishing
the atomic number include chemical separations, measurement of the X-ray
spectrum accompanying a nuclear decay process, or establishment of a genetic
relationship between the unknown new nucleus and some known nuclide. As the
quest for new elements focuses on still heavier species, the probability of producing
the new elements has decreased, and one has had to devote increasing attention to
Figure 15.6 Plot of the excitation energy of the completely fused species formed from a
given target–projectile combination. Reactions are assumed to take place at the interaction
barrier.
15.3 ELEMENT SYNTHESIS 437
the problem of detecting a few atoms of a new species amid a background of many
orders of magnitude more of other atoms. Thus, modern attempts to make new
heavy-element atoms usually involve some kind of “separator.”
An example of a modern separator is the velocity filter SHIP (Fig. 15.7) at the
GSI in Darmstadt, Germany. In this separator, nuclear reaction products
(from the target wheel) undergo different deflections (in crossed electric and
magnetic fields) according to their electric and magnetic rigidity which will
depend on whether they are fission fragments, scattered beam particles, or the
desired heavy-element residues. The efficiency of the separator is 50% for
heavy-element residues, while transfer products and scattered beam nuclei are
rejected by factors of 1014
and 1011
, respectively. The heavy recoil atoms
are implanted in the silicon detectors. Their implantation energy and position are
correlated with any subsequent decays of the nuclei to establish genetic relationships
to known nuclei.
15.4 HISTORY OF TRANSURANIUM ELEMENT DISCOVERY
The first scientific attempts to prepare the elements beyond uranium were performed
by Enrico Fermi, Emilio Segre, and co-workers in Rome in 1934, shortly after the
existence of the neutron was discovered. This group of investigators irradiated
uranium with slow neutrons and found several radioactive products, which were
thought to be due to new elements. However, detailed chemical studies by Otto
Hahn and Fritz Strassman in Berlin showed these species were isotopes of the
known elements created by the fission of uranium into two approximately equal
parts (see Chap. 11). This discovery of nuclear fission in December of 1938 was
thus a by-product of man’s quest for the transuranium elements.
Figure 15.7 Velocity filter SHIP. (Figure also appears in color figure section.)
438 THE TRANSURANIUM ELEMENTS
With poetic justice, the actual discovery of the first transuranium element came as
part of an experiment to study the nuclear fission process. Edwin McMillan, working
at the University of California at Berkeley in the spring of 1939, was trying to
measure the energies of the two recoiling fragments from the neutron-induced
fission of uranium. He placed a thin layer of uranium oxide on one piece of
paper. Next to this he stacked very thin sheets of cigarette paper to stop and
collect the uranium fission fragments. During his studies he found there was
another radioactive product of the reaction—one that did not recoil enough to
escape the uranium layer, as did the fission products. He suspected that this
product was formed by the capture of a neutron by the more abundant isotope of
uranium, 238
92 U. McMillan and Philip Abelson, who joined him in this research,
showed in 1940, by chemical means, this product is an isotope of element 93,
239
93 Np, formed in the following sequence:
238
92 U þ 1
0n À! 239
92 U þ g
239
92 U(t1=2 ¼ 23:5 min ) À!
bÀ
239
93 Np(t1=2 ¼ 2:36 d)
Neptunium, the element beyond uranium, was named after the planet Neptune
because this planet is beyond the planet Uranus for which uranium is named.
Plutonium (named after the planet Pluto, following the pattern used in naming
neptunium) was the second transuranium element to be discovered. By bombarding
uranium with charged particles, in particular, deuterons (2
H), using the 60-in. cyclotron
at the University of California at Berkeley, Glenn T. Seaborg, Edwin McMillan,
Joseph W. Kennedy, and Arthur C. Wahl succeeded in preparing a new isotope of
neptunium, 238
Np, which decayed by b2
emission to 238
Pu, that is,
238
92 U þ 2
1H À! 238
93 Np þ 2n
238
93 Np(t1=2 ¼ 2:12 d) À!
bÀ
238
94 Pu(t1=2 ¼ 87:7 y)
Early in 1941, 239
Pu, the most important isotope of plutonium was discovered by
Kennedy, Segre, Wahl, and Seaborg. 239
Pu was produced by the decay of 239
Np,
which in turn was produced by the irradiation of 238
U by neutrons, using the reaction
discovered by McMillan
238
92 U þ 1
0n À! 239
92 U þ g
239
92 U(t1=2 ¼ 23:5 min ) À!
bÀ
239
93 Np(t1=2 ¼ 2:35 d) À!
bÀ
239
94 Pu(t1=2 ¼ 24,110 y)
This isotope, 239
Pu, was shown to have a cross section for thermal neutron-induced
fission that exceeded that of 235
U, a property that made it important for nuclear
weapons, considering that it could be prepared by chemical separation as compared
to isotopic separation that was necessary for 235
U.
The next transuranium elements to be discovered, americium and curium
(Am and Cm; Z ¼ 95 and 96, respectively), represent an important milestone in
15.4 HISTORY OF TRANSURANIUM ELEMENT DISCOVERY 439
chemistry, the recognition of a new group of elements in the periodic table, the actinides.
According to the periodic table of the early 20th century, one expected americium
and curium to be eka-iridium and eka-platinum, that is, to have chemical
properties similar to iridium and platinum. In 1944, Seaborg conceived the idea
that all the known elements heavier than actinium (Z ¼ 89) had been misplaced
in the periodic table. He postulated that the elements heavier than actinium might
form a second series similar to the lanthanide elements (Fig. 15.1), called the
actinide series. This series would end in element 103 (Lr) and, analogous to the
lanthanides, would show a common oxidation state of þ3.
Once this redox property was understood, the use of a proper chemical
procedure led quickly to the identification of an isotope of a new element. Thus, a
new a-emitting nuclide, now known to be 242
96 Cm (half-life 162.9 d), was identified
by Seaborg, Albert Ghiorso, and Ralph James in the summer of 1944 in the
bombardment of 239
94 Pu with 32-MeV helium ions:
239
94 Pu þ 4
2He À! 242
96 Cm þ 1
0n
The bombardment took place in the Berkeley 60-in. cyclotron, after which the target
material was shipped to the Metallurgical Laboratory at Chicago for chemical separation
and identification. A crucial step in the identification of the a-emitting
nuclide as an isotope of element 96, 242
96 Cm, was the identification of the known
238
94 Pu as the a-decay daughter of the new nuclide.
The identification of an isotope of element 95, by Seaborg, Ghiorso, James, and
Leon Morgan in late 1944 and early 1945, followed the identification of this isotope
of element 96 (242
Cm) as a result of the bombardment of 239
94 Pu with neutrons in a
nuclear reactor. The production reactions, involving multiple neutron capture by
plutonium, are
239
94 Pu þ 1
0n À! 240
94 Pu þ g
240
94 Pu þ 1
0n À! 241
94 Pu þ g
241
94 Pu(t1=2 ¼ 14:4 y) À!
bÀ
241
95 Am(t1=2 ¼ 432:7 y)
241
95 Am þ 1
0n À! 242
95 Am þ g
242
95 Am(t1=2 ¼ 16:0 h) À!
bÀ
242
96 Cm
The years after World War II led to the discovery of elements 97–103 and the
completion of the actinide series. While the story of the discovery of each of
these elements is fascinating, we shall, in the interests of brevity, refer the reader
elsewhere (see References) for detailed accounts of most of these discoveries. As
an example of the techniques involved, we shall discuss the discovery of element
101 (mendelevium).
440 THE TRANSURANIUM ELEMENTS
The discovery of mendelevium was one of the most dramatic in the sequence of
transuranium element syntheses. It marked the first time in which a new element was
produced and identified one atom at a time. By 1955, scientists at Berkeley had prepared
an equilibrium amount of about 109
atoms of 253
99 Es by neutron irradiation of
plutonium in the Materials Testing Reactor in Idaho. As the result of a “back of the
envelope” calculation done by Ghiorso during an airplane flight, they thought it
might be possible to prepare element 101 using the reaction
253
99 Es þ 4
2He À! 256
101Md þ 1
0n
The amount of element 101 expected to be produced in an experiment can be
calculated using the formula
N101 ¼
NEssf(1 À eÀlt
)
l
where N101 and NEs are the number of element 101 atoms produced and the number
of 253
99 Es target atoms, respectively, s is the reaction cross section (estimated to be
210227
cm2
), f is the helium ion flux (ffi1014
particles/s), l is the decay constant
of 256
101Md (estimated to be %1024
s21
), and t is the length of each bombardment
(%104
s).
N101 ¼
(109
)(10À27
)(1014
)(1 À eÀ(10À4
)(104
)
)
10À4
% 1 atom
Thus, the production of only one atom of element 101 per experiment could be
expected!
Adding immensely to the complexity of the experiment was the absolute necessity
for the chemical separation of the one atom of element 101 from the 109
atoms
of einsteinium in the target and its ultimate, complete chemical identification by
separation with the ion exchange method. This separation and identification
would presumably have to take place in a period of hours, or perhaps even one
hour or less, since the half-life of new nuclide was unknown. Furthermore, the
target material had a 20-d half-life, and one needed a nondestructive technique
allowing reuse of the target material.
The definitive experiments were performed in a memorable, all-night session, on
February 18, 1955. To increase the number of events that might be observed at one
time, three successive 3-h bombardments were made, and, in turn, their transmutation
products were quickly and completely separated by the ion exchange method.
Some of the nuclide 253
99 Es was present in each case so, together with the 246
98 Cf produced
from 246
96 Cm also present in the target [via the 244
Cm (4
He, 2n) reaction], it was
possible to define the positions in which the elements came off the column used to
contain the ion exchange resin. Five spontaneous fission counters then were used to
count simultaneously the corresponding drops of solution from the three runs.
15.4 HISTORY OF TRANSURANIUM ELEMENT DISCOVERY 441
A total of five spontaneous fission counts was observed in the element 101
position, while a total of eight spontaneous fission counts were also observed in
the element 100 position. No such counts were observed in any other position.
The original data are presented in Figure 15.8.
The synthesis of the transactinides is noteworthy from a chemical and a nuclear
viewpoint. From the chemical point of view, rutherfordium (Z ¼ 104) is important
as an example of the first transactinide element. From Figure 15.1, we would expect
rutherfordium to behave as a Group 4 (IVB) element, such as hafnium or zirconium,
but not like the heavy actinides. Its solution chemistry, as deduced from chromatography
experiments, is different from that of the actinides and resembles that of
zirconium and hafnium. More recently, detailed gas chromatography has shown
important deviations from expected periodic table trends and relativistic quantum
chemical calculations.
The work on the discovery and identification of elements 104–106 was controversial
and contentious due, in part, to the difficulty of the experiments. Looking
back now, the following series of experiments clearly identified these elements.
Ghiorso et al. (1969) produced isotopes of element 104 in experiments at
Berkeley in 1969. The nuclear reactions involved were
249
98 Cf þ 12
6 C À! 257
104Rf(t1=2 % 3:8 s) þ 41
0n
249
98 Cf þ 13
6 C À! 259
104Rf(t1=2 % 3:4 s) þ 31
0n
The atomic numbers of these isotopes were identified by detecting the known No
daughters of these nuclei. The group suggested the name of rutherfordium (chemical
symbol Rf) for element 104 in honor of Lord Ernest Rutherford.
Figure 15.8 Original elution data corresponding to the discovery of mendelevium,
February 18, 1955. The curves for einsteinium-253 (given the old symbol E253
) and
californium-246 are for a-particle emission. (Dowex 50 ion exchange resin was used, and
the eluting agent was ammonium a-hydroxyisobutyrate.)
442 THE TRANSURANIUM ELEMENTS
Contemporaneously with the Berkeley experiments, Zvara et al. (1969, 1970),
working at Dubna, produced 3.2 + 0.8 s 259
104 by the 242
94 Pu(22
10Ne, 5n) reaction.
The chloride of this spontaneously fissioning activity was shown using gas chromatography
to be slightly less volatile than Hf, but more volatile than the actinides.
An international group of reviewers (Barber et al., 1992) has determined that the
Berkeley and Dubna groups should share the credit for the discovery of element
104 and has suggested the name of rutherfordium for element 104.
In 1970, Ghiorso et al. (1970) reported the observation of an isotope of element
105 produced in the reaction
249
98 Cf þ 15
7 N À! 260
105Db(t1=2 ¼ 1:5 s) þ 41
0n
The Z and A of this isotope were established by correlations between the parent
260
Db and its daughter 256
Lr. They suggested the name of hahnium (chemical
symbol Ha) for this element in honor of the German radiochemist Otto Hahn,
co-discoverer of fission.
In a series of experiments occurring at a similar time, Druin et al. (1971) identified
a mother–daughter pair from the decay of 260,261
Db formed in the reaction
243
Am(22
Ne, 4–5n). A name of nielsbohrium (chemical symbol Ns) was suggested
for this element in honor of the Danish physicist Niels Bohr by the Russian group.
An international group (Barber et al., 1992) has suggested that credit for this discovery
also be shared, and subsequently the name of dubnium (chemical symbol Db)
was assigned to this element.
Element 106 was first synthesized by Ghiorso et al. (1974) at Berkeley in 1974
using the reaction
249
98 Cf þ 18
8O À! 263
106(t1=2 ¼ 0:8 s) þ 41
0n
The nuclide was identified by genetic links to its daughters 259
Rf and 255
No. This
synthesis was reconfirmed in 1993 by Gregorich et al. (1994). Element 106 has
been named seaborgium (symbol Sg) after one of the authors of this book. Glenn,
the co-discoverer of plutonium and nine other transuranium elements, said upon
this occasion (Fig. 15.9): “It is the greatest honor ever bestowed upon me—even
better, I think, than winning the Nobel Prize.”
In 1981, G. Mu¨nzenberg et al. (1981), working in Darmstadt at the velocity filter
SHIP, identified the isotope 262
Bh produced in the “cold fusion” reaction:
209
83 Bi þ 54
24Cr À! 262
107Bh(t1=2 ¼ 102 ms) þ 1
0n
This nuclide was identified by genetic links to its daughters 258
Db, 254
Lr, 250
Md,
250
Fm, and 246
Cf. The cross section reported for this reaction was 200 pb (approximately
1/5,000,000 of the production cross section assumed in the discovery
of Md). This element was named bohrium (chemical symbol Bh) in honor of
Niels Bohr.
15.4 HISTORY OF TRANSURANIUM ELEMENT DISCOVERY 443
In 1984, Mu¨nzenberg et al. (1987), working at Darmstadt, produced three atoms
of 265
Hs using the cold fusion reaction 208
Pb(58
Fe, 1n). 265
Hs was identified by
genetic links to its daughters 261
Sg and 257
Rf. The half-life of this nucleus was
1.8 ms, and it decayed by a-particle emission. The production cross section was
20 pb. At a similar time, Oganessian et al. (1984) reported the production of
263– 265
Hs in the reactions 209
Bi(55
Mn, n), 206
Pb(58
Fe, n), 207
Pb(58
Fe, n), and
208
Pb(58
Fe, n). This group reported observation of spontaneous fission and a
decays of the granddaughter and great-great-great-granddaughters. Because of this
weaker identification, credit for this discovery (Barber et al., 1992) was assigned
to the Darmstadt group, which suggested the name of hassium (chemical symbol
Hs) in honor of the region of Germany, Hesse, in which the work was done.
In 1982, Mu¨nzenberg et al. (1982, 1984) reported the observation of one atom of
element 109 formed in the reaction 209
Bi(58
Fe, n). The decay of this atom was
weakly correlated to its daughters 262
Bh, 258
Db, and 258
Rf. The production cross
section was 10 pb. This discovery was confirmed by the later observation of more
Figure 15.9 Glenn Seaborg points out seaborgium in the periodic table. (Figure also
appears in color figure section.)
444 THE TRANSURANIUM ELEMENTS
atoms at Darmstadt and the observed production of the descendant 246
Cf at Dubna.
The discoverers suggested the name of meitnerium (chemical symbol Mt) in honor
of Lise Meitner.
In 1991, Ghiorso et al. (1995a, 1995b) studied the reaction of 209
Bi with 59
Co.
They found one event that they associated with the production of 267
110 (267
Ds).
The evaporation residue formed decayed by the emission of an 11.6-MeV a particle
4 ms after implantation. Their evidence for the formation of 267
110 was weakened
by the inability (due to malfunctioning electronics) to detect the decay of the
daughter 263
Hs, although the decay of other members of the decay chain was
observed. Further work was not pursued due to the closure of the accelerator
involved. This observation was probably correct, although the evidence presented
is not strong enough to justify the claim of element discovery.
There is no doubt that Hofmann et al. (1995a), working in Darmstadt in 1994,
observed the production of several atoms (nominally four—see below) of 269
Ds in
the reaction 208
Pb(62
Ni, n). This nuclide was identified by genetic links to its daughters
265
Hs, 261
Sg, 257
Rf, and 253
No. The neutron-rich isotope of element 110, 271
Ds,
was produced later (1998) (Hofmann, 1998) using the reaction 208
Pb(64
Ni, n). Nine
atoms were observed and identified. This latter reaction was used by workers in
Berkeley (Ginter, 2003) and RIKEN (K. Morita et al., Nucl. Phys. A734, 101
(2004).) to confirm the discovery of element 110. The very n-rich isotope of element
110, 273
Ds, was observed by Lazarev et al. (1996) using the reaction 244
Pu(34
S, 5n).
The name darmstadtium (chemical symbol Ds) has been given to this element.
An unfortunate footnote to this chapter in the history of the discovery of the
heaviest elements is the revelation (Hofmann et al., 2002) that one of the decay
chains reported by Hofmann et al. (1995a) for 269
Ds was “spuriously created,” the
result of human error or scientific misconduct. As disturbing as this finding is, it
should not detract from the other correctly identified decay chains.
The first production of an isotope of element 111, 272
111 (t1/2 1.5 ms) was by
Hofmann et al. (1995b) in 1994 using the reaction 209
Bi(64
Ni, n). Three decay chains
were observed in 1994. Subsequently, three more decay chains corresponding to the
decay of 272
111 were reported (Hofmann et al., 2002) in 2002. The decay of 272
111
is convincingly linked to its daughters 268
Mt, 264
Bh, 260
Db, and 256
Lr. This discovery
was confirmed by experiments at Berkeley (Folden et al., 2004) and RIKEN
(Morita, 2004). The name roentgenium (Rg) has been suggested for this element.
In 1995, Hofmann et al. (1996) reported the formation of 277
112 in the reaction
208
Pb(70
Zn, n). Two decay chains were reported connecting 277
112 to its daughters
273
Ds, 209
Hs, 265
Sg, 261
Rf, and 257
No. Unfortunately, one of these decay chains, the
first one “observed,” is now known (Hofmann et al., 2002) to also have been “spuriously
created” by human error or scientific misconduct. A subsequent experiment
(Hofmann et al., 2002) in 2000 resulted in the observation of an additional decay
chain for 277
112. These results have been confirmed by experiments at RIKEN.
It is now widely believed (S. Hofmann, private communication) that the same
individual was responsible for both false reports, that is, one of the 269
Ds decay
chains and one of the 277
112 decay chains along with false reports (see below) of
the synthesis of 293
118 and its decay products.
15.4 HISTORY OF TRANSURANIUM ELEMENT DISCOVERY 445
In 2004, Morita et al. reported the synthesis of a single atom of element 113 in the
70
Zn þ 209
Bi reaction. The production cross section was 55 fb, a remarkably small
value requiring several months of bombardment.
In 1999, Oganessian et al. (1999a, 1999b, 2000) reported the successful synthesis
of five atoms of element 114 using the 48
Ca þ 242,244
Pu reaction. The long half-lives
associated with these atoms (s-min) represent the possible approach to the long
sought “island of stability” of superheavy nuclei near Z ¼ 114 and N ¼ 184. (All
previous heavy nuclei with Z ! 110 have decayed with millisecond half-lives.)
The experimental group, consisting of scientists from Dubna and the U.S. Lawrence
Livermore National Laboratory reported the following observations:
48
20Ca þ 244
94 Pu À! 289
114(t1=2 % 21 s) þ 31
0n
48
20Ca þ 244
94 Pu À! 288
114(t1=2 % 1:9 s) þ 41
0n
48
20Ca þ 242
94 Pu À! 287
114(t1=2 % 5:4 s) þ 31
0n
All these nuclei were observed to decay by a sequence of emitted a particles with the
decay chains ending in spontaneous fission. Because these nuclei are very n-rich,
their descendents have not been characterized before. Other laboratories have not
confirmed these exciting results.
In an extension of this work, Oganessian et al. (2001) reported the successful
synthesis of three atoms of element 116 using the reaction
48
20Ca þ 248
96 Cm À! 292
116(t1=2 % 0:05 s) þ 41
0n
Each observed decay sequence involved the observation of three energetic
a-particle decays followed by a spontaneous fission.
In an unpublished report in 2002, Oganessian et al. (http://159.93.28.88/flnr/
index.html) reported the synthesis of two atoms of element 118, using the reaction
249
Cf(48
Ca, 3n) 294
118 (Oganessian et al., Phys. Rev. C64, 021604 (2004)). In 2004,
Oganessian et al. reported the successful synthesis of element 115 (and by a-decay,
its daughter element 113) in the reaction 245
Am(48
Ca, xn)2912x
115. These events
are still awaiting confirmation.
In the midst of all the exciting advances in heavy-element science in the period
from 1994 to 2002, there was a dark chapter, the element 118 fiasco that occurred in
Berkeley. In 1999, Ninov et al. (1999) reported the successful synthesis of three
atoms of element 118 using the reaction 208
Pb(86
Kr, n) 293
118. The evidence was
stunning, consisting of three decay chains involving highly correlated high-energy
a-particle decays after the implantation of a putative 293
118. The reported production
cross section was 2 pb, a number later revised to 7 pb. The result was
quite unexpected because the empirical systematics of cold fusion cross sections
(Fig. 15.4) would have predicted femtobarn cross sections for this reaction.
Other laboratories were not able to reproduce this work (Hofmann and
Munzenberg, 2000; Stodel et al., 2001; Morimoto, 2001), and eventually the
446 THE TRANSURANIUM ELEMENTS
Berkeley group reported their inability to reproduce the original observation
(Gregorich et al., Eur. J. Phys. A18, 633 (2003)). A subsequent investigation
(Gilchriese et al., 2003) revealed the original data had been fabricated by one
individual, who was later connected to similar instances of fraud at Darmstadt in
the work with elements 110 and 112 mentioned above. From these episodes, one
learns that “science works,” fraud will be found, and the traditional method of
independent confirmation of important findings is reaffirmed.
15.5 SUPERHEAVY ELEMENTS
Up to 1970, it was thought that the practical limit of the periodic table would be
reached at about element 108. By extrapolating the experimental data on heavyelement
half-lives, we concluded that the half-lives of the longest-lived isotopes
of the heavy elements beyond about element 108 would be so short (,1026
s)
due to spontaneous fission decay that we could not produce and study them
(Fig. 15.10). However, in the late 1960s and early 1970s, nuclear theorists, using
techniques developed by Vilen Strutinsky and Wladyslaw Swiatecki, predicted
1024
Longest lived isotopes
of transuranium elements
1 000 000 years
1 000 years
1 year
1 day
1 hour
1 minute
1 second
Decay by losing
alpha particles
1021
1018
1015
1012
109
106
103
101
10–3
10–6
10–9
10–12
94 98 102
Atomic number
Halflife(seconds)
106 110
Decay by
spontaneous
fission
Figure 15.10 Half-lives of the longest-lived isotope of each element versus atomic number
Z circa 1970. (Figure also appears in color figure section.)
15.5 SUPERHEAVY ELEMENTS 447
that special stability against fission would be associated with proton number
Z ¼ 114 and neutron number N ¼ 184. These superheavy elements were predicted
to have half-lives of the order of the age of the universe. They were predicted to
form an “island” of stability separated from the “peninsula” of known nuclei
(Fig. 15.11a).
We now know these predictions were wrong, in part. While we believe there are a
group of “superheavy” nuclei whose half-lives are relatively long compared to lower
Z elements, we do not believe they form an island of stability. Rather, we picture them
as a continuation of the peninsula of known nuclei (Fig. 15.11b). We also believe that
their half-lives are short compared to geologic time scales. Therefore, they do not
exist in nature. The most stable of the superheavy nuclei, those with Z ¼ 112,
N  184, are predicted to decay by a-particle emission with half-lives of 20 days.
Figure 15.11 (a) Allegorical representation of heavy nuclide stability, circa 1975, showing
the superheavy island. (b) Modern plot of the predicted half-lives of the heaviest nuclei. Note
there is a peninsula connecting the known and superheavy nuclei.
448 THE TRANSURANIUM ELEMENTS
The principal problem associated with the superheavy nuclei is not their possible
existence (which is considered relatively certain) but rather how to make them. Literally
hundreds of synthesis reactions have been proposed, and several have been
tried. Up to recently, all have failed because either the formation of the fused
system is too improbable or its excitation energy is too large, resulting in too
small a probability for formation of the product nuclei. For example, the most
widely studied synthesis reaction is the 48
Ca þ 248
Cm reaction. Using Figures
15.5 and 15.6 as guides, we roughly estimate the fusion cross section to be
3 Â 10232
cm2
, the excitation energy 30 MeV (survival probability 1026
), and
a predicted formation cross section of 10238
cm2
. With modern technology, this
would correspond to one atom per month. This production rate is at the limit of
modern experiment technology.
However, as indicated in Section 15.4, observations of the synthesis of elements
114 and 116 (and possibly element 118) have been reported with low (pb) cross
sections. In addition, instead of following the trends shown in Figure 15.10, the
half-lives of the longest-lived known isotopes of elements 106–112 are reported
to be approximately milliseconds to seconds, an enhancement of orders of magnitude
in their lifetimes. At present, none of these observations have been verified
by independent measurements.
Some have taken the viewpoint that, without the special stability associated with
nuclear shell structure, elements as light as Z ¼ 106–108 would have negligibly
short half-lives. The mere existence of these nuclei with millisecond half-lives is
said to be a demonstration that we have already made superheavy nuclei, according
to this view. The shell stabilization of these nuclei, which are deformed, is due to the
special stability of the N ¼ 162 configuration in deformed nuclei. (The “traditional”
superheavy nuclei with Z  114, N ¼ 184 were calculated to have spherical shapes.)
Our best theoretical predictions of the expected half-lives of elements 110–120
are shown in Figure 15.12. Clearly one expects new regions of very heavy nuclei
with half-lives that are substantially longer than those observed to date, but how
to make them remains to be seen.
15.6 CHEMISTRY OF THE TRANSURANIUM ELEMENTS
The chemical behavior of the transuranium elements is interesting because of its
complexity and the insights offered into the chemistry of the lighter elements.
The placing of these manmade elements into the periodic table (Fig. 15.1) represents
one of the few significant alterations of the original periodic table of Mendelyeev.
Since so little is known about the chemistry of the transactinide elements, one has
the unique opportunity to test periodic table predictions of chemical behavior
before the relevant experiments are done.
The actinide and known transactinide elements are transition elements, that is,
they have partially filled f or d electronic orbitals. As such, they are metals. Like
other transition metals, most of them are sufficiently electropositive to dissolve in
mineral acids. However, there is an important distinction that separates the actinide
15.6 CHEMISTRY OF THE TRANSURANIUM ELEMENTS 449
elements from the other transition elements, including the transactinide elements.
The partially filled d orbitals of most transition elements extend out to the boundary
of the atoms and are influenced greatly by (or can influence) the chemical environment
of the atom or ion. Thus, the chemical properties of elements with partially
filled d orbitals are highly complex and seem to vary somewhat irregularly as one
passes from element to element. But the 5f orbitals of the actinides are better screened
from the chemical environment of the atom or ion by the higher lying s and p shell
electrons, and thus there is a greater similarity in chemical properties among the
actinides compared to the other transition elements. (Correspondingly, the 4f orbitals
of the lanthanides are even better screened than the 5f actinide orbitals, and the
chemical behavior of the lanthanides is even more homologous.) The greater extension
of the 5f orbitals (relative to the 7s and 7p orbitals) compared to the lanthanide
4f orbitals allows some covalency in actinide bonding not seen with the lanthanides.
As the atomic number Z of the nucleus increases, the electrons become more
tightly bound. As their binding energy increases, so does their velocity. For electrons
in the 1s shell, the average velocity is roughly Z atomic units. (The speed of light, c,
is 137.035 au.) Thus, for Z ¼ 90, the velocity is about 90c/137 or 0.66c. The
Schro¨dinger equation is no longer appropriate in this case and one must use the
fully relativistic treatment of Dirac.
Figure 15.12 Predicted half-lives of the transmeiterium nuclei (Mo¨ller et al., 1997).
450 THE TRANSURANIUM ELEMENTS
The solution of the Dirac equation for the hydrogenlike atoms leads to wave functions
that are products of radial and angular factors similar to the solutions of the
Schro¨dinger equation that give the familiar atomic orbitals. The angular factors
are shown in Figure 15.13. Each state is specified by four quantum numbers
whose meaning is slightly different than for the Schro¨dinger equation. They are n,
the principal quantum number with values of 1, 2, 3, . . . , l, the azimuthal quantum
number with values 0, 1, 2, . . . , n 2 1, denoted by s, p, d, f, g; j, the angular momentum
quantum number with values of l + 1
2 (usually denoted as a subscript to l) and m,
the magnetic quantum number, taking on half-integer values from 2j to þj. Thus,
the three p, the five d, and the seven f orbitals are no longer degenerate and split into
one p1/2, and two p3/2, two d3/2, and three d5/2, three f5/2, and four f7/2 levels, with
the occupancy of each being 2j þ 1. This is called the “spin–orbit” splitting. The
orbital shapes are given by j and m with orbitals of the same j and m having the
same shape. What is surprising to the traditional chemist in Figure 15.13 is that
the p orbitals are shaped like a sphere (p1/2), a toroid or doughnut (p3=2, m ¼ 3
2),
and a dog-bone (p3=2, m ¼ 1
2). One notes that the state with the highest m value for
a given j value always has a doughnut-shaped distribution, while the lowest value
of m corresponds to a distribution stretched along the z axis with no nodes. States
of intermediate m are multilobed toroids.
The effect of using relativistic rather than nonrelativistic quantum mechanics to
obtain the predicted atomic orbitals is threefold: (a) a contraction and stabilization of
the s1/2 and p1/2 shells, (b) the splitting of the energy levels due to the spin–orbit
coupling, and (c) an expansion (and destabilization) of the outer d and all f shells.
These effects are of approximately equal magnitude and all increase as Z2
. In
Figure 15.14, we show the magnitude of these effects for uranium. The chemical
Figure 15.13 Pictorial representation of the relativistic orbital shapes.
15.6 CHEMISTRY OF THE TRANSURANIUM ELEMENTS 451
consequences of these effects have been well documented for elements such as gold
where, for example, the yellow color is due to relativistic effects. (Nonrelativistic
quantum mechanics predicts gold and silver to have similar colors.)
The ionic radii of the M3þ
and M4þ
ions of the actinides decrease with increasing
positive charge of the nucleus (the actinide contraction) (Fig. 15.15). This contraction
is due to the successive addition of electrons in an inner f shell where the
incomplete screening of the nuclear charge by the added f electron leads to a
contraction of the outer valence orbital. Because the ionic radii of ions of the
same oxidation state are generally similar (Fig. 15.15), the ionic compounds of
the actinides are isostructural.
The comparable energies of the 5f, 6d, 7s, and 7p orbitals and their spatial overlap
will lead to bonding involving any or all of them. Thus, complex formation is an
important part of actinide chemistry. The most stable oxidation states of the actinides
and their solution chemistry will depend on the ligands present also because
of the small differences between the energy of the electronic levels relative to
chemical bond energies.
The known oxidation states of the actinide elements are shown in Table 15.3. The
lower oxidation states are stabilized by acid while the higher oxidation states are
more stable in basic solutions. In solution, the 2þ, 3þ, and 4þ species are present
as metal cations, while the higher oxidation states are present as oxo-cations,
MO2
þ
, and MO2
2þ
. The most common oxidation state is þ3 for the transplutonium
elements like the lanthanide elements. Relativistic quantum mechanics predicts the
ground state of lawrencium (Lr) to be 5f14
7s2
7p1/2 and not 5f14
7s2
6d1
. This might
lead to a stable þ1 oxidation state for lawrencium, but experiments designed to
look for this state have not observed it. An upper limit for the reduction potential
of E8 , 0.44 V for the Lr3þ
/Lr1þ
half-reaction has been determined.
Figure 15.14 Comparison of the predicted (nonrelativistic and relativistic) energies of the
valence electronic levels for the uranium atom.
452 THE TRANSURANIUM ELEMENTS
The redox chemistry of the actinide elements, especially plutonium, is complex
(Katz et al., 1980). Disproportionation reactions are especially important for the þ4
and þ5 oxidation states. Some of the equilibria are kinetically slow and irreversible.
All transuranium elements undergo extensive hydrolysis with the þ4 cations reacting
most readily due to their large charge/radius ratio. Pu (IV) hydrolyzes extensively
in acid solution and forms polymers. The polymers are of colloidal
dimensions and are a serious problem in nuclear fuel reprocessing.
Hydrolysis is actually a special type of complex ion formation. The large positive
charge associated with transuranium cations that leads to hydrolysis is also the
Figure 15.15 Variation of the ionic radii of trivalent lanthanide and actinide ions with
increasing Z.
TABLE 15.3 Oxidation States of the Actinide Elementsa
Atomic Number: 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103
Element: Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr
Oxidation States
1?
(2) (2) (2) (2) 2 2 2
3 (3) (3) 3 3 3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 (4) 4?
5 5 5 5 5 5? 5?
6 6 6 6 6?
7 (7) 7?
a
The most common oxidation states are underlined, unstable oxidation states are shown in parentheses.
Question marks indicate species that have been claimed but not substantiated.
Source: From Katz et al. (1980).
15.6 CHEMISTRY OF THE TRANSURANIUM ELEMENTS 453
driving force for the interaction of nucleophiles with the transuranium cations.
Water is only one example of a nucleophilic ligand. Other nucleophilic ligands
present in solution may replace water molecules directly bound to the metal
cation to form inner sphere complexes or alternatively, they may displace water
molecules only from the outer hydrate shell to form outer sphere complexes.
Because of the competition between water and other ligands for positions in the
inner coordination sphere of the central transuranium atom, it is not surprising that
the stability of the complexes formed with a given ligand decreases, in the order
M4þ
. MO2
2þ
. M3þ
. MO2
þ
. (Note that the strength of complexation does
not depend simply on the net cation charge but rather the charge density seen by
the anion or ligand as it approaches the metal. In the case of MO2
2þ
, the effective
charge is about 3.3 rather than 2.) Although there is some variation within the
given cation types, the general order of complexing power of different anions is
FÀ
. NO3
À
. ClÀ
. ClO4
À
for singly charged anions and CO2
2À
. C2O4
2À
.
SO4
2À
for doubly charged anions.
The actinide cations are “hard acids,” that is, their binding to ligands is described
in terms of electrostatic interactions, and they prefer to interact with hard bases such
as oxygen or fluorine rather than softer bases such as nitrogen or sulfur. The actinide
cations do form complexes with the soft bases but only in nonaqueous solvents.
As typical hard acids, the stabilities of the actinide complexes are due to favorable
entropy effects. The enthalpy terms are either endothermic or very weakly
exothermic and are of little importance in determining the overall position of the
equilibrium in complex formation.
The formation of complexes could be thought to be a three-step process:
M(aq) þ X(aq) ! ½M(H2O)nXŠ(aq) ! ½M(H2O)XŠ(aq) ! MX(aq)
The first step is diffusion controlled while in the second step an “outer sphere”
complex is formed with at least one water molecule intervening between the
ligand and the metal atom. In the third or rate-determining step, a direct connection
between the metal and ligand is established with the formation of an “inner sphere”
complex. The process could terminate after the second step if the ligand cannot
displace the water. Actinides form inner and outer sphere complexes, although in
most cases the stronger inner sphere complexes are formed. The halide, nitrate, sulfonate,
and trichloroacetate ligands form outer sphere complexes of the trivalent
actinides while fluoride, iodate, sulfate, and acetate form inner sphere complexes.
It should be noted that the study of the chemistry of the elements with Z . 100 is
very difficult. These elements have short half-lives and the typical production rates
are about one atom/experiment. The experiments must be carried out hundreds of
times, and the results summed to produce statistically meaningful results.
The elements Lr–112 are expected (nonrelativistically) to be d-block elements
because they are expected to involve the filling of the 6d orbital. However, relativistic
calculations have shown that rutherfordium prefers a 6d 7p electron configuration
rather than the 6d2
configuration expected for nonrelativistic behavior and a simple
extrapolation of periodic table trends. This prediction also implies that RfCl4 should
454 THE TRANSURANIUM ELEMENTS
be more covalently bonded than its homologs HfCl4 and ZrCl4. In particular, the
calculations show RfCl4 to be more volatile than HfCl4, which is more volatile
than ZrCl4 with bond dissociation energies in the order RfCl4 . ZrCl4 . HfCl4.
(The periodic table extrapolations would predict the volatility sequence Zr Cl4 .
HfCl4. RfCl4.)
The first aqueous chemistry of rutherfordium showed that it eluted from liquid
chromatography columns as a 4 þ ion, consistent with its position in the periodic
table as a d-block element rather than a trivalent actinide. Gas chromatography of
the rutherfordium halides has shown the volatility sequence ZrCl4 . RfCl4 . HfCl4
with a similar sequence for the tetrabromides (Fig. 15.16). Thus, rutherfordium does
not follow the expected periodic table trend nor is its behavior in accord with relativistic
calculations.
The aqueous chemistry of dubnium has also shown unexpected trends. Dubnium
does not behave like its homolog tantalum in aqueous solutions but is similar to
niobium or the pseudo-Group 5 element, protactinium, under certain conditions.
For example, dubnium did not extract from methylisobutylketone under conditions
where tantalum is extracted but niobium is not. The extraction of dubnium, niobium,
tantalum, and protactinium from 6 M HCl solutions by amines agreed with relativistic
calculations. Gas-phase thermochromatography of NbBr5, TaBr5, and DbBr5
shows NbBr5 and TaBr5 to behave similarly, while DbBr5 is less volatile. Just the
30 40
–60
–70
–80
–90
–100
–110
–120
–130
50 60 70 80
Atomic number
ZrCl4
ZrBr4
HfCl4
HfBr4
RfCl4
RfBr4
∆H0
ads/(kJ/mol)
Volatility
90 100 110
Figure 15.16 Adsorption enthalpies, DHa, on SiO2 for Group 4 tetrachlorides and
tetrabromides. [From K. E. Gregorich, In Radiochemistry of Rutherfordium and Hahnium,
Proc. “The Robert A. Welch Foundation. 41st Conference on Chemical Research—The
Transactinide Elements,” Houston, Texas, Oct. 27–28 (1997), p. 95.]
15.6 CHEMISTRY OF THE TRANSURANIUM ELEMENTS 455
opposite trend was predicted by relativistic calculations. Thus, the chemistry of
dubnium and rutherfordium deviates significantly from periodic table trends, a
fact that is partly explained by relativistic calculations.
The study of the chemistry of seaborgium is remarkable for its technical difficulty
as well as the insight offered. In an experiment carried out over a 2-year period, 15
atoms of seaborgium were identified. From this experiment, one concluded that the
volatility sequence MoO2Cl2 . WO2Cl2 % SgO2Cl2 was followed. This observation
agreed with both the extrapolations of periodic table trends and relativistic
calculations. In an aqueous chemistry experiment, three atoms of seaborgium
were detected, showing seaborgium to have a hexavalent character expected of a
Group 6 element. The most stable oxidation state of seaborgium is þ6, and, like
its homologs molybdenum and tungsten, seaborgium forms neutral or anionic
oxo- or oxohalide compounds.
To study the chemistry of elements 107 (Bh) and 108 (Hs) one must be able
to produce isotopes of these elements where half-lives are long enough for chemical
studies. 269
Hs is reported (Schadel, 2002) to have t1/2 14 s, 270
Hs has t1/2 ¼ 2–7 s,
and 267
Bh is reported to have a half-life of 17 s. Because of the small probability of
producing these nuclei, methods for chemical study must be very sensitive.
Among the projected methods of study, liquid–liquid extraction and gas-phase
thermochemistry are thought to be the most viable.
Thermochromatographic measurements have indicated that Bh is less
volatile than Re, which is less volatile than Tc, in agreement with periodic table
trends.
The chemistry of hassium has been studied recently using the formation of
chemically stable, volatile HsO4, a property of Group 10 (VIII) elements. Thermochromatography
has shown HsO4 to be less volatile than OsO4, a result in agreement
with some relativistic predictions but not others (Fig. 15.17).
Figure 15.17 Thermochromatogram of HsO4 and OsO4. [From GSI Nachrichten 1/02.]
(Figure also appears in color figure section.)
456 THE TRANSURANIUM ELEMENTS
15.7 ENVIRONMENTAL CHEMISTRY OF THE
TRANSURANIUM ELEMENTS
With the large annual production of neptunium, plutonium, and the higher actinides
in the nuclear power industry, there has been increasing concern about the possible
release of these elements to the environment. This concern has been heightened by
the nuclear reactor accidents at Three Mile Island and Chernobyl. Coupled with the
prospect of cleaning up the detritus of the nuclear weapons programs of the major
nations and the general lack of a publicly acceptable method of long-term disposal
of nuclear waste, there is considerable interest in the environmental chemistry of the
transuranium elements.
Plutonium is clearly the most significant transuranium element in the environment.
The plutonium present in the environment is due primarily to atmospheric
testing of nuclear weapons, secondarily to the disintegration upon reentry of satellites
equipped with 238
Pu power sources, and, lastly, to the processing of irradiated
fuel and fuel fabrication in the nuclear power industry and the plutonium production
program. Some major radionuclide releases are summarized in Table 15.4. During
the period from 1950 to 1963, about 4.2 tons of plutonium (mostly a mixture of
239
Pu and 240
Pu) was injected into the atmosphere as a result of nuclear weapons
testing. Because of the high temperatures involved, most of this plutonium was
thought to be in the form of a refractory oxide. Most of this plutonium has been redeposited
on Earth with concentrations being highest at the midlatitudes. Of the
350,000 Ci of 238
Pu and 239
Pu originally injected into the atmosphere, about
1000 Ci remained in 1989. Approximately 9.7 Â 106
Ci of 241
Pu were also injected
into the atmosphere during weapons testing. When this completely decays (t1/2 of
241
Pu is 14.4 y), a total of 3.4 Â 105
Ci of 241
Am will be formed. There is an
additional 1.4 tons of plutonium deposited in the ground (1989) due to surface
and subsurface nuclear weapons testing. Approximately 16,000 Ci of 238
Pu were
TABLE 15.4 Events Leading to Large Injections of Radionuclides into the Atmosphere
Source Country Time Radioactivity (Bq)a
Important Nuclides
Hiroshima &
Nagasaki
Japan 1945 4 Â 1016
Fiss. prod. Actinides
Atmospheric
weapons tests
USA
USSR
1963 2 Â 1020
Fiss. prod. Actinides
Windscale UK 1957 1 Â 1015 131
I
Chelyabinsk
(Kysthym)
USSR 1957 8 Â 1016
Fiss. prod. 90
Sr, 137
Cs
Three Mile
Island
USA 1979 1 Â 1012
Noble gases, 131
I
Chernobyl USSR 1986 2 Â 1018 137
Cs
a
1 becquerel (Bq) is one disintegration/s.
Source: From G. Choppin, J. O. Liljinzin, and J. Rydberg, Radiochemistry and Nuclear Chemistry,
Pergamon, London, 1994.
15.7 ENVIRONMENTAL CHEMISTRY OF THE TRANSURANIUM ELEMENTS 457
injected into the atmosphere when a satellite containing an isotopic power source
disintegrated over the Indian Ocean in 1964. The Chernobyl accident caused the
release of 800 Ci 238
Pu, 700 Ci of 239
Pu and 1000 Ci of 240
Pu, representing
3% of the reactor core inventory. This activity was dispersed over large areas
of the former Soviet Union and Europe. The amount of plutonium in the environment
due to fuel reprocessing is small.
Over 99% of the plutonium released to the environment ends up in the soil and in
sediments. The global average concentration of plutonium in soils is 5 Â 1024
to
2 Â 1022
pCi/g dry weight, with most of the plutonium being near the soil
surface. The concentrations of plutonium in natural waters are quite low, with an
average concentration being 1024
pCi/L that is, 10218
M. (Greater than 96%
of any plutonium released to an aquatic ecosystem ends up in the sediments. In
these sediments, there is some translocation of the plutonium to the sediment
surface due to the activities of benthic biota.) Less than 1% (and perhaps closer
to 0.1%) of all the plutonium in the environment ends up in the biota. The concentrations
of plutonium in vegetation range from 1025
to 2%, with concentrations in
litter and animals ranging from 1024
to 3% and 1028
to 1%, respectively. None
of these concentrations has been observed to cause any discernible effect.
Despite the extremely low concentrations of the transuranium elements in water,
most of the environmental chemistry of these elements has been focused on their
behavior in the aquatic environment. One notes that the neutrality of natural
water (pH ¼ 5–9) results in extensive hydrolysis of the highly charged ions
except for Pu(V) and a very low solubility. In addition, natural waters contain organics
as well as micro- and macroscopic concentrations of various inorganic species
such as metals and anions that can compete with, complex, or react with the transuranium
species. The final concentrations of the actinide elements in the environment
are thus the result of a complex set of competing chemical reactions such as
hydrolysis, complexation, redox reactions, and colloid formation. As a consequence,
the aqueous environmental chemistry of the transuranium elements is significantly
different from their ordinary solution chemistry in the laboratory.
In natural waters, hydrolysis is the primary factor affecting concentration. The
tendency to hydrolyze follows the relative effective charge of the ions. This is
known to be
An4þ
. AnO2þ
2 . An3þ
. AnOþ
2
(where An represents an actinide element).
The hydrolysis reaction can be written as
x Anmþ
þ y OHÀ ÀÀ! Anx(OH)y
The hydrolysis products can be monomeric, polynuclear, or colloidal.
A number of strongly complexing inorganic anions are present in natural waters,
such as HCOÀ
3 =CO2À
3 , ClÀ
, SO2À
4 , PO3À
4 , and so on. The complexation order of these
458 THE TRANSURANIUM ELEMENTS
anions is
CO2À
3 . SO2À
4 . PO3À
4 . Cl .Á Á Á
Also present in many natural waters are humic/fulvic acid, citric acid, and the like.
These organics also can complex actinides. In Figure 15.18, we show the relative
stability constants for the first complexation reaction of various ligands with actinides
of different oxidation states. Clearly, the carbonate and humate ions along
with hydrolysis dominate the chemistry. The tetravalent actinide ions will tend
toward hydrolysis reactions or carbonate complexation rather than humate/fulvate
formation.
The aquatic solution chemistry of the actinides is also influenced by pH and redox
potential (Eh). The approximate ranges of pH and Eh for natural waters are shown in
Figure 15.19. The pH varies from 4 to 9.5 and Eh from 2300 to þ500 mV. In these
Figure 15.18 Comparison of complexation stability constants for the interaction of various
ligands with different actinide oxidation states (Kim, 1986).
15.7 ENVIRONMENTAL CHEMISTRY OF THE TRANSURANIUM ELEMENTS 459
pH and Eh ranges, neptunium and plutonium can be present in several different
oxidation states while americium and curium will be trivalent. In the oxidizing
environment of surface waters, Np(V), Pu(IV), Pu(V), and Am(III) will be the dominant
species, while in the reducing environment of deep groundwater, other species
may be present (Fig. 15.20).
Colloids are always present in natural waters containing the transuranium
elements. (Colloids are defined as particles with sizes ranging from 1 to 450 nm.
These particles form stable suspensions in natural waters.) Colloids of the transuranium
elements can be formed by hydrolysis of transuranium ions, or by the sorption
of transuranium elements on the “naturally occurring colloids.” The naturally occurring
colloids include such species as metal hydroxides, silicate polymers, organics
(such as humates), and the like. The mobility of the transuranium elements in an
aquifer is determined largely by the mobility of its pseudocolloids, that is, those colloidal
species formed by the adsorption of the transuranium ions upon the naturally
occurring colloids.
The speciation of the transuranium elements in waters is thus a complex function
of hydrolysis, colloid formation, redox reactions, and complexation with available
ligands. The solubility (mobility) is, thus, highly dependent on the particular
aquatic environment and its characteristics.
However, bearing in mind these caveats, we can make certain generalizations
about the behavior of the actinide elements in natural waters. Americium and
curium remain in the þ3 oxidation state over the natural range of environmental
conditions. For plutonium, Pu(III) is unstable to oxidation at environmental
acidities, and so the other three states are observed with the dominant oxidation
state in natural waters being Pu(V). [Humic materials cause a slow reduction of
Figure 15.19 A Pourbaix diagram showing the ranges of pH and Eh values in natural waters.
460 THE TRANSURANIUM ELEMENTS
Figure 15.20 Redox diagram for U, Np, Pu and Am in 1M HClO4 at 258C (Choppin,
Liljenzin, and Rydberg, 2002).
15.7 ENVIRONMENTAL CHEMISTRY OF THE TRANSURANIUM ELEMENTS 461
Pu(V) to Pu(IV) so that Pu(IV) is found to be important in waters containing significant
amounts of organic material.] Under reducing conditions, neptunium should be
present as Np(IV) and behave like Pu(IV); under oxidizing conditions, NpOþ
2 will be
the stable species. In marine waters, Pu(IV) and the transplutonium elements will
tend to undergo hydrolysis to form insoluble hydroxides and oxides. However,
these elements can also form strong complexes with inorganic anions
(OHÀ
, CO2À
3 , HPO2À
4 , FÀ
, and SO2À
4 ) and organic complexing agents that may be
present. The speciation and solubility of these elements are largely determined by
hydrolysis and formation of carbonate, fluoride, and phosphate complexes. Stable
soluble species include Pu(V, VI) and Np(V), although under most conditions the
actinides will form insoluble species that concentrate in the sediments.
Pu(IV), which forms highly charged polymers, strongly sorbs to soils and
sediments. Other actinide III and IV oxidation states also bind by ion exchange to
clays. The uptake of these species by solids is in the same sequence as the order
of hydrolysis: Pu . Am(III) . U(VI) . Np(V). The uptake of these actinides by
plants appears to be in the reverse order of hydrolysis: Np(V) . U(VI) .
Am(III) . Pu(IV), with plants showing little ability to assimilate the immobile
hydrolyzed species. The further concentration of these species in the food chain
with subsequent deposit in humans appears to be minor. Of the 4 tons of plutonium
released to the environment in atmospheric testing of nuclear weapons, the total
amount fixed in the world population is less than 1 g [of this amount, most
(99.9%) was inhaled rather than ingested].
PROBLEMS
1. Predict the aqueous solution chemistry of element 114. What is the expected oxidation
state? By extrapolating periodic table trends, estimate the first ionization
potential of element 114 [see Nash and Bursten (1999)].
2. Suppose you want to synthesize the nucleus 271
Mt using the 37
Cl þ 238
U reaction.
Estimate the production cross section for this reaction. What is the expected halflife
of this nucleus? What is the expected decay mode?
3. What is the expected relative population of the þ4, þ5, and þ6 oxidation states
of Pu in seawater (Eh ¼ 290 mV)?
4. The reported discoveries of elements 114 and 116 do not connect the observed
decays to the region of known nuclei. Devise an experimental program to
make this connection or to establish the Z of the 114 and 116 nuclei.
5. It has been proposed to do a Stern–Gerlach experiment with Lr atoms to
better understand the electron configuration of Lr. (a) Describe the original
Stern–Gerlach experiment and what it demonstrated about the quantization
of electron spin. Why was an inhomogeneous magnetic field used instead of
a uniform field? (b) Using nonrelativistic quantum mechanics, one predicts
the electron configuration of Lr to be [Rn] 5f14
6d1
7s2
. Using relativistic
462 THE TRANSURANIUM ELEMENTS
quantum mechanics, the predicted ground-state electron configuration is [Rn]
5f14
7p1/27s2
. Predict the expected pattern of spots one would observe when
one passes a beam of neutral Lr atoms through an inhomogeneous magnetic
field assuming (a) the nonrelativistic prediction is correct and (b) assuming the
relativistic prediction is correct. (c) More sophisticated relativistic calculations
predict a spacing of 1 eV between the ground state [Rn] 5f14
7p1/27s2
electron
configuration and an electronic excited state with the configuration [Rn]
5f14
6d1
7s2
. If the temperature of an assembly of Lr atoms is 300 K, predict the
relative population of the two configurations.
6. In the discussion of Figures 15.5 and 15.6, a rough rule of estimating the survival
probability as (EÃ
/10) Â 1022
was given. Using the equations presented in
Chapter 11, calculate Gn/Gf (the survival probability) for 254
No with
Bf ¼ 7.1 MeV and an excitation energy of 50 MeV. Compare this estimate to
the rough rule.
7. Consider the reaction 48
Ca þ 206
Pb ! 252
No þ 2n. Assume the energy of the
48
Ca projectile in the lab system was 217 MeV. Using the equations in Chapters
10 and 11, calculate the formation cross section for this reaction. Compare your
calculation with the rough estimate made using Figures 15.5 and 15.6.
8. Estimate the total decay power (W) produced in a sample of 10 g of 238
Pu.
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I. I. Uleuhla, A. H. Wapstra, and D. H. Wilkinson. Prog. Part. Nucl. Phys. 29, 453 (1992).
Druin, V. A., A. G. Demin, Yu. P. Kharitonov, G. N. Akap’ev, V. I. Rud,
G. Y. Sung-Chin-Yang, L. P. Chelnokov, and K. A. Gavrilov. Yad. Fiz. 13, 251 (1971).
Folden, C. M. et al. Phys. Rev. Lett. 93, 212702 (2004).
Ghiorso, A., M. Nurmia, J. Harris, K. Eskola, and P. Eskola. Phys. Rev. Lett. 22, 1317 (1969).
Ghiorso, A., M. Nurmia, K. Eskola, J. Harris, and P. Eskola. Phys. Rev. Lett. 24, 1498 (1970).
Ghiorso, A., J. M. Nitschke, J. R. Alonso, C. T. Alonso, M. Nurmia, G. T. Seaborg, E. K.
Hulet, and R. W. Lougheed. Phys. Rev. Lett. 33, 1490 (1974).
Ghiorso, A., et al. Nucl. Phys. A583, 861 (1995a).
Ghiorso, A., et al. Phys. Rev. C51, R2293 (1995b).
Gilchriese, M., A. Sessler, G. Trilling, and R. Vogt. Lawrence Berkeley National Laboratory
Report, LBNL-51773 (2003).
Ginter, T. N., et al. Phys. Rev. C 67, 064609 (2003).
Gregorich, K. E., M. R. Lane, M. F. Mohar, D. M. Lee, C. D. Kacher, E. R. Sylwester, and
D. C. Hoffman. Phys. Rev. Lett. 72, 1423 (1994).
Gregorich, K. E., et al. Eur. Phys. J. A 18, 633 (2003).
Hofmann, S. Rep. Prog. Phys. 61, 639 (1998).
Hofmann, S., and G. Mu¨nzenberg. Rev. Mod. Phys. 72, 733 (2000).
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Hoffmann, S., et al. Z. Phys. A 350, 277 (1995a).
Hofmann, S., et al. Z. Phys. A 350, 281 (1995b).
Hofmann, S., et al. Z. Phys. A 354, 229 (1996).
Hofmann, S., et al. Eur. Phys. J. A 14, 147 (2002).
Katz, J. J., G. T. Seaborg, and L. Morss. Chemistry of the Actinide Elements, 2nd ed.,
Chapman and Hall, London, 1980.
Kim, J. I. Chemical Behaviour of Transuranic Elements in Natural Aquatic Systems, in
Handbook on the Physics and Chemistry of the Actinides, Vol. 4, A. J. Freeman and
C. Keller, Eds., North-Holland, Amsterdam, 1986, pp. 413–456.
Lazarev, Y. A., et al. Phys. Rev. C 54, 620 (1996).
Mo¨ller, P., J. R. Nix, and K. L. Kratz. At. Data Nucl. Data Tables 66, 131 (1997).
Morimoto, K. AIP Conf. Series, 561, 351 (2001).
Mu¨nzenberg, G., et al. Z. Phys. A 300, 107 (1981).
Mu¨nzenberg, G., et al. Z. Phys. A 309, 89 (1982).
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Mu¨nzenberg, G., et al. Z. Phys. A 317, 235 (1984); G. Mu¨nzenberg, et al., Z. Phys. A 328,
49 (1987).
Nash, C. S. and B. E. Bursten. J. Phys. Chem. A 103, 402 (1999).
Ninov, V., et al. Phys. Rev. Lett. 83, 1101 (1999).
Oganessian, Y. T., et al. Z. Phys. A 319, 215 (1984).
Oganessian, Y. T., et al. Phys. Rev. Lett. 83, 3154 (1999a).
Oganessian, Y. T., et al. Nature 400, 242–245 (1999b).
Oganessian, Y. T., et al. Phys. Rev. C 62, 041604 (2000).
Oganessian, Y. T., et al. Phys. Rev. C 63, 011301 (2001).
Schadel, M. J. Nucl. Radiochem. Sci. 3, 113 (2002).
Stodel, C., et al. AIP Conf. Series 561, 311 (2001).
Zvara, I., Yu. T. Chuburicov, R. Tsaletka, and M. R. Shalaevskii. Radiokhimiya 11, 163
(1969).
Zvara, I., Yu. T. Chuburkov, V. Z. Belov, G. V. Buklanov, B. B. Zakhvataev, T. S. Zvarova,
O. D. Maslov, R. Caletka, and M. R. Shalaevsky. J. Inorg. Nucl. Chem. 32, 1885 (1970).
BIBLIOGRAPHY
Seaborg, G. T., and W. Loveland. The Elements Beyond Uranium, Wiley, New York, 1990,
and the many references contained therein. The material in this chapter represents a
condensation of this book.
Seaborg, G. T., and E. G. Valens. Elements of the Universe, Dutton, New York, 1958.
Scha¨del, M., ed. The Chemistry of Superheavy Elements, Kluwer, Dordrecht, 2003.
The Transactinide Elements. Proceedings of the 41st Robert Welch Conference on Chemical
Research, Welch, Houston, 1998. The best modern summary of many aspects of the study
of the heaviest chemical elements.
464 THE TRANSURANIUM ELEMENTS
CHAPTER 16
NUCLEAR REACTOR CHEMISTRY
One of the most important applications of nuclear and radiochemistry is in the area
of nuclear power. Chemistry and chemical processes are intimately involved in
reactor operation, the preparation and processing of reactor fuel, and the storage
and ultimate disposal of radioactive waste. In this chapter, we shall examine some
of the most important chemistry associated with nuclear power.
16.1 INTRODUCTION
The basic principles behind nuclear reactors and the current design of light-water
reactors are discussed in Chapter 14. The two types of light-water reactors, the pressurized
water reactor (PWR) and the boiling water reactor (BWR) are shown schematically
in Figure 14.2. In these light-water reactors, the reactor fuel is enriched to
3% in 235
U, and ordinary water is used as the coolant and moderator. In a
PWR, the reactor core is encased in a steel pressure vessel filled with water under
a pressure of 150 bar or greater. The water does not boil due to this high pressure
even though the temperature of the water is as high as 3508C. It passes through a heat
exchanger where its heat is used to boil water in the secondary coolant loop. The
steam is dried and used to drive a turbine, then condensed and returned to the
boiler. The water in the primary loop usually contains boron (as 0.025 M boric
acid, H3BO3) to control the reactivity of the reactor.
465
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
In a BWR, the water passing through the reactor core is allowed to boil because
it is at a lower pressure, 75 bar. The steam produced is passed through a turbine,
condensed, and returned to the reactor.
In PWRs, the fuel is UO2, enriched typically to 3.3% 235
U; while for BWRs, the
fuel is UO2, enriched to 2.6%. (Natural uranium is 0.72% 235
U). The fuel elements
are clad in Zircaloy, a zirconium alloy that includes tin, iron, chromium, and nickel
that prevents fission product release and protects them against corrosion by the
coolant. The control rod material in BWRs is B4C, while PWRs have Ag–In–Cd
or Hf control materials.
The nuclear fuel cycle is a set of steps in the processing of the reactor’s fissile
materials that begins with the mining of uranium and extends through the final disposition
of the waste from the reactor. These steps are referred to as a cycle because
it is possible that the material taken from the reactor after use can be recycled. A
schematic diagram of the nuclear fuel cycle is shown in Figure 16.1.
The diagram in Figure 16.1 shows two possible paths for this cycle, that is, with
and without fuel reprocessing. The majority of reactors in the world and all U.S.
reactors operate with a once-through cycle without reprocessing. Some countries,
particularly France, do fuel reprocessing with reuse of the plutonium from spent
fuel. The portions of the cycle, that precede the introduction of the fissile material
into the reactor are referred to as the front end of the cycle, while the back end
includes those steps that occur after the removal of the fuel from the reactor. The
details of this cycle and the chemistry involved are discussed below.
16.2 FISSION PRODUCT CHEMISTRY
In the chemistry of the fuel cycle and reactor operations, one must deal with the
chemical properties of the actinide elements, particularly uranium and plutonium
and those of the fission products. In this section, we focus on the fission products
and their chemistry. In Figures 16.2 and 16.3, we show the chemical composition
and associated fission product activities in irradiated fuel. The fission products
include the elements from zinc to dysprosium, with all periodic table groups
being represented.
The inert gases (Group 18) are represented by isotopes of Kr and Xe. These isotopes
are generally short-lived and will decay before fuel reprocessing. As inert
gases, they are unreactive and consequently they are isolated using cryogenic
techniques.
The alkali metals (Group 1) are represented by Rb and Cs where long-lived 137
Cs
(t1/2 ¼ 30 y) is the most important nuclide. Their solution chemistry is that of the
generally soluble þ1 cation. The alkaline earths (Group 2) are represented by the
high-yield nuclides 140
Ba, 90
Sr, and 91
Sr. These nuclides can be separated using
ion exchange or solvent extraction or gravimetric techniques. The 28-y 90
Sr is an
important radiation hazard in aged spent fuel, while 12.8-d 140
Ba frequently determines
the shielding requirements during fuel for the 10- to 100-d cooling period.
466 NUCLEAR REACTOR CHEMISTRY
Group 3 contains Y and the lanthanides. These elements are chemically similar.
They can be separated from one another by ion exchange, while their separation
from U and Pu can be done using solvent extraction with tri-butyl phosphate
(TBP). These elements have soluble nitrates, chlorides, and sulfates while their fluorides
and hydroxides are insoluble. LaF3 is frequently used as a carrier for this
group. They form stable complexes with strong chelating agents such as DPTA,
EDTA, and the like. The Group 4 element in fission product mixtures is zirconium
whose chemistry is that of the þ4 oxidation state. The principal nuclide of interest is
Figure 16.1 Schematic diagram of the nuclear fuel cycle.
16.2 FISSION PRODUCT CHEMISTRY 467
the 63-d 95
Zr and its 35-d daughter, 95
Nb. The chemistry of Zr can be tricky as it
readily forms complexes and does form colloids, which can lead to poor separation
factors.
The Group 5 element niobium occurs as a decay product, 95
Nb, of 95
Zr. Like its
zirconium parent, 95
Nb forms colloids and is readily hydrolyzed. The principal
Group 6 fission product is 99
Mo (t1/2 ¼ 67 h), which is important at short cooling
times because of its high yield but is insignificant in aged fuel. The most important
Group 7 fission product is the long-lived (2.1 Â 105
y) 99
Tc. Its chemistry is that of
the pertechnate ion TcO4
2
. The elements of Groups 8, 9, and 10 are important
because of their activity, and in the case of ruthenium and its multiple oxidation
states and the slow interconversion kinetics, the chemistry can be troublesome.
RuO4 can be volatilized leading to the loss of Ru in radiochemical procedures.
The chemistry of the Group 11 element, Ag, is straightforward (þ1 cation,
forming insoluble compounds) and the Groups 12 and 13 elements, Zn, Ga, Cd,
and In have low yields and small activities. The Groups 14 and 15 elements, Ge,
As, Sn, and Sb also have low activities in aged fuel. The Group 16 element, Te,
is present in the form of 30- to 100-d activities, but most interest is focused on
Figure 16.2 Chemical composition of the fission products in irradiated fuel as a function of
decay time after a 2-month irradiation. [From J. Prawitz and J. Rydberg, Acta. Chem. Scand.
12, 393 (1958).]
468 NUCLEAR REACTOR CHEMISTRY
78-h 132
Te, which decays to short-lived 132
I, which is volatile and can be released in
rapid processing of fuel. The halogens, Br and I, are not important in fuel reprocessing
due to their short half-lives but can be important in reactor operation and
accidents due to their volatility.
Figure 16.3 Principal fission product activities in irradiated fuel as a function of decay time
after a 2-month irradiation. [From J. Prawitz and J. Rydberg, Acta. Chem. Scand. 12, 385
(1958).]
16.2 FISSION PRODUCT CHEMISTRY 469
16.3 RADIOCHEMISTRY OF URANIUM
16.3.1 Uranium Isotopes
Natural uranium is 99.274 atom % 238
U, 0.7205 atom % 235
U, and 0.0056 atom %
234
U. The 234/238 ratio is exactly the ratio of their half-lives as expected for nuclei
in secular equilibrium. The isotope 233
U is produced by neutron capture on 232
Th,
followed by b2
decay. 232
U is a short-lived (t1/2 ¼ 72 y) nuclide that is a contaminant
in 233
U samples (from fast neutron reactions). The daughters of 232
U are
hard g-ray emitters that make working with 232
U containing samples difficult.
236,237,239
U are produced by neutron captures on 235
U and 238
U. 236
U is long-lived
but 237,239
U are short-lived and decay to 237
Np and 239
Pu, respectively.
16.3.2 Metallic Uranium
Metallic uranium can exist in three different solid phases with differing densities,
depending on temperature. At room temperature, the a phase is observed with a
density of 19.07 g/cm3
and a melting point of 11328C. Metallic uranium is a very
reactive metal that is silvery in color. (Frequently, a surface oxide layer makes
metallic uranium look black.) Uranium powder is pyrophoric. When uranium
metal is cut or scratched in the laboratory, a shower of sparks is sometimes observed
due to the creation of small particles that ignite. Uranium metal with an oxide
coating will burn at 7008C to form U3O8. Uranium reacts with hot water to
produce UO2 and UH3. In reactors, uranium is alloyed with zirconium to resist
corrosion and radiation damage. Metallic uranium can be produced by the reduction
of UF4, for example, by reaction with magnesium
UF4 þ 2Mg À! 2MgF2 þ U
16.3.3 Uranium Compounds
Uranium exists in the þ3, þ4, þ5, and þ6 oxidation states. The þ5 state disproportionates
to the þ4 and þ6 states and is of little importance. Trivalent uranium
reduces water and therefore there is no stable aqueous chemistry of U3þ
although
compounds do exist.
The most important uranium compounds are the oxides. UO2 is the compound
used in reactor fuel. It is a stable refractory material that is brown-black in color
and is nonreactive with H2O. It has density of 10.97 g/cm3
and can be prepared
by the reduction of UO3 with hydrogen. U3O8 (UO2
. 2UO3) is a green-black solid
that occurs in the mineral pitchblende. It has a density of 8.38 g/cm3
, is soluble
in HNO3, and can be prepared by oxidizing UO2 or reducing UO3. UO3 is a
yellow-orange solid (“orange oxide”) and is important as an intermediate in the production
of UO2 or UF6.
Uranium hydride, UH3, is a reactive black powder. It is a powerful reducing agent
and is pyrophoric. A mixture of uranium and zirconium hydrides is used as the fuel
470 NUCLEAR REACTOR CHEMISTRY
for the TRIGA research reactors. The resulting very large negative temperature
coefficient of reactivity provides a large safety margin for reactor operation.
Uranium halides exist in the þ3 oxidation state (UF3, UCl3, UBR3, UI3), the þ4
oxidation state (UF4, UCl4, UBr4, and UI4), the þ5 oxidation state (UF5, UCl5), and
the þ6 oxidation state (UF6, UCl6).
UF4 (“green salt”) is an intermediate in the production of U and UF6. It can be
made by reacting UO2 with excess HF as
UO2 þ 4HF () UF4 þ 2H2O
or, for laboratory use, by the reaction
U4þ
þ 4FÀ
À! UF4 #
(Uranium tetrafluoride precipitated from aqueous solutions exists as UF4
. 5H2O,
and it is difficult to remove the waters of hydration so that the dry reaction above
is preferred.) UF4 is frequently used to make accelerator targets of uranium by
vacuum volatilization.
UF6 (“hex”) is the only readily available uranium compound that is volatile at
room temperature. It is a colorless solid that is used in the uranium enrichment
process. It sublimes at room temperature without melting. UF6 is rapidly hydrolyzed
by water and is a fluorinating agent. This latter property means that one must carefully
choose the materials to contain UF6.
16.3.4 Uranium Solution Chemistry
The solution chemistry of uranium is that of the þ4 and þ6 oxidation states, that is,
U4þ
and UO2
2þ
. The formal reduction potential of uranium in aqueous solution (i.e.,
1 M HClO4) is
UO2
2+ +0.063 UO2
+ +0.58 U4+ −0.031 U3+ −1.70 U
+0.32
The U(IV) chemistry is similar to that of Th4þ
, except for the difference in the
charge/radius ratio of the ions. U4þ
solutions are green in color, stable, and
slowly oxidized by air to UO2
2þ
. Solutions of U4þ
are generally prepared by
reduction of solutions of the uranyl (UO2
2þ
) ion. U(IV) forms complexes with
many anions (C2O4
22
,C2H3O2
2
,CO3
22
, Cl2
, and NO3
2
). The chlorides and bromides
of U(IV) are soluble while the fluorides and hydroxides are insoluble. In aqueous
solution, U(IV) hydrolyzes via the reaction,
U4þ
þ H2O À! U(OH)3þ
þ Hþ
K ¼ 0:027
The U(VI) can be prepared by dissolving UO3 in acid or U metal in HNO3.
Solutions of the uranyl ion show a characteristic yellow-green color and are very
16.3 RADIOCHEMISTRY OF URANIUM 471
stable. U(VI) shows complex solution equilbria due to the occurrence of hydrolysis,
which leads to [(UO2)2OH]3þ
, [(UO2)2(OH)2]2þ
, and [(UO2)3(OH)4]2þ
mixtures.
Because of hydrolysis reactions, aqueous solutions of uranyl salts are slightly
acidic. Addition of base to uranyl solutions results in precipitation beginning at a
pH between 4 and 7.
Uranyl ions form complexes in solutions with most anions. Uranyl sulfate and
carbonate complexes are especially strong and are used in extracting uranium
from its ores. Of great practical importance are the complexes of the uranyl ions
with nitrate that are soluble in organic liquids such as alcohols, ethers, ketones,
and esters. One of the most important of these reactions is that involving the extraction
of uranyl nitrate into TBP (the Purex process):
UO2þ
2 (aq) þ 2NOÀ
3 (aq) þ 2TBP(org) () UO2(NO3)2 Á2TBP(org)
Neglecting activity coefficients, the coefficient for the distribution of uranium
between the organic and aqueous phases is written as
D ¼
½UO2(NO3)2 Á 2TBPŠorg
UO2þ
2
 Ã
aq
¼ K(NO3)2
aq(TBP)2
org
where K is the equilibrium constant (K  15–60). One can use the concentration of a
salting agent such as NO3
2
to control the extraction process. (In the Purex process,
these salting agents ultimately leave the cycle with the fission products, contributing
to the problems of dealing with these wastes.)
16.4 NUCLEAR FUEL CYCLE—THE FRONT END
The nuclear fuel cycle (Fig. 16.1) begins with the mining of uranium ore. Uranium is
by no means rare. Its overall abundance in Earth’s crust is 4 ppm (which is more
abundant than Ag, Hg, Bi, or Cd). There are 104
tonnes of uranium in Earth’s
crust. The problem is one of concentration in that most uranium deposits contain
,0.001% uranium.
16.4.1 Mining and Milling
Uranium ore can be classified as high grade (1–4% U), medium grade (0.1–0.5%
U), and low grade (,0.1% U). In the high-grade deposits, in Zaire and Canada,
uranium is found as pitchblende or uranite, materials of general composition
xUO2
. yUO3 where 0 , y/x , 2. The medium-grade ores are found in places
such as the Colorado plateau of the United States, where uranium is incorporated
in carnotite (K2O . 2UO3
. V2O5
. xH2O) or autunite (CaO . 2UO3
. P2O5
. xH2O).
Low-grade sources include the gold ore residues of South Africa, seawater where
the nominal uranium concentration of 3 ppb corresponds to a reservoir of 1010
tonnes of uranium, or the fertilizer by-products of the phosphate fields of Florida
472 NUCLEAR REACTOR CHEMISTRY
and Idaho, Tennessee shale, or the lignites of Wyoming and the Dakotas. The
average uranium content of the ores used in the nuclear fuel cycle in the United
States in recent years is 0.24%.
After mining, the uranium must be concentrated before further operations are
carried out. This is done in the mills, which are located near the mines. Here the
uranium ore content is increased from a few tenths of a percent (in the ore) to
85–95% (in a semirefined concentrate known as “yellowcake”), while eliminating
other elements that are present (the “tailings”).
While very high grade pitchblende deposits can be concentrated by physical techniques
involving their specific gravity, most concentration of uranium involves
chemical leaching from the ore. The ore is prepared for leaching by crushing and
roasting (to destroy organic material). It is then subjected to chemical leaching followed
by a separation of the liquids and solids, followed by concentration/purification
and final product recovery.
The normal choice (80% of all U.S. ores) for chemical leaching is acid leaching
with sulfuric acid (due to its low cost and great availability). For the ores that have
high carbonate content, an alkaline leach with Na2CO3 is used to prevent high acid
concentration.
The problem in acid leaching is that only U(VI) dissolves in H2SO4. Any U(IV)
present must be oxidized to U(VI) prior to leaching. The chemical steps can be
summarized as
FeS2, Fe þ H2SO4 À! Fe2þ
þ H2 þ SO2À
4
2Fe2þ
þ MnO2 þ 4Hþ
À! 2Fe3þ
þ Mn2þ
þ 2H2O
or
6Fe2þ
þ NaClO3 þ 6Hþ
À! 6Fe3þ
þ NaCl þ 3H2O
leading to
UO2 þ 2Fe3þ
À! UO2þ
2 þ 2Fe2þ
UO3 þ H2SO4 À! UO2þ
2 þ SO2À
4 þ H2O
UO2þ
2 þ 2SO2À
4 () ½UO2(SO4)2Š2À
() ½UO2(SO4)3Š4À
The final product thus appears in three chemical forms, UO2
2þ
, [UO2(SO4)2]22
, and
[UO2(SO4)3]42
.
In alkaline leaching of high carbonate materials, one takes advantage of the
unique solubility of [UO2(CO3)3]42
. One begins with finely divided material and
must also deal with the oxidation of any U(IV) that is present. The basic reaction
(pun intended) is
UO2 þ (oxidant, Ox) þ 3CO2À
3 À! ½UO2(CO3)3Š4À
þ Ox2À
16.4 NUCLEAR FUEL CYCLE—THE FRONT END 473
After leaching, a concentration/purification is done to get rid of other materials
leached from the ore. This can be done by ion exchange or solvent extraction. In the
ion exchange method, three steps are employed: (a) the absorption of uranium from
the leach liquor onto the resin, (b) the selective elution of uranium from the resin,
and (c) the regeneration of the resin. Anion exchange is the preferred method of
ion exchange with the relevant chemical reactions for acid leach being
½UO2(SO4)3Š4À
þ 4RX () R4½UO2(SO4)3Š þ 4XÀ
and for alkaline leach
½UO2(CO3)3Š4À
þ 4RX () R4½UO2(CO3)3Š þ 4XÀ
The eluant is usually 1 M NO3
2
in the form of NH4NO3. The physical method for
carrying out the extraction can involve: (a) a fixed resin bed, (b) a “resin-in-pulp”
technique where resin in baskets is passed through a stream of pulp or slurry from
the leach process, or (c) a moving bed of resin.
Concentration/purification by solvent extraction usually involves four steps:
(a) extraction of uranium from the leach liquor in a solvent, (b) scrubbing to remove
impurities from the solvent, (c) stripping to remove uranium from the solvent, and
(d) regeneration of the solvent. The solvent phase in solvent extraction will contain
the extractant that complexes uranium to make it soluble in the organic phase, a
diluent, an inexpensive material to dilute the extractant, and a modifier to improve
the solubility of the extractant in the diluent. Typical extractants are amines with isodecanol
acting as a modifier to improve the amine solubility in a diluent such as kerosene.
The typical chemistry of the extraction would involve the reactions
2R3N(org) þ H2SO4 À! (R3NH)2SO4(org)
2(R3NH)2SO4(org) þ ½UO2(SO4)3Š4À
(aq) À! (R3NH)4½UO2(SO4)3Š(org)
þ 2SO2À
4 (aq)
with the stripping reactions being
(R3NH)4½UO2(SO4)3Š(org) þ 4NH4OH À! 4R3N(org) þ 4H2O þ ½UO2(SO4)2Š2À
þ 4NHþ
4 þ SO2À
4
Following solvent extraction, uranium is precipitated from the solution by the
addition of gaseous ammonia with the yellowcake product (Fig. 16.4) being collected,
packaged in 55-gal drums, and shipped to a refinery for further purification and
conversion to UF6 for enrichment.
Alkaline leach mills will use NaOH for final product recovery (in the form of
sodium uranate, Na2U2O7). Often these uranium ore concentrates (yellowcake)
are transformed into U3O8 by drying at 2008C to remove water and calcining, that
is, heating until decomposition.
474 NUCLEAR REACTOR CHEMISTRY
16.4.2 Refining and Chemical Conversion
Following milling, the yellowcake is shipped for refining and conversion. In most
refineries, the uranium ore concentrates are purified by solvent extraction and
then converted to UF6 for enrichment. (A schematic diagram for this process is
shown in Fig. 16.5.)
The yellowcake is dissolved in nitric acid. Uranyl nitrate is separated from metallic
impurities in the dissolver by solvent extraction with TBP in some organic
hydrocarbon, such as hexane. The organic extract is scrubbed with 1 M nitric
acid, and uranium is stripped from the organic phase by back-extraction with 0.01
M HNO3. The uranyl nitrate–hexahydrate, UO2(NO3)2
. 6H2O or UNH is converted
to UO3 (orange oxide) in two steps, a concentration by evaporation and a denitrification
by heating. UO3 is reduced to UO2 (brown oxide) using hydrogen via the
overall reaction
UO3 þ H2 À! UO2 þ H2O
UO2 is hydrofluorinated via the reaction
UO2 þ 4HF À! UF4 þ 2H2O
using anhydrous HF gas to produce green salt, UF4. This green salt can be burned in
F2 to produce UF6. UF4 can also be reduced to metallic uranium.
16.4.3 Enrichment
As discussed earlier, natural uranium is 0.72 atom % 235
U, and the fuel used in lightwater
reactors is typically 3% 235
U. This means the refined uranium must be
enriched in the lighter 235 isotope prior to fuel fabrication. This can be done by a
Figure 16.4 Ammonium diuranate (yellowcake) after solvent extraction. (Photo courtesy of
UIC.) (Figure also appears in color figure section.)
16.4 NUCLEAR FUEL CYCLE—THE FRONT END 475
variety of physical approaches that take advantage of the small mass difference
between 235
U and 238
U. We shall discuss four of these methods, gaseous diffusion,
gas centrifuges, electromagnetic separation, and laser enrichment.
In gaseous diffusion, one takes advantage of the fact that in a gas, the lighter molecules
have a higher velocity. If we assume the average kinetic energy of all gas
molecules at a given temperature is the same, then we can write
Average kinetic energy ¼ mv2
=2
For 235
UF6 and 238
UF6, we have
m235v2
235 ¼ m238v2
238
v235
v238
¼
m238
m235
 1=2
¼
314
311
 1=2
¼ 1:0043
Figure 16.5 Schematic diagram of the refining and conversion of uranium ore concentrates.
(From Benedict et al., 1981.)
476 NUCLEAR REACTOR CHEMISTRY
If we send a stream of UF6 gas into a vessel with porous walls (Fig. 16.6), then the
lighter 235 molecules will pass through the pores slightly more frequently (due to
the greater number of impacts on the walls per time). The maximum separation
factor is 1.0043, but, in practice, this is not achieved and the less enriched part of
the gas stream is recycled. Typically, if one starts with natural uranium (0.72%
235
U) and with tails depleted to 0.3%, about 1200 enrichment stages are needed
to get a 4% enrichment. In practice, membranes are used with several million 10to
100-nm pores/cm2
, and the process must be carried out at elevated temperatures,
dealing with UF6, a strong fluorinating agent that decomposes upon contact with
water. Most enrichment is done using this technique.
In gas centrifuges (Fig. 16.7), one takes advantage of the fact that the centrifugal
force will push the heavier 238
UF6 molecules to the wall of the centrifuge, and the
gas near the center will be enriched in 235
UF6. The gas flow pattern allows the
heavier gas to be collected near the top and the lighter gas near the bottom. Separation
is more efficient than in the gaseous diffusion method, requiring only about 10
stages to enrich 235
U from 0.72 to 3% with a 0.2% tail.
In electromagnetic separation, one uses the fact that when ions of differing mass
with the same charge state are accelerated through a potential difference, their
kinetic energy is the same, and the radius of curvature in a magnetic field is proportional
to
ffiffiffiffiffiffiffiffiffiffi
mass
p
. Thus, the material to be separated is ionized, accelerated, and
passed through an analyzing magnet that bends the isotopes into separate beams,
depositing them onto collectors for each isotope. The technology is straightforward,
but the cost is high for ionizing macroscopic amounts of material, and so this technique
has largely been abandoned, although it was used recently in Iraq in an attempt
to obtain enriched uranium for nuclear weapons.
In laser enrichment, one takes advantage of the fact that the atomic energy levels
of different isotopes differ slightly (called the isotope shift). This is due to the differing
reduced electron masses for the different isotopes and the change in the overlap
between the wave functions of the inner electrons and the nucleus, due to isotopic
Figure 16.6 Schematic diagram of the operation of a gaseous diffusion cell. (From Leuze,
1981.)
16.4 NUCLEAR FUEL CYCLE—THE FRONT END 477
differences in nuclear radii. Lasers can be tuned to excite 235
U atoms, but not 238
U
atoms, to higher energy levels. The excited atoms are then ionized with another laser
beam. The resulting ionized 235
U atoms can be separated electromagnetically. To
date, the feasibility of this technique has been demonstrated, but there are no
commercial applications.
16.4.4 Fuel Fabrication
The enriched UF6 is converted into UO2 at the fuel fabrication plants. The UF6 is
reacted with water to produce a solution of UO2F2 and HF:
UF6(g) þ 2H2O À! UO2F2 þ 4HF
Ammonium hydroxide is added to the uranyl fluoride solution to quantitatively
precipitate ammonium diuranate
2UO2F2 þ 6NH4OH À! (NH4)2U2O7 þ NH4F þ 3H2O
This product is collected, calcined in air to produce U3O8, and heated with hydrogen
to make UO2 powder. The UO2 powder is pressed into pellets, which are sintered,
Figure 16.7 Schematic diagram of a gas centrifuge. (From Leuze, 1981.)
478 NUCLEAR REACTOR CHEMISTRY
ground to size, and loaded into zircalloy tubing filled with helium. The tubes are
sealed and assembled into fuel bundles.
16.5 NUCLEAR FUEL CYCLE—THE BACK END
At one point in the history of nuclear power, the concept of reprocessing spent
reactor fuel to recover its plutonium content to fuel other reactors was considered
central to reactor development. The idea of an energy source that could generate
its own fuel was very appealing. But, as outlined earlier, most fuel is not reprocessed/recycled
but used in a “once-through” manner. The reasons for this are
complex, that is, no shortage in near-term uranium supply, low uranium prices,
some technical problems in reprocessing, and a concern that reprocessing would
make plutonium too readily available for use in weaponry. Nonetheless there are
operating plants for reprocessing reactor fuel, and the overall fate of spent fuel
continues to be of great concern and interest.
16.5.1 Properties of Spent Fuel
Periodically, a portion of the fuel in a nuclear reactor is removed and replaced with
fresh fuel. In the past, the average lifetime of fuel in the reactor was 3 years with
one-third of the fuel being removed each year. More recently, attempts are being
made to extend fuel lifetimes.
Initially, the radioactivity levels of the irradiated fuel are very high (Fig. 16.8).
Chemically, the 235
U content has been reduced from its initial 3 to 1% while the
Figure 16.8 Radioactivity and heat output of spent fuel as a function of time. (From
Bodansky, 1996.)
16.5 NUCLEAR FUEL CYCLE—THE BACK END 479
238
U content has been reduced from 97 to 94% (Fig. 16.9). 239
Pu and other Pu
isotopes are produced along with the fission products.
The original plan for this spent fuel was to store it for about 150 d and then
transfer it to other facilities for disposal as waste or reprocessing. In the United
States, this transfer has not occurred, and most of the fuel has remained in
cooling ponds at the reactor sites, for times that have exceeded 20 y. In this case,
a closer look at the properties of irradiated fuel is justified.
In Figure 16.8, we show the activity and heat output for spent fuel, beginning one
year after removal from the reactor. (During the first year, the activity drops to 1% of
its initial value and drops another factor of 5 in the first 10-y period.) The “waste
disposal problem” begins at the 10-y point. Interpreted in this light, the dominant
activities over the long term in unseparated waste are 90
Sr, 137
Cs, 241
Pu, 241
Am
(from the decay of 241
Pu), 239,240
Pu, 99
Tc, and 237
Np. The activity level of the
waste decreases faster than the heat output due to the shorter half-lives of the
Figure 16.9 Changes in the chemical composition of irradiated fuel. (From Murray, 2003.)
480 NUCLEAR REACTOR CHEMISTRY
b emitters, which do not contribute as much to the heat output as the longer-lived a
emitters. The heat output of the waste is sufficiently large to require care in waste
storage.
16.5.2 Fuel Reprocessing
Fuel reprocessing has three objectives: (a) to recover U or Pu from the spent fuel for
reuse as a nuclear reactor fuel or to render the waste less hazardous, (b) to remove
fission products from the actinides to lessen short-term radioactivity problems and in
the case of recycle of the actinides, to remove reactor poisons, and (c) to convert the
radioactive waste into a safe form for storage. Fuel reprocessing was/is important in
the production of plutonium for weapons use.
The Purex process is used for almost all fuel reprocessing today. Irradiated UO2
fuel is dissolved in HNO3 with the uranium being oxidized to UO2(NO3)2 and the
plutonium oxidized to Pu(NO3)4. A solution of TBP in a high-boiling hydrocarbon,
such as n-dodecane, is used to selectively extract the hexavalent UO2(NO3)2 and the
tetravalent Pu(NO3)4 from the other actinides and fission products in the aqueous
phase. The overall reactions are
UO2þ
2 (aq) þ 2NOÀ
3 (aq) þ 2TBP(org) À! UO2(NO3)2 Á2TBP(org)
or
Pu4þ
(aq) þ 4NOÀ
3 (aq) þ 2TBP(org) À! Pu(NO3)4(TBP)2(org)
[These equilibria can be shifted to the right (i.e., improved extraction) by increasing
the TBP concentration in the organic phase or increasing the [NO3
2
](aq).] In a
second step, the TBP solution is treated with a dilute nitric acid solution of a reducing
agent, such as ferrous sulfamate or U(IV), which reduces the plutonium to a
trivalent state but leaves the uranium in a hexavalent state. Plutonium will then
transfer to the aqueous phase, leaving uranium in the organic phase. The uranium
is stripped from the organic phase.
The only fission fragments that extract during the Purex process are Zr, Ru, Nb,
and Tc, with the most troublesome being Zr and Ru. Zr forms a number of complex
species with the most important being [Zr(NO3)4
. 2TBP]. The formation of this
complex is inhibited by the addition of F2
whereby
Zr(NO3)4 Á2TBP þ 6FÀ
() ZrF2À
6 þ 4NOÀ
3 þ 2TBP
An overall schematic view of the Purex process is shown in Figure 16.10.
Detailed descriptions of the process can be found in Benedict and co-workers
(1981) and Wymer and Vondra (1981). We shall briefly summarize the important
steps. The first step is to prepare the irradiated fuel for dissolution by mechanically
chopping it into small pieces (1–5 cm). This opening of the cladding causes the
16.5 NUCLEAR FUEL CYCLE—THE BACK END 481
release of 10% of the Kr and Xe fission products as well as some 3
H and volatile
fission products. These off-gases are combined with those from the dissolution step.
In the dissolution step, the fuel pieces are dissolved in near boiling 10 M HNO3.
This step, which takes a few hours, dissolves the uranium, plutonium, and fission
products, leaving the cladding to be recovered. The Kr and Xe are recovered
from the off-gas of steam, air, and NOx. The chemical reactions for the dissolution
of uranium involve processes like
3UO2 þ 8HNO3 À! 3UO2(NO3)2 þ 2NO þ 4H2O
and
UO2 þ 4HNO3 À! UO2(NO3)2 þ 2NO2 þ 4H2O
The plutonium is oxidized to Pu(IV) and Pu(VI), while the neptunium ends up in the
pentavalent or hexavalent states. Small amounts of plutonium and fission products
Irradiated
fuel
Off-gases
Off-gases
Gaseous
effluent16. Off - gas
treatment
15. NOx
absorption
Preparation
1. for
dissolution
2. Fuel
dissolution
Fuel &
Cladding
Cladding
hulls
Recovered
nitric
acid
Dissolver
solution
3. Feed
preparation
4. Primary
decontam-
ination
5. Uranium-
plutonium
partition
Nitric
acid
Chemicals
to adjust
valence and
acidity
Solvent,
tributyl phosphate
in hydrocarbon
Stripped
solvent
8. Uranium
purification
9. Conversion
to UO3
UO3
10. Conversion
to UF6
14. Solvent
cleanup
7. Conversion
to oxide
Purified
plutonium
nitrate
6. Plutonium
purification Recycle
for Pu
recovery
Stripped
solvent
to 4
Impure
plutonium
nitrate
Impure uranium &
plutonium in
solvent
Low-level
waste
High-level
aqueous waste
Feed to solvent
extraction
Recovered
nitric acid
11.
Nitric acid
recovery
Concentrated
high-level
aqueous
waste 12. Interim
liquid waste
storage
13. Waste
solidification
Solid
high-level
waste
Cleaned
solvent
Stripped
solvent
to cleanup
Plutonium
reductant
Cleaned
solvent
Purified
uranyl
nitrate
Low-level
waste
Recycle
for U
recovery
Recovered
nitric acid
Pure UF6 Pure PuO2
Impure
uranyl
nitrate
Figure 16.10 Schematic diagram of the Purex process. (From Benedict et al., 1981.)
482 NUCLEAR REACTOR CHEMISTRY
may not dissolve, and they can be leached with acid solutions containing the
oxidant Ce4þ
.
The off-gas treatment involves primarily iodine, krypton, and xenon. There are a
variety of processes for capturing the iodine and disposing of it. Kr and Xe are captured
by either cryogenic techniques or selective absorption, such as absorption in
chlorofluoromethane. Most of the off-gas volume is due to Xe (800 L/Mg fuel)
with the activity being mostly 10.7-y 85
Kr (11,000 Ci/Mg fuel).
The dissolver solution is treated with chemicals to adjust the acidity, valence, and
concentrations of the species involved. The HNO3 concentrations are 2–3 M, the
UO2(NO3)2 concentrations are 1–2 M, and the Pu is stabilized as Pu(IV) using
N2O4 or hydroxylamine. In these and subsequent manipulations of these solutions,
attention must be given to criticality control. This is done by regulating the solution
geometry, the concentrations of fissile materials, and by the addition of neutron
absorbers such as Gd.
The primary separation of plutonium and uranium from the fission products
involves a solvent extraction with 30 vol % TBP at room temperature. The activity
levels in this separation are quite high (1700 Ci/L for the fission products) and the
aqueous waste, which contains 99þ
% of the fission products, is a high-level waste.
Am and Cm are not extracted and Np is partially extracted. Because of the high radiation
levels, there are radiolysis problems with TPB, leading to solvent degradation.
Primary products of the radiolysis of TBP are the dibutyl- and monobutylphosphoric
acids along with phosphoric acid. These degradation products are removed in the
solvent purification steps.
Following decontamination of the uranium/plutonium from the fission products,
the plutonium is separated from the uranium. This is done by reducing the Pu(IV) to
nonextractable Pu(III), leaving uranium in the hexavalent state. In the older Purex
plants, this was done using Fe2þ
while the newer plants add U4þ
. The plutonium
thus ends up in an aqueous phase while the uranium remains in the organic phase.
Uranium is back-extracted (and thus removed from the organic phase) with 0.01 M
HNO3. It is purified by a series of solvent extraction cycles until the Pu/U ratio is
,1028
and the total bg activity is less than twice that of aged natural uranium.
16.6 RADIOACTIVE WASTE DISPOSAL
Radioactive waste management began with the advent of nuclear energy and has
been studied since then, with the expenditure of billions of dollars. Despite this
Herculean effort, great uncertainty remains about when and how, many aspects of
waste disposal, especially high-level waste, will be understood and dealt with
effectively.
16.6.1 Classification of Radioactive Waste
The simplest way to classify radioactive waste is by its physical state, that is,
whether it is a gas, liquid, or solid. Gaseous waste arises from gas evolution,
16.6 RADIOACTIVE WASTE DISPOSAL 483
during nuclear fuel reprocessing and by activation of air during reactor operation.
The principal gaseous activation product is 1.8-h 41
Ar, which is usually dispersed
into the atmosphere from a stack whose height ensures safe ground-level concentrations
of the released gas.
The off-gases from fuel reprocessing are the largest contributors to the gaseous
waste. The fission products krypton and xenon escape when the fuel elements dissolve.
Molecular iodine and ruthenium tetraoxide can also be released. Iodine and
ruthenium are removed from the waste stream by trapping. Radio xenon has
mostly decayed after a cooling time of about one year, while 85
Kr is trapped cryogenically.
While small quantities of 14
C are formed in reactors, the release of this
14
C as CO2 is an important component (1
2) of the public dose due to the fuel cycle.
The most important liquid wastes are the high-level effluents, containing fission
products from fuel reprocessing. They contain .99% of the fission products in the
fuel with small quantities of U and Pu. Medium-level liquid waste has an activity
of 4 GBq/L and results from various steps in fuel reprocessing. Low-level
(,0.1 GBq/m3
) waste is treated or concentrated. Liquid organic waste is usually
incinerated or chemically destroyed.
Solid waste comes from the mining and milling of uranium ore and the sludge
from spent fuel storage. It also includes contaminated equipment and structures.
High-level solid waste includes the hulls from the dissolving of spent fuel, ion
exchange resin, and the like.
Radioactive waste may also be classified as to origin (defense or commercial
waste), the material present (transuranium waste, spent fuel), or the level of radioactivity
present (high, medium, low). The principal categories of waste using this
classification scheme are: high-level waste (HLW) resulting from spent fuel reprocessing
and consisting mainly of fission products and a small portion (,0.5%) of the
original U and Pu; spent nuclear fuel (SNF), which is irradiated fuel that has not
been reprocessed; transuranic waste (TRU), which is a-emitting waste with
.100 nCi/g of the transuranium nuclei; low-level waste (LLW), which is waste
with small amounts of radioactivity (non-TRU) arising from a variety of sources;
and mill tailings, a special type of LLW that contains a-emitting radionuclides
resulting from uranium mills. This finely ground sandy material contains U daughters.
Waste may contain hazardous waste (i.e., carcinogens, flammable materials,
etc.) as well as radioactive waste. Such waste is referred to as mixed waste.
16.6.2 Amounts and Associated Hazards
The volume and activity of this waste, as of 1996, is summarized in Table 16.1.
The largest volume, but lowest activity, is the mill tailings, which have a specific
activity of 1.8 nCi/g, mostly due to the 238
U decay series daughters (230
Th, 226
Ra,
222
Rn, 210
Pb, 210
Bi, 210
Po, etc.). Of special concern is gaseous 222
Rn. Most of the
activity associated with radioactive waste is in the unprocessed spent fuel. The military
high-level waste tends to have larger volumes than the commercial HLW
because the latter has been compacted. Most of the commercial spent fuel is
484 NUCLEAR REACTOR CHEMISTRY
stored at the reactor sites while most of the military/USDOE waste is stored at a few
major production facilities.
One measure of the hazard associated with this waste is the water dilution volume
(m3
). The water dilution volume is the volume of water needed to dilute a radionuclide
to its maximum permissible concentration in water. A plot of the water
dilution volume (WDV) for spent fuel is shown in Figure 16.11.
Though the activity of the spent fuel falls by more than a factor of 1000 in the first
thousand years (Fig. 16.3), the WDV falls more slowly. This is due to the hazards
posed by the long-lived a emitters in the spent fuel. For times greater than 500
years, the actinide radio toxicity prevails. On a time scale of 104
–106
years, the
WDV values approach those of the original ore used to make the reactor fuel.
(There are limitations to this measure of hazard because of how radionuclides
enter the biosphere and are concentrated.)
16.6.3 Storage and Disposal of Nuclear Waste
What does one do with the radioactive waste described in the previous section?
Clearly, the most important component of the waste is the spent fuel. Currently,
most spent fuel assemblies are held in cooling ponds at the reactor sites, although
one cannot do this indefinitely. In a few reactor sites, dry storage of the spent fuel
is used. The fuel rods are transferred to special casks when the heat output and
activity are such that air cooling will suffice.
Because a permanent disposal strategy has proven illusive due to technical and
political considerations, plans have been made for interim storage facilities where
TABLE 16.1 1996 U.S. Radioactive Waste Inventory
Type/Origin
Volume
(103
m3
)
Activity
(MCi)
Military
HLW 347 853
LLW 3,474 16
SNF (MTHM)a
2,483 9,980
TRU 238 1.3
Mill tailings 28,000 0.08
Commercial
HLW 2 26
LLW 1,782 8
SNF (MTIHM)a
34,252 12,537
Mill tailings 11,870 0.03
a
By convention, the quantity of spent fuel is reported in mass units
(MTHM, or MTIHM, metric tons of heavy metal or metric tons of
initial heavy metal).
Source: Integrated Data Base Report-1996, DOE/RW-0006, Rev.
13, U.S. Dept. of Energy, Washington, DC, 1997.
16.6 RADIOACTIVE WASTE DISPOSAL 485
the fuel is stored in a retrievable manner until a permanent storage facility is
developed.
The favored method for permanent storage of radioactive waste is deep geologic
repositories. This option is the only option for unprocessed spent fuel assemblies and
for most HLW. (An alternative, supplemental strategy discussed below is to remove
some of the actinides in the HLW by chemical separations prior to geologic storage.)
In general terms, the goal of long-term waste storage is to isolate the radioactive
waste from humans and the environment. The prevailing design strategy for waste
repositories is that of multiple barriers (Fig. 16.12).
The first barrier is the form of the waste, which will immobilize the radioactive
materials. The waste form should not be damaged by heat or radiation nor be
attacked by groundwater. The waste is placed in a steel canister, which is resistant
to leaching. The canister is surrounded by packing materials that prevent radioactivity
from escaping, and the entire repository is backfilled with a material
that absorbs or resists chemical intrusion. The final barrier is the host medium
that separates the repository from the surrounding area.
The host medium can be bedded salt, salt domes, granite, basalt, or volcanic tuff.
Each medium has advantages and disadvantages with regard to resistance to water
intrusion, site availability, and political considerations. Each medium can work and
the most important factor is how the local site is designed. An additional factor is the
Figure 16.11 Water dilution volume for radionuclides in PWR spent fuel. (From National
Research Council, A Study of the Isolation System for Geologic Disposal of Radioactive
Waste, NAS, Washington, 1983.)
486 NUCLEAR REACTOR CHEMISTRY
position of the repository relative to the groundwater table, with most repositories
being below the water table. The United States has chosen the Yucca Mountain
region in Nevada, near the Nevada nuclear weapons test site, as its location for permanent
geologic storage. The Yucca Mountain area features welded volcanic tuff as
the host medium. Welded volcanic tuff is a material of low permeability, and the
Yucca Mountain site is above the water table.
The biggest concern with respect to radionuclides in a waste repository is their
movement in the groundwater. Attention is focused on the horizontal motion of
the groundwater not the upward motion, as the repositories are several hundred
meters below the surface. The repositories and their boundaries and locations are
such that the biosphere is 10–100 km away from the center of the repository. The
velocity of the groundwater in typical geologic media proposed for repositories
ranges from 0.01 to 10 m/y. The velocity of radionuclides in groundwater is
smaller than the velocity of the groundwater due to sorption phenomena. This sorption
can be expressed by a retardation factor R where R is the ratio of the groundwater
velocity to the average radionuclide velocity. R can be expressed by an
approximation as
R % 1 þ 10Kd
where Kd is the ratio of the radionuclide concentration in the rock of the repository to
that in the groundwater. Retardation factors for radionuclides in geologic media of
interest as repository sites range from 1 to 3000 and depend strongly on the ion being
sorbed (Saling and Fentimann, 2001). For 239
Pu, R ¼ 200 for volcanic tuff. The
Yucca Mountain groundwater velocities are 0.025 m/y, meaning that 239
Pu will
decay before it migrates 6 km. A similar conclusion can be reached for the transplutonium
nuclei and 90
Sr. From this standpoint, special concern is given to 99
Tc and
237
Np, whose long half-lives can allow geologic transport.
Figure 16.12 Schematic representation of the multiple barrier waste disposal strategy.
(From Murray, 2003.)
16.6 RADIOACTIVE WASTE DISPOSAL 487
Spent Fuel The largest single radioactive waste disposal problem is the spent
fuel from military and commercial reactors. As discussed earlier, the spent fuel
from commercial reactors is stored in water ponds at the reactor sites. The spent
fuel storage facility consists of a cooling and cleanup system for the water along
with equipment to safely transfer the fuel rods from the reactor to the storage
area. A typical pool will have a volume of 400,000 gal. The water will contain
2000 ppm boron that acts as a neutron absorber and will be maintained at a temperature
of ,708C.
The long-term fate of the spent fuel is geologic storage. The spent fuel assemblies
are packaged in canisters with a stabilizing material (powder or sand) in the canister.
The fuel canisters are placed in the geologic storage site and covered by a backfill to
impede water movement.
High-Level Waste The high-level waste (HLW) consists primarily of liquid
waste from fuel reprocessing. It contains all the fission products from the spent
fuel along with all the neptunium and transplutonium nuclei and less than 1% of
the uranium and plutonium. HLW is intensely radioactive with a high heat output
(Fig. 16.8). The hazard potential of this waste remains high for much longer than
the time scale of nuclear energy use. In addition to its radiological hazard, it is
very corrosive, being up to 7 M in HNO3 and containing 250 g/L salt. This
waste is ultimately to be stored in geologic repositories after solidification of the
liquid waste. The volume of the solidified waste is modest in that a commercial
nuclear power reactor plant running for 1 GW-year will produce about 2 m3
of
solidified waste after reprocessing.
The liquid waste is stored for at least 6 y prior to solidification to reduce the decay
heat (Fig. 16.8) by a factor of 10 or more. The first U.S. military fuel reprocessing
wastes were stored as neutralized waste in mild steel tanks at the Hanford reservation
in eastern Washington. These steel-lined, reinforced-concrete tanks were
500,000–1,000,000 gal in capacity with provisions for removal of waste heat and
radiolysis products. Corrosion of several tanks occurred with the release of waste.
Fortunately, the soil around these tanks retarded nuclide transport. A better (and
more expensive) design for storage tanks was implemented at the Savannah River
site in South Carolina consisting of a second steel tank inside of a Hanford-style
tank. The storage of acid waste in these tanks has not encountered the corrosion
problems seen with the Hanford tanks.
The solidification of waste takes place in two steps, a calcining step and an incorporation
of the calcined material into borosilicate glass. Calcining can be done in
various ways but primarily involves the removal of volatile products like water or
NO3
2
and conversion of all species in the HLW to solid stable oxides. The oxides
are then mixed with SiO2, B2O3, and the like to make a borosilicate glass, which
is then prepared for geologic storage (Fig. 16.13).
Transuranic Waste Transuranic waste (TRU) results from fuel reprocessing
and fuel fabrication facilities, the production of nuclear weapons, and the decommissioning
of nuclear reactors or fuel cycle facilities. TRU includes clothing,
488 NUCLEAR REACTOR CHEMISTRY
equipment, and the like from reprocessing facilities along with plutonium and other
transplutonium elements removed in fuel reprocessing. Because of the relatively low
activity levels of this waste, the primary processing steps prior to storage involve
volume reduction. In the United States, this waste is stored in an interim storage
facility, the Waste Isolation Pilot Plant (WIPP), an underground salt-bed facility
near Carlsbad, New Mexico.
Low-Level Waste Low-level waste (LLW) consists of contaminated dry trash,
paper, plastics, protective clothing, organic liquids such as liquid scintillation
samples, and the like. LLW is produced by any facility that handles radioactive
materials such as nuclear power plants, medical facilities, colleges, and so forth.
In the United States, commercial LLW is sent to one of three disposal sites
(Barnwell, South Carolina, Richland, Washington, and Clive, Utah). Due to the
limited size of these sites (and similar disposal sites through the world) and
steeply escalating costs for waste disposal, the primary goal of LLW treatment
prior to disposal is volume reduction, either by incineration or compaction, followed
Figure 16.13 Schematic diagram of the final steps in putting vitrified waste into a geologic
repository. (Figure also appears in color figure section.)
16.6 RADIOACTIVE WASTE DISPOSAL 489
by immobilization. For noncombustible solids, volume reduction can be achieved by
mechanical disassembly, crushing, melting, or dissolution. For noncombustible
liquids, evaporation, calcination, filtration, or concentration on ion exchange
resins are used for volume reduction. Combustible material is oxidized to ashes.
Mechanical techniques of volume reduction reduce volume by 5–10 times, while
combustion reduces volume by 50–100 times. The products of volume reduction
are immobilized using absorbents (vermiculite, clay, etc.), cement, or salt matrices.
Mill Tailings The tailings from uranium mining and milling contain all the
daughters of uranium present in the original ore. The mill tailings thus have
70% of the original activity of the ore. This activity decays with the half-life of
230
Th, 8 Â 104
y. The radiological hazard is mostly in the 226
Ra and its daughter,
222
Rn. The inert gas 222
Rn can escape from the tailings and can create a radiological
inhalation risk. Other radionuclides, such as 230
Th or 226
Ra, can be leached from the
tailings by water, being transferred to the biosphere. The tailings themselves are
finely divided sandy material that can be dispersed by wind. Remediation of sites
containing mill tailings involves covering the tailings by a clay or earth overburden
that is 3–8 m thick. This overburden reduces Rn release, minimizes leaching, and
prevents wind dispersal.
Partitioning of Waste As discussed earlier, for a variety of reasons, the nuclear
fuel cycle is operated in a “once-through” mode in the United States and for the
majority of reactors worldwide. The long-lived transuranium nuclides are thus
mixed with the shorter-lived fission products in the HLW and SNF. As concern
about the lack of long-term, widely accepted programs to deal with this waste
mounts, attention is being focused on partitioning (chemically separating) the transuranium
nuclei from the waste and transmuting it to nuclides of shorter half-life by
irradiation in high neutron fluxes. These waste destruction steps also can be incorpor-
atedintoadvanceddesignreactorsystemsoradvancedfuelcycledesigns.Othertargets
of partitioning are troublesome fission products such as 99
Tc, 129
I, 90
Sr, and 134,137
Cs.
At the outset, one should comment that there is a division (Benedict et al., 1981;
Bodansky, 1996) among scientists as to whether the gains in reduced radiotoxicity of
the waste offset the additional cost and risk of further treatment and handling of the
waste. Removal of the actinides from a waste repository gives a small reduction in
risk because the actinides migrate so slowly through the repository. Several
additional problems would be created by partitioning and transmutation such as: a
greater volume of wastes, the need for large, expensive processing facilities, an
increase in the neutron flux coming from the waste, necessitating increased shielding,
and the possibility of additional releases of radioactivity to the environment. A
U.S. study has suggested 99.9% removal of the actinides and 95% removal of Tc and
I is a minimum requirement to justify the effort.
Nonetheless, the United States and other nations have been developing additional
chemical separation processes to bring about these goals. Strontium and cesium can
be removed from HLW by extraction with crown ethers, such as ditertiarybutyldicyclohexanone-18-crown-6,
which can also extract 99
Tc. In the United States,
490 NUCLEAR REACTOR CHEMISTRY
a variation of the Purex process is being developed to reprocess spent fuel. This modification
of the Purex process is such that only U and Tc are extracted from the fission
products and TRU nuclei (Np, Pu, Am, Cm). It is called the UREX process (uranium
extraction). It is intended to recover .99.9% of the U, .95% of the Tc and leave
.99.9% of the TRU nuclei in the acid waste. All chemicals used in the process are
converted to gases in subsequent processing to minimize waste. A Purex separation
is carried out using 30% TBP in kerosene working on a 1 M HNO3 solution of
spent fuel and then acetohydroxamic acid (AHA), an analog of hydroxylamine, is
used to reduce Np(VI) to nonextractible Np(V) and to complex Pu(IV) and Np(IV),
preventing their extraction. Figures 16.14 and 16.15 show the flowsheet for the
UREX process and its place in the total, new proposed treatment of spent fuel.
Other work on pyrochemical processes involving electrochemical reduction steps
in molten salt baths is also underway.
Transmutation Transmutation is the term used in connection with waste
management to describe a nuclear process in which long-lived nuclides are
changed into shorter-lived nuclides by nuclear reactions. Possible sources of bombarding
particles are neutrons from reactors or accelerators that have been configured
to produce large neutron fluxes, by reactions, such as spallation. Fast
neutrons are preferred as the bombarding particle because some of the relevant isotopes
of the actinides Np, Am, and Cm are not thermally fissionable. 129
I and 99
Tc,
on the other hand, are readily destroyed in thermal fluxes. Transmutation can also
be used to destroy plutonium from dismantled nuclear weapons. Of course, the
Extractant
30 vol% TBP
Extractant
30 vol% TBP
Feed
290 g/LU
Pu, Np, FP
1 M HNO3
Scrub
0.3 M HNO3
0.47 M AHA
Scrub*
Tc Strip
6 M HNO3
Tc Strip
Re-extract
U
Raffinate
Pn, NP, Am
Cm & FP
Tc Product
Tc
5.5 M HNO3
*Extraction and scrub sections
controlled to <25°C
U/Tc Extraction*
U Strip
0.01 M HNO3
U Strip
60°C
Spent Solvent
To Solvent
Washing
U Product
60–70 g/LU
0.5 M HNO3
Figure 16.14 Flowsheet for the UREX process.
16.6 RADIOACTIVE WASTE DISPOSAL 491
resulting fission or reaction products from transmutation are radioactive and do
constitute radioactive waste, albeit with shorter half-lives.
Current attention is centered on methods using charged particle accelerators
for transmutation. In a typical scheme, an accelerator is used to produce 10- to
200-mA beams of 1-GeV protons. The proton beam strikes a liquid lead or lead–
bismuth eutectic target giving rise to 30–40 fast neutrons/proton, via spallation.
These fast (predominantly 1- to 10-MeV neutrons) are slowed down in a graphite
moderator surrounding the Pb target. Some designs produce fast (75–225 keV) neutrons
in the moderator while other designs thermalize the neutrons. The actinides
and fission products to be transmuted are dissolved in molten salts or other media
allowing high heat transfer in channels passing through the moderator. The
neutron flux in the channels is 1015
–1016
n/cm2
-s.
Typical heat production in the moderator-fuel blanket is 750–1500 MW. The
excess heat is used to generate electricity that helps to pay for the operation of
the facility. The transmuted material will have 20% of the original plutonium
and minor actinides of the input material and will contain significant fission
product activities. This transmuted material can be put into geologic storage, reducing
the long-term hazard of the repository material. The overall feasibility of this
accelerator transmutation of waste (ATW) has not been established yet.
16.7 CHEMISTRY OF OPERATING REACTORS
A complex set of chemical processes occurs during the operation of a nuclear
reactor. Up to now, we have concentrated our attention on the chemical, physical,
Spent
Nuclear
Fuel
Chop-
Leach
Process
UREX Solvent
Extraction
Process
Conversion
to Oxide
Oxide
Reduction
to Metal
Electrometallurgical
Separation Process
Waste
Treatment
ATW Fuel
Assembly
Fabrication
Disposal as LLW
or
Storage
ATW
Transmutation
Target
Fabrication
Hull
Cleanup
Nitrate
Solution
Iodine
Tc
Tc
TRU
Uranium
Np
Raffinate
(TRU, FP)
TRU, FP
Oxides
TRU, FP
FP
FP
TRU
Figure 16.15 Schematic representation of the new schemes for reprocessing of SNF.
(Figure also appears in color figure section.)
492 NUCLEAR REACTOR CHEMISTRY
and nuclear processes occurring in the fuel and their consequences. Now we turn our
attention to the coolant, the moderator, and the reactor materials and the changes in
them due to the hostile chemical environment. The reactor environment has coolant
temperatures up to 3508C, pressures of 75–150 atm, and intense neutron and g
radiation. Further complications are introduced by the two phases present in
BWRs, and the use of chemical shims like B to control the reactivity of PWRs.
Both reactor types exhibit potentially serious possibilities of corrosion. (One
PWR in Ohio was closed for several years to rapair the steel pressure vessel that
was corroded by boric acid.)
16.7.1 Radiation Chemistry of Coolants
About 2% of the total neutron and g-ray energy released in a nuclear reactor is
deposited in the cooling water which leads to decomposition of the water itself.
The situation is of special concern in BWRs where the neutron and g-ray fluxes
near the core are 109
R/h of neutrons and 108
R/h of photons. (There is a hydrogen
gas overpressure in PWRs suppressing to some extent water decomposition or
oxygen gas production.) Due to radiolysis of the coolant water, this energy
deposit produces an oxidizing environment with 100–300 ppb oxygen and hydrogen
peroxide, with lesser concentrations in the vapor phase. The primary process is
H2O À! H†
þ OH†
These radicals react to form H2 or H2O2 by
H†
þ H†
À! H2
H†
þ OH†
À! H2O
OH†
þ OH†
À! H2O2
Molecular O2 is generated by
H2O2 þ OH†
À! HO2
†
HO2
†
þ OH†
À! H2O þ O2
HO2
†
þ HO2
†
À! H2O2 þ O2
H2O2 þ HO2
†
À! H2O þ †
OH þ O2
and destroyed by
O2 þ H†
À! HO2
†
16.7.2 Corrosion
The oxygen and radiolysis products attack the outer layers of the stainless steel or
nickel-based alloys used in the reactor structure, forming a thin oxide layer on
these components. Corrosion products are released from this thin oxide layer by
16.7 CHEMISTRY OF OPERATING REACTORS 493
the cooling water and become activated as they pass near the reactor core. These
products can deposit on fuel surfaces, in coolant channels, and on reactor materials.
These deposits are referred to as crud (Chalk River unidentified deposits). In PWRs
the chemical composition of the crud is NixFeyO4 where x/y ¼ 0.25 and x þ y ¼ 3.
In PWRs, the crud that also contains boron compounds causes a phenomenon known
as axial offset anomaly (AOA), which is an unexpected deviation from predictions of
the core axial power distribution during operation. Because it is not well understood,
AOA limits the operating power of several PWRs, at great cost to the utilities.
Attempts are made to minimize corrosion by controlling the chemistry of the
coolant water. Adding 7
LiOH raises the pH to 8. One can use oxygen scavengers
such as hydrazine to reduce the oxygen concentration.
16.7.3 Coolant Activities
When corrosion products are deposited on the fuel surfaces, they are activated by
neutron capture. Some of the most prominent of these activities are 55
Fe, 63
Ni,
60
Co, 54
Mn, 58
Co, and 59
Fe. These radionuclides will then be found in the reactor
coolant.
Fission products can be released from defects in the fuel rods or from tramp
uranium on the fuel cladding. Of special importance are the volatile fission products
131– 135
I (89
Kr, 137,138
Xe in BWR steam). Cations include the Sr and Cs isotopes,
which are present along with 129
I and 99
Tc. One can use the ratio of short/longlived
isotopes such as the ratio of 133
I/131
I to determine the source of the fission
product release, by assuming the short-lived species can only result from tramp
fuel or large cracks in the fuel assembly.
Impurities in the water and water activation products also contribute to the radioactivity
of the coolant water. Tritium is produced as a low yield (0.01%) fission
product that can diffuse out of the fuel, by activation of boron or 6
Li impurities in
PWRs. 24
Na and 38
Cl are produced by neutron activation of water impurities. In
BWRs, the primary source of radiation fields in the coolant and steam systems
during normal operations is 7.1s 16
N. This nuclide is produced by 16
O(n, p)16
N reactions
from fast neutrons interacting with the coolant water. This 16
N activity can
exist as NO2
2
, NO3
2
in the coolant and NH4
þ
in the steam.
PROBLEMS
1. Define or describe the following terms or concepts: (a) crud, (b) axial offset
anomaly, (c) accelerator transmutation of waste, (d) UREX process, (e) LLW,
(f) TRU, (g) HLW, (h) SNF, (i) Purex process, (j) back end of the fuel cycle,
and (k) pyroprocessing.
2. Define or describe the following terms or concepts: (a) yellowcake, (b) orange
oxide, and (c) green salt.
494 NUCLEAR REACTOR CHEMISTRY
3. Since UO2 can be converted directly to UF6, why is it first converted to UF4 and
then to UF6?
4. Write balanced chemical equations for three different methods to produce
metallic uranium.
5. Nitric acid readily dissolves UO2. Why doesn’t hydrochloric acid?
6. Given a G value of 1.59 for the production of hydrogen gas by the irradiation of
TBP with 1-MeV electrons, calculate the rate of hydrogen gas evolution in a
liter of TBP irradiated for 1 h at a dose rate of 200 W/L. Assume STP
conditions.
7. Draw a flowsheet for the Purex process like Figure 16.10. Estimate the relative
volumes of all streams in the process using data from the references cited in the
text.
8. Discuss the disposal of the following examples of radioactive waste:
a. Water solutions containing 1 mCi of 3
H from a research lab
b. Gas escaping from the dissolution of 1 kg of irradiated reactor fuel
c. Ion exchange resin used to purify the cooling water of a 1-MW research
reactor
9. An amount of 500 g of natural uranium is irradiated in a neutron flux of
1013
n/cm2
-s for 1 y. What is the heat output of this material after cooling for
1 week? One month? One year?
10. What is the theoretical maximum separation factor for separation by gaseous
diffusion of 3
He and 4
He?
11. If a gaseous diffusion plant produces uranium with a 235/238 ratio of 5, what is
the expected 234/235 ratio in the resulting material?
12. Discuss quantitatively the relative merits of using LiOH, NaOH, or KOH for pH
control in reactor coolant water.
13. In reactors based on a Th fuel cycle, 233
Pa is produced. Discuss the radiochemistry
of this radionuclide.
REFERENCES
Benedict, M., T. H. Pigford, and H. W. Levy. Nuclear Chemical Engineering, 2nd ed.,
McGraw-Hill, New York, 1981.
Bodansky, D. Nuclear Energy—Principles, Practices and Prospects, AIP. Woodbury, NY,
1996.
Murray, R. L. Understanding Radioactive Waste, 5th ed., BMI, Columbus, 2003.
REFERENCES 495
Saling, J. H. and A. W. Fentimann, Eds. Radioactive Waste Management, 2nd ed., Taylor &
Francis, New York, 2001.
Wymer R. G. and B. L. Vondra. Light Water Reactor Nuclear Fuel Cycle, CRC, Boca Raton,
FL, 1981.
BIBLIOGRAPHY
Bowman, C. D. Accelerator-Driven Systems for Nuclear Waste Transmutation, Ann. Rev.
Nucl. Part. Sci. 48, 505 (1998).
Choppin, G., J. O. Liljenzin, and J. Rydberg. Radiochemistry and Nuclear Chemistry, 3rd ed.,
Butterworth-Heinemann, Woburn, 2002.
Eisenbud, M. Environmental Radioactivity, 3rd ed., Academic, Orlando, 1987.
Glasstone, S. Sourcebook on Atomic Energy, 3rd ed., Van Nostrand, New York, 1967.
Lin, C. C. Radiochemistry in Nuclear Power Reactors, NAS-NS-3119, National Academy
Press, Washington, 1996.
National Research Council, A Study of the Isolation System for Geologic Disposal of
Radioactive Wastes, National Academy Press, Washington, 1983.
Peterson, S. and R. G. Wymer. Chemistry in Nuclear Technology, Addison-Wesley, Reading,
MA, 1963.
Roberts, L. E. J. Radioactive Waste Management, Ann. Rev. Nucl. Part. Sci. 40, 79 (1990).
496 NUCLEAR REACTOR CHEMISTRY
CHAPTER 17
INTERACTION OF RADIATION
WITH MATTER
17.1 INTRODUCTION
At this point we have described nuclear transitions and reactions that produce
various forms of nuclear rad\iation. The radiation propagates out from the originating
nucleus and interacts with other matter along its path. These interactions with
external matter allow us to observe the radiation, and its effects, and to determine
the nature of the transition inside the nucleus. The interaction of radiation with
matter is also the cause of chemical, physical, and biological changes that
concern the public at large. We will specifically address the operating principles
of radiation detectors in the next chapter, but first we will consider the fundamental
interactions of nuclear radiation with matter.
It should be clear that radiation (with the exception of neutrons) primarily interacts
with bound electrons. For example, a silicon atom contains 14 electrons in a
sphere with a radius of 0.12 nm that presents a geometrical cross section of
4.5 Â 10220
m2
. The nucleus at the center of the sphere has a radius of 3.6 fm with
a geometrical cross section of 4.1 Â 10229
m2
. The geometrical probability to strike
the electrons in an atom is something like 9 orders of magnitude higher than that to
strike the nucleus. A scattered electron leaves the original atom and creates an ion
pair. The interaction of a single particle of nuclear radiation can lead to tens or hundreds
of thousands of ion pairs, and so nuclear radiation is generally called ionizing radiation.
497
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
From the starting point that we should consider the interaction of the radiation
with electrons, we can divide nuclear radiation into four classes as indicated in
Table 17.1. The overall scattering of the particle from an electron will be dominated
by the mass and the charge on the particle. Thus, massive charged particles will tend
to scatter the small mass electrons widely without losing much energy, while the collision
of an electron with another electron will lead to energy sharing, and a photon
can be scattered or even absorbed by a single bound electron. Neutrons only interact
very weakly with electrons through their small magnetic moment and predominately
interact with nuclei. (As will be discussed later, neutrons are very penetrating and
difficult to detect due to the small probability of striking a nucleus.) As a general
rule, all of the interactions end up in releasing energetic electrons. The heavy
charged particles and recoiling atoms scatter electrons. The electrons interact with
matter to create moving charged particles, while neutrons create recoiling nuclei
and photons create moving electrons.
We will discuss the interaction of each class of radiation starting with the simplest,
those charged particles that are more massive than electrons. We will consider
what happens as they pass through various types of matter. The radiation generally
penetrates through many, many atomic layers, so we can generally assume that the
atoms are randomly distributed in space. This assumption is certainly true for liquids
and gases, but we usually think of solids as having a regular crystal or lattice structure.
The solid material that we encounter in everyday life certainly has a lattice
structure on the microscopic scale, but nearly all materials are polycrystalline on
a larger scale. Thus, the types of radiation that we will generally consider will cross
many crystal boundaries in normal materials. The exceptions are single crystals of
silicon, germanium, or other special materials that are used in semiconductorbased
radiation detectors that rely on their special electronic properties. We have
to be aware of the orientation of the crystal axes in these devices relative to the
propagation direction of the radiation.
Also before starting the discussion, we should define a quantity called the areal
density. We have said that ionizing radiation will pass through a significant
amount of material and sometimes will even pass completely through an object.
We can imagine that the solid piece of material can be compressed into a thin,
two-dimensional sheet as far as the radiation is concerned. The origins of this
concept lie in early experiments with a rays in which very thin sheets of metals
were necessary. We can easily measure the length x and breadth y of such thin
foils, but it is very difficult to measure the thickness z without destroying the foil.
On the other hand, we can determine the mass m of the foil rather easily and then
TABLE 17.1 Particle Classes for Interaction with
Matter
Charged Particles Uncharged Particles
Protons, heavy ions Neutrons
Electrons g rays
498 INTERACTION OF RADIATION WITH MATTER
calculate the thickness using the density of the material. For example, if the foil is
made from a pure chemical element, so that the density is known, then:
Linear thickness ; z ¼
m
x  y
 
r
¼
areal density
density
where r is the density (and has dimensions of mass per unit volume, of course). The
measured quantity of mass per unit area is often used to characterize thin foils of
pure materials. This quantity is called the areal density. Typical dimensions for
materials that we might use in experiments are mg/cm2
. Thus, if the density of a
typical metal is in the range of 5–10 g/cm3
, the thickness of a typical foil would
be on the order of fractions of a millimeter.
Example Problem A ream of a certain type of standard letter-sized paper
(in the United States) was found to have a mass of 2.26 kg. What is the areal
density of one sheet of this paper?
Solution Recall that a ream of paper contains 500 sheets, so that the areal
density of one sheet is simply:
A ¼ total mass=surface area
A ¼ 2260 g=½(500 Â 8:5 in. Â 11 in.) Â (2:54 cm=in.)2
Š
 1000 mg=g ¼ 7:5 mg=cm2
17.2 HEAVY CHARGED PARTICLES (A ! 1)
We can imagine the progress of any ionizing radiation through material as a series of
straight-line segments between scattering events. The scattering events primarily
involve electrons. The total path is thus made up of these line segments, and the
overall trajectory of the particle in the material will depend on the kinematics of
these scattering events. Elastic scattering is, of course, governed by the conservation
of momentum and energy, so we should expect that the mass of the particle will play
a large role in determining the overall features of the trajectory of the particle.
The tracks of a few heavy charged particles that have stopped in a photographic
film are shown in Figure 17.1. The photographic film, called a nuclear emulsion, is
sensitive to the ionization that is caused by the charged particles as they move.
Normally, visible photons “expose” photographic film by creating photoelectrons,
and the ionization is converted into an image through the development process.
All of the heavy charged ions that we have to consider have positive charges. As
can be seen in Figure 17.1, energetic charged ions move through material on essentially
straight trajectories, giving up or losing kinetic energy through collisions with
the atomic electrons of the material. Only rarely by comparison is an ion scattered by
the Coulomb potential of a nucleus, and even more rarely does a nuclear reaction
take place. Nuclear reactions are excluded when the initial kinetic energy of the
17.2 HEAVY CHARGED PARTICLES (A ! 1) 499
heavy charged particle is lower than the Coulomb barrier (as discussed in Chapter
10). Thus, the ions interact with an extremely large number of electrons, and we
can examine the average behavior of the ions as they pass through material.
The rate at which charged particles lose energy as they travel through a given
material is called the stopping power of that material. The stopping power is
made up from two parts, the electronic stopping power due to the interaction with
the atomic electrons of the material and the nuclear stopping power. Thus
À
dE
dx
¼ Selectronic þ Snuclear % Selectronic (17:1)
because the electronic stopping power is always much larger than the nuclear stopping
power. (We should note that some authors use the term “nuclear stopping” not
for nuclear processes but rather for the atomic scattering processes of neutralized
cores at the very end of their range.) Notice that the minus sign on the rate to indicate
that the ions are losing kinetic energy. The nuclear stopping power is not zero, of
course, because we know that nuclear reactions do take place even if they are
rare. The stopping powers are functions of the mass, charge, and velocity of the
ion, the atomic number, and density of the medium.
Niels Bohr suggested that the energy loss rate could be estimated in a very simple
picture as the series of impulses delivered to individual electrons by the ion. Imagine
an ion moving on a straight-line trajectory past an electron (see Fig. 17.2). A net
impulse to the electron will occur in the direction perpendicular to the trajectory
Figure 17.1 Trails of ionization left by heavy charged particles (initial energy of 10 MeV/
nucleon) as they penetrate through a photographic plate (nuclear emulsion). The ions interact
with the atomic electrons in the emulsion creating ion pairs to “expose” the emulsion and
the tracks become visible after the film is developed. Notice the straight-line tracks. (From
Knoll, 2000.)
500 INTERACTION OF RADIATION WITH MATTER
of the ion because any impulse due to the approaching ion will be cancelled by that
of the receding ion. It can be shown that the energy gained by a single electron and
thus lost by an ion in one encounter depends on the impact parameter, b, as:
DE(b) ¼
2q2
e4
mev2b2
(17:2)
where q is the charge of the ion (sometimes equal to the atomic number), v is the
velocity of the ion, and me is the mass of the electron. This expression can be converted
to a differential expression by multiplying by the electron number density, Ne,
times the volume element:
ÀdE(b) ¼ DE(b)Ne dV ¼ DE(b)Ne2p db dx (17:3)
by using the cylindrical coordinates of impact parameter and taking x along the ion’s
path. This expression should not be integrated from b ¼ 0 to b ¼ 1 but only over
the range bmin to bmax that are appropriate to the initial assumptions with regard
to the ion and the electron, so that:
À
dE
dx
¼
4pq2
e4
mev2
Ne ln
bmax
bmin
(17:4)
The minimum impact parameter will correspond to those collisions in which the
maximum amount of kinetic energy is transferred to the electron. Due to conservation
of momentum, the maximum electron energy is Wmax ¼ 1
2
À Á
me(2gv)2
where we
have included the relativistic factor g due to the low mass of the electron. Recall that
g ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1 À b2
s
and b ¼
v
c
Thus, substituting into the expression for the energy loss at a given impact
parameter:
DE(bmin) ¼
2q2
e4
mev2b2
min
¼ 2g2
mev2
(17:5)
Figure 17.2 Trajectory of a moving ion past an electron. (From Leo, 1987.)
17.2 HEAVY CHARGED PARTICLES (A ! 1) 501
we find that
bmin ¼
qe2
gmev2
(17:6)
The maximum impact parameter has to be estimated from different considerations.
The basis of this process is that the ion rapidly moves past the electron
and delivers a sharp impulse to the electron. The electrons are bound in atoms
and thus are orbiting with their own characteristic frequencies or time scales.
Thus, the time for the ion to cross the atom should be less than the average time
for an electron orbit; otherwise the collision will not be adiabatic or “rapid.” The
time for the ion to move past can be estimated as the ratio of the impact parameter
to the ion’s velocity, the average orbital time for an electron will clearly depend on
the chemical element, as there will be an average radius and velocity, thus
bmax
gv
Re
ve
¼ f(Z) (17:7)
where f(Z) is a function of the atomic number of the stopping material. So that we
can combine these two limits into the expression for the stopping power or energy
loss rate to get Bohr’s classical formula:
À
dE
dx
 
Bohr
¼
4pq2
e4
mev2
Ne ln
g2
mev3
f(Z)
qe2
(17:8)
This expression has been superseded by the expression derived by Bethe and Bloch
based on momentum transfer in a quantum mechanically correct formalism. Their
expression with the expanded form of the electron number density is
À
dE
dx
 
Bethe–Bloch
¼ 4pNAr2
e mec2
r
Zq2
Ab2
ln
Wmax
I
 
À b2
!
(17:9)
where NA is Avogadro’s number, re is the classical radius of the electron, r is the
density of the stopping medium with atomic number, Z, mass number, A, and
ionization potential, I. Finally, Wmax is the maximum energy transfer, encountered
above. The structure is very similar to the classical formula, as should be expected,
but it includes an extra term in the logarithm. Various formulas are available to give
the average variation of the ionization potential for the chemical elements. For
example, the expressions:
I=Z ¼ (12 þ 7ZÀ1
) eV Z , 13
¼ (9:76 þ 58:8ZÀ1:19
) eV Z ! 13
(17:10)
are the results from one empirical fitting of the data, but one should realize that the
variation could be quite complicated due to the filling of the atomic shells.
502 INTERACTION OF RADIATION WITH MATTER
Example Problem Evaluate the stopping power of beryllium metal for 18
08þ
ions with a kinetic energy of 540 MeV (E/A ¼ 30 MeV) using the Bethe–
Bloch formula.
Solution Finding some necessary constants, the density of beryllium metal is
1.85 g/cm3
and Z ¼ 4 so that the ionization potential can be estimated as:
I=Z ¼ (12 þ 7Z1
) eV ¼ 13:75 eV
I ¼ 55 eV
The values of b and g for the ion can be obtained from the relativistic expressions
(derived elsewhere):
b ¼ ½1 À (m2
0=(m0c2
þ (E=A)Š1=2
b ¼ ½1 À (931:5=(931:5 þ 30)Š1=2
¼ 0:1766
g ¼ ½1=(1 À b2
)1=2
Š ¼ 1:01598
The value of Wmax is can be evaluated as:
Wmax ¼ 2mec2
(gb)2
¼ 2 Â 0:511 MeV(1:01598 Â 0:1766)2
¼ 0:03291 MeV
Finally, the expression with the constants evaluated is
À
dE
dx
 
Bethe–Bloch
¼ 0:3071
MeV † cm2
g
r
Zq2
Ab2
ln
Wmax
I
 
À b2
!
À
dE
dx
 
Bethe–Bloch
¼ 0:3071
MeV † cm2
g
(1:85 g=cm3
)
Â
82
9(0:1766)2
ln
0:03291 MeV
55 Â 10À6 MeV
 
À 0:17662
!
¼ 518:0 MeV  ½ln (598:4) À 0:03119Š ¼ 3:3 GeV=cm
Notice that the answer indicates that an ion with only 540 MeV of kinetic energy
will lose all its energy and stop before it travels a fraction of a centimeter. As discussed
later, we need to integrate this expression to determine the predicted range.
The modern form of the stopping power includes two corrections. The first
correction applies at high energies at which polarization of electrons by the electric
field of the moving ion tends to shield distant electrons; this correction depends on
the electron density; it is subtractive and given the symbol d. The second correction
applies at low energies when the collisions are no longer adiabatic, similar to the
limit applied by Bohr. This correction is termed the shell correction as it depends
17.2 HEAVY CHARGED PARTICLES (A ! 1) 503
on the orbital velocities of the electrons. It is also a subtractive term and given the
symbol C. If we evaluate all the constants, then the modern form is
À
dE
dx
 
Bethe–Bloch
¼ 0:3071
MeV † cm2
g
r
Zq2
Ab2
ln
Wmax
I
 
À b2
À
d
2
À
C
Z
!
(17:11)
which has the dimensions of MeV/cm when the usual form of the density in g/cm3
is used. The actual evaluation of this function is complicated due to the detailed variation
of the ionization potential and the two correction terms. The reader is referred
to more detailed discussions (Leo, 1987) for actual formulas for the correction
factors. In practice, several computer codes and detailed tables of the stopping
powers are available. In addition, some authors divide through by the density, r,
and report the mass stopping power:
À1
r
dE
dx
with dimensions of MeV-cm2
/g, which is convenient for combining materials.
If we look at the form of these equations for the stopping power, we would see
that they all have a part that depends on the moving ion and another part that
depends on the stopping medium. If we concentrate on the part that depends on
the ion, we find that
À
dE
dx
/
q2
v2
ln (g2
v2
)g(Z) (17:12)
in which we can convert the factors of v2
into kinetic energy, E, by suitably applying
factors of 1
2 mion. The function g(Z) collects all the variation on the absorbing
medium. The revised expression shows that the energy loss rate will be proportional
to the mass of the ion:
À
dE
dx
/
Aq2
2E
ln
g2
2E
A
 
g(Z) (17:13)
and inversely proportional to the kinetic energy. At low ion velocities
(E=A , 10 MeV=A), the ln (g2
2E=A) term is approximately constant and
À
dE
dx
/
Aq2
E
(17:14)
Thus, a more energetic ion will tend to lose energy at a lower rate than a less
energetic ion. Be careful to note that we have ignored the relativistic terms, g2
and b2
, in the parentheses, which produce a minimum in the complete function
near b 0.96 and a small rise at higher velocities. (Particles with b 0.96 are
called minimum ionizing particles.) The proportionality of the stopping power
504 INTERACTION OF RADIATION WITH MATTER
on the mass and square of the charge of the ion for a given kinetic energy provides
the basis for a very effective particle identification technique using thin
silicon semiconductor detectors, as discussed in Chapter 18.
We are now in a position to examine the slowing down of a charged particle as it
penetrates into material. Kinetic energy is lost through scattering electrons away
from the essentially straight-line path of the ion. If the initial kinetic energy of
the ion is a few MeV/A or higher, the rate at which kinetic energy is dissipated
slowly increases as the ion penetrates into the material. For example, the stopping
power of beryllium metal for a very energetic 40
Ar ion is shown in Figure 17.3
along with the residual energy of the ion. (The thickness scale can be converted
into a linear distance by dividing by the density in appropriate units.) Notice that
the stopping power is relatively constant over most of the ion’s path. The kinetic
energy of the ion uniformly decreases as it moves through the material. However,
two changes occur as the velocity of the ion approaches the Bohr velocity of the
atomic electrons,
vBohr ¼
ZhÀ
mena0
¼ 5:51 Â 10À3 Zc
n
The energy loss rate begins to increase dramatically as b ! 0, but, more importantly,
the charge state on the ion starts to decrease as the ion captures orbital
electrons causing the rate to drop. As indicated in Figure 17.3, the ion rapidly
loses energy at the end of its range and stops rather suddenly. The energy loss
for an a particle near the end of its range is shown in Figure 17.4. The resulting
5000
0
ELAB
dE/dX
30
20
10
0
0 2000
Distance (mg/cm2)
4000
Figure 17.3 Energy loss rate as a function of thickness for a 40
Ar projectile in beryllium
metal is shown on the scale to the right for an ion that enters the foil at the very high
energy of 8 GeV. The remaining kinetic energy of the ion is shown on the left scale. Note
that the ion penetrates approximately 21 mm into the metal.
17.2 HEAVY CHARGED PARTICLES (A ! 1) 505
peak in the energy loss function just before the end of the charged particle’s range
is called the Bragg peak. The fact that charged particles deliver a significant fraction
of their kinetic energy at the end of their range makes charged particles
useful for radiation therapy.
All of these expressions for the stopping power only apply to pure chemical
elements. The stopping power of a compound or any complicated mixture will
depend on the overall density and the relative numbers of electrons from each
chemical element. Recognizing that the ionizing radiation will usually move
through macroscopic distances, we can use an averaging procedure called
Bragg’s rule. The average mass stopping power is
1
r
dE
dx
 
total
¼
w1
r1
dE
dx
 
1
þ
w2
r2
dE
dx
 
2
þ
w3
r3
dE
dx
 
3
þ Á Á Á (17:15)
where w1, r1, and so forth refer to the fraction by mass of element 1 in the entire
mixture and its elemental density. The sum ranges over all the elements in the
mixture. Thus, if the mixture was a pure compound, then we would combine the
numbers of each element in the molecular formula. If the mixture had several
components, then we would combine the masses of each element from all the
components and so on to get an overall mass stopping power.
Figure 17.4 Density of ionization along the path of an a particle stopping in air is shown.
The Bragg peak in the ionization density is evident. [From M. G. Hollaway and M. S.
Livingston, Phys. Rev. 54, 18 (1938).]
506 INTERACTION OF RADIATION WITH MATTER
Example Problem What is the rate of energy loss of an 8-MeV a particle in
air? Assume air is 21% oxygen and 79% nitrogen:
Solution
1
r
dE
dx
 
total
¼
X
i
wi
1
ri
dE
dx
 
i
where r is the total density, and ri the density of the ith element. For oxygen:
1
r
ÀdE
dx
 
oxygen
¼ 0:3070
MeV † cm2
g
Zq2
Ab2
ln
wmax
I
 
À b2
h i
neglecting any correction terms
I=Z ¼ (12 þ 7ZÀ1
) eV ¼ 12:875
I ¼ 103 eV
b ¼ 1 À
m0c2
m0c2 þ E=A
!1=2
¼ 0:04629
g ¼ ½1=(1 À b2
)Š1=2
¼ 1:0011
wmax ¼ 2 Â 0:511(gb)2
¼ 0:002194 MeV
1
r
ÀdE
dx
 
oxygen
¼ 0:3070
22
16(0:04629)2
ln
0:002194
103 Â 10À6
 
À (0:04629)2
!
À1
r
dE
dx
 
oxygen
¼ 109:4 MeV=g=cm2
For nitrogen
I=Z ¼ (12 þ 7ZÀ1
) eV ¼ 13
I ¼ 91 eV
À1
r
dE
dx
 
nitrogen
¼ 0:3070
22
14(0:04629)2
ln
0:002194
91 Â 10À6
À (0:04629)2
!
¼ 130:2 MeV=g=cm2
For air
À1
r
dE
dx
 
air
¼ 0:21
À1
r
dE
dx
 
oxygen
þ 0:79
À1
r
dE
dx
 
nitrogen
¼ 0:21(109:4) þ 0:79(130:2) ¼ 125:8 MeV=g=cm2
17.2 HEAVY CHARGED PARTICLES (A ! 1) 507
One of the implications of the stopping power formulas for heavy charged
particles is that all particles of a given type will follow the same energy loss
pattern in a given material. More specifically, the example shown in Figures 17.3
started with 8 GeV 40
Ar ions in Be. However, these curves display the expected
result for all 40
Ar ions with kinetic energies less than 8 GeV. The energy loss
rate for an ion with 4 GeV or even 4 MeV can be read off the graph by finding
the position at which the residual energy of the ion is equal to the required
energy. This may seem a trivial point, but it has the more subtle meaning that
all the ions will follow exactly the same energy loss pattern, within the limits
of the statistical process, if we ignore nuclear Coulomb scattering. Formally, we
can write that the amount of kinetic energy lost, DE, in a finite thickness, Dx, of
material is
DE(Dx) ¼
dE
dx
 
Dx (17:16)
and the statistical variation in the energy lost d DE would be evidenced as a width in
the measured value that is called the amount of energy straggling. In a colloquial
expression, the ions are said to straggle through the material, and the width of the
energy distribution is due to this straggling. Thus, as ions pass through matter, the
spread in their energies increases (Fig. 17.5). This can be represented mathematically
as a Gaussian distribution:
N(E) dE
N
¼
1
ap1=2
exp À
(E À E)2
a2
!
(17:17)
where the straggling parameter a, which is the half-width at (1/e)th height, is given
by the expression
a2
¼ 4pq2
e4
Nex0 1 þ
kI
mev2
ln
2mev2
I
 !
(17:18)
where k is a constant (about 4
3) and the thickness of the absorber the ion has
penetrated is rx0.
The range or distance that a heavy charged particle will travel in a material can be
obtained by integrating the energy loss rate along the path of the ion. In the approximation
that the ion follows a straight-line trajectory, then the range for a given
kinetic energy, R(T), would be given by the integral:
R(T) ¼
ðT
0
À
dE
dx
 À1
dE (17:19)
where the function dE/dx is the appropriate function for the ion in the material.
There are two difficulties in applying this simple integral, the ions will suffer a
different number of collisions with atomic electrons, and, more importantly, the
508 INTERACTION OF RADIATION WITH MATTER
ions will undergo some scattering from the Coulomb fields of the atomic nuclei. The
multiple Coulomb scattering leads to an effect that the ion’s trajectory is not straight
but rather is made up from a series of straight line segments. Thus, the apparent
range or the projection of the range onto the initial velocity vector of the ion will
not be a single value but rather will consist of a statistical distribution of values.
Thus, the distribution of ranges is due to range straggling. It is important to note
that size of range straggling will grow as an ion penetrates into material because
it will literally add-up, in contrast to the energy straggling mentioned above. The
range of an ion and its fluctuations are integral quantities, whereas the energy loss
rate and its fluctuations are differential quantities. It is still true that Coulomb and
nuclear scattering are relatively rare, so that the range straggling for typical ion energies
in metals is on the order of a few percent of the range. The qualitative features of
the range distribution and the attenuation curve for a typical heavy charged particle
are shown in Figure 17.6. Heavy charged particles penetrate uniformly into matter
with essentially no attenuation in intensity until they are nearly at rest; at this point
the intensity of moving ions rapidly drops to zero.
Figure 17.5 Energy spectrum of 3-MeV protons: (a) before and (b) after passing through a
3.3-mg/cm2
gold foil. [From L. P. Nielson, Dan. Mat. Fys. Medd. 33 (6), 1961.]
17.2 HEAVY CHARGED PARTICLES (A ! 1) 509
The calculated range–energy curves for some low-mass charged particles in
silicon are shown in Figure 17.7. We can see from the integral form of the range
as a function of initial kinetic energy, given above, that R / aEb
. The exponent
should be of order 2 at low energies where the energy loss rate is dominated by
the 1/b2
or 1/E term. The range–energy relationships are very useful in determining
the kinetic energies of particles by measuring the attenuation curves. More
recently, range–energy relationships are used to identify charged particles that
are detected in silicon semiconductor telescopes as they emerge from nuclear reactions.
The scaling rules that apply to the stopping power for different ions in a given
medium can be extended to the range–energy relationship. For example, given the
range of ion 1 at an initial kinetic energy T1, the range of ion 2 with a different mass,
charge, and kinetic energy is
R2(T2) ¼
M2
M1
q2
1
q2
2
R1 T2
M1
M2
 
(17:20)
Notice that we have to scale the range of the known, first particle at the appropriate
energy of the unknown, second particle.
As a final, more practical, point about the stopping power and ranges of charged
particles, we should consider the best method to calculate the amount of energy
deposited in a thin foil. Clearly, the ion will slow down as it passes through the
material so that the energy loss rate will change as the particle passes through the
foil. Thus, we should use the average energy loss rate but notice that the function
is not linear, so that we will need a technique to determine the average. Two
cases can be identified: thin foils in which the initial, average, and final energy
Figure 17.6 Intensity distribution or attenuation curve is shown as a function of absorber
thickness for a typical energetic heavy ion penetrating into a metal. The effect of range
straggling is indicated by the Gaussian distribution of ranges. (From Leo, 1987.)
510 INTERACTION OF RADIATION WITH MATTER
loss rates are nearly the same and thick foils in which the particle undergoes a substantial
energy loss. In the former example of a thin foil, we can use the expression
written above that:
DE(Dx) ¼
dE
dx
 
Dx
and we should verify that the final rate is approximately equal to the initial rate:
dE
dx
 
initial
¼
dE
dx
 
final
If the energy loss rates are not substantially different, then we can use the initial
rate to obtain the average in a successive approximation procedure. For the case
Figure 17.7 Range–energy curves for some charged particles in silicon. Note the data has
the form R ¼ aEb
with a similar exponent for all ions. (From Knoll, 2000.)
17.2 HEAVY CHARGED PARTICLES (A ! 1) 511
of substantial slowing, we can recall that the range relations come from the integration
of dE/dx and thus provide the average energy loss rate that we should
use. The technique relies on determining the ranges of ions in graphs or tables of
ranges as follows: Imagine that an incident particle with an energy E0 passes
through some material with thickness t. These are the “known quantities.” The particle
will emerge from the foil with an energy E1, which we would like to determine.
We can find the total range of the ion in the material from tables, R0. The particles
that emerge from the foil will have a residual range equal to R0 2 t. We can then use
the range table or graph to determine E1 that corresponds to the range R1 ¼ R0 2 t.
The slowing down and averaging of the energy loss rate will be contained in the
range function and do not have to be explicitly evaluated.
Example Problem Imagine that a beam of 40
Ar ions at 400 MeV (10 MeV/A)
is incident on a 18.5-mg/cm2
beryllium foil (0.1 mm thick). Do the ions pass
through the foil, and, if they do, what is their residual kinetic energy?
Solution Using a standard reference for stopping power, the tables of
Northcliffe and Schilling (1970), we find for these ions that dE/dx ¼
9.597 MeV-cm2
/mg in beryllium. Thus, for our first estimate of the energy lost:
DE  (dE=dx)initial Dx ¼ 9:567 Â 18:5 ¼ 177:5 MeV
giving a residual energy of 400 2 177.5 ¼ 222 MeV, which is almost half the
initial kinetic energy. The ions will pass through the foil, but this estimate of
energy loss is probably too low. Recall that the ions lose more energy per distance
traveled as they slow down. Checking, we see that the energy loss rate for these
ions at 178 MeV is substantially larger, that is, dE/dx ¼ 15.3-MeV-cm2
/mg.
Thus, this is not a “thin” foil for these ions.
We can use the range technique with information in the same table, for 40
Ar
ions with E0 ¼ 400 MeV:
R(400 MeV) ¼ 28:278 mg=cm2
R(E1) ¼ 28:278 À 18:5 ¼ 9:8 mg=cm2
This range lies between the tabulated values, and, by linear interpolation between
the range values for E ¼ 160 and 200 MeV, one finds that E1  185 MeV. Thus,
just using the initial energy loss rate gives a substantial error.
For the practicing nuclear chemist, range–energy tables or relationships are
among the most commonly used tools. The largest collection of data on stopping
powers and ranges of ions in matter is that of Ziegler and Biersack1
in the
form of the computer programs SRIM/TRIM. Subsets of these tables exist for
1
Check out http://www.srim.org. This website and the references cited therein represent the largest and
most widely used compilation about the stopping of energetic ions in matter. The computer programs
SRIM and TRIM found these are used widely to estimate stopping powers, ranges, and straggling.
512 INTERACTION OF RADIATION WITH MATTER
low-energy heavy ions interacting with matter (Northcliffe and Schilling, 1970), a
particles interacting with matter (Williamson et al., 1966) and for energetic heavy
ions (Hubert et al., 1990).
For the most commonly encountered heavy charged particles, the a particles
from radioactive decay, some semiempirical range–energy rules are used. For the
range of a particles in air, Rair, we have
Rair (cm) ¼ ½0:005Ea (MeV) þ 0:285ŠE3=2
a (MeV) (17:21)
or
Rair (mg=cm2
) ¼ 0:40E3=2
a (MeV) (17:22)
so that the range of a 7-MeV a particle in air is about 5.9 cm. For a pure element
with 10 , Z , 15, we have
RZ
Rair
¼ 0:90 þ 0:0275Z þ (0:06 À 0:0086Z) log10
Ea
4
 
(17:23)
where RZ is the range in a pure element of atomic number Z expressed in mg/cm2
,
Rair is the range in air in mg/cm2
, and Ea is the a-particle energy in MeV. [For
Z , 10, substitute 1.00 for the term (0.09 þ 0.0275Z). For Z . 15, replace the
term RZ by (RZ-0.005Z).] For compounds or mixtures, the range in the compound
or mixture, RC in mg/cm2
, is given as
1
RC
¼
X
i
pi
Ri
(17:24)
where pi is the weight fraction of the ith element in the mixture or compound and Ri
is the range of an a particle of this energy in the ith element.
Example Problem What is the range of an 8-MeV a particle in air?
Solution
Rair (cm) ¼ (0:005Ea (MeV) þ 0:285)E3=2
a (MeV)
¼ (0:005 Á 8 þ 0:285)83=2
¼ 7:4 cm
What is the range of this same a particle in Al?
RZ
Rair
¼ 0:90 þ 0:0275 Z þ (0:06 À 0:0086Z) log10
Ea
4
Rair (mg=cm2
) ¼ 0:40E3=2
a ¼ 0:0510 mg=cm2
RAl ¼ 0:0510 0:90 þ 0:0275 Á 13 þ (0:06 À 0:0086 Á 13) log10
8
4
!
¼ 0:065 mg=cm2
17.2 HEAVY CHARGED PARTICLES (A ! 1) 513
Application of these formulas show that the ranges of decay a particles in
solids are very short. A sheet of paper will stop the a particles from most radioactive
sources. a-Emitting nuclei are not external radiation hazards, but, because
of their high LET, they do represent significant inhalation or ingestion hazards.
17.3 ELECTRONS
The passage of energetic electrons through matter is similar to that of heavy charged
particles in that the Coulomb interaction plays a dominant role. However, three clear
differences can be easily seen: The incident electrons are generally relativistic
particles (notice that 1 MeV of kinetic energy corresponds to nearly twice the rest
mass of an electron, 0.511 MeV); the scattering is predominantly between identical
particles and repulsive; and the interactions with nuclei are attractive, and the
direction of the electron can be dramatically changed, even reversed, in a collision
with a heavy nucleus. A fourth difference that is not so obvious is that a fraction
of the kinetic energy is lost through the radiative process of bremsstrahlung.
Bremsstrahlung (the German word can be literally translated as “braking radiation”)
is a general process in which electromagnetic radiation is emitted whenever a
charged particle undergoes a substantial acceleration. The scattering of electrons,
particularly to large angles, corresponds to a classical acceleration that creates/
requires the emission of bremsstrahlung. By comparison, very few heavy charged
particles undergo large accelerations as they slow down in material.
Summarizing this overview of the possible interaction mechanisms for fast
electrons in material, we find that the rate of energy loss in a material is
À
dE
dx
 
electron
¼ Selectronic þ Sradiative (17:25)
where the electronic stopping power is similar to the electronic interaction between
charged particles and electrons discussed above, whereas the radiative stopping
power is specific to electrons. The electronic stopping power for electrons is written as:
Selectronic ¼ À
dE
dx
 
electronic
¼
2pZe4
rN
mev2
ln
mev2
E
2I2(1 À b2
)
 
À ln 2 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À b2
q
À 1 þ b2
 
þ (1 À b2
) þ
1
8
1 À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À b2
q 2!
(17:26)
using the same definitions as for the stopping power of heavy charged particles. On the
other hand, the expression for the radiative stopping power is
Sradiative ¼ À
dE
dx
 
radiative
¼
(Z þ 1)Ze4
rNE
137 m2
ec4
4 ln
2E
mec2
 
À
4
3
!
(17:27)
514 INTERACTION OF RADIATION WITH MATTER
and given that b % 1 contains an extra factor of ZÃ
E in the term in front of the parenthesis.
With some algebra and for a typical electron energy, one can show that the ratio of
the two contributions to the stopping power depends on the atomic number of the
material, Z, and the electron kinetic energy E:
Sradiative
Selectronic
%
ZE
800 MeV
(17:28)
which indicates that the radiative contribution is only significant for large atomic
numbers (Z  80–90) and high electron energies (E  10–100 MeV). Typical b particles
from radioactive sources are emitted with only 1–10 MeV of kinetic energy,
often much less, and the radiative contribution to the stopping power is very small.
The bremsstrahlung spectrum is smooth and continuous, ranging from zero energy
up to the electron energy due to the random distribution of electron scattering angles.
The important feature that the electrons are occasionally scattered to large angles
during the penetration of material causes each electron to follow a tortuous path. A
beam of electrons will not have a fixed range in the sense of a beam of heavy charged
particles. In fact, given the identical nature of the particles involved in the scattering
process, obtaining the range distribution is problematic. Moreover, the primary or a
secondary electron can be scattered backwards and emitted from the material. A
schematic plot of the range distribution is shown in Figure 17.8 for a monoenergetic
source of electrons. The falloff of the intensity with penetration depth starts
immediately as the electron enters the material and then gradually approaches zero.
This figure emphasizes the fact that the concept of range cannot be applied in
a simple way to energetic electrons. As an approximation, the electron range is
taken as the extrapolation of the linear portion of the attenuation curve to zero.
As shown in Figure 17.9 the product of this definition of the range times the
density of a material is a smooth function of incident electron energy for a wide
range of materials. Such behavior comes from the fact that the range has a strong
dependence on the electron density through the electronic stopping and a weak
dependence on the atomic number of the material through the ionization potential
and radiative stopping.
Many related measurements have been made of the range distribution of electrons
emitted in b2
decay. These studies were particularly important before
Figure 17.8 Schematic attenuation curve for an energetic electron in a solid material. (From
Knoll, 2000.)
17.3 ELECTRONS 515
solid-state detectors were available. The measurements have shown that the combination
of the Fermi energy distribution of electrons from the decay with the sloping
range distribution leads to an approximately exponential attenuation of the b-decay
electrons. This can be expressed as
Nt ¼ N0eÀmt
(17:29)
where Nt is the number of b particles transmitted through a thickness t. The absorption
coefficient m can be related to the endpoint energy Emax of the b spectrum as:
m(m2
=kg) ¼ 1:7EÀ1:14
max (17:30)
where Emax is given in MeV.
Example Problem What fraction of the b particles emitted by 32
P
(Emax
b ¼ 1.71 MeV) will be stopped by a sheet of Al of thickness 1 mm?
Figure 17.9 Product of the range of an energetic electron in a solid material with the density
as a function of the incident kinetic energy is shown. (From Knoll, 2000.)
516 INTERACTION OF RADIATION WITH MATTER
Solution The mass absorption coefficient is 1.7(1.71)21.14
¼ 0.922 m2
/kg. The
fraction of these b particles that are transmitted is Nt/N0 ¼ e2mt
¼
exp[20.922 m2
/kgÂ(1 Â 1023
m) Â (2.7 Â 103
kg/m3
)] ¼ exp[22.489] ¼ 0.083.
The fraction absorbed ¼ 1.0 2 0.083 ¼ 0.917.
In the distant past, measurements of the attenuation of the b spectrum from a
newly discovered isotope were used to identify the energy of the b decay. Recently,
the attenuation of strong sources has been used to monitor the thickness of materials
during manufacturing processes. Notice that the monitoring can be continuous,
nondestructive, and a physical probe does not need to “touch” the material being
measured.
The backscattering of energetic electrons from materials is a feature that is extremely
rare for heavy charged particles. Backscattering is primarily due to multiple
interactions with (heavy) nuclei that significantly alter the direction of the incident
electron that is enhanced by the fact that two energetic electrons can be created when
the incident electron scatters from an atomic electron. The coefficient of backscattering
is used to quantify the probability that an electron will emerge “backwards”
from a surface that is irradiated with electrons. The coefficient is a
function of the energy of the incident electron and the atomic number of the absorber.
Formally, the coefficient is the fraction of time that an electron is emitted from
the surface of a material following the entry of an energetic electron. The coefficient
is approximately 0.5 for E , 1 MeV electrons in gold, approximately 0.3, 0.04,
and 0.1 for copper, aluminum, and carbon, respectively. It falls below 0.1 for
E ¼ 10 MeV electrons in gold, below 0.05 for copper, and to near zero in aluminum
and carbon.
The bremsstrahlung radiation from electron beams has important practical applications
even though it is a small contribution to the stopping power. Bremsstrahlung
forms the basis for operation of X-ray tubes and other “controllable” high-fluence
sources of radiation. Such devices collide an electron beam with an energy of the
order of 10–50 keV with a large electrode, usually made out of a heavy element
like tungsten or tantalum. The electrons penetrate the electrode and the bulk of
their kinetic energy is lost through electron scattering and eventually creates heat.
However, a small fraction of the incident energy is converted into electromagnetic
energy in the X-ray region. This is called “thick-target” bremsstrahlung because the
incident electrons are completely stopped inside the material. If we assume that the
bremsstrahlung is independent of electron energy, then the fraction of the electron
energy would be frad ¼ 0.0014ZE from the expression above. The observed fraction
is about a factor of 2 lower, which would be consistent with simply taking the average
energy of the electron in the material to be one-half the initial energy. We should note
that some of the energetic electrons can create inner shell atomic vacancies in the
atoms that make up the lattice. These vacancies will be filled by discrete K and L
X-ray transitions. These sharp lines will add to the continuous bremsstrahlung
spectrum and will depend, of course, on the atomic number of the material.
Another mechanism for electron energy loss in matter is the emission of
Cerenkov radiation. When a beam of fast moving charged particles with a velocity
17.3 ELECTRONS 517
v near the speed of light c enters another medium with index of refraction n, the particle
velocity will exceed the speed of light in the new medium (which is c/n). The
electron radiates the “excess” energy in the form of a blue-white light call Cerenkov
radiation. This light is localized in a cone of half-angle u around the direction of
motion of the electron such that cos u ¼ c/nv.
17.4 ELECTROMAGNETIC RADIATION
As photons move through material, they only interact or “scatter” in localized or discrete
interactions, and they do not interact at long distances, that is, they are not
subject to the Coulomb or nuclear forces. This behavior is in clear contrast to the
long-range interactions felt by charged particles. Thus, as a beam of photons propagates
through any material, the intensity of the beam will decrease as the photons
that interact are removed, but the energy of all the noninteracting photons will
remain constant. The photons will interact in ways that predominantly release fast
moving electrons, low-energy photons will interact only once and give rise to a
single primary electron, energetic photons can interact several times and give
rise to a few primary electrons. The most energetic photons can create a matter–
antimatter pair of electrons that induce a cascade of secondary electrons.
The energy of the noninteracting photons remains constant so that the probability
that a photon will interact in a fixed thickness of material will also remain constant
regardless of the photon energy. This leads immediately to an exponential attenuation
of electromagnetic radiation that is called the Beer–Lambert law. The law
was applied to the absorption of visible light but applies to all electromagnetic
radiation. The derivation of the exponential attenuation law is similar to the derivation
of the exponential decay law of radioactive nuclei and will not be repeated
here. (The analogy is that the probability of radioactive decay is constant in a
given time interval.)
The general expression for the attenuation of photons is
I ¼ I0eÀmx
(17:31)
where I and I0 are transmitted and incident intensities, respectively, x is the thickness,
and m is an energy-dependent total linear absorption coefficient that
depends on the nature of the material. The mean free path of the photons in the
material is simply l ¼ 1/m, and we can define a half-thickness as x1/2 ¼ ln(2)/m
in analogy to the radioactive half-life. The exponential nature of the attenuation
means that the intensity of the transmitted radiation does not go to zero, although
it can be made arbitrarily small. The mass attenuation coefficient is obtained by
dividing the linear attenuation coefficient by the density of the material, m/r. The
mass attenuation coefficient is independent of the physical state of the absorber
and represents the fact that the fundamental interactions can be expressed in
terms of cross sections per atom. Extensive tabulations and figures, such as
Figure 17.10, are available for the mass attenuation coefficients of photons with
energies in the range of 0.01 to 10 MeV.
518 INTERACTION OF RADIATION WITH MATTER
Example Problem Estimate the fraction of 1.0 MeV photons that will be transmitted
through a lead absorber that is 5 cm thick (the thickness of “lead bricks”
commonly used in radiation shields).
Solution The transmitted fraction is simply
f ¼
I
I0
¼ eÀmx
where x is 5 cm and m can be obtained from the mass attenuation coefficient in
Figure 17.10. Reading the value from Figure 17.8, m0/r ¼ 0.07 cm2
/g for the
Figure 17.10 Mass attenuation coefficient for energetic photons in lead. (From Evans,
1955.)
17.4 ELECTROMAGNETIC RADIATION 519
total attenuation. m0 is equal to m0/rÃ
r, of course. Thus:
f ¼
I
I0
¼ eÀ(m0=r)rx
¼ eÀ(0:07 cm2
=g)Ã11:35 g=cm3
Ã5 cm
¼ eÀ0:795Ã
5
¼ 0:019
Approximately 2% will be transmitted. Notice that the half-thickness for these
photons in lead, x1/2 ¼ ln 2/m is 0.87 cm.
Concentrating on photon energies that are associated with the differences between
nuclear energy levels, those in the region from 10 keV to 10 MeV, we find that only
three types of interactions play a role in attenuating a photon beam. These mechanisms
are shown schematically in Figure 17.11. Each photon that interacts with an
atom via any one of the mechanisms will be lost from beam. The type of interaction
is random, but their relative probabilities depend on the photon energy. Photoelectric
absorption dominates at low energies, Eg , 0.1 MeV in the heaviest elements, while
Compton scattering is most important at intermediate energies, whereas pair
production has an absolute threshold at 1.022 MeV and only is important for the
highest energy photons. We will consider each of these processes in turn.
17.4.1 Photoelectric Effect
The photoelectric effect was originally described by Albert Einstein and helped to
establish the quantized nature of light. The photoelectric effect has many extremely
Figure 17.11 Schematic diagrams of (top to bottom) photoelectric effect, Compton effect,
and pair production.
520 INTERACTION OF RADIATION WITH MATTER
important applications, for example, the detection of visible light by photocells and
the photovoltaic conversion of sunlight. The photoelectric effect converts a single
photon into a single free electron. When the photon interacts with a bound electron,
the photon can be completely absorbed and the electron emerges with a kinetic
energy (KE) that corresponds to the photon energy, hn, minus the electron
binding energy (BE), KEe2 ¼ hn 2 BE (see Fig. 17.11). Photocells use a semiconductor
like silicon for the absorbing material, and the electrons released by
visible light have relatively small kinetic energies and are collected as a photocurrent.
In the present application of the absorption of a nuclear photon, a fast
electron is usually created in the bulk of a solid medium because the binding
energy is often small compared to the photon energy. The fast electron goes on to
lose its kinetic energy by scattering through the material (as discussed above). Conservation
of momentum requires that the electron be bound in an atom (that could be
in a lattice) that recoils.
The cross section or probability of the photoelectric effect is on the order of the
square of the atomic size for photons in the keV region and decreases rapidly with
increasing photon energy. The cross section also has a strong dependence on the
atomic number of the absorbing material, as there is a sharp increase in the cross
section at each threshold for the emission of bound electrons. As an example, the
heavy-element lead (Z ¼ 82) has K, L, and M (principal quantum numbers,
n ¼ 1, 2, and 3) binding energies of approximately 88, 15, and 3 keV, respectively,
which provide strong photoelectric absorption for photons in this energy region. The
sharp increase in the photoelectric cross section can be seen as the sharp peaks on
the left in Figure 17.10. The overall probability for photoelectric absorption
follows the very rough expression:
sphotoelectric / Zn
=E7=2
g (17:32)
where the exponent n is between 4 and 5. This expression only includes the dramatic
effects of the electron binding energies in an overall way and is not meant to replace
the measured values.
We should note that the photoelectric effect often leaves an inner shell vacancy in
the atom that previously contained the “ejected” electron. This vacancy will be filled
by an atomic transition, called fluorescence, and generally produces an X-ray
photon. In an interesting twist of fate, the X-ray photon will have an energy that
is just below the sharp rise in the attenuation coefficient due to conservation of
momentum and can often escape from the absorber. Recall that the direction of
the fluorescence photon will be uncorrelated with the direction of the incident
photon and a fraction will be emitted “backwards” from the absorber. The absorber
will thus emit its own characteristic X-rays when it is irradiated with high-energy
photons.
In g-ray spectroscopy lead shields are commonly used. This can result in the production
of Pb X-rays that can interfere with the measurement of low-energy photons.
Lining the Pb shields with layers of Al and Cu that absorb the Pb X-rays and other
subsequent radiation ameliorates these problems.
17.4 ELECTROMAGNETIC RADIATION 521
17.4.2 Compton Scattering
If the energy of the incident photon exceeds the typical binding energies of the innermost
atomic electrons, the probability of photoelectric absorption drops below the
probability that the photon will scatter from an electron leading to a scattered
electron and a lower energy photon. This process is called Compton scattering.
A schematic diagram of this process is shown in Figure 17.12.
From the conservation of momentum in the x direction we have
p ¼ p0
cos u þ pe cos u (17:33)
From conservation of momentum in the y direction, we have
0 ¼ Àp0
sin u þ pe sin u (17:34)
Assuming the collision is elastic, conservation of energy gives us
E ¼ E0
þ Te (17:35)
Combining these equations (see Problems) and utilizing the relativistically correct
expressions for the energy and momentum of the electron
p2
e ¼
1
c2
½Te(Te þ 2mec2
)Š (17:36)
we get
l0
À l ¼
h
mec
(1 À cos u) (17:37)
where l0
and l are the wavelengths of the scattered and incident g rays. The quantity
h=mec is called the Compton wavelength of the electron and is equal to 2426 fm.
Figure 17.12 Schematic diagram of Compton scattering.
522 INTERACTION OF RADIATION WITH MATTER
Note that the shift in g-ray energy is independent of the incident energy. The
expressions for the energies of the scattered photon and electron are given in
Table 17.2.
It is clear that the minimum energy of the scattered g ray occurs when u ¼ 1808
(cos u ¼ 21). In this case, we have
Emin
g1 ¼
mec2
2
1
1 þ
mec2
2Eg
0
B
@
1
C
A
% 255 keV (17:38)
In this case, the electron energy, Te will be maximum and Te ¼ Eg 2 0.255 MeV. If
we consider all scattering angles u, then the distribution of scattered electron kinetic
energies is as shown in Figure 17.13. The sharp peak at Eg 2 0.255 MeV is called
the Compton edge. The minimum energy photon, E % 225 keV, will be a noticeable
TABLE 17.2 Compton Scattering at Fixed Angles
ug (degrees) Emitted Photon Energy (hn0
) Electron Kinetic Energy
0 hn 0
90 hn
mc2
mc2 þ hn
 
 hn À mc2
180 hn
mc2
mc2 þ 2hn
 
hn À mc2
=2
hv
hv
Ec
E
Compton continuum
“Compton
edge”
Before After
hv¢
q
q = 0
dN
dE
q = p
O
e-
eFigure
17.13 Top: schematic version of Compton scattering. Bottom: schematic variation
of the distribution of electron kinetic energies produced in Compton scattering. (From
Knoll, 2000.)
17.4 ELECTROMAGNETIC RADIATION 523
component in g-ray spectra resulting from the interaction of a photon from a radioactive
source with the lead shield surrounding the detector, resulting in a backscattered
photon (E  225 keV), which strikes the radiation detector (the “backscatter
peak”).
The Compton scattering cross section per electron of the stopping material is
independent of Z, and thus the cross section per atom goes as Z. For energies
about 0.5 MeV, it varies roughly as 1/Eg.
17.4.3 Pair Production
Whenever the energy of the initial photon exceeds the rest mass of two electrons,
1.022 MeV, the process of pair production is possible. During the process of pair
production, the initial photon interacts with the Coulomb field of a nucleus and is
converted into an electron and a positron, a matter–antimatter pair, that shares
the initial energy of the photon (see Fig. 17.11). Conservation of energy and momentum
in the Coulomb field cause the pair of electrons to move forward along the
initial direction of the photon with a small opening angle. The pair of particles
will interact with the electrons and nuclei in the remaining material as described
above.
The process of bremsstrahlung observed in electron stopping is closely related to
the process of pair production. From a schematic standpoint, in the first case, a
moving electron interacts with the Coulomb field of an atom, making a transition
between two energy states and a (bremsstrahlung) photon is emitted. In the
second case, a photon is destroyed by interaction with an atomic Coulomb field
and a pair of electrons is created. The probability of pair production has an absolute
threshold of 1.022 MeV, that is, this process cannot take place if the photon has a
lower energy. The cross section increases relatively rapidly and saturates above
10 MeV, as indicated in Figure 17.10. The variation of the pair-production
cross section with photon energy is complicated, but the cross section depends on
the square of the atomic number of the absorber. For large photon energies,
spair / Z2
ln (Eg=mc2
). Pair production is the predominant attenuation process for
high-energy photons.
Pair production has a threshold energy of 1.022 MeV because two particles are
created, one electron and one positron. Thus, some energy is “stored in” or “used
to create” the mass of the pair. Notice the total electric charge is conserved
because the electron charge is 21e and the positron charge is þ1e. One of the
unique features of this process is that the energy that went into the creation of the
two particles will be “released” when the positron comes to rest and annihilates
with an electron. The annihilation process is
eþ
þ eÀ
À! g þ g
in which the two g rays have exactly the same energy, mc2
¼ 0.511 MeV, and are
emitted at 1808, or back to back. The axis along which the two g rays are emitted
524 INTERACTION OF RADIATION WITH MATTER
will be random with respect to the initial direction of the incident photon because the
positron will undergo a slowing down process involving multiple scattering with
atoms and atomic electrons. In the final phase of the process, the positron captures
a single electron and forms a neutral species called positronium. Therefore, the
characteristic annihilation radiation (photons with an energy of 0.511 MeV) can
escape from the absorber whenever pair production can occur.
In summary, photons pass through material until they interact individually with
the atoms or nuclei in the material. Depending on the energy of the photons the interaction
will be predominately pair production (high energy), Compton scattering, or
photoelectric absorption (low energy). The relative importance of these processes is
summarized in Figure 17.14 as a function of atomic number and photon energy. In
each interaction, the photon ionizes the material, creating one or two fast moving
electrons and leaving a positive ion. Pair production gives two fast moving electrons—one
positive, one negative, Compton scattering gives one fast moving
electron and a lower energy photon, and the photoelectric effect gives one fast
moving electron. The fast electrons have a much higher rate of ionization than
the photons, and the general features of the interactions of these electrons with
the material have been described above. The “path” of the scattered photon will
be erratic when Compton scattering is the predominant process. g Rays from
nuclear decay processes tend to have energies on the order of 1 MeV. From
Figure 17.14 we expect these photons to interact via Compton scattering in all
materials. The first interaction will give an electron and a lower energy photon.
The interaction probability of the secondary photon will usually be higher than
that of the primary photon and will often result in a photoelectric absorption.
Recall also that, depending on the energy of the primary photon, the absorber will
weakly emit lower energy photons such as annihilation radiation, backscatter
radiation, or fluorescence X-rays.
Figure 17.14 Summary of the relative importance of the three mechanisms by which
photons interact with matter. The curves indicate the locations in the atomic number–
photon energy plane at which the cross section for Compton scattering is equal to that for
photoelectric absorption, left side, or is equal to that for pair production, right side.
17.4 ELECTROMAGNETIC RADIATION 525
17.5 NEUTRONS
We will now briefly consider the propagation of neutrons through material. Neutrons
are the most penetrating radiation for the simple reason that their only significant
interaction is with nuclei via the strong force. (Neutrons only have a very small
interaction with electrons through their magnetic dipole moment that can be
ignored.) As we discussed at the beginning of this chapter, nuclei are very much
smaller than atoms, and so the probability that a fast neutron will interact with
(strike) a nucleus is very, very small. On the other hand, neutrons cause significant
radiation damage because all of their interactions cause nuclear recoil and many lead
to nuclear transmutations.
A neutron will move through material along a straight line with a constant energy
until it encounters a nucleus and induces a nuclear reaction. Thus, neutron attenuation
follows an exponential law similar to that for photons. Written in terms of
an energy-dependent attenuation length, mE we have
I ¼ I0eÀmEx
(17:39)
where x is a linear dimension and I0 is the incident intensity. The attenuation length
is the inverse of the mean free path, l, and is proportional to the total nuclear reaction
cross section:
mE ¼
1
lE
¼ N0sTotal(E) (17:40)
where N0 is a constant that gives the total number of nuclei per unit volume in the
material. The total nuclear reaction cross section is a characteristic of each isotope in
the absorbing material and has the dimensions of an area. If we have a monoisotopic
element such as gold or bismuth, then we will only have to account for the energy
dependence of the neutron. If the material contains several isotopes such as silver
(107
Ag and 109
Ag), nickel (five isotopes), or is a compound NaF (one isotope of
each element), and so forth, then the effective cross section will be the numberweighted
cross section:
saverage ¼ f1 sTotal(E)1 þ f2 sTotal(E)2 þ f3 sTotal(E)3 þ Á Á Á (17:41)
where the constants, fi, are the fraction by number of each isotope in the sample.
Example Problem Calculate the average thermal neutron capture cross section
and the mean free path for LiF, a solid crystalline material at room temperature
with a density of 2.635 g/cm3
and a molar mass of 25.94 g/mol. Lithium has two
stable isotopes 6
Li (7.5%) and 7
Li (92.5%) with thermal neutron capture cross
sections of sthermal ¼ 39 mb and 45 mb, respectively. Fluorine is monoisotopic,
19
F, with sthermal ¼ 9.6 mb.
Solution
saverage ¼ f1 sTotal(E)1 þ f2 sTotal(E)2 þ f3 sTotal(E)3
526 INTERACTION OF RADIATION WITH MATTER
Notice that half the atoms are fluorine and half the atoms are lithium, but the
lithium atoms are split unevenly between A ¼ 6 and A ¼ 7. The fractions of
each isotope must reflect this distribution.
saverage ¼ 0:075 Â 0:5 Â 39 mb þ 0:925 Â 0:5 Â 45 mb þ 1:0 Â 0:5 Â 9:6 mb
saverage ¼ 27:1 mb
Rearranging the equation above relating the mean free path and the total reaction
cross one has
lthermal ¼
1
N0sthermal
N0 ¼
NAr
molar mass
¼
6:022 Â 1023
=mol  2:634 g=cm3
25:94 g=mol
N0 ¼ 6:11 Â 1022
cmÀ3
And finally for the mean free path:
lthermal ¼
1
6:11  1022 cmÀ3  27:1 mb  1  10À27 cm2=mb
lthermal ¼ 604 cm
Thus, the average thermal neutron travels more than 6 m in solid LiF before
undergoing a nuclear capture reaction! Note the total reaction cross section
will be larger and the neutrons will most likely scatter before being captured.
Neutrons can interact with matter via a number of different reactions,
depending on their energy. The following are among the most important of these
reactions:
1. Elastic scattering, A(n, n)A, which is the principal interaction mechanism for
neutrons.
2. Inelastic scattering, A(n, n0
)AÃ
, where the product nucleus AÃ
is left in an excited
state. To undergo inelastic scattering, the incident neutron must have sufficient
energy to excite the product nucleus, generally about 1 MeV or more.
3. Radiative capture, A(n, g)A þ 1. As discussed earlier, this cross section shows
a 1/n energy dependence, and this process is important for low-energy
neutrons.
4. Fission, A(n, f), which is most likely at thermal energies but occurs at all energies
where the neutron binding energy exceeds the fission barrier height for
fissile nuclei.
17.5 NEUTRONS 527
5. Knockout reactions, such as (n, p), (n, a), (n, t), and so forth, which are
maximum for neutrons of eV–keV energy but occur at higher energies.
The total neutron interaction cross section, stotal(E), is the sum of the various
reaction cross sections:
stotal ¼ selastic þ sinelastic þ scapture þ Á Á Á (17:42)
One of the technologically most important interactions of neutrons with matter is
their loss of energy (“slowing down”) by a series of elastic collisions. Let us
consider the case where particle 1 of mass m1, speed v1, collides with particle 2,
mass m2, at rest. After the collision, the particles will have speeds v0
1 and v0
2 in the
lab system.
m1 v1
v0
c m2
m2,v2
′
m1, v1
′
θ2
θ1
Before After
In the center of mass (cm) system, we have
m1
V1 = v1 – v0 V2 = v0
m2
V2 = v0
V1 = v1 – v0
Before After
After the collision, the relationship between the cm and the lab systems is
V2 = v0
V1 = v1 – v0
v0
v2′
v1′
v0
θ2
θ1
θ
v0 ¼
M1
M1 þ M2
v1 ; velocity of cm (17:43)
528 INTERACTION OF RADIATION WITH MATTER
If we designate the kinetic energy of particle 1 after the collision at T0
1, we have
T0
1 ¼
1
2
m1(v0
1)2
(17:44)
T0
1 ¼
1
2
m1(V1 þ v0)2
¼
1
2
m1(V2
1 þ v2
0 þ 2V1 v0 cos u) (17:45)
Thus, T0
1 will have a maximum value for u ¼ 08 and a minimum value for u ¼ 1808.
We have
T0
1(max) ¼
1
2
m1(V2
1 þ v2
0 þ 2V1v0) ¼
1
2
m1v2
1 ¼ T1
T0
1(min) ¼
1
2
m1(V2
1 þ v2
0 À 2V1v0) ¼
1
2
m1(V1 À v0)2
¼
1
2
m1(v1 À 2v0)2
¼ T1
m1 À m2
m1 þ m2
 2
(17:46)
For the special case where particle 1 is a neutron and particle 2 is a proton,
m1 % m2:
T0
1(min) ¼ 0
T0
1(max) ¼ T1
It follows that
T0
1 ¼
T1(1 þ cos u)
2
u1 þ u2 ¼ 908
v1 ¼ v2 ¼ v0
(17:47)
If we assume the angular distribution of the scattered neutrons is isotropic
in the cm system, then the probability of a neutron scattering into and solid
angle dV, P(dV), is a constant given by
P(dV) ¼
dV
4p
(17:48)
where the solid angle dV is given in steradians. Substituting in for dV, assuming
spherical symmetry, we have
P(dV) ¼
2p sin u du
4p
¼
1
2
sin u du
(17:49)
17.5 NEUTRONS 529
When a neutron is scattered into an angular interval (u to u þ du), its energy
is changed from T1 to the interval (T0
1 to T0
1 þ dT0
1). Here
dT0
1 ¼ Àm1V1v0 sin u du (17:50)
Thus, we have for the probability of scattering into an energy interval dT0
1:
P(dT0
1) ¼ P(dV)
¼
1
2
sin u du
¼
dT0
1
2m1V1v0
(17:51)
We have an equal probability of scattering into each energy interval. For neutrons
scattering off hydrogen
T0
1(min) ¼ 0
T0
1(average) ¼
T1
2
After n collisions
T0
1(average) %
1
2
 n
T1 (17:52)
Thus, to reduce a 1-MeV neutron to thermal energies ( 1
40 eV) would require
about 25 collisions.
17.6 RADIATION EXPOSURE AND DOSIMETRY
Up to this point we have taken a very microscopic view of the propagation of beams
of particles through material. We have described the degradation of the intensity and
the energies of the beams in terms of individual interactions. Now we will take a
more macroscopic view from the standpoint of the absorber.
As we have seen, the passage of radiation through material causes ionization of the
atoms and molecules. The creation of free electrons, recoiling positive ions, and in
some cases transmuted nuclei can disrupt the chemical structure of the material. It
is important to note that the effect of the passage of a single particle through a macroscopic
object will usually cause a minimal overall effect due to the enormous number
of atoms present in the object. Special devices are necessary to observe the ionization
from individual particles as described in the next chapter on radiation detectors.
Physical changes in an everyday object will be observed when it has been exposed
to large amounts of radiation. At the same time we should realize that certain
materials will be relatively immune to the ionization caused by photons and fast
530 INTERACTION OF RADIATION WITH MATTER
electrons, whereas neutron irradiation of the same material generally will have a
substantial effect. For example, a metal lattice is characterized by delocalized electrons,
and the local ionization caused by a Compton scattering or photoelectric
absorption will be quickly neutralized (or repaired). Neutron absorption by a metal
will generally lead to b decay and the transmutation of one atom into the neighboring
element. On the other hand, local ionization created by photons and electrons in
insulating materials like glasses will persist for a long time. Similarly, lattice
defects caused by atomic recoil have to be removed by annealing, but atomic
recoil has little effect in liquids (and none in gases). Thus, we can see that the
effects of radiation on a material will depend in great detail on the type and
amount of radiation, and on the physical and chemical nature of the material being
irradiated.
We have seen that the neutral forms of radiation, photons and neutrons, are very
penetrating and can pass through layers of material without interacting. In these
cases, we need to distinguish between the amount of radiation exposure and
the amount of energy absorbed by the material. Photons such as X-rays,
bremsstrahlung, and g rays play an important role in nuclear medicine, but they
are not strongly absorbed by tissue. The exposure to these photons is not equal to
the dose. In the case of highly ionizing radiation such as charged particles, the
exposure will correspond to the absorbed energy except for very thin materials
that allow the particle to escape. In order to characterize radiation effects, we
need to know the amount of energy absorbed by the material, which is called the
absorbed dose.
The unit of radiation exposure is the roentgen (R). It is a historical unit of the
exposure and characterizes the radiation incident on an absorbing material
without regard to the character of the absorber. The unit was formalized in 1928
as “The amount of radiation which produces one electrostatic unit of ions (esu),
either positive or negative, per cubic centimeter of air at standard temperature and
pressure.” Translated in modern units:
1 roentgen ¼ 2:58 Â 10À4
coulomb=kg-air ¼ 0:3336 nC=cm3
at STP
This value corresponds to an absorbed energy of approximately 8.8 mJ/kg using the
effective ionization energy of 34 eV per ion pair in air. The roentgen is most often
used to describe the intensity of a photon source such as a medical X-ray machine
or other irradiator. The exposure should be measured at some distance from the
source so that the radiation field is uniform compared to the dimensions of the
detector. The detector is usually an ion chamber filled with dry air that is sensitive
to pico-coulombs of charge.
As studies of the effects of exposure to all types of radiation went on, it became
clear that these effects were correlated with the amount of absorbed energy, which is
generally less than the exposure. In 1962 the “rad” was formally defined as a special
unit of energy called the “radiation absorbed dose” with a value of 100 ergs per gram
of absorbing material. The rad is a convenient physical standard that correlates well
17.6 RADIATION EXPOSURE AND DOSIMETRY 531
with chemical and biological effects of radiation, whereas the roentgen defined in
terms of an air ionization measurement was left for exposure. More recently, the
gray (Gy) was introduced as the SI unit for the absorbed dose:
1 Gy ¼ 1 J=kg ¼ 100 rad ¼ 6:24 Â 1012
MeV=kg
so that one centigray is exactly one rad. One gray corresponds to a relatively large
amount of energy to be absorbed from a radiation source per unit mass. For example,
1 Gy in water is 18 mJ/mol. Modern dosimeters routinely measure doses at the few
millirad (mr) level or few tens of mGy.
The acronym “kerma” for “kinetic energy released in absorbing material” has
been used to conceptually connect the energy deposited by ionizing radiation with
the radiation field. It is defined to include the kinetic energy, which is locally
absorbed from products of interaction with the particular medium such as
Compton electrons, photoelectrons, and pair production while excluding the
energy, which is not locally absorbed, from Compton-scattered photons, characteristic
fluorescence radiation, and annihilation photons. The kerma is defined as:
K ¼ fEmx=r
where f is the particle fluence (m22
), E is the energy of the radiation, mx is a
linear energy attenuation coefficient (energy/m), and r is the density (kg/m3
).
The dimensions of the kerma are thus J/kg or gray. The concept of kerma in
air is very close to the practical definition of the roentgen as a unit of exposure,
times a factor for the amount of energy necessary to create an ion pair. However,
bremsstrahlung photons are lost in the secondary-particle equilibrium condition
assumed in the definition of the roentgen, but they represent a small effect
under most conditions.
Just as the effect of radiation on a specific material depends on the dose or amount
of absorbed energy in contrast to the exposure, the effect of radiation on biological
systems depends on the energy density and not just the energy. One can imagine that
a biological system could survive the formation of a single ion pair and the following
chemical transformations from a single photoelectric event. However, if a large
number of direct ionization events take place in a small volume due to the
passage of a heavy charged particle through some biological material, the resulting
chemical changes could be profound. The important parameter is called the “linear
energy transfer,” or LET, which is very close to the specific energy loss, 2dE/dx,
discussed earlier in this chapter. The value of LET for a given particle is smaller than
the value of 2dE/dx because the LET does not include the radiative energy loss
term, as the bremsstrahlung radiation is not absorbed “locally.” Recall that the
radiative energy loss term was only significant for high-energy electrons. Typical
values of LET for photons and fast electrons are a few MeV/mm but are one or
two orders of magnitude larger for heavy charged particles.
532 INTERACTION OF RADIATION WITH MATTER
The concept of dose equivalent has developed over time to quantify the more
damaging effects of high LET radiation. The original definition of the absorbed
dose in rads was multiplied by a quality factor, Q . 1. The quality factor increased
with increasing LET. The historical unit for dose equivalent is called a rem for
“roentgen equivalent man,” and measurements of dose equivalents to biological
systems, especially people, are most commonly reported in millirem (mrem).
With the more recent SI dose unit of gray, a new SI unit of dose equivalent
(or as it is now called, equivalent dose) was introduced called the sievert (Sv).
The sievert and rem are different by a factor of 100 in the same way as the rad
and gray:
1 Sv ¼ 100 rem or 10 mSv ¼ 1 mrem
The equivalent dose in Sv ¼ absorbed dose in grays  wR (radiation weighting
factor, formerly the quality factor). The absorbed dose for low LET radiation, b
and g rays, is taken as having a radiation weighting factor of unity, wR ¼ 1. The radiation
weighting factor has been defined to increase in proportion to the log of the
LET. Thus, the radiation weighting factor for a particles in tissue is about 20.
The factor for neutrons takes an intermediate value due to the high probability for
scattering protons in tissue. A listing of radiation weighting factor values for
various types of radiation is shown in Table 17.3.
Notice that the dose has a strict definition of energy per unit mass of the absorber
and, in principle, can be measured for a given radiation at a certain energy in a
specific material. The equivalent dose is a relative unit in that a radiation weighting
factor is applied to a measured quantity. The dose can be measured from ionization
in an electronic radiation detector; the equivalent dose must take into account the
type of radiation causing the ionization.
PROBLEMS
1. Calculate dE/dx for a 10-MeV a particle interacting with aluminum.
2. At what kinetic energy does an electron have the same energy loss as a 6-MeV
a particle interacting with aluminum?
3. Calculate dE/dx for an 8-MeV a particle interacting with Mylar.
TABLE 17.3 Radiation Weighting Factors for Various Radiations
g b
Proton
(.2 MeV) a
Fast n
(2–20 MeV) Thermal n
Radiation weighting factor 1 1 5 20 10 5
PROBLEMS 533
4. It has been said very approximately that a particles and protons having the same
speed have approximately the same range in matter. Why is this false? Which
has the longer range and why?
5. Verify that the minimum ionization for heavy charged particles takes place at
b  0.96.
6. Thin nickel foil is used to slow down monoenergetic 10-MeV protons. What is the
maximum thickness that can be used if one wants the straggling to be ,1% of the
mean transmitted energy? What is the mean transmitted energy in this case?
7. Calculate the thickness of aluminum foil needed to degrade a beam of 10 MeV/
nucleon 12
C ions to 3 MeV/nucleon.
8. Calculate the energy loss of a 6-MeV a particle in passing through 50 mg=cm2
of natural nickel?
9. Assuming no energy losses occur, calculate the heating of a 500-mg/cm2
foil of
208
Pb when bombarded with 1 particle microampere of 86
Kr19þ
ions.
10. Devise a way, using measurements of dE/dx and E, to build a particle identification
system.
11. Calculate the range in aluminum of: (a) an 80-MeV 80
Br ion, (b) a 12-MeV a
particle, and (c) a 1-MeV electron.
12. A GM counter window is made of mica, NaAl3Si3O10(OH)2, with a thickness of
2 mg/cm2
. (a) What is the minimum energy b2
particle that can penetrate this
window? (b) What is the minimum energy a particle that can penetrate this
window?
13. Repeat the calculation outlined in Problem 12 for skin of thickness 1 mm with an
average density of 1 g/cm3
. Assume skin is 65% O, 18% C, 10% H, and 7% N.
14. Particles of kinetic energy 400 MeV are incident on a medium of index of
refraction of 1.888. One observes Cerenkov radiation with an opening angle
of u ¼ cos21
(0.55). What are the particles?
15. How far does a 32
P b2
particle (Emax ¼ 1.7 MeV) penetrate in P-10 counter gas?
16. Suppose you have a sample that contains radionuclides that emit 1 MeV b
particles and 1 MeV g rays. Devise an attenuation technique that would allow
you to count the g rays without interference from the b particles.
17. What is the fractional attenuation of a beam of 1 MeV photons in 2.5 cm of Pb?
18. Prove that a photon with Eg . 1.022 MeV cannot undergo pair production in
free space.
19. Lead is thought to be a “better” absorber of photons than aluminum. At what
g-ray energies is the mass absorption coefficient of lead greater than that of
aluminum? Why?
534 INTERACTION OF RADIATION WITH MATTER
20. A 1-MeV photon undergoes Compton scattering through angles of 08, 908, and
1808. What is the energy of the scattered photon in each case?
21. What is the mean free path of a 0.1-, a 1.0-, and a 3.0-MeV photon in NaI?
22. Calculate the mean free path of a 200-keV photon in water.
23. How much lead shielding will it take to reduce the radiation exposure level to
,10 mrem/h 1 ft from a 5-mCi 60
Co source?
24. Prove the scattering angle is 908 for A þ A elastic scattering.
25. Consider a particle with mass m1 scattering elastically from a particle (at rest)
with mass m2. If m1 . m2, show that the scattering angle cannot exceed
sin21
(m2/m1).
26. In graphite, how many collisions are necessary to reduce the kinetic energy of a
1-MeV neutron to thermal energies? What is the approximate time scale for this
process?
REFERENCES
Evans, R. D. The Atomic Nucleus, McGraw-Hill, New York, 1955.
Hubert, F., R. Bimbot, and H. Gauvin. At. Data and Nucl. Data Tables, 46, 1 (1990).
Knoll, G. F. Radiation Detection and Measurement, 3rd ed., Wiley, New York, 2000.
Leo, W. R. Techniques for Nuclear and Particle Physics Experiments, Springer, Berlin, 1987.
Northcliffe, L. C. and R. P. Schilling. At. Data and Nucl. Data Tables, A7, 233 (1970).
Williamson, G. F., J. P. Boujot, and J. Picard. CEA—R03A2, 1966.
BIBLIOGRAPHY
Harvey, B. G. Introduction to Nuclear Physics and Chemistry, 2nd ed., Prentice-Hall,
Englewood Cliffs, NJ, 1969. The same information as Meyerhof from a chemist’s
perspective.
Lieser, K. H. Nuclear and Radiochemistry: Fundamentals and Applications, VCH,
New York, 1997. Covers a number of the practical aspects of the subject that are important
to radiochemists.
Marmier, P. and E. Sheldon. Physics of Nuclei and Particles, Vol. I, Academic, New York,
1969. Detailed treatment of many of the important concepts. Still a useful reference.
Meyerhof, W. E. Elements of Nuclear Physics, McGraw-Hill, New York, 1967. A simplified
treatment that captures the essential details.
Tsoulfanidis, N. Measurement and Detection of Nuclear Radiation, 2nd ed., Taylor and
Francis, Washington, DC, 1983. Many detailed numerical examples.
BIBLIOGRAPHY 535
CHAPTER 18
RADIATION DETECTORS
A fundamental feature of nuclear processes is that the energy released is generally
larger than the binding energies of atomic electrons. Any emitted particles will have
sufficient energy to ionize atoms. Nuclear radiation is called ionizing radiation,
therefore, and detecting this ionization allows us to observe nuclear processes. Radiations
that interact with matter via the electromagnetic force, that is, electrons,
charged particles, and photons, can directly ionize or excite atoms. These radiations
are readily detected. Neutrons interact with nuclei only via the nuclear force and are
detected through indirect or secondary ionization processes.
We should note that though the energy released in nuclear processes is several
(even many) orders of magnitude larger than atomic binding energies, the total
number of ion pairs that can be created when radiation interacts with matter is
small on a macroscopic scale. For example, typical electron binding energies are
about 10 eV. If the total energy available from a 1-MeV nuclear decay was completely
converted into electron/ion pairs, then the total number of pairs would be 105
corresponding to a charge of 10214
C. Even this estimate of the charge created is
optimistic because it is unlikely that all of the energy will create ion pairs. (The
“effective” ionization energy of most gases is about 35 eV/ion pair because some
ion pairs recombine.)
To measure the radiation, the primary ionization must be preserved and not be
lost to recombination or scavenging by electronegative atoms. Metals are not
useful for creating radiation detectors, therefore. On the other hand, the created
ions must be mobile so that they can be collected. This rules out insulating materials
537
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
in most cases. The small electrical signals must be amplified to be observed, and so
electronic instrumentation plays a role in modern nuclear chemistry.
Although the various types of radiation detectors differ in many respects, several
common criteria are used to evaluate the performance of any detector type. The
criteria used for this purpose are as follows:
1. Sensitivity of Detector What types of radiation will the detector detect? For
example, solid scintillation detectors are normally not used to detect a particles
from radioactive decay because the a particles cannot penetrate the
detector covering.
2. Energy Resolution of Detector Will the detector measure the energy of the
radiation striking it, and if so, how precisely does it do this? If two g rays of
energies 1.10 and 1.15 MeV strike the detector, can it distinguish between
them?
3. Time Resolution of Detector or Its Pulse-Resolving Time How high a counting
rate will be measured by the detector without error? How accurately and
precisely can one measure the time of arrival of a particle at the detector?
4. Detector Efficiency If 100 g rays strike a detector, exactly how many will be
detected? Each detector discussed here will be evaluated using these basic
criteria.
In this chapter we will consider the techniques developed to detect and quantitatively
measure how much ionization and/or excitation is caused by different nuclear
radiations. As all radiation creates ionization and/or excitation, we will separate the
discussion of detection methods according to the general techniques used to collect
and amplify the results of the interaction of the primary radiation with matter
rather than by the type of radiation. These detection methods can be classified as:
(a) collection of the ionization produced in a gas or solid, (b) detection of secondary
electronic excitation in a solid or liquid scintillator, or (c) detection of specific
chemical changes induced in sensitive emulsions.
A brief summary of these detector types is as follows:
Gas Ionization Several detector types take advantage of the ionizing effect of
radiation on gases. The ion pairs so produced can be separately collected. When a
potential gradient is applied between the two electrodes in a gas-filled ion
chamber, the positively charged molecules move to the cathode and the negative
ions (electrons) move swiftly to the anode, thereby creating a measurable pulse.
Such pulses can be readily measured by the associated devices as individual
events or integrated current.
Ionization in a Solid (Semiconductor Detectors) In a semiconductor radiation
detector, incident radiation interacts with the detector material, a semiconductor
such as Si or Ge, to create hole–electron pairs. These hole–electron pairs are
collected by charged electrodes with the electrons migrating to the positive electrode
538 RADIATION DETECTORS
and the holes to the negative electrode, thereby creating an electrical pulse. Such
pulses contain information on the type, energy, time of arrival, and number of particles
arriving per unit time. The important features of semiconductor detectors are
their superior energy resolution due to a lower ionization potential and compact size.
Solid Scintillators Some of the energy of ionizing radiation can be transferred to
fluor molecules (i.e., compounds that can produce fluorescence) in a crystalline solid.
The absorbed energy causes excitation of orbital electrons in the fluor. Deexcitation
causes the emission of the absorbed energy as electromagnetic radiation in the visible
or near-ultraviolet region (scintillations). Observing these weak scintillations
visually under certain circumstances is possible (see Deeper Look), but visual
observation is normally not a feasible detection method. Instead, a photomultiplier
tube close to the solid fluor is employed. In the photomultiplier, the photons are
converted to photoelectrons, which are greatly amplified by secondary electron
emission through a series of electrodes (dynodes) to cause a sizable electrical
pulse. Thus, the original excitation energy is transformed into a measurable pulse.
Liquid Scintillators This detection mechanism is quite similar in principle to
the preceding one. Here, however, the radioactive sample and the fluor are the
solute in a liquid medium, usually a nonpolar solvent. The energy of nuclear radiation
first excites the solvent molecules. This excitation energy eventually appears
as photons emitted from the fluor following an intermediate transfer stage. The
photons are detected by means of a photomultiplier arrangement.
Nuclear Emulsions The process involved here is a chemical one. Ionizing radiation
from a sample interacts with the silver halide grains in a photographic emulsion
to cause a chemical reaction. Subsequent development of the film produces an image
and so permits a semiquantitative estimate of the radiation coming from the sample.
Deeper Look—Visual Detection of Radiation There are very few cases in which
nuclear radiation can be directly observed by humans. The eye is not sensitive to
photons in the X-ray and g-ray regions. The energy per decay is small so that
only large sources generate enough energy that they warm up and glow. An exception
is the Cerenkov radiation emitted by very energetic electrons from the b decay
of fission products and Compton-scattered electrons from g decay of these
fragments in nuclear reactors. During operation, the reactor produces very
neutron-rich fission products that rapidly emit very penetrating b particles and
Compton-scattered electrons. These particles can leave the fuel rods and enter the
reactor coolant. The energetic electrons are relativistic and travel with velocities
near the speed of light. However, because the speed of light is lower in liquid
water than in a vacuum, the electrons emit characteristic blue photons—called
Cerenkov radiation—as they adjust their speed downward. Large power reactors
do not have viewing ports, but the cores of research reactors are usually visible.
The cores are surrounded by an eerie blue glow when the reactors are operating.
RADIATION DETECTORS 539
18.1 DETECTORS BASED ON IONIZATION
Many detectors have been developed to collect and amplify the primary ionization
created by nuclear particles. In principle, the careful measurement of this ionization
provides the most information about the particle and its energy. The devices with the
highest resolution are these detectors based upon ionization. Broadly speaking,
ionization-based detectors have the common feature that the incident radiation
creates ion pairs in an active volume of the device. An electric field is applied to
the active volume to separate the charge pairs and sweep the ions to the electrodes.
Ionization-based detectors have mostly used gases as the active medium. Very
few devices use liquids because extremely pure materials are needed to preserve
the primary ionization. Gas-filled detectors are easy to construct and operate, but
the density of the stopping material is low. The effective ionization potential is
large, typically 20 eV. Semiconductors are 103
times denser than gases and
have lower ionization potentials, 2 eV, but producing large volumes of suitably
pure material is expensive. Liquids also have high densities, however, successful
devices have only been made with liquefied rare gases, liquid argon, and xenon.
The impurity level has limited these devices.
18.1.1 Gas Ionization Detectors
As an energetic charged particle passes through a gas, its electrostatic field will dislodge
orbital electrons from atoms sufficiently close to its path. In each case, the
negatively charged electron dislodged and the more massive positive ion comprising
the remainder of the atom form an ion pair. The minimum energy (in electron volts)
required for such ion pair formation in a given gas is called the ionization potential.
This value differs markedly for different gases and is dependent on the type and
energy of the charged particle. A more meaningful value is the average energy
lost by the particle in producing one ion pair, which is nearly independent of particle
energy and type (and is about 35 eV).
The rate of energy loss will depend on the energy and type of charged particle as
discussed in Chapter 17. The a particles will create intense ionization (2104
–105
ion pairs/cm of path length), whereas b2
particles will produce 102
–103
ion
pairs/cm, and the passage of g rays will result in 1–10 ion pairs/cm.
Ion Chambers How can we use this primary ionization to produce a detectable
signal? The first class of devices to be discussed is the pulse-type ion chambers. A
sketch of such a device (a parallel-plate ion chamber) is shown in Figure 18.1. Note
that one electrode has been connected to the negative terminal of the voltage source,
making it the cathode, while the other electrode acts as the anode.
If a 3.5-MeV a particle traverses the chamber, intense ionization will occur along
its short path. Since about 35 eV are expended, on the average, in forming an ion pair
in air, the 3.5-MeV a particle could form approximately 1 Â 105
such ion pairs before
dissipating all its energy. Because of the potential on the chamber electrodes, these
ions migrate rapidly to the respective electrodes. The less massive electrons move
540 RADIATION DETECTORS
very quickly to the anode and produce a rapid buildup of charge there. (Because the
positive ions move about 1000 times slower than the electrons, their effect can be neglected
for the moment.) The time for collection of the electron charge is about 0.1 to
1 ms depending on the volume of the chamber and the potential gradient. The magnitude
of this charge due to the electrons can be calculated as follows:
One eÀ
charge ¼ 1:6 Â 10À19
C
105
eÀ
 1:6  10À19
C=eÀ
¼ 1:6 Â 10À14
C (18:1)
The collected charge flows through the external circuit as a surge, or pulse. If a 20-pF
capacitor is used, the potential of the pulse, V(t), is found as follows:
V ¼
Q
C
(18:2)
where Q is the charge (coulombs) and C is the capacitance (farads). For this case,
V ¼ 0.0008 V. The precise measurement of such small pulses is difficult and sensitive
low-noise electronic modules are needed to measure the signals accurately. Note that
in these ion chambers, there is no amplification of the primary ionization.
The discussion up to now has not been completely accurate in that the effect of
the positive ions on charge collection has been totally neglected. In practice, the
positive ions are troublesome. Although they move very slowly to the cathode, as
they move, they induce a charge on the negative electrode. If no correction is
made for this induced charge, the size of the output pulse will depend on the position
of the particle track in the chamber volume. A simple method for eliminating this
positive ion induction is the addition of a grid to the ionization chamber, as
shown in Figure 18.2.
Figure 18.1 Schematic representation of a parallel-plate ionization chamber in which one
ion pair has been formed. V is the voltage source, R denotes resistor, and C denotes the
capacitor. (From O’Kelley, 1962.)
18.1 DETECTORS BASED ON IONIZATION 541
The grid is charged positively with respect to the cathode, but it is less positive
than the anode or collecting electrode. The grid acts to shield the collecting electrode
from the effects of the positive ions and accelerates electrons toward the anode. An
“internal sample” can be placed on the cathode or high-voltage electrode.
In many applications, instead of recording pulses from each particle that strikes
an ionization chamber, the charge from several events is integrated or added. The
total current from the chamber is then measured as a function of time. These
devices are generally useful for high radiation field measurements. For example,
if one 3.5-MeV a particle produces 105
ion pairs, if we have 107
particles/s entering
the chamber, we will produce 1012
ion pairs/s, producing a current of 1027
A, which
can be readily detected.
The electronic signals from the passage of individual particles through an ion
chamber can be accurately measured, along with the energy deposited in the gas
as a function of position inside the volume. The rate of ionization is a characteristic
of the nature of the radiation as discussed in Chapter 17. For example, devices with
multiple or segmented anodes have been constructed to take samples of the rate of
ionization (Fig. 18.3). Bragg curve counters determine the relative ionization along
the path of the particle by measuring the time distribution of ions as they arrive at the
anode. These detectors require sophisticated electronic readout to measure the ionization
collected as a function of time which encodes the position at which a particle
passed through the gas volume. Therefore, Bragg curve detectors and segmented
anode ionization chambers are usually only used to detect charged particles from
nuclear reactions.
The most sophisticated gas ionization detector is the time projection chamber
(TPC). It is a large, gas-filled cylinder with a negative high-voltage electrode at
its center and an external magnetic field. Electrons produced when ionizing radiation
passes through the chamber drift toward the end of the cylinder under the influence
of axial magnetic and electric fields. The location where the electrons hit the end of
the chamber is measured by a set of anode wires. The arrival time of the electrons
and the total charge deposited are also recorded. The original particle trajectory
through the chamber can be reconstructed from this information.
Figure 18.2 Schematic diagram of a gridded ionization chamber. (From O’Kelley, 1962.)
542 RADIATION DETECTORS
Gas-filled ionization counters that collect the primary electrons on a wire, as
opposed to a plate, can internally amplify the initial ionization. The cylindrical electric
field can be very large near thin wire anodes (50 mm), causing the primary electrons
to be accelerated past the point at which they create a secondary ionization
cascade (Fig. 18.4). The secondary ions so formed are accelerated by the prevailing
potential gradient, thereby producing still more ionization. Thus, from a few primary
ion pairs, a geometrical increase results in a veritable torrent of negative ions moving
toward the chamber electrodes. The process described is known as gas amplification;
the flood of ions produced is termed the Townsend avalanche, in honor of the
discoverer of this phenomenon. Because of gas amplification, most of the electrons
are collected at the anode within a microsecond or less from the entrance of a single
(a)
(b)
E
(c)
+HV
+HV
+HV
+HV
DE
Figure 18.3 Schematic diagram of an ion chamber that drifts the ionization perpendicular to
the particle’s path is shown. In this case the anode is segmented and the relative rate of
ionization along the path can be determined. The device also contains a Frisch grid
between the anode and chamber to improve the pulse–shape response of the device.
(c) The schematic version of a detector that drifts the ionization along the particle’s path,
called a Bragg counter, is shown. The time distribution of the output signal will contain
information on the relative rate of ionization all along the particle’s path.
18.1 DETECTORS BASED ON IONIZATION 543
charged particle into the chamber. A strong pulse is thereby formed, fed into the
external circuit, and is directly measured after only low amplification.
As the potential gradient between the electrodes of the ionization chamber is
further increased, the number of electrons, mostly secondary, reaching the anode
rises sharply for a given original ionization event from a charged particle.
Eventually, a potential will be reached at which the chamber undergoes continuous
discharge and is no longer usable as a detector. There are two distinct potential
regions between the ion chamber region (no amplification) and continuous discharge
that are useful for gas ionization devices. They are called the proportional region and
Geiger–Mu¨ller region.
Proportional Counters In detectors operating in the proportional region, the
number of ions that form an output pulse is very much greater than, yet proportional
to, the number formed by the initial ionization. Gas amplification factors of about
103
–104
are generally obtained. The amplification factor is primarily dependent
on the composition of the chamber filling gas and the potential gradient. At a
given potential, the amplification factor is the same for all ionizing events. Consequently,
if an a particle traversing the ionization chamber causes 105
primary ion
pairs, with an amplification factor of 103
, a charge equivalent to 108
electrons
would be collected at the anode. An incident b particle, on the other hand, producing
only 103
ion pairs, would, after amplification by the factor of 103
, result in a
collected charge equivalent to only 106
electrons.
As with simple ionization chambers, then, it is possible to differentiate between a
and b particles in the proportional region based on pulse size. This is one advantage
of operating a detector in the region. Because the amplification factor in the
proportional region is so heavily dependent on the applied potential, highly stable
high-voltage supplies are necessary.
The avalanche of electrons in proportional detectors is collected only on part
of the anode wire. Furthermore, only a small fraction of the gas volume of the
Figure 18.4 Schematic illustration of the Townsend avalanche in a gas ionization device.
The avalanche occurs very close to the wire in reality. (From Knoll, 2000.)
544 RADIATION DETECTORS
ionization chamber is involved in the formation of ions. These factors result in a very
short dead time, that is, the interval during which ion pairs from a previous ionization
event are being collected and the chamber is rendered unresponsive to a
new ionizing particle. Ionization chambers operating in the proportional region
are thus inactivated for only 1–2 ms following each ionization event. Dead times
as low as 0.2–0.5 ms can be achieved, but if a proportional counter is used for
spectroscopy purposes, the average time between pulses should be 100 ms or
greater due to the slower operation of the external amplifiers and other electronics.
Some practical designs for proportional counters are shown in Figure 18.5. In the
cylindrical detector, a very thin window of split mica or Mylar plastic covers one end
of the tube. It can be so thin (down to 150 mg/cm3
) that the absorption of a particles
by the window is not extensive. An even more efficient arrangement is found with
the hemispherical detector, where the radioactive sample can be introduced directly
into the detector chamber. In the hemispherical detector, one detects 50% of all the
Figure 18.5 (a) Diagram of a 2p gas flow proportional counter. The sample is introduced by
sliding out the bottom of the chamber. (b) A 4p gas flow proportional counter for absolute
counting. (From Wang et al., 1975.)
18.1 DETECTORS BASED ON IONIZATION 545
particles emitted by the source. Such windowless detectors are widely used for
a- and weak b-particle counting.
With either ultrathin end-window or windowless detectors, a certain amount of air
leaks into the counting chamber. Both the oxygen and the water vapor of the air
reduce the detection efficiency because they scavenge the electrons creating slow
moving negative ions. Detectors of this variety, therefore, must be purged with an
appropriate counting gas before counting is started and must be continually flushed
at a lower flow rate during the counting operation. Consequently, such chambers
are often called gas flow detectors. The operating potential of the chamber is determined,
largely, by the gases used for this purpose. Argon, methane, a 90% argon–
10% methane mixture, known as P-10 gas, or a 4% isobutane–96% helium
mixture, known as Q-gas, are some commonly used counting gases.
Other high-purity gases and gas mixtures are used to fill the detectors. Often
argon is used for its relatively high density, but fluorocarbons like CF4 and C2F6
and hydrocarbons like isooctane and isobutane are also used in devices designed
to detect charged particles.
The electronic instrumentation necessary for the operation of the proportional
counter is shown in Figure 18.6. Pulses from the detector pass through a preamplifier
and amplifier, where they are shaped and amplified. Emerging from the amplifier,
the pulses go to a discriminator. The discriminator is set so as not to trip on noise
pulses but rather to trip on radiation pulses of any larger size. The number of discriminator
pulses produced is recorded by the scaler.
When the count rate of a sample emitting both a and b particles is determined
over the voltage range of a proportional detector and the data are plotted, the
results are as seen in Figure 18.7. The characteristic curve for a proportional detector
exhibits two plateaus. The plateau at the lower voltage represents a radiation
alone because, in this voltage range, only the a particles, with their much greater
specific ionization, produce pulses large enough to trigger the discriminator. Not
only may the a particles thus be counted separately from accompanying b radiation
at this potential, but also the background radiation counting rate (primarily cosmic
rays and g rays) is extremely low, on the order of a few counts per hour, as well.
Figure 18.6 Schematic block diagram of the components of a proportional counter. (From
Wang et al., 1975.)
546 RADIATION DETECTORS
As the potential gradient in the chamber is increased, the amplification factor
becomes correspondingly greater. Eventually the primary ions produced by even
the most energetic b particles are amplified sufficiently to produce pulses large
enough to be recorded. This point represents the b threshold. Further increases in
potential gradient allow even the pulses from the weaker b particles to be registered.
The b plateau in the operating voltage has now been reached. The count rate here
actually represents a plus b radiation. A good proportional counter has a b
plateau slope of less than 0.2% per 100 V. The efficiency of proportional detectors
for g radiation is so low that they are seldom used for g counting. Often, in discussions
of the proportional counter, one forgets to mention that the proportional
counter is an excellent spectrometer (i.e., an energy-measuring instrument) for
low-energy radiation, such as X-rays.
At still higher potential gradients, the gas amplification factor may reach 108
.
Now even a weak b particle or g ray can create sufficient ion pairs to completely
saturate the available “ion space” in the chamber. Consequently, the size of the
charge collected on the anode no longer depends on the number of primary ions produced,
and, thus, it is no longer possible to distinguish between the various types of
radiation. This potential level is called the Geiger–Mu¨ller region, after the German
physicists who first investigated it. Ionization chambers operated in this potential
region are commonly called Geiger–Mu¨ller (G–M) detectors. Since the maximal
gas amplification is realized in this region, the size of the output pulse from the
detector will remain the same over a considerable voltage range until continuous discharge
occurs. This fact makes it possible to use a less expensive high-voltage
supply than that required for proportional detectors.
Use of a very high amplification factor in the Geiger–Mu¨ller region is not
without problems. One is the longer dead time of the chamber. Following the
passage of an ionizing particle through a detector, an electron avalanche occurs
along the entire anode wire, resulting in a cylindrical sheath of positive ions
around the anode. The number of such positive ions per pulse will be one or more
Figure 18.7 Characteristic curve for a proportional counter. (From Wang et al., 1975.)
18.1 DETECTORS BASED ON IONIZATION 547
orders of magnitude greater than that in chambers operated in the proportional
region. To be neutralized, the positive ions must migrate to the cathode wall.
Being much more massive than the electrons, these ions move at a slower velocity
in the electrical field. During this migration, the chamber is unresponsive to any new
ionizing particles passing through it. Thus, the dead time of a detector operated in
the Geiger–Mu¨ller region is from 100 to 300 ms or more.
A correction for “dead time loss” can be made from the value of the dead time, t.
If the true counting rate is n (t ¼ 0), and m is the measured rate, we have
n ¼
m
1 À mt
(18:3)
So, if we measure a counting rate of 1000 cps with a dead time of 250 ms, the true
counting rate is
N ¼
1000
1 À (1000 Â 250 Â 10À6Þ
¼ 1333 cps
with the measured counting rate being low by 33%.
Another important problem is the perpetuated chamber ionization resulting from
complications associated with the discharge of positive ions at the cathode wall of a
detector. As a result, we need a means of terminating or quenching the perpetual ionization
in the detector. This is done by introducing polyatomic organic compounds
or halogen gases into the counter gas. These molecules absorb energy by collisions
and dissociate into neutral species, which stops the discharge. Since they are
destroyed during quenching, the lifetime of a G–M tube is typically limited to
108
–109
pulses.
One of us (GTS), as a graduate student in 1936, was involved with the discovery
of the quenching phenomenon in Geiger–Mu¨ller counters. He and a fellow graduate
student, David C. Grahame, were plagued with erratic Geiger–Mu¨ller counter behavior
until they discovered the beneficial effect of water vapor, which was introduced
by accident into the argon gas in their counter. They found that reliable behavior also
followed the admixture of small amounts of other gases, such as ammonia and
natural gas. They did not publish a description of this discovery and were quite interested
to read in 1937 the publication by Trost [Z. Phys. 105, 399 (1937)] of his observation
that ethyl alcohol had a similar quenching effect leading to reliable operations
of such counters.
18.1.2 Semiconductor Detectors (Solid-State Ionization Chambers)
As mentioned at the beginning of this section, the primary ionization must be collected
to make a direct measurement of the energy of nuclear radiation. Condensed
phases have higher densities than gases and so provide more efficient stopping of
the radiation per unit length. However, metals allow rapid recombination of the electron/positive
ion pairs and insulators inhibit the collection of the charge. Therefore,
only semiconductors have been used extensively for radiation detectors. Metals and
548 RADIATION DETECTORS
insulators (like concrete) are used extensively in radiation shielding and some transparent
inorganic crystals have a special sensitivity to radiation, which is discussed
below.
Silicon and germanium are the most common semiconductors used to construct
“solid-state ionization chambers.” These materials must be extremely pure to
observe the primary ionization (105
electrons) and, as we will see below, germanium
devices must be cooled to reduce the thermal noise to observe the signals. The
properties of small-scale devices based on Group III/Group V materials, for
example, GaAs, have been studied, but no large-scale applications have been
made. The size and shape of the available semiconductors have grown over time
but are still severely limited by production techniques and the availability of
high-purity material.
Early solid-state devices relied on observing the ionization in intrinsic semiconductors.
Early devices were impractical due to the requirement of extremely pure
material. Modern devices are based on semiconductor junction diodes. These
diodes have a rectifying junction that only allows the flow of current in one direction.
Incident radiation creates ionization inside the bulk of the diode and creates
a pulse of current in the opposite direction to the normal current flow through a
diode that is straightforward to detect.
To understand how semiconductor radiation detectors operate, it is necessary to
review a little of the basic chemistry of semiconductors. Consider a typical Group IV
element, such as Si or Ge. It will crystallize in the diamond lattice structure, as
shown in Figure 18.8. Each silicon atom is bound by four electron pair bonds to adjacent
silicon atoms. The electrons are not free to migrate through the crystal, and
therefore pure silicon is a poor conductor of electricity. The electron energy
levels of silicon are shown in Figure 18.9. The electron energy levels of the
valence electrons are so close together that they form a nearly continuous “band”
of energies, known as the valence band. In pure silicon, there is a region of energies
above the valence band in which there are no allowed energy levels. This energy
region is called the forbidden gap and corresponds to 1.08 eV for silicon. Just
above the forbidden gap is the conduction band, another band of energies that
Figure 18.8 Schematic view of the crystal lattice of Si. The dots represent electron pair
bonds between the Si atoms. (From Wang et al., 1975.)
18.1 DETECTORS BASED ON IONIZATION 549
allows free electron migration through the crystal, that is, the conduction of electricity.
Suppose we replace a silicon atom in the silicon lattice with a Group V atom,
such as phosphorus. Then we will have the situation depicted in Figure 18.10.
Phosphorus has five valence electrons. After forming four electron pair bonds to
the adjacent silicon atoms, there is one electron left over. This leftover electron
will be very loosely bound to the phosphorus atom and will be easily removed to
conduct electricity through the crystal.
In terms of our diagrams of the crystalline–electron energy levels, we have the
situation shown in Figure 18.11. The “extra” phosphorus electron occupies a
“donor level” very close to the conduction band and is easily promoted into this conduction
band. Silicon containing Group V impurities, such as phosphorus, is called
n-type silicon because the species that carries charge through the crystal is negative.
What happens when a Group III atom, like boron, replaces an atom in the silicon
lattice? The situation is shown in Figure 18.12. Boron has three valence electrons
and can form electron pair bonds with three of its neighbors. It has no electron to
pair up with the electron on the fourth silicon atom. We say we have an electron
hole in the silicon lattice.
In terms of our energy level diagrams, we have the situation illustrated in
Figure 18.13. The hole occupies an energy level very close to the valence band
(an acceptor level) and can be easily promoted into the valence band. (Promotion
Figure 18.10 Schematic diagram of a typical n-type impurity in a silicon crystal lattice.
(From Wang et al., 1975.)
Figure 18.9 Schematic diagram of the energy levels of crystalline silicon. (From Wang
et al., 1975.)
550 RADIATION DETECTORS
of a hole into the valence band simply means that an electron in the valence band and
a hole in the acceptor level switch places, so that a hole is created in the valence
band.)
We must realize (unphysical as it may sound) that a hole in the valence band can
conduct electricity as well as an electron in the conduction band. How does this
work? Consider Figure 18.14. Imagine that electron 1 moves to fill hole 0. This
step creates a hole at position 1. Electron 2 moves to fill this hole, leaving a hole
at position 2. Electron 3 fills the hole at position 2, leaving a hole at position 3,
and so forth. Thus, as the hole moves to the right in Figure 18.14, negative
charge is moving toward the left. Since electricity is the movement of charge, the
motion of the hole corresponds to the flow of electricity. Silicon containing
Group III impurities is said to be p-type silicon because of the positive charge
carriers (the holes).
A silicon–semiconductor–radiation detector of a layer of p-type silicon in
contact with a layer of n-type Si is shown in Figure 18.15. What happens when
this p-n junction is created? The electrons from the n-type silicon will migrate
across the junction and fill the holes in the p-type silicon to create an area around
the p-n junction in which there is no excess of holes or electrons. (We say that a
depletion region has been formed around the junction.) Imagine that we apply a
positive voltage to the n-type material and a negative voltage to the p-type material
(the junction is said to be reverse biased). The electrons will be “pulled farther
Figure 18.11 Schematic diagram of the energy levels of crystalline silicon with a donor
impurity. (From Wang et al., 1975.)
Figure 18.12 Schematic diagram of a silicon crystal lattice with a p-type impurity in it.
(From Wang et al., 1975.)
18.1 DETECTORS BASED ON IONIZATION 551
away” from the junction by the positive voltage on the n-type material, thus creating
a much thicker depletion region around the p-n junction. The exact thickness of the
depletion region, d, is given by
d / (rV)2
(18:4)
where r is the resistivity of the silicon and V is the magnitude of the applied reversebias
voltage. Note that the depletion of the depletion region can be varied at will by
changing the voltage applied to the detector.
The depletion region acts as the sensitive volume of the detector. The passage of
ionizing radiation through this region will create holes in the valence band and electrons
in the conduction band. The electrons will migrate to the positive charge on the
n side, while the holes will migrate to the negative voltage on the p side, thereby
creating an electrical pulse at the output of the device.
The average energy necessary to create a hole–electron pair in silicon is
3.6 eV. [This average energy is about three times the forbidden gap energy
(1.1 eV) because most electrons are promoted from deep in the valence band to
high in the conduction band.] The energy required to create a hole–electron pair
is independent of particle charge and mass, thus causing semiconductor detector
response to be independent of particle type. If we remember that the average
Figure 18.13 Schematic diagram of the energy levels of silicon with a p-type impurity.
(From Wang et al., 1975.)
Figure 18.14 Schematic diagram of a silicon crystal lattice with a p-type impurity boron, at
one lattice point. The hole is labeled 0, while the electrons are denoted as 1, 2, etc. (From
Wang et al., 1975.)
552 RADIATION DETECTORS
energy to create an ion–electron pair in a gas ionization device was 35 eV, then
we see that, for the same energy deposit in the detector, we get 35/3.6 % 10
times more charged pairs. If we note that the energy resolution of a detector, DE/E,
is proportional to N21/2
where N is the number of charge pairs formed, we can
see that the energy resolution of a semiconductor is approximately 101/2
¼ 3.2
times better than the energy resolution of a gas ionization detector. [Furthermore,
as we will see later, the average g-ray energy deposit required to liberate one
photoelectron at the cathode of a photomultiplier tube is %1000 eV, then we can
say that the resolution of a semiconductor detector is (1000/3.6)2
% 17 times
better than that of a scintillation detector.] More detailed considerations show that
the observed resolution can be smaller than the estimate based on the statistics of
ion pair formation due to correlations between processes giving rise to ion pair
formation.
For some semiconductor detectors, germanium is used instead of silicon for the
detector material. The reasons for this substitution are as follows: (a) The average
energy needed to create a hole–electron pair in germanium is 2.9 eV rather than
the 3.6 eV necessary for Si. Thus, the energy resolution for germanium should be
(3.6/2.9)2
¼ 1.1 times better than silicon. (b) The atomic number of germanium
(32) is much higher than that of silicon (14), leading to increased probability of
g-ray interaction with the detector material. Consequently, germanium is preferred
to silicon for g-ray detection. The forbidden gap is so small, however, for germanium
(0.66 eV) that room temperature thermal excitation leads to the formation of
hole–electron pairs in the solid. Therefore, germanium detectors must be operated
at liquid nitrogen temperature (77 K) to prevent this thermal electron noise from
overwhelming the small signals from the primary ionization.
The silicon-based solid-state detectors fall into three general categories:
surface barrier devices, PIN diodes, and Si(Li) (pronounced “silly”) devices.
These detectors are used to measure short-ranged radiation: charged particles in
the first two cases and low-energy g rays and X-rays in the third case. The detector
Figure 18.15 Schematic diagram of a p-n junction. (From Wang et al., 1975.)
18.1 DETECTORS BASED ON IONIZATION 553
consists of a thin layer of silicon material (often 200 mm thick, but thicknesses
from 5 mm up to 5 mm are available). An electric field (typically V/mm) is
applied in the direction opposite to the “normal” flow of current through the
diode. Radiation creates electron–hole pairs that are swept to the electrodes by
the electric field and induce a current signal. These signals are amplified in an
external circuit.
Silicon-surface barrier (SSB) detectors consist of a thin cylindrical piece of high
resistivity (103
V cm) pure n-type silicon with a thin gold contact on one side and an
aluminum contact on the other (Fig. 18.16). The gold contact is a thin layer through
which the radiation enters the silicon. Just under the gold is an oxide layer that forms
the semiconductor junction (or barrier). The gold layer is sensitive to physical wear,
and the oxide layer can be depleted by extended exposure to vacuum. The oxide
layer is also very sensitive to organic molecules but can be reconstructed with
proper treatment. During use these detectors must be shielded from visible light
as electron–hole pairs can be created by photons that enter the silicon through the
thin gold contact. The gold and oxide layers are also thin to reduce the amount of
kinetic energy lost by the particle before it enters the active silicon region. These
layers make up a dead layer that can be significant in a spectroscopy. Recently,
SSBs with very thin and uniform dead layers have been created with ion-implanted
junctions. A thin layer of boron ions is implanted near the surface of n-type silicon to
form the junction.
“Ruggedized” detectors are available in which the radiation enters the silicon
through the thicker and light-tight aluminum contact. The bulk of the material is
p-type silicon and a negative bias is applied to the gold contact so that the entry
(a)
(b)
P-layer
Intrinsic-layer
N-layer
Thin Au
Oxide layer
Thick Al
+HV
Figure 18.16 Representation of silicon diodes, simple SSB and segmented.
554 RADIATION DETECTORS
window can remain at ground potential. Another design of a “rugged” surface
barrier detector replaces the gold contact with a thicker nickel contact. The nickel
is resilient enough to be wiped clean.
Silicon surface barrier detectors have found widespread application in a-particle
spectroscopy and in nuclear reaction studies. These detectors can be used in stacks to
identify particles uniquely. Consider the situation in which a penetrating ion with a
total kinetic energy (KE) passes through a thin SSB detector and is stopped in a
second thicker SSB detector. Such a stack of detectors is called a silicon detector
telescope and provides two signals, the energy lost in the thinner detector, DE,
and the remainder, KE 2 DE. The rate of energy loss for a charged particle is
approximately given by the simple expression:
dE
dX

mZ2
KE
(18:5)
where m is the ion’s mass, Z its nuclear charge, for a given kinetic energy. When the
first detector is thin, dE/dX is approximately constant and DE ffi (dE/dX) DX. Thus,
DE will be a hyperbolic function of KE for each ion with a different value of mZ2
.
Therefore, the components of a mixture of penetrating charged particles can be
identified by their relative values of ionization. Stacks of SSBs with several thin
detectors are used when unambiguous particle identification through redundant
measurements is necessary. Variations in the thickness of the silicon in the manufacturing
process place a practical limit on the particle identification. Thickness
variations as small as l mm are currently achievable.
The typical resolution of a single detector is 20 keV but depends on the detector
geometry, in particular, on the detector capacitance. Notice that SSB detectors
have parallel electrodes separated by a thin dielectric; the capacitance of such
an object will increase with increasing area and with decreasing thickness. Thus,
thin large-area devices will have the largest capacitance and thus the poorest
resolution.
Silicon PIN diodes are a more recent class of detectors that have become available,
in large part, due to the growth of the semiconductor industry. These devices
are made up with a p-type layer on one side of an intrinsic silicon wafer and an
n-type layer on the opposite, therefore a p-I-n sandwich. The detectors are available
in a much larger range of sizes and shapes than surface-barrier detectors. For
example, 25-cm2
devices with single or multiple specially shaped contacts are
routinely available. The contacts on the front and back of PIN diodes can have
different shapes and sizes. Consider a 5-cm  5-cm rectangular wafer that has
two horizontal electrical contacts on the front and two vertical contacts on the
back. The divided contacts define four quadrants of silicon that are electrically
separate. A particle that enters the detector will generate two signals, one on the
front and another on the back that uniquely identifies the quadrant of silicon.
Rectangular devices are often used with 16 stripes on the front and 16 stripes on
the back and provide very accurate position measurements in nuclear reaction
studies.
18.1 DETECTORS BASED ON IONIZATION 555
All solid-state detectors are damaged by long exposures to charged particles. The
threshold dose for observable damage (in particles/cm2
) is 108
for fission fragments,
109
for a particles, 1012
for fast neutrons, and 1013
for electrons. Notice
that the particles generally come to rest in the silicon and stay there. The lattice
will be disrupted and poisoned by the presence of many stopped particles. If the
particles have the same energy (same range), then all of the particles will stop in
a very narrow band of the silicon and can create a dead layer inside the silicon.
Lithium-drifted silicon detectors have been developed for measurements of b
particles. Electrons are more penetrating than heavier charged particles and so
these devices, usually called Si(Li) detectors have to be much thicker than heavyion
detectors. Si(Li) detectors are commonly 5 mm thick. Such large volumes of
very pure silicon are not readily available; thus, the technique of drifting lithium
ions into the bulk material to compensate for internal lattice defects in p-type
silicon has been developed. A layer of lithium metal is applied to the surface and
some atoms diffuse into the bulk silicon. The lithium atoms readily donate an electron
into the conduction band and become ions. A bias can be applied to the silicon
that causes the lithium ions to migrate from the surface through the lattice. The
migrating ions will be trapped by negative impurities in the lattice, thus “compensating”
for the effect of the impurity. The lithium ions retain their high mobility
in the lattice and the detectors have to be stored with a small retaining bias if
they are stored for long periods at room temperature.
Si(Li) detectors are favored over Ge(Li) detectors for b detection because of their
low g sensitivity and their lower (by 1
3 to 1
2) backscattering. The energy resolution
of Si(Li) detectors for electrons is 1–2 keV for electron energies up to
1000 keV. The detection efficiency of Si(Li) detectors for b particles ranges from
one-half that of a gas counter for a low-energy b emitter like 14
C to greater than
that of a gas counter for an energetic b emitter such as 32
P. The background of
these detectors is exceptionally low because of their small size for a given stopping
power, and they do not require any peripheral gas supply, and the like. Very good
energy resolution for X-ray detection is possible. A resolution of 180 eV for the
5.9-keV Mn Ka X-ray have been obtained with Si(Li) detectors, whereas the best
energy resolution available from a scintillation detector is about 1000 eV.
All of the silicon detectors can be cooled to reduce the thermal noise that produces
a background under all the induced signals. The thermal noise is created by random
fluctuations that promote an electron across the bandgap into the conduction band
resulting in an electron–hole pair. The number of promoted electrons will be proportional
to a Boltzmann function containing the bandgap D and the temperature T:
Nthermal / eD=kT
(18:6)
For practical reasons, silicon detectors are usually cooled from room temperature
down to approximately 2208C; cooling below 2608C is not useful because the
system noise becomes dominated by the external electronic circuit. Temperatures
below 2208C are not used also because the internal physical stresses from differences
556 RADIATION DETECTORS
in the thermal expansion coefficients of the construction materials become important.
Specially prepared detectors are recommended before cooling to the lower
temperatures.
We can estimate the factor by which the thermal noise will be reduced with the
Boltzmann expression:
Nthermal(T ¼ À208
)
Nthermal(T ¼ þ258)
 e253=298
¼ 2:3 (18:7)
Cooled silicon detectors are particularly useful in experiments in which the measured
particles are expected to cause significant damage to the crystal lattice during the
experiment. If the detector is not cooled, the thermal noise will dramatically
change during the measurement and the detector resolution will decrease with time.
Germanium detectors have the highest resolution of any direct ionization devices.
This is due to the small bandgap of germanium of 0.73 eV (at 80 K) and effective
ionization potential of 2.95 eV that allows the creation of many ion pairs for a
given amount of radiation. The bandgap is also small enough that the number of
electron–hole pairs created by thermal fluctuations causes a very significant electronic
noise. The noise is reduced by enclosing the germanium and the first stage
of the amplification circuit in a cryostat and cooling both to liquid nitrogen
temperature.
Other things being equal, the size of the signals produced in a germanium diode
compared with a silicon diode should be larger by the inverse ratio of the effective
ionization potentials, 3.76/2.95  1.27 (at 80 K). However, the thermal noise will
be larger in proportion to a Boltzmann exponential distribution with the bandgaps,
so at the same temperature the noise in the germanium will be larger by the factor:
eÀDGe=kT
eÀDSi=kT
¼ e1:16=0:73
¼ 4:9 (18:8)
These facts would appear to favor the use of silicon detectors strongly. However, the
“stopping power” of matter for photons is much lower that than for charged particles
giving photons long penetration depths in all materials. Moreover, the probability of
a photoelectric interaction with an atom, which contributes significantly to the
absorption of the full energy of photons, increases in proportion to Z5
. This
makes high-Z materials more effective total absorbers of photons. Also from the
practical standpoint, manufacturing techniques have been developed to produce
very high purity germanium crystals that are much larger than silicon crystals.
Germanium detectors are used almost exclusively to detect g radiation. Energetic
photons can easily penetrate the cryostat, and the high resolution of germanium
detectors are well suited to the very precise energies of the g rays emitted by the
deexcitation of nuclear levels. There are two main classes of germanium detectors,
those that use lithium compensated material, called Ge(Li) (pronounced like jelly),
which have been supplanted by intrinsic germanium, also called high-purity
germanium.
18.1 DETECTORS BASED ON IONIZATION 557
Ge(Li) detectors are similar in principle to the Si(Li) detectors described above.
Large ingots (1 L) of p-type germanium material are prepared in a relatively pure
state. Lithium metal is applied to the surface, some atoms diffuse into the lattice,
donate their electron, and the ions are subsequently drifted through the material.
The mobility of lithium ions is much higher in germanium than in silicon. This
allows very effective compensation of the impurities in large volumes of germanium
but also allows the rapid loss of compensation by the lattice at room temperature.
The hallmark of Ge(Li) detectors is that the germanium crystals must be kept at
liquid nitrogen temperatures for their entire useful lifetime. If the detectors are
allowed to warm up, even for a very short time, the lithium compensation is lost
to some extent and their high resolution is degraded. The crystals can be redrifted
but the manufacturer must do this.
The shape of the germanium crystals is generally cylindrical with the lithium
applied to the outer surface and drifted in toward the center. The lithium contact produces
an n-type region on the surface of the crystal, the bulk becomes intrinsic
through compensation, and a small p-type region is deliberately left undrifted
to produce a P-I-N diode structure. The detector thus has coaxial p- and n-type “electrodes”
and is sometimes called a coax detector. A reverse bias is applied to the
germanium diode, as to the silicon detectors, and the small current pulses from
the primary ionization events are collected and amplified. The requirement that
Ge(Li) detectors must be kept very cold during their entire life span has spurred
the development of germanium purification techniques. Large volumes of intrinsic
germanium material can now be produced in sufficient quantity to produce PIN
diode detectors without lithium drifting. These devices can be stored at room temperature
and only need be cooled when they are used. The n-type region is prepared
by lithium diffusion without drifting and an extremely thin p-type region is prepared
by implanting boron ions. The crystals are generally cylindrical with coaxial
electrodes. Other shapes, such as thin planar detectors for low-energy g rays and
X-rays and crystals with hollow wells with large geometrical efficiency are
available.
For g-ray detection, the detector of choice is the Ge detector because of its higher
Z. The most spectacular feature of the Ge detector is the superior energy resolution.
An energy resolution of 1.75 keV for the 1332-keV g ray of 60
Co is routinely
obtained [compared with the typical 90–100 keV for a 3 Â 3 in. NaI(Tl) detectors].
What this means in terms of the ability to resolve g-ray spectra is shown in
Figure 18.17.
18.2 SCINTILLATION DETECTORS
In a scintillation detector, a fraction of the energy deposited by the primary radiation
in the detector is converted to light that, in turn, is converted into an electrical
signal. Conceptually, the process can be divided into the scintillation process
itself (energy ! light), the collection and conversion of the light into electrons
558 RADIATION DETECTORS
and the multiplication of the electrons to make a macroscopic signal. We will divide
our discussion similarly.
As discussed in Chapter 17, as radiation interacts with matter, it will lose energy
by ionizing or exciting matter. As we have seen, only a few materials have the right
properties to allow the collection of the primary ionization from nuclear radiation.
If the ionization is not preserved and collected, the electron/positive ion pairs are
expected to recombine eventually. During this recombination, the energy used to
separate the charges will be reemitted to the surroundings, very often as lattice
vibrations and heat. Occasionally, a triplet electronic excited state is populated,
and the energy from such states is released as visible photons. This emission
process is well known as atomic or molecular fluorescence and is called scintillation
when it is caused by exciting radiation. (We will use the term “visible light” loosely
in our discussion. The wavelengths of the fluorescent photons from excited
electronic states are characteristic of the material and range from UV to red.)
These visible, secondary photons can be easily detected and amplified with
photomultiplier tubes.
Figure 18.17 g-Ray spectrum of neutron-activated lung tissue as measured with a 3 Â 3 in.
NaI(Tl) detector and a Ge detector. (From Cooper, 1971.)
18.2 SCINTILLATION DETECTORS 559
The details of the scintillation process are complicated and depend very much on the
molecular structure of the scintillator. In organic crystals, the molecules of the organic
solid are excited from their ground states to their electronic excited states (see
Fig. 18.18). The decay of these states by the emission of photons occurs in about
1028
s (fluorescence). Some of the initial energy absorbed by the molecule is dissipated
as lattice vibrations before or after the decay by photon emission. As a result, the crystal
will generally transmit its own fluorescent radiation without absorption.
There are three common types of organic scintillator. The first type is a pure
crystalline material, such as anthracene. The second type, the liquid scintillator,
is the solution of an organic scintillator in an organic liquid, such as a solution of
p-terphenyl in toluene (%3 g solute/L solution). The third type is the solution of
an organic scintillator, such as p-terphenyl, in a solid plastic, such as polystyrene.
All these organic scintillators are characterized by short fluorescence lifetimes,
that is, 2–3 ns. This allows their use in high-count rate situations or for fast time
measurements. The light output is modest, being 10–50% of that of NaI(Tl) (see
below). Because of the low Z of the organic scintillators, they are primarily used
in the detection of heavy charged particles or electrons. The plastic scintillators
are easily machined into a variety of shapes and/or made as thin films. Energy
resolutions of about 10–14% are possible. Liquid scintillators are used to assay
low-energy b emitters, like 14
C or 3
H. The radioactive material is dissolved or suspended
in the scintillator solution. Another application involves the use of liquid
scintillators for large volume (several m3
) detectors where the liquid scintillator
has been “loaded” with a neutron absorbing material, such as gadolinium. The
gadolinium captures neutrons producing e2
and g rays that are detected by the
scintillator.
The scintillation process in inorganic scintillators differs from that in organic
scintillators. Consider the structure of an ionic crystal, as shown in Figure 18.19.
When an energetic electron passes through the crystal, it may raise valence electrons
from the valence band to the conduction band. The electron vacancy in the valence
Figure 18.18 Schematic view of the scintillation mechanism in organic crystals. (From
Wang et al., 1975.)
560 RADIATION DETECTORS
band resulting from this ionization is called a hole, in the valence band. The electron
in the conduction band and the hole in the valence band can migrate independently
through the crystal.
Alternatively, another process called excitation can occur by which a valence
band electron is excited to an energy level lower than the conduction band. The electron
remains bound to the hole in the valence band. This neutral electron–hole pair
is called an exciton, and it can move through the crystal also. Associated with the
exciton is a band of energy levels called the exciton band (see Fig. 18.19).
The presence of lattice defects and/or intentionally placed impurities in the alkali
halide crystal will cause the formation of local energy levels in the forbidden gap,
called traps or activator centers. Figure 18.19 shows the energy levels of an alkali
halide crystal, including the activator centers and traps. (Atomic thallium is a
common activator for alkali halide crystals.)
Excitons, holes, and electrons produced by the interaction of radiation with the
crystal wander through the crystal until they are trapped at an activator center or
trap. Migration of an exciton in a crystal may be thought of as a 6- to 8-eV
excited iodide ion, I2Ã
, transferring its energy to an adjacent stable I2
, which, in
turn, becomes excited. Thus, energy may be transferred from I2
to I2
in the
crystal lattice to final capture by either an activator center or crystal impurity.
By exciton capture or hole–electron capture, the activator centers are raised
from their ground state to an excited state. The deexcitation of this activator
center by emission of light occurs in a time about 0.3 ms. Hence the energy deposited
by the radiation in the scintillator is emitted as light by the activator center (Tl).
The amount of light emitted by the entire crystal is directly proportional to the
amount of energy deposited in the crystal by the incident radiation. The fraction
of the deposited energy converted into fluorescence photons is small, about 10%.
Thallium-activated sodium iodide [NaI(Tl)] is the most widely used inorganic
scintillator. This material is used extensively to detect g rays because it is relatively
inexpensive, has a high stopping power for photons, and is rugged and easy to use.
The fluorescence light output has a relatively slow decay time of almost 230 ns,
limiting the count rate in such detectors. The energy resolution of NaI(Tl) detectors
is rarely better than 6% for the 1332-keV g ray of 60
Co (as compared to the 0.13%
typically seen with Ge detectors). NaI(Tl) detectors are very efficient for detecting
Figure 18.19 Electronic energy levels in an ionic crystal. (From Wang et al., 1975.)
18.2 SCINTILLATION DETECTORS 561
g radiation (with typical detection efficiencies of 1–10%). The efficiency of a 3-in.
diameter right cylinder that is 3 in. long is the reference standard for g-ray detectors.
Other inorganic scintillators of note are bismuth germanate (Bi4Ge3O12, BGO)
which is a high Z, high-density material. Its low-light output [10–20% that of
NaI(Tl)] limits its use to situations where a high efficiency (with poorer resolution)
is needed. Barium fluoride (BaF2) is a high Z material with a fast light output
(t , 1 ns) with reduced light output. It is used in situations where its high
density, high Z, and fast timing are important.
How is the light emitted by the scintillator converted into an electrical signal? To
answer this question, let us consider a schematic diagram of a typical scintillation
detector (Fig. 18.20). The photons of visible light emitted by the activator centers,
Figure 18.20 (a) Cutaway diagram of a typical solid scintillation detector. (b) Diagram of a
scintillation detector illustrating schematically the way in which light from the scintillator is
transmitted to a photomultiplier tube. A typical wiring diagram is shown for the 10-stage
photomultiplier operated with a positive high-voltage supply. (From O’Kelley, 1962.)
562 RADIATION DETECTORS
such as Tlþ
ions in a NaI(Tl) crystal, pass through the transparent fluor substance
and out through a clear window to impinge on an adjacent photocathode. The
typical photocathode is composed of a thin, photosensitive layer (commonly a
cesium–antimony alloy) on the inner surface of the end of the photomultiplier
tube. Here impinging photons, particularly those having wave lengths between
3000 and 6000 A, are absorbed, with a consequent emission of photoelectrons.
The number of photoelectrons ejected is slightly less than, but directly proportional
to, the number of incident photons. Such a burst of photoelectrons resulting from a
single g-ray interaction in the crystal is still far too weak to be registered directly.
Amplification occurs by means of a series of electrodes, called dynodes, spaced
along the length of the photomultiplier tube (see Fig. 18.20). Each dynode is maintained
at a higher potential (usually about 50 V higher) than the preceding one. The
photoelectrons emitted from the photocathode are focused by an electric field to hit
the first dynode. Striking the dynode surface, they cause the secondary emission of a
larger number of electrons. This new burst of electrons is guided by the potential
gradient to the second dynode in the series, where a still larger number of electrons
are dislodged. This electron-multiplying process continues at each dynode until at
last the collecting anode is struck by 105
–106
electrons for each original photoelectron
ejected from the photocathode. Thus, the size of the output pulse from
the photomultiplier is directly related to the quantity of energy dissipated by the
incident g-ray photon in the fluor.
To present a quantitative example of the energy conversions involved in scintillation
detection, we trace the results of the interaction of a single 1.17-MeV g ray
from 60
Co with a thallium-activated sodium iodide crystal [Nal(TI)]:
1. If 20% of the energy of the g ray results in exciton production in a fluor
crystal, and if it is assumed that 7 eV is needed to produce an exciton, then
approximately 33,000 excitons could result from this g-ray photon.
2. Assuming that only 10% of the excitation events result in the production of
photons of visible light seen by the adjacent photocathode, this would mean
that about 3300 photons would reach the photocathode.
3. This number of photons striking a photocathode with a conversion efficiency
of 10% would eject approximately 330 photoelectrons.
4. The successive dynodes of a photomultiplier operated at an overall gain of
106
could then amplify this quantity of photoelectrons so that 3.3 Â 108
electrons would be collected at the photomultiplier anode, or a charge of
5 Â 10211
C.
5. This charge could then be transformed by a preamplifier circuit with a capacitance
of 30 pF into an output pulse of 1.8 V. A pulse of this size would be
capable of directly triggering a scaler.
60
Co emits two g rays per disintegration. The other g ray has an energy of 1.33 MeV.
Following the preceding calculations, this 1.33-MeV g ray would result in an output
pulse of about 2.05 V from the detector. The statistical uncertainty in the pulse
18.2 SCINTILLATION DETECTORS 563
heights would be determined by the step in the chain with the greatest uncertainty
(Step 3). In this case, we would expect a resolution of %6% so that the two pulse
heights would be barely resolved.
18.3 NUCLEAR TRACK DETECTORS
The passage of highly ionizing radiation through an insulating solid leaves a wake of
destruction in the material. In covalently bonded materials, the chemical structure of
the material along the track can be significantly and permanently changed by the
passage of a single energetic ion. Certain polymeric (plastic) materials and the
mineral mica (a form of silicon dioxide) are particularly sensitive to such radiation
damage. The original radiation damage remains localized on the molecular scale but
is not visible without enhancement. However, the track can be expanded by chemical
etching from the molecular scale (nanometers) up to the microscopic scale
(micrometers).
Nuclear track detectors are very simple and very efficient detectors of rare events
that produce highly ionizing radiation. Carefully prepared and scanned track detectors
have been used to identify individual rare decays. The detectors are integrating
in that the damage caused by a track is not spontaneously repaired. The drawback to
track detectors is that the tracks are small and can only be observed with a microscope.
In the past, scanning by eye was extremely labor intensive and prone to
error. Modern computer-controlled scanning has improved the speed and reliability
of the analysis. Plastic track detectors that are sensitive to a particles are used extensively
in commercial radon detectors.
Chemical etching of the material takes place on all surfaces that are exposed
to the etching solution. The exposed surfaces of the material are eroded along
with the material along the track. Therefore, the rate of etching has to be carefully
controlled to get the maximum amount of information from the track. Notice
that etching of a uniform track will generally form a circular cone because the
material will be more easily removed from the surface than from deep along the
track. Mica tracks are diamond-shaped due to the lattice structure as opposed to
being circular.
Nuclear emulsions are closely related “track detectors” that trace their origins to
the original discovery of radiation by Becquerel. Nuclear emulsions are very finegrained
photographic film. The film is “exposed” by the passage of radiation
through it, and the grains of AgC1 are activated by the ionization. The film is developed,
and, with careful handling and microscopic observation, the track or path of
individual particles can be traced. Occasionally, a particle interacts with a nucleus
in the emulsion, creating many fragments or particles, and the tracks of the reaction
products can be traced. The emulsion is also sensitive to the rate of ionization and
the nature of the particle in each track can often be determined. On the other hand,
most people are familiar with the shadow images of skeletal features taken with
X-rays. The X-rays are absorbed and scattered more efficiently by the heavy
elements in bones (essentially calcium) than by the light elements in soft tissue
564 RADIATION DETECTORS
(carbon, oxygen, hydrogen) and create a shadow. The grains in the emulsion are
then exposed by the transmitted X-rays and are developed to form the negative
image.
18.4 NUCLEAR ELECTRONICS AND DATA COLLECTION
As we have seen, essentially all of the nuclear radiation detectors produce electronic
pulses in response to the interaction of some ionizing radiation. These signals are
processed by standardized nuclear instrumentation modules (NIM) electronics to
count the number of pulses or to more fully analyze the size or even the shape of
the signal. In addition, computer-based electronics in the CAMAC (computer automated
measurement and control) system are used to measure the time relationships
of pulses, the pulse heights, and the signal shapes. The signals are recorded and
stored by computers for later analysis. An important feature of scientific studies
with radioactivities and with nuclear beams is that the data must be collected as
rapidly as possible usually during a very limited time. A radioactive source will
decay away after being produced and cannot be “stopped” because the scientist is
not ready to use it. Similarly, the nuclear reactions induced by particle beams
take place in a very short time and must be recorded when they occur. Then after
a set of events has been collected “on-line,” the data are analyzed “off-line.”
We will give a very brief overview of the kinds of modules used. CAMAC and
NIM electronics fall into three broad categories: linear electronics that maintain a
linear relationship to the size of the initial signal, logic circuits that provide only
a standard (or single sized) pulse indicating that a given logical condition was
met, and data acquisition modules to measure the signals and record the data.
One should realize that with modern high-density electronics the functions that
we will describe can correspond to a single electronic module or may be condensed
into a single integrated circuit. Therefore, we will only describe the functions performed
by the electronic modules and not specific equipment.
The output of most detectors is an electrical pulse that carries information about
the energy deposited in the detector, the time of the interaction, and the like. Linear
electronics are described as modules that preserve and extract information about the
energy deposit in the detector from the detector signal. An overview of these
modules and their function is given in Table 18.1. A typical pulse-height analysis
system is shown in Figure 18.21. The signal from the detector is given a preliminary
amplification and shaping by a preamp before being sent through a coaxial cable
to a linear amplifier. This is done to prevent noise in the cable from destroying
the tiny detector signal. In the amplifier the signal is further amplified and
shaped before analysis. The height of the pulse is related to the energy deposited
in the detector. The analog-to-digital converter (ADC) converts the signal from
the amplifier into digital data (a number of standard pulses) thus measuring its
size. The ADC could be contained on a plug-in card in a personal computer
(used to measure the distribution of pulses from a single detector monitoring a
radioactive source), or it might be one of many identical ADC units in a CAMAC
18.4 NUCLEAR ELECTRONICS AND DATA COLLECTION 565
module (used to record the signals from many detectors monitoring nuclear
collisions simultaneously).
Logic modules are used to monitor the counting rate of single detectors and the
relative times at which radiation is detected. A fast signal derived from the detector
itself, the preamplifier, or from a timing-filter amplifier is sent to a discriminator.
TABLE 18.1 Summary of Common Pulse-Processing Modules
Input Output
Linear input–
linear output
Preamplifier Linear charge pulse from
detector
Linear tail pulse
Linear amplifier Linear tail pulse Amplified and shaped linear pulse
Biased amplifier Shaped linear pulse Linear pulse proportional to amplitude
of input pulse that lies above input
bias level
Pulse stretcher Fast linear pulse Conventional shaped linear pulse of
amplitude equal to input pulse
Sum amplifier Two or more shaped linear
pulses
Shaped linear pulse with amplitude
equal to the sum of the coincident
input pulses
Delay Fast linear or shaped linear
pulse
Identical pulse after a fixed time delay
Linear gate Shaped linear pulse and
gate pulse
Linear pulse identical to linear input if
gate pulse is supplied in time overlap
Linear input–
logic output
Integral
discriminator
Shaped linear pulse Logic pulse if input amplitude exceeds
discriminator level
Single-channel
analyzer
Shaped linear pulse Logic pulse if input amplitude lies
within acceptance window
Time pickoff Fast linear or shaped linear
pulse
Logic pulse synchronized with some
feature of input pulse
Logic input–
linear output
Time to amplitude
converter
Logic start and stop pulses
separated by a time Dt
Shaped linear pulse with amplitude
proportional to Dt
Logic input–
linear output
Coindence Logic pulses at two or more
inputs
Logic pulse if pulses appear at all
inputs within a time interval Dt
Anticoincidence Logic pulses at two inputs Logic pulse only if pulse appears at one
input without pulse at second input
within time Dt
Scaler Logic pulses One logic pulse for N input pulses
566 RADIATION DETECTORS
The discriminator produces an output pulse with a fixed shape (generally square) and
size when the input signal crosses a reference. Discriminators usually have multiple
identical output signals. The logic pulses can be sent to a scaler that simply counts
the number of pulses, to a count rate meter to monitor radiation rates or doses, and to
a time-to-amplitude converter (TAC) to measure the relative times of arrival of two
or more logic signals.
18.5 NUCLEAR STATISTICS
Radioactive decay is a random process. The number of nuclei in a sample of radioactive
material that decay in any time period is not a fixed number but will differ,
usually, for various time periods. This point can be readily shown by making
repeated measurements of the activity of a long-lived radionuclide, each for the
same time duration. The results of such an experiment might be as shown in
Table 18.2. Note that in these measurements there is a large range of activity
values with a clustering near the center of the range. We can plot these data
(Table 18.2) as a distribution function, by “binning” the data (Fig. 18.22). We can
now ask ourselves if we can understand this distribution function. Statisticians
have given us mathematical models that describe these and other similar distribution
functions. As a background for our discussion of how to extract the maximum
amount of information from these data, let us consider some of these models.
The most general model to describe radioactive decay is the binomial distribution.
For a process that has two outcomes (success or failure, decay or no
decay), we can write for the distribution function P(x)
P(x) ¼
n!
(n À x)!x!
px
(1 À p)nÀx
(18:9)
where n is the number of trials where each trial has a probability of success p and
P(x) is the predicted probability of getting x successes. Applying this distribution
to radioactivity, P(x) might be taken as the probability of getting x counts in a
given time interval and p ¼ lDt where Dt is a time short compared with the
Figure 18.21 Schematic diagram of a simple pulse height analysis system for nuclear
spectroscopy. (From Wang et al., 1975.)
18.5 NUCLEAR STATISTICS 567
TABLE 18.2 Typical Sequence of Counts of a Long-Lived
Sample (170
Tm)a
Measurement
Number cp0.1 m xi 2 xm (xi 2 xm)2
1 1880 218 324
2 1887 211 121
3 1915 17 289
4 1851 247 2209
5 1874 224 576
6 1853 245 2025
7 1931 33 10899
8 1886 232 1024
9 1980 82 6724
10 1893 25 25
11 1976 78 6084
12 1876 222 484
13 1901 3 9
14 1979 81 6561
15 1836 262 3844
16 1832 266 4536
17 1930 32 1024
18 1917 19 361
19 1899 1 1
20 1890 28 64
a
We are indebted to Prof. R. A. Schmitt for providing these data.
Figure 18.22 Distribution of counts of a long-lived sample.
568 RADIATION DETECTORS
measurement time and the half-life. Note that x and n are both integers. Typical
binomial distribution functions are shown in Figure 18.23.
The binomial distribution function is cumbersome and a simplification can be
made. If the probability of success p is small (p ( 1) (the measurement time is
very short compared with the half-life), we can approximate the binomial distribution
by the Poisson distribution. The Poisson distribution is written as
P(x) ¼ (xm)x
exp (Àxm)=x! (18:10)
where
xm ¼ pn (18:11)
Thus, we have a simplified distribution characterized by one parameter, xm compared
to two parameters in the binomial distribution. The Poisson distribution is an asymmetric
distribution as shown in Figure 18.24. Besides being a more tractable function
to use, the Poisson distribution has certain important properties that we will use in analyzing
radioactivity data. Let us consider a parameter, the variance, s2
, which
expresses something about the width of the distribution of values about the mean, xm.
For a set of N measurements, we can calculate s2
as:
s2
¼
PN
i¼1 xi À xmð Þ2
N À 1
(18:12)
Figure 18.23 Binomial distributions for various values of N and p. (From Leo, 1987.)
18.5 NUCLEAR STATISTICS 569
(For the data of Table 18.2, xm ¼ 1898, s ¼ 44.2.) For a binomial distribution
s2
¼ np(1 À p) (18:13)
which is cumbersome to use. But, for a Poisson distribution, we can show that
s2
¼ xm (18:14)
s ¼ (xm)1=2
(18:15)
Applying these equations to the data of Table 18.2, we get sPois ¼ 43.6. This illustrates
the important point that these distribution functions are models, not physical
laws, and when they are applied to finite data sets, their predictions may deviate
from observation.
The Poisson distribution can be applied also to describe the action of detectors.
For example, suppose the interaction of a g-ray photon with an inefficient scintillator
produced, on average, 3.3 photoelectrons from the photocathode. The probability
of producing no photoelectrons (not seeing the event) is given by the Poisson
distribution as:
P(0) ¼ exp(À3:3) ¼ 3:7% (18:16)
Thus, 3.7% of the events will be missed due to “statistical fluctuations.”
A further simplification of the parent binomial distribution occurs when the
number of successes is relatively large, that is, we get more than about 30 counts
in a measurement. Then, the binomial distribution can be represented as a normal
or Gaussian distribution. Here we write
P(x) ¼
1
ffiffiffiffiffiffiffiffiffiffiffi
2pxm
p exp À
x À xmð Þ2
2xm
 
(18:17)
0.6
0.4
0.2
0 2 4
m = 0.5 0.6
0.4
0.2
0 2 4 6
m = 1
0.2
0.1
0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 18 20
m = 4
0.1
m = 10
Mean
Figure 18.24 Poisson distribution for various values of xm. (From Leo, 1987.)
570 RADIATION DETECTORS
This analytical approximation is symmetric (Fig. 18.25). As shown in Figure 18.25,
68.3% of the measured values lie within +1s of the mean, xm. Furthermore 95.5%
of all measurements lie within +2s of the mean and 99.7% lie within +3s of the
mean. The full width at half maximum (FWHM) is 2.35s.
Thus, for a single measurement of a count rate of 100, we would estimate that
s ¼ 10. We could say, with a 68.3% chance of being correct that the true rate
was between 100 2 10 ¼ 90 and 100 þ 10 ¼ 110. With 95.5% certainty, we
could say the true rate lies between 80 and 120. Generalizing, we can quote the
results of a measurement as x + ns where n is related to the probability that an
infinite number of measurements would give a value within the quoted range. For
n ¼ 0.6745, 1, 1.6449, 1.96, 2, 2.5758, 3, the “confidence limits” are 50, 68.3, 90,
95, 95.5, 99, and 99.7%, respectively. Commonly, people will quote the results of
a measurement as x + s. One should remember that doing so means one will be
wrong 31.7% of the time, that is, the mean count rate will be outside x + s.
If this risk is not acceptable, one should pick a greater confidence level, that is,
2s, 3s, and so forth.
Another distribution function of interest relates to the distribution of time intervals
between successive counts. We know the average time between counts is
(1/count rate). The distribution of time intervals is given by the interval distribution.
This distribution (applicable to all random events) states that for a process with an
average time between events tm, the probability of getting a time t between successive
events is
I(t) ¼
1
tm
exp
Àt
tm
 
dt (18:18)
Figure 18.25 Plot of the normal distribution function showing the mean xm and 1s and 2s
points. (From Wang et al., 1975.)
18.5 NUCLEAR STATISTICS 571
For radioactive decay
tm ¼
1
l
(18:19)
This distribution function is shown in Figure 18.26. Note the most probable time
between events is zero. Random events (counts, natural disasters, etc.) occur in
“bunches.”
Let us summarize how we describe the statistical uncertainty in measurements of
radioactivity. If we measure the activity of a sample (þbackground) as 64 counts in
1 min, then we estimate
S þ B ¼ 64 cpm
with an uncertainty sSþB
sSþB ¼ 8 cpm
What if a second measurement with no sample showed a background of 10 counts in
1 min? We would then estimate
B ¼ 10 cpm
sB ¼ (10)1=2
¼ 3:2 cpm
What is the net sample rate and its uncertainty? This raises the general question of
calculating the uncertainty in the result of some mathematical operations on an
uncertain number. If we consider two independently determined numbers and
their uncertainties (standard deviations), A + sa, B + sb, we can write down, as
Figure 18.26 Plot of the frequency of occurrence of a time interval t between successive
counts, where the average rate is 1 cpm. (From Wang et al., 1975.)
572 RADIATION DETECTORS
shown in Table 18.3, some rules for the uncertainty in the result of some common
mathematical operations.
As an example of the use of the relations outlined in Table 18.3, we would
calculate that for our sample and background counting case,
Net rate ¼ (sample þ background) À (background)
¼ 64 À 10 ¼ 54 cpm
Uncertainty in net rate ¼ (82
þ 3:22
)1=2
¼ 8:6 cpm
Up to now we have carefully restricted our discussion of nuclear statistics to cases
where l-min counts were taken. If the number of counts recorded in 1 min was x,
then the counting rate has been quoted as x + (x)1/2
cpm. Suppose, however, that
we recorded 160 counts in 5 min. What would be the standard deviation of the
average counting rate (in cpm)? The best estimate of the mean number of counts
in the 5-min period would by 160 + (160)1/2
that is, 160 + 13 counts. The
average rate would be 160/5 + 13/5 ¼ 32+3 cpm. In general, therefore, the rate
R is given as:
R ¼
number of counts recorded
measurement time
¼
x
t
The standard deviation of the rate, sR, is
sR ¼
(x)1=2
t
¼
(RÃ
t)1=2
t
¼
R
t
 1=2
(18:20)
Thus, for the preceding example we could have calculated directly that
sR ¼
R
t
 1=2
¼
32
5
 1=2
¼ 3
Often we wish to compute the average of two numbers, x1 and x2, both of which have
an uncertainty denoted by their standard deviations s1 and s2, respectively. The best
average of these two numbers is not the simple average but the weighted average xm
TABLE 18.3 Uncertainties for Some Common Operations
Operation Answer Uncertainty
Addition A þ B (s2
A þ s2
B)1=2
Subtraction A 2 B (s2
A þ s2
B)1=2
Multiplication A Â B A Â B½(sA=A)2
þ (sB=B)2
Š1=2
Division A/B A=B½(sA=A)2
þ (sB=B)2
Š1=2
18.5 NUCLEAR STATISTICS 573
given by
xm ¼
x1
s2
1
þ
x2
s2
2
 0
1
s2
1
þ
1
s2
2
 
xm ¼
x1 þ wx2
1 þ w
where
w ¼
s1
s2
 2
In short, each number is weighted by the inverse of its standard deviation squared.
For the weighted average of N values, xi, with standard deviation, si, we have
xm ¼
XN
iÀ1
xi=s2
i
À Á
XN
i¼1
1=s2
i
À Á
The uncertainty or standard deviation of x is given by
sxm
¼
1
XN
i¼1
1=s2
i
À Á
0
@
1
A
1=2
For example, suppose that we make two independent measurements of an
activity, obtaining results of 35 + 10 cpm and 46 + 2 cpm. The weighted average
of the two measurements is
w ¼ (10=2)2
¼ 25
xm ¼
½(35)ð1 þ (25)(46)ފ
1 þ 25
ffi 46 cpm
The standard deviation of the weighted average is
sx ¼
½100 þ (25)2
(4)Š
262
& '1=2
ffi 2:0
Thus, we would say that the average rate was 46+2 cpm.
18.5.1 Rejection of Abnormal Data
In our discussions so far, we have only considered the uncertainty in the experimental
data due to the randomness of radioactive decay. But there may also be
574 RADIATION DETECTORS
systematic error that contributes to the overall uncertainty in the data. As a result,
when we make repeated measurements of a sample activity under seemingly identical
situations, we will find occasionally one measurement that differs from the
others by a large amount. If included in the average, this abnormal observation
may cause significant error. When are we justified in rejecting such data? One
criterion for rejecting such data is to reject suspected values that deviate from the
mean by more than 2s or 3s. The probabilities of occurrence of such deviations
are 4.5 and 0.27%, respectively.
What about the question of whether a detector or counting system is working
properly? For example, the data in Table 18.2 do not exactly match a Poisson or
normal distribution. Was the counting system malfunctioning? One parameter
that we can calculate that will help us answer such questions is x2
(chi squared).
Formally,
x2
¼
PN
i¼1 xi À xmð Þ2
xm
For the data in Table 18.2, we calculate that x2
¼ 37194/1898 ¼ 19.60.
Figure 18.27 shows the properties of the x2
distribution in terms of p, the probability
that a random sample of N values from a Poisson distribution. For the data of
Table 18.2, x2
/v ¼ 1.03, which is acceptable. To be suspicious of the data x2
/v
should have been .1.7 or ,0.6.
Figure 18.27 Plot of the chi-squared distribution.
18.5 NUCLEAR STATISTICS 575
18.5.2 Setting Upper Limits When No Counts are Observed
Suppose your experiment failed to detect a type of decay you were seeking: What
can you say about its occurrence? The simplest answer is what is termed the
one-event upper limit. You assume that you had detected one event, and calculate
the resulting decay rate, cross section, and the like, taking into account detection
efficiencies, solid angles, and so forth.
A more sophisticated answer can be obtained by using the properties of a Poisson
distribution. The probability of observing n events if the mean value is m is given as:
r
n
m
 
¼
mn
n!
¼ eÀm
The probability of observing 0 events in a time period T for a process with mean
rate l is
r(0=lT) ¼ eÀlT
It can be shown that the upper limit on the rate (when zero counts are observed), l0,
is given by
l0 ¼ À
1
T
ln(1 À CL)
where CL is the confidence limit you wish to attach to your upper limit. (If you want
to quote an upper limit with 95% confidence, then CL ¼ 0.95.)
Example Suppose you are attempting to observe a process that should produce
1 decay per week, and you observe no counts (with a 100% efficient detector) in
4 weeks. At the 95% confidence level, the upper limit on the decay rate is
l0 ¼ À 1
4 ln (1 À 0:95) ¼ 0:75=week
PROBLEMS
1. What detector would you chose to detect the following? Please give your reasoning.
(a) 0.1 mCi 32
P? (b) A mixture of 0.1 mCi 3
H and 0.5 mCi 14
C. (c) A mixture
of 10 nCi each of 235
U and 238
U. (d) 50 Bq 60
Co.
2. An advertisement for a high-purity Ge detector quoted its relative efficiency as
being 200%. Is that possible? Please explain.
3. How would you detect 10-MeV neutrons?
4. How many counts would you have to collect to have 90% confidence in the
quoted counting rate?
576 RADIATION DETECTORS
5. How would you detect the individual b particles, g rays, and delayed neutrons
from a fission product mixture?
6. What is the maximum allowed counting rate with a scintillation detector with a
0.25-ms dead time if you can only tolerate a 3% counting loss?
7. Given five measurements of the counting rate of a long-lived sample that are
2712, 2753, 2708, 2739, and 2726. Do a chi-squared test to see if these data
are consistent with a Poisson distribution.
8. How many counts do you need to collect to have a 2% uncertainty in your
result?
9. Sketch a diagram of the nuclear instruments you would need to make logical
AND, OR, NAND, and NOR circuits.
10. If a sample has an average counting rate of 16 counts/h and you read out the
data every hour for each of 1000 h, how many times will you get 2, 8, 16,
and 32 counts?
REFERENCES
Cooper, J. A. Applied Ge(Li) Gamma Ray Spectroscopy, BNWL-SA-3603, 1971.
Knoll, G. F. Radiation Detection and Measurement, 3rd ed., Wiley, New York, 2000.
Leo, W. R. Techniques for Nuclear and Particle Physics Experiments, Springer, Berlin, 1987.
O’Kelley, G. D. Detection and Measurement of Nuclear Radiation, NAS-NS-3105, 1962.
BIBLIOGRAPHY
Birks, J. B. Scintillation Counters, McGraw-Hill, New York, 1953. The classic work on scin-
tillators.
Evans, R. D. The Atomic Nucleus, McGraw-Hill, New York, 1955. Noted for its discussion of
statistics.
Kleinknecht, K. Detectors for Particle Radiation, 2nd ed., Cambridge, 1998. An excellent discussion
from a high-energy physics perspective.
Tsoulfanidis, N. Measurement and Detection of Radiation, 2nd ed., Taylor and Francis,
Washington, DC, 1995.
Wang, C. H., D. L. Willis, and W. D. Loveland. Radiotracer Methodology in the Biological,
Environmental and Physical Sciences, Prentice-Hall, Englewood Cliffs, NJ, 1975. Noted
for its discussion of liquid scintillation counting, this book served as a model for much of
the discussion of this chapter.
Wilkinson, D. H. Ionization Chambers and Counters, Cambridge University Press,
Cambridge, UK, 1950. The classic discussion of ionization chambers, proportional
counters, and Geiger–Mu¨ller counters.
BIBLIOGRAPHY 577
CHAPTER 19
RADIOCHEMICAL TECHNIQUES
Radiochemistry is defined as “the chemical study of radioactive elements, both
natural and artificial, and their use in the study of chemical processes” (Random
House Dictionary, 1984). Operationally, radiochemistry is defined by the activities
of radiochemists, that is, (a) nuclear analytical methods, (b) the application of radionuclides
in areas outside of chemistry, such as medicine, (c) the physics and chemistry
of the radioelements, (d) the physics and chemistry of high-activity-level
matter, and (e) radiotracer studies. We have dealt with several of these topics in
Chapters 4, 13, 15, and 16. In this chapter, we will discuss the basic principles
behind radiochemical techniques and some details of their application.
Because of the small amounts of material involved, the presence of radioactivity
that implies certain regulatory and safety concerns and the frequent need to deal with
short-lived nuclei, these techniques are not the same as ordinary chemical techniques.
Specialized techniques have evolved from the early part of the 20th
century when chemistry was a principal tool in identifying the basic nature of radioactive
decay through the extensive use of chemistry in the Manhattan Project in
World War II to the present, “high-tech” character of many radiochemical manipulations.
These techniques are quite important for they are often the key to a successful
experiment, even though they may get scant mention in descriptions of the
experiment. Often the successful application of these techniques requires careful,
painstaking attention to detail, frequent practice to develop the necessary manipulative
skills, and a thorough knowledge of the underlying scientific principles.
579
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
In addition to the discussions of these topics in textbooks such as this, there are
excellent textbooks that focus primarily on radiochemistry (Adloff and Guillamont,
1984; Lieser, 1997; Ehmann and Vance, 1991).
19.1 UNIQUE ASPECTS OF RADIOCHEMISTRY
Radiochemistry involves the application of the basic ideas of inorganic, organic,
physical, and analytical chemistry to the manipulation of radioactive material.
However, the need to manipulate radioactive materials imposes some special constraints
(and features) upon these endeavors. The first of these features is the
number of atoms involved and the solution concentrations. The range of activity
levels in radiochemical procedures ranges from pCi to MCi. For the sake of discussion,
let us assume an activity level, D, typical of radiotracer experiments of 1 mCi
(¼ 3.7 Â 104
dis/s ¼ 3.7 Â 104
Bq), of a nucleus with mass number A  100. If we
assume a half-life for this radionuclide of 3 d, the number of nuclei present can be
calculated from the equation
N ¼
D
l
¼
(1 mCi)(3:7 Â 104
dps=mCi)(3 d)(24 h=d)(3600 s=h)
ln 2
where l is the decay constant of the nuclide (¼ ln 2/t1/2). Then
N  1:4 Â 1010
atoms
Mass of sample ¼ 2:3 Â 10À12
g
This tiny quantity of material, if prepared as an aqueous solution of volume 1 L,
would have a concentration of 10214
mol/L. This simple calculation demonstrates
a number of the important features of radiochemistry, that is, (a) the manipulation of
samples involving infinitesimal quantities of material, (b) the power of nuclear
analytical techniques (since 1 mCi is a significant, easily detectable quantity of
radioactivity), and (c) in an extension of the calculation, since the decay of a
single atom might occur by a-particle emission (with 100% detection efficiency),
the ability to do chemistry one atom at a time.
The small number of atoms involved in some radiochemical procedures can alter
the expected behavior. Although time-dependent processes obeying first-order
kinetics are not changed by changes in concentration, the same is not true of
second-order kinetics. For example, at 1022
M, isotopic exchange between U(IV)
and U(VI) has a lifetime of 2 h, whereas at 10210
M, the same lifetime is
400 d. Another example is Np(V), which is unstable with respect to disproportionation
and yet mCi/L solutions of NpO2
þ
are stable. The extreme dilution in some
solutions can mean that equilibrium is not reached due to kinetic limitations.
Fallout plutonium, present in the aqueous environment at concentrations of
10218
–10217
M, has not reached equilibrium in over 40 y.
580 RADIOCHEMICAL TECHNIQUES
In addition to the limitations posed by kinetics or thermodynamics, there are
certain practical problems associated with very low solution concentrations. An
important problem is the adsorption of tracer levels of radioactivity on the surfaces
of laboratory glassware. Glass has an ion exchange capacity of 10210
mol/cm2
along with a similar number of chemisorption sites. A 100-mL beaker can thus
absorb 1028
mol, which is significant if the concentration of the tracer is
1026
M. One suppresses this absorption by having high [Hþ
] (thus blocking
adsorption sites), by treating glass surfaces with nonadsorbing silicone coatings,
or by the use of holdback carriers (see below).
Conventional analytical techniques generally operate at the part per million or
higher levels. Some techniques such as laser photo acoustic spectroscopy are
capable of measuring phenomena at the 1028
–1026
mol/L level. The most sensitive
conventional analytical techniques, time-resolved laser-induced fluorescence, and
ICP-MS are capable of measuring concentrations at the part per trillion level, that
is, 1 part in 1012
, but rarely does one see detection sensitivities at the single atom
level as routinely found in some radioanalytical techniques. While techniques
such as ICP-MS are replacing the use of neutron activation analysis in the routine
measurement of part per billion concentrations, there can be no doubt about the
unique sensitivity associated with radioanalytical methods.
Along with the unique sensitivity and small quantities of material associated with
radiochemistry, there is the need to comply with the regulations governing the safe
use and handling of radioactive material. This task is a primary focus in the design
and execution of radiochemical experiments and is often a significant factor in the
cost of the experiment. Because so many of these rules are site specific, they are not
treated in this chapter.
There are some chemical effects that accompany high specific activities that are
unique to radiochemistry and are worth noting. Foremost among these are the
chemical changes accompanying radioactive decay. The interaction of ionizing radiation
from a radioactive source with air can result in the generation of ozone and the
nitrous oxides, which can lead to corrosion problems. Sources containing Ra or Rn
produced from the decay of heavier elements, such as U, will emanate Rn gas as the
decay product of Ra. The decay products of gaseous Rn are particulates that deposit
on nearby surfaces, such as the interior of the lungs, leading to contamination problems.
In high-activity aqueous solutions, one can make various species such as the
solvated electron, eÀ
aq, hydroxyl radicals, OH†
, as well as the solvated proton, H3Oþ
.
The hydroxyl radical, OH†
, is a strong oxidizing agent with
OH†
þ eÀ
À! OHÀ
E0 ¼ 2:8 V
while the solvated electron, eÀ
aq, is a strong reducing agent
eÀ
aq þ Hþ
À!
1
2
H2 E0 ¼ 2:77 V
Solutions involving high activity levels will change their redox properties as a
function of time. For example, all the atoms in a 100-Bq/mL (1027
mol/L) solution
19.1 UNIQUE ASPECTS OF RADIOCHEMISTRY 581
of 239
Pu will undergo a redox change in a period of one year. In general, it is hard to
keep high specific activity solutions stable. Reagents, column materials, and the like
can suffer radiation damage also. In radiotracer studies, the self-decomposition
(radiolysis) of 3
H or 14
C-labeled compounds can lead to a variable concentrations
and variable number of products.
Many of these effects of radioactive decay can be treated quantitatively using “G
values.” Historically, the G value was defined as the number of molecules or species
decomposed or formed per 100 eV of absorbed energy. A newer (SI) definition of
the G value is the number of moles of molecules or species formed or decomposed
per Joule of energy absorbed. (Note that 1 mol/J ¼ 9.76 Â 106
molecules/100 eV.)
The G values depend on the radiation and the medium being irradiated and its physical
state. Table 19.1 shows some typical G values for the irradiation of neutral liquid water.
The actual final products of radiolysis are the result of a complex set of chemical
reactions. Detailed quantitative estimates of product yields are, therefore, more
complicated and beyond the scope of this book. The reader is referred to other textbooks
that discuss how these estimates are made (Woods et al., 1999; Mozumder,
1999).
Radioactive decay also causes chemical transmutations. The daughter nucleus in
a or b decay is a different chemical element than the mother nucleus, but it is in the
same chemical environment as the mother nucleus. Change of oxidation state or
bonding is a possibility.
In a decay, one expects all chemical bonds to the decaying atom to be broken as
the recoil energy of the daughter nucleus exceeds chemical bond energies. Surprisingly,
the oxidation state of the daughter nucleus is frequently that of the parent
nucleus after all electronic and atomic rearrangements have taken place. (An
obvious exception is when the daughter cannot exhibit the parent’s oxidation state
such as the a decay of U(VI) as UO2
2þ
, where the thorium daughter does not
exhibit the þ6 oxidation state.)
In b2
decay, especially for low-energy b2
emitters like 14
C or 3
H, the effects on
chemical bonding are modest. So if we have
14
CH4 À! 14
NHþ
4
the b2
decay can be considered an oxidizing process. In fact, b2
decay (of 83
SeO4
22
,
and 242
AmO2
þ
) was used successfully to prepare new higher oxidation states (of
83
BrO4
2
and 242
CmO2
2þ
) of some elements. In electron capture or internal conversion
decay, there are massive rearrangements of the atomic electrons, which makes these
considerations more complicated.
TABLE 19.1 Product Yields (mmol/J) in Irradiated Neutral Water
Radiation G(2H2O) G(H2) G(H2O2) G(eaq) G(H†
) G(†
OH)
g and fast electrons 0.43 0.047 0.073 0.062 0.28 0.0027
12-MeV a particles 0.29 0.12 0.11 0.028 0.056 0.007
582 RADIOCHEMICAL TECHNIQUES
Some tracers (usually cations) in solution behave as colloids rather than true solutions.
Such species are termed radiocolloids and are aggregates of 103
–107
atoms,
with a size of the aggregate in the range 0.1–500 nm. They are quite often formed
during hydrolysis, especially of the actinides in high oxidation states. One can differentiate
between real radiocolloids and pseudocolloids, in which a radionuclide is
sorbed on an existing colloid, such as humic acid or Fe(OH)3. Formation of real colloids
can be prevented by using solutions of low pH or by addition of complexing
agents. The chemical behavior of these radiocolloids is difficult to predict, as the
systems are not at equilibrium.
There are certain unique features to the chemical separations used in radiochemistry
compared to those in ordinary analytical chemistry that are worth noting. First
of all, high yields are not necessarily needed, provided the yields of the separations
can be measured. Emphasis is placed on radioactive purity, expressed as decontamination
factors rather than chemical purity. Chemical purity is usually expressed as
the ratio of the number of moles (molecules) of interest in the sample after separation
to the number of all the moles (molecules) in the sample. Radioactive
purity is usually expressed as the ratio of the activity of interest to that of all
the activities in the sample. The decontamination factor is defined as the ratio of
the radioactive purity after the separation to that prior to the separation. Decontamination
factors of 105
–107
are routinely achieved with higher values possible. In the
event that the radionuclide(s) of interest are short-lived, then the time required for
the separation is of paramount importance, as it does no good to have a very pure
sample in which most of the desired activity has decayed during the separation.
As indicated above, frequently the amount of material involved in a radiochemical
procedure is quite small. To obviate some of the difficulties associated with this,
a weighable amount (mg) of inactive material, the carrier, is added to the procedure
at an early stage. It is essential that this carrier and the radionuclide
(tracer) be in the same chemical form. This is achieved usually by subjecting the
carrier þ tracer system to one or more redox cycles prior to initiating any chemical
separations to ensure that the carrier and tracers are in the same oxidation state.
Carriers frequently are stable isotopes of the radionuclide of interest, but they
need not be. Nonisotopic carriers are used in a variety of situations. Scavengers
are nonisotopic carriers used in precipitations that carry/incorporate other radionuclides
into their precipitates indiscriminately. For example, the precipitation of
Fe (OH)3 frequently carries, quantitatively, many other cations that are absorbed
on the surface of the gelatinous precipitate. Such scavengers are frequently used
in chemical separations by precipitation in which a radionuclide is put in a
soluble oxidation state, a scavenging precipitation is used to remove radioactive
impurities, and then the nuclide is oxidized/reduced to an oxidation state where it
can be precipitated. In such scavenging precipitations, holdback carriers are introduced
to dilute the radionuclide atoms by inactive atoms and thus prevent them from
being scavenged.
It is certainly possible, although usually more difficult, to do carrier-free radiochemistry
in which one works with the radionuclides in their low, tracer-level
concentrations. Such carrier-free radiochemistry is used when the presence of the
19.1 UNIQUE ASPECTS OF RADIOCHEMISTRY 583
additional mass of carrier atoms would lead to problems of sample thickness (a
emitters), biological side effects (radiopharmaceuticals), or where high specific
activities are needed (synthesis of labeled compounds). (Formally, specific activity
is the activity per mass unit, such as mCi/mg or mCi/mmol, etc.)
19.2 AVAILABILITY OF RADIOACTIVE MATERIAL
To do radiochemistry, one needs radioactive materials. As indicated in Chapter 3,
radionuclides may be classified as primordial (remnants of nucleosynthesis), cosmogenic
(being continuously generated by the action of cosmic rays with the upper
atmosphere) or anthropogenic (made by humans). Most of the radionuclides used
in radiochemistry are of the latter type, that is, made artificially in response to perceived
needs. In Table 4.1, we summarized the commonly used radionuclides and
their method of preparation. As indicated in this table, a large number of these
nuclides can be made by neutron irradiation in a nuclear reactor using (n,g)
reactions. Such nuclei are, of course, not carrier-free, are largely b2
emitters and
have low specific activities. Charged-particle-induced reactions, using cyclotrons,
are used to synthesize neutron-deficient nuclei, that decay by EC or bþ
emission.
The short-lived nuclei used in PET or other procedures in nuclear medicine fall
into this category.
The transuranium nuclei are a special class of radionuclides, being made by both
reactor irradiation and production in charged particle accelerators. In Table 19.2, we
summarize the properties and available amounts for research in the United States by
qualified individuals. One should also note that while large quantities of 239
Pu are
available, it is classified as a special nuclear material because of its use in weaponry,
and very strict regulations govern the possession and use of this nuclide (along with
233
U and 235
U).
19.3 TARGETRY
As indicated above, a combination of reactor and cyclotron irradiations is used to
prepare most radionuclides. While many of these radionuclides are available commercially,
some are not. In addition, nuclear structure, nuclear reactions, and
heavy-element research require accelerator or reactor irradiations to produce
short-lived nuclei or to study the dynamics of nuclear collisions, and so on. One
of the frequent chores of radiochemists is the preparation of accelerator targets
and samples for reactor irradiation. It is this chore that we address in this section.
The first question to be addressed in preparing accelerator targets or samples for
irradiation is the question of impurities and/or other chemical constituents of the
sample. For neutron irradiation, one generally prefers metals or nonactivable
inorganic salts. The salts usually include nitrates, sulfates, and the like but not
halides (especially chlorides due to the activation of chlorine) nor sodium nor potassium
salts. In general, one avoids materials that undergo radiolysis, although it is
584 RADIOCHEMICAL TECHNIQUES
possible, with suitable precautions, to irradiate materials, such as gasoline, oil, and
other flammable materials in reactors. Liquid samples can be irradiated in reactors
easily, but one must generally pay attention to pressure buildup in the irradiation
container due to radiolytic decomposition of water. Unless purged with nitrogen,
water will contain dissolved argon, which will activate to form 41
Ar, and the radioactive
atoms will be released upon opening the irradiation container. For irradiations
of a few hours in moderate flux (1012
n/cm2
s) reactors, the samples to be irradiated
may be heat sealed in polyethylene vials, usually using double encapsulation.
For long irradiations or higher fluxes (1015
n/cm2
s) encapsulation in quartz is
needed. One never uses Pyrex or other boron-containing glasses due to their high
cross sections for neutron absorption. When irradiating larger samples, one must
pay attention to self-shielding in the samples. For example, the flux reduction in a
0.5-mm Au foil is about 27% due to self-absorption. However, in irradiations of
most liquid samples or geological samples, these self-absorption corrections can
be neglected.
Preparation of the targets for charged particle irradiations requires more effort
due to the large rate of energy loss of charged particles in matter. In general, material
to be irradiated must be in vacuum, thus making the irradiation of liquids and gases
TABLE 19.2 Availability of Transuranium Element Materials
Nuclide t1/2 Decay Mode
Amounts
Available
Specific Activity
(dpm/mg)
237
Np 2.14 Â 106
y a, SF(10210
%) kg 1565.0
238
Pu 87.7 y a, SF(1027
%) kg 3.8 Â 107
239
Pu 2.41 Â 104
y a, SF(1024
%) kg 1.38 Â 105
240
Pu 6.56 Â 103
y a, SF(1026
%) 10–50 g 5.04 Â 106
241
Pu 14.4 y b, a(1023
%) 1–10 g 2.29 Â 108
242
Pu 3.76 Â 105
y a, SF(1023
%) 100 g 8.73 Â 103
244
Pu 8.00 Â 107
y a, SF(0.1%) 10–100 mg 39.1
241
Am 433 y a, SF(10210
%) kg 7.6 Â 106
243
Am 7.38 Â 103
y a, SF(1028
%) 10–100 g 4.4 Â 105
242
Cm 162.9 d a, SF(1025
%) 100 g 7.4 Â 109
243
Cm 28.5 y a, (0.2%) 10–100 mg 1.15 Â 108
244
Cm 18.1 y a, SF(1024
%) 10–100 g 1.80 Â 108
248
Cm 3.40 Â 105
y a, SF(8.3%) 10–100 mg 9.4 Â 103
249
Bk 320 d b,a(1023
%),
SF(1028
%)
10–50 mg 3.6 Â 109
249
Cf 350.6 y a, SF(1027
%) 1–10 mg 9.1 Â 106
250
Cf 13.1 y a, SF(0.08%) 10 mg 2.4 Â 108
252
Cf 2.6 y a, SF(3.1%) 10–1000 mg 1.2 Â 109
254
Cf 60.5 d SF, a(0.3%) mg 1.9 Â 1010
253
Es 20.4 d a, SF(1025
%) 1–l0 mg 5.6 Â 1010
254
Es 276 d a 1–5 mg 4.1 Â 109
257
Fm 100.5 d a, SF(0.2%) 1 pg 1.1 Â 1010
19.3 TARGETRY 585
more difficult. Solids must be in the form of thin foils or deposits on thin backing
material. Typical backing materials are carbon, aluminum, beryllium, and titanium.
The typical measure of thickness of accelerator targets is in units of areal density
(mass/unit area, i.e., mg/cm2
). The thickness expressed in units of areal density
(mg/cm2
) is the linear thickness (cm) multiplied by the density (mg/cm3
). One
can weigh very thin samples and determine their area and thus their areal density.
Typical thicknesses of accelerator targets are 0.1–5 mg/cm2
but depend, of
course, on the rate of energy loss of the irradiating ion in passing through the
target material. Target backings are frequently 10–100 mg/cm2
. Such thicknesses
qualify as being “thin,” that is, easily breakable and require special preparation
techniques.
Because of the high rate of energy loss of charged particles in matter, one must
pay attention to cooling the targets or in some way dissipating the energy deposited
in the target material as the beam nuclei pass through it. For example, consider the
irradiation of a 0.5-mg/cm2 208
Pb target by 450-MeV 86
Kr ions. Each Kr ion passing
through the target deposits 8.1 MeV in the target. If the Kr beam intensity is
1 particle-mA (6.24 Â 1012
ions/s), then the rate of energy deposit in the target is
5 Â 1013
MeV=s % 8:1 J=s. If the foil has an area of 2 cm2
, it would have a
mass of 1 g. The specific heat of Pb is 0.130 J/g/8C. Thus, in the absence of any
cooling, the temperature of the target would rise 8.18/s, and the foil would soon
melt. Since the foil will generally be in vacuum, without further intervention, it
would only cool radiatively, which will not suffice. Heat transfer from such a foil
can occur by clamping it to the front of a cooled block (remembering now that the
entire beam energy will be dumped into the cooling block). Alternately, a jet of a
light gas such as helium can be used to cool the backside of the foil or the
particle beam can be spread over a larger area foil thus reducing the temperature
increase.
Over the years, a number of specialized techniques have been developed for the
preparation of accelerator targets. These techniques are also used in the preparation
of thin sources for counting, such as those used in a or b spectroscopy. The first and
simplest technique for depositing a target material on a backing foil is by evaporation
of a solution of the desired material on the foil. Generally, this is a poor
choice as the solute tends to deposit at the edges of the dried droplet, leading to variations
of up to a factor of 100 in thickness over the area of the deposit. Uniformity
can be improved by using a spreading agent such as insulin to coat the surface of the
backing material prior to evaporation.
A method that was widely used in the past is electrospraying. A solution of the
nuclide to be deposited is prepared in a volatile, nonconducting liquid like acetone,
alcohol, and the like. A capillary is drawn out to a fine point such that no liquid can
escape under normal conditions and filled with the solution. A fine wire is threaded
though the capillary to within a few millimeters of the tip. A high voltage (3–10 kV)
is applied between the wire in the tube and the backing material on which the deposit
is to be made. One gets a spray of charged drops that are collected on the backing
material, placed 1 cm from the capillary. The volatile solvent evaporates
leaving a uniform film. The deposit can be calcined.
586 RADIOCHEMICAL TECHNIQUES
Another widely used technique for preparing thin deposits on a backing material
is electrodeposition. Two types of electrodeposition are commonly used: (a) the
direct deposit of a metal on a cathodic surface by reduction or (b) precipitation of
a cationic species in an insoluble form on an electrode. This latter technique is
widely used to deposit actinides and lanthanides. A 10- to 100-mL aqueous solution
of the actinide or lanthanide is mixed with 15 mL of isopropyl alcohol and placed
in a plating cell (Fig. 19.1). The inorganic material forms a positively charged
complex in which the inorganic molecule is surrounded by a cluster of solvent molecules.
A high voltage (600 V) is applied between a rotating anode and the cathodic
backing material. The positively charged complex is attracted to the cathode of
the cell. The lanthanides/actinides precipitate as hydrous oxides near the cathode,
which is a region of high pH. The alcohol is withdrawn from the cell and the
deposit is dried and calcined. This technique is called molecular plating because
the film is not that of the metal but some molecular form of it. Deposit thicknesses
are restricted to ,0.5 mg/cm2
but the deposition is rapid and quantitative and
allows the use of active metals such as Al as backing foils (Parker and Slatis, 1966).
Vacuum deposition is a well-established technique for making very uniform
deposits of non-refractory materials on a backing material. In Figure 19.2, we
show a typical simple evaporation apparatus.
The material to be evaporated is placed in a sample holder. Frequently, these
sample holders are indented strips of W, Ta, or Mo, or wire baskets of the same
metals, or carbon crucibles. These sample holders can be heated resistively by
passing a large current through them, thus melting and then volatilizing the material.
Alternatively, the sample holder can be bombarded by low-energy electrons to heat
the sample. The entire process takes place in vacuum. Under reduced pressure, most
Figure 19.1 Schematic diagram of molecular plating cell. (From Wang et al., 1975.)
19.3 TARGETRY 587
materials melt readily and then evaporate. The substrate on which the vapors from
the heated sample condense is placed some distance from the source of evaporating
material. The area of the deposit may be defined by collimators. The deposits produced
by vacuum evaporation are very uniform, but the process is not efficient, with
,1% of the sample material being deposited in typical applications. Self-supporting
deposits can be prepared if the substrate on which the vapors condense is coated with
a release agent prior to evaporation.
The thicknesses of these thin targets can be measured using a variety of techniques.
The simplest and most reliable method is weighing. One weighs a known
area of target material and computes its areal density. Such a technique will give
the average deposit thickness but no information about its uniformity or composition.
The uniformity of targets or thin foils can be measured with a-particle
thickness gauges. In such devices, a collimated beam of low-energy a particles
passes through the foil whose thickness is to be measured. Changes in count rate
are noted as the beam scans over the area of the target or foil. To get absolute thicknesses
from such devices, calibration with foils of known thickness is needed.
This technique works best when the energy of the a particle is so low as to
barely pass through the foil. In this case, small changes in thickness are magnified
in the observed count rate. X-ray fluorescence or neutron activation analysis can
be used to determine the elemental composition of the target or source material.
Figure 19.2 Simple schematic diagram of a vacuum deposition apparatus. (From Wang
et al., 1975.)
588 RADIOCHEMICAL TECHNIQUES
Approximate values of the target thicknesses can also be obtained by noting the
energy loss of monoenergetic ions as they pass through a foil. The thicknesses are
calculated from the observed dE/dx and empirical stopping power relationships.
19.4 MEASURING BEAM INTENSITY AND FLUXES
Measurement of charged particle beam intensities is largely done using physical
methods, although some older radiochemical methods are used occasionally. The
most common techniques to measure the intensity of a charged particle beam is
through the use of a Faraday cup. The beam is stopped in an electrically isolated
section of beam pipe referred to as a Faraday cup (Fig. 19.3).
The collected charge is measured with an electrometer that functions as a current
integrator. The beam intensity is just the current divided by the charge on each ion.
Care must be taken regarding the loss of secondary electrons. (The beam will liberate
secondary electrons in the material in which it stops.) If these electrons escape,
their positive ion partners will add to the positive charges of the stopped ions, thus
causing an overestimate of the beam current. The Faraday cup is thus made as a long
cylinder to inhibit electron escape geometrically, and a magnetic field is applied to
the cup along with a suppressing voltage to further prevent electron loss.
When the energy of the charged particle beam is too large to easily stop the beam
in a Faraday cup, the beam intensity is frequently monitored by a secondary ionization
chamber. These ion chambers have thin entrance and exit windows and
measure the differential energy loss when the beam traverses them. They must be
calibrated to give absolute beam intensities. If the charged particle beam intensity
is very low (,106
particles/s), then individual particles can be counted in a
plastic scintillator detector mounted on a photomultiplier tube.
When performing irradiations with neutrons or high-energy protons, it is
common to measure the beam intensity using a monitor reaction. A thin foil of a
Figure 19.3 Schematic diagram of a Faraday cup. (From Lefort, 1968.)
19.4 MEASURING BEAM INTENSITY AND FLUXES 589
pure element is placed in the irradiating flux near the target and irradiated simultaneously
with the target. Then both the reaction products from the target foil and
the monitor foil are collected and counted. The flux is calculated using a known
cross section for the monitor reaction.
Assuming that the monitor and target foils are exposed to the same irradiating
flux, we have, for the activity of the monitor and target foils, Amon and Atgt, respectively,
at the end of the irradiation:
Atgt ¼ Nrsrf(1 À eÀl
r
t
)
Amon ¼ Nmsmf(1 À eÀl
m
t
)
where Ni, si, li are the number of target atoms, cross section, and product decay
constant, respectively, for the ith reaction. The cross section for the reaction of
interest, sr, is
sr ¼
Atgt
Nrf(1 À eÀlrt)
Substituting for f in terms of the monitor reaction, we get
sr ¼
Atgt
Amon
†
Nr
Nmon
†
(1 À eÀlmt
)
(1 À eÀlrt)
† smon
If the irradiation is long enough to produce saturation activities in both the target
and monitor foils, we have
sr ¼
Atgt
Amon
†
Nr
Nmon
† smon
For high-energy protons, typical monitor reactions are 27
Al ! 24
Na,
27
Al ! 22
Na, 12
C ! 7
Be, 27
Al ! 18
F, 197
Au ! 149
Tb, and 12
C(p,pn)11
C where
the arrows indicate a complex set of reaction paths leading from the initial
nucleus to the product nucleus. Care must be taken in the case of reactions producing
24
Na or 11
C to correct for secondary neutron-induced reactions that produce these
nuclides. In high-energy reactions, the loss of recoils from the monitor or target
foils can be corrected for by irradiating a stack of three identical foils and only
counting the center foil. The forward-going recoils from the first foil enter the
second foil and compensate for its forward recoil loss. The backward recoils from
the third foil enter the second foil and compensate for its backward recoil loss.
The measurement of neutron fluxes by foil activation is more complicated
because the neutrons are not monoenergetic and the monitor cross sections are
energy dependent. The simplest case is monitoring slow neutron fluxes. Radiative
capture (n,g) reactions have their largest cross sections at thermal energies and
are thus used in slow neutron monitors. Typical slow neutron activation detectors
are Mn, Co, Cu, Ag, In, Dy, and Au. Each of these elements has one or more odd
A isotopes with a large thermal (n,g) cross section, 1–2000 barns. The (n,g)
590 RADIOCHEMICAL TECHNIQUES
reaction products have half-lives ranging from minutes to hours. The activation
cross sections generally vary as 1/v, although some nuclides have resonances in
the capture cross sections for neutrons with energies between 1 and 1000 eV. A
correction for such resonance capture can be made by irradiating the monitor foils
with and without a Cd cover. The (n,g) cross section for Cd below 0.4 eV is very
large and is small for energies above this, and thus very few low-energy neutrons
penetrate the Cd cover. Irradiation of a foil without a Cd cover will cause reactions
induced by both thermal and resonance neutrons, while the Cd-covered foil will just
respond to resonance neutrons.
One can also use so-called threshold monitor detectors where the activating reaction
has an energy threshold, such as the (n,a), (n,p), and (n,2n) reactions. By exposing
a set of threshold detectors (involving different reactions with different
thresholds) to a neutron flux, one can determine the relative amounts of different
energy groups in the neutron spectrum. Further information about the use of activation
detectors to measure neutron fluxes can be found in the textbooks by Knoll
(2000) and Tsoulfanidis (1995).
19.5 RECOILS, EVAPORATION RESIDUES (EVRs),
AND HEAVY RESIDUES
In a nuclear reaction, the momentum transfer to the struck nucleus is not negligible.
If an A ¼ 100 nucleus fuses completely with a 100-MeV a-particle projectile, the
kinetic energy of the completely fused system is 4 MeV. A similar fusion of an
A ¼ 100 nucleus with a 100-MeV 16
O projectile will give the completely fused
system an energy of 13.8 MeV. These energies are extremely large compared to
chemical bond energies. Depending upon the position in the target foil where the
nuclear reaction takes place, some or all of these recoiling nuclei may escape
from the target foil. These recoil nuclei, which are usually radioactive, can be collected
or studied using physical or radiochemical techniques. In reactor irradiations,
these recoils produce “contamination” on the surface of irradiation containers.
When these heavy recoil nuclei are the result of a complete fusion of the projectile
and target nuclei, they are usually called evaporation residues because they
result from a deexcitation of the primary complete fusion product by particle evaporation
(emission). In intermediate energy and relativistic nuclear collisions, the
momentum transfer to the target nucleus is much less, and the energy of the recoiling
nucleus is 5–100 keV/nucleon. Such recoils are usually called heavy residues.1
1
It should be noted that in this discussion we are tacitly assuming “normal” reaction kinematics with the
lighter nucleus being the projectile (that is in motion) and the heavier collision partner being at rest in the
laboratory system. In reactions studied using inverse kinematics with a heavier projectile striking a lighter
target nucleus, the momentum of the recoiling heavy nucleus is approximately the same as that of the projectile
nucleus. In inverse kinematics reactions, the energies of the EVRs or heavy residues are large and
their spatial and energy distribution is compressed accordingly. Collection of these recoils is relatively
easy, but high resolution is needed because of the spatial and energy compression.
19.5 RECOILS, EVAPORATION RESIDUES (EVRs), AND HEAVY RESIDUES 591
There are a variety of ways to collect the recoiling heavy products of a nuclear
reaction. One radiochemical technique is the so-called thick target–thick catcher
method. Here a target foil whose thickness exceeds the average range of the
recoils is surrounded by catcher foils of C or Al or some other material whose thickness
exceeds the range of the recoiling product nucleus, which will not lead to
production of the nuclide of interest. The average range of the recoiling product,
kRl, (which can be related to its total kinetic energy) is given as
kRl ¼
NcW
Nc þ Nw
where Nc is the number of recoils that escape from the target, Nw is the number that
remains in the target, and W is the thickness of the target. The fraction of product
nuclei that recoil into the forward catcher foil, F, and the fraction that recoil into
the backward catcher foil, B, can be used to deduce something about the relative
velocity imparted to the recoiling nucleus by the initial projectile–target interaction,
v, and the isotropic velocity kick, V, given to it by successive momentum kicks by
sequential particle emission. Formally, we define the quantity h where h ¼ v=V.
Also h can be related to F and B:
h ¼
(F=B)1=2
À 1
(F=B)1=2
þ 1
So the thick target–thick catcher method can lead to crude information about the
kinematics of the nuclear reaction under study (Harvey, 1960). This technique
can be used to advantage in the study of intermediate energy and relativistic
nuclear collisions where the energy of the heavy residues is low (10–100 keV/
nucleon). In this case, most of the residues stop in the target foil and cannot be
studied any other way.1
For the study of recoils in low-energy and some intermediate-energy reactions,
one can use a thin target (,0.5 mg/cm2
). The energy loss of the recoils in emerging
from these targets is negligible or calculable and tolerable. With thin target
irradiations, one can stop the recoiling nuclei in a catcher foil, which can be
counted, perhaps after intervening chemical separations to isolate the products of
interest. The “catcher foil” can take the form of a tape or rotating wheel that can
rapidly transport the activity to a remote, low background location for counting.
Alternatively, the catcher foil can take the form of a stack of thin foils that stops
the products. These foils, upon disassembly and counting, can be used to construct
a differential range distribution for products of interest. Catcher foils or stacks of
foils can be mounted at various angles with respect to the incident beam and can
be used to measure product angular or energy distributions. These catcher foil techniques
are now only used to study reactions with very low cross sections where their
use provides a high detection sensitivity.
The catcher foil can take the form of a jet of rapidly moving gas, a helium jet. The
atoms produced in a nuclear reaction recoil out of a thin target and are stopped in
21 atm of helium gas in the target chamber. The gas contains an aerosol, typically
592 RADIOCHEMICAL TECHNIQUES
an alkali halide or more recently, carbon clusters, to which charged reaction product
recoils attach themselves via van der Waals forces. The helium gas (and the aerosol
particles) escapes through a small orifice to a vacuum chamber, with the gas achieving
sonic velocity. The gas and aerosol can be transported for substantial distances in
thin capillary tubes. The aerosol particles (and the attached atoms) are collected by
allowing the gas stream to strike a collector surface. The resulting deposit can be
counted directly or dissolved for further chemical processing. The aerosol loaded
gas stream (jet) can also be used to transport the atoms through a thin capillary a
distance of several meters in a few seconds. If the aerosol particles are carbon clusters,
they can be burned in oxygen, freeing the transportal activity. If the carrier gas
is extremely pure helium gas, then the residues will remain ionized and can be collected
using electrostatic devices. A number of systems have been developed and
used to collect reaction products as ions without involving aerosols in the carrier gas.
The principal limitation of the isolation devices discussed previously (tapes, jets,
etc.) is that the reaction product must be stopped and mechanically transported to
radiation detectors before product identification can occur. This restricts their use
to studies of nuclei whose t1
2
. 1 ms. For detection and identification of species
whose t1
2
, 1 ms, one employs instruments based upon direct magnetic and/or electrostatic
deflection of target recoils. The most spectacularly successful of these
devices is the velocity filter SHIP (Fig. 15.7). Evaporation residues produced in
compound nucleus reactions emerge from the target and pass through a thin
carbon foil that has the effect of equilibrating the ionic charge distribution of the
residues. The ions then pass through two filter stages consisting of electric deflectors,
dipole magnets, and a quadruple triplet for focusing. The solid angle of acceptance
of the separator is 2.7 msr with a separation time for the reaction products of
2 ms with a total efficiency of collecting evaporation residues of 20% for
Aproj . 40. Since complete fusion evaporation residues have very different
velocities and angular distributions than targetlike transfer and deep-inelastic products
(a factor of 2 difference in velocity between transfer products and evaporation
residues) and beam nuclei, the separator with its +5% velocity acceptance
range and narrow angular acceptance very effectively separates the evaporation residues
from the other reaction products and the beam. Following separation, the residues
pass through a large-area time-of-flight detector and are stopped in an array of
position-sensitive detectors. From their time of flight and the energy deposited as
they stop in the position-sensitive detectors, a rough estimate of their mass may
be obtained (dA/A 0.01). The final genetic identification of the residues is made
by recording the time correlations between the original position signals from the
detectors and subsequent decay signals from the same location (due to a or spontaneous
fission decay) and/or signals from g or X-ray detectors placed next to the
position-sensitive detector. This device for studying heavy-element evaporation
residues was used in the discovery of elements 107–112 (Chapter 15).
An alternate method of recoil collection and separation from other reaction products
is the gas-filled separator. Unlike the vacuum separators SHIP at GSI or the
Fragment Mass Analyzer at ANL that can only accept a limited range of chargeto-mass
ratios, a gas-filled separator is a magnetic separator that is filled with a
19.5 RECOILS, EVAPORATION RESIDUES (EVRs), AND HEAVY RESIDUES 593
low pressure (1 torr) gas (usually H or He). Evaporation residues emerging from
the target undergo charge changing collisions in the gas and quickly equilibrate to a
common average charge state. The change in recoil charge states with gas pressure
in a gas-filled magnet is shown in Figure 19.4. At a pressure of 1 torr, all recoils
have a common average charge state. This charge state will differ from the charge of
the projectile nuclei or other reaction products. This charge equilibration allows a
very efficient collection of the recoils but with a loss of the detailed mass selection
obtained in vacuum separators like SHIP that use both electric and magnetic fields
for separation. The appropriate magnetic rigidity to collect a given product nuclide
with a gas-filled separator can be roughly described by the simple formula
Br ¼
mv
qave
¼
mv
(v=v0)eZ1=3
¼
0:0227A
Z1=3
Tm
where m, Z, and v are the mass, atomic number, and velocity of the recoiling ion,
respectively, and v0 is the Bohr velocity, 2.18 Â 106
m/s. [In reality, qave is a sensitive
function of the atomic structure of the recoiling ion and the gas (Ghiorso et al.,
1988; Oganessian et al., 2001).] The magnetic rigidity is proportional to the recoil
velocity so that recoils of the same charge and velocity are focused. The primary
Distance along Focal Plane (cm)
MAGNETIC RIGIDITY Br (T-m)
50 60 70 80
0
200
400
0
200
400
0
500
1500
0.75 0.80 0.85
B=1.270 T
25+
Torr
10-6
24+
23+
22+
26+
27+
2¥10-3 Torr
2¥10-2 Torr
Counts/Channel
(a)
(b)
(c)
PN2
=
0.90
1000
(d)
(e)
(f )
PN2
=
50 60 70 80
0
200
400
0
100
200
300
100
200
0.5 Torr
2.0 Torr
8.0 Torr
300
0.85
58Ni E=350MeV
B=1.426 T
0.90 0.95 1.001.05
(g)
(h)
(i )
(j)
(k)
40 50 60 70 80
0
200 24 Torr
20 Torr
16 Torr
0
200
0
200
0
200
0
300
600
PN2
=
B=1.312 T
8.0 Torr
12 Torr
0.75 0.80 0.85 0.90
Figure 19.4 Changes in focal plane distributions in a gas-filled magnet for the reaction
involving 350-MeV 58
Ni ions. (From Paul, 1989.)
594 RADIOCHEMICAL TECHNIQUES
beam is separated from the recoils right after the target by a dipole bending magnet
followed by two quadrupole magnets that focus the beam onto the focal plane. For
the study of heavy-element production reactions, collection efficiencies of 25–50%
can be achieved.
19.6 RADIOCHEMICAL SEPARATION TECHNIQUES
In the study of nuclear reactions, nuclear structure, and the heaviest elements, one frequently
needs to chemically separate the nuclide(s) of interest from other radioactive
species that are present. This is done by performing radiochemical separations that
involve the conventional separation techniques of analytical chemistry adapted to
the special needs of radiochemistry. For example, radiochemical purity is generally
more important than chemical purity. When dealing with short-lived nuclides,
speed may be more important than yield or purity. The high cost of radioactive
waste disposal may require unusual waste minimization steps. As noted earlier, radiochemical
separations need not be quantitative. One only needs to know the yield.
Because of the availability of modern high-resolution counting equipment, such as
Ge g-ray spectrometers, modern radiochemical separations frequently are designed
only to reduce the level of radioactive impurities in the sample rather than producing
a pure sample. [The counting instrumentation is used to “isolate” the nuclide(s) of
interest from other nuclides.] Thus, modern procedures sometimes are similar to
qualitative analysis schemes, breaking products into chemically similar groups and
using instrumentation to further separate the group members. A recent review summarizes
some newer developments of relevance to radiochemistry (Bond et al., 1999).
19.6.1 Precipitation
The oldest, most well-established chemical separation technique is precipitation.
Because the amount of the radionuclide present may be very small, carriers are frequently
used. The carrier is added in macroscopic quantities and ensures the radioactive
species will be part of a kinetic and thermodynamic equilibrium system.
Recovery of the carrier also serves as a measure of the yield of the separation. It
is important that there is an isotopic exchange between the carrier and the radionuclide.
There is the related phenomenon of co-precipitation wherein the radionuclide
is incorporated into or adsorbed on the surface of a precipitate that does
not involve an isotope of the radionuclide or isomorphously replaces one of the
elements in the precipitate. Examples of this behavior are the sorption of radionuclides
by Fe(OH)3 or the co-precipitation of the actinides with LaF3. Separation
by precipitation is largely restricted to laboratory procedures and apart from the
bismuth phosphate process used in World War II to purify Pu, has little commercial
application.
As a practical matter, precipitation is usually carried out in hot, dilute aqueous
solutions to allow the slow formation of large crystals. The pH of the solution
is chosen to minimize colloid formation. After precipitation, the precipitate is
19.6 RADIOCHEMICAL SEPARATION TECHNIQUES 595
washed carefully to remove impurities, dissolved, and reprecipitated to cause further
purification. The precipitate is collected by filtration (Fig. 19.5).
The filter paper is supported by a glass frit clamped between two glass tubes. The
precipitate is washed finally with acetone or alcohol to dry it. The precipitate is
chosen to have a known stoichiometry to allow calculation of the yield of the separation
and should not absorb water or CO2 so that an accurate weight can be
obtained. (The filter paper used in the filtration must be treated with all the reagents
beforehand, dried, and weighed so that any material loss in filtration is minimized.)
19.6.2 Solvent Extraction
Separation by liquid–liquid extraction (solvent extraction) has played an important
role in radiochemical separations. Ether extraction of uranium was used in early
weapons development, and the use of tributyl phosphate (TBP) as an extractant
for U and Pu was recognized in 1946, resulting in the commercial PUREX
process for reprocessing spent reactor fuel (see Chapter 16). In recent years, there
has been a good deal of development of solvent extraction processes for the
removal of transuranic elements, or 90
Sr, and 137
Cs from acidic high-level waste.
Laboratory demonstrations of the TRUEX process that uses the neutral extractant
CMPO [octyl(phenyl)-N,N-diisobutylcarbamoyl-methylphosphine oxide] to separate
the transuranium elements from acidic high-level waste have been successful.
More recently crown ethers have been used as specific extractants for Sr and Cs.
Figure 19.5 Schematic diagram of a filtration apparatus used in radiochemistry.
596 RADIOCHEMICAL TECHNIQUES
In solvent extraction, the species to be separated is transferred between two
immiscible or partially miscible phases, such as water and a nonpolar organic
phase. To achieve sufficient solubility in the organic phase, the species must be in
the form of a neutral, nonhydrated species. The transfer between phases is achieved
by selectively complexing the species of interest causing its solubility in water to
decrease with a concomitant increase in its solubility in the organic phase.
A hydrated metal ion (MZþ
) will always prefer the aqueous phase to the organic
phase. To get the metal ion to extract, some or all of the inner hydration sphere must
be removed. The resulting complex must be electrically neutral and organophilic,
that is, have an organic “surface” that interacts with the organic solvent. This can
be done by:
1. Forming a neutral complex MAZ by coordination with organic anions A2
.
2. Replacing water in the inner coordination sphere by large organic molecules B
such that one forms MBZþ
N , which is extracted into the organic phase as an ion
association complex (MBN)Zþ
LZÀ
x .
3. Forming metal complexes of form MLZÀN
N with ligands (L) such that they
combine with large organic cations RBþ
to form ion pair complexes
(RBþ
)NÀZ(MLN)NÀZ
.
The extracting agents are thus divided into three classes: polydentate organic
anions A2
, neutral organic molecules B, or large organic cations RBþ
.
Polydentate organic anions, which form chelates (ring structures of four to seven
atoms) are important extracting agents. Among these are the b-diketonates, such as
acetylacetonate, the pyrazolones, benzoylacetonate, and thenoyltrifluoroacetone
(TTA), with the extraction increasing strongly through this sequence. Representing
the organic chelating agent as HA, the overall reaction involved in the chelate
extraction of a metal ion, Mnþ
, is
Mnþ
(aq) þ nHA(o) ÀÀ! MAn(o) þ nHþ
(aq)
When an aqueous solution containing extractable metal ions is brought into contact
with an organic phase containing chelating agent, the chelating agent dissolves in
the water phase, ionizes, complexes the metal ion, and the metal chelate dissolves
in the organic phase. The low solubility of the metal complexes and their slow
rates of formation limit the industrial use of this type of anionic extraction.
However, a number of organophosphorus compounds are efficient extractants as
they and their complexes are very soluble in organic solvents. The most important of
these are monobasic diethylhexylphosphoric acid (HDEHP) and dibutylphosphoric
acid (HDBP). The actinide MO2þ
2 ions are very effectively extracted by these
reagents as are the actinide (IV) ions.
Among the neutral extractants, alcohols, ethers, and ketones have been used
extensively. The most famous example of these is the extraction of uranyl nitrate
into diethyl ether, the process used in the Manhattan Project to purify the uranium
19.6 RADIOCHEMICAL SEPARATION TECHNIQUES 597
used in the first reactors. In one of the early large-scale processes (the Redox
process) to recover uranium and plutonium from irradiated fuel, methyl isobutyl
ketone was used to extract the actinides as nitrates.
The most widely used neutral extractants, however, are the organophosphorus
compounds, of which the ester, TBP, is the most important. TBP forms complexes
with the actinide elements thorium, uranium, neptunium, and plutonium by bonding
to the central metal atom via the phosphoryl oxygen in the structure
(C4H9O)3Pþ
OÀ
The overall reactions are
MO2þ
2 (aq) þ 2NOÀ
3 (aq) þ 2TBP(o)$MO2(NO3)2 † 2TBP(o)
or
M4þ
(aq) þ 4NOÀ
3 (aq) þ 2TBP(o)$M(NO3)4(TBP)2(o)
These equilibria can be shifted to the right, increasing the degree of extraction by
increasing the concentration of uncombined TBP in the organic phase or by increasing
the concentration of [NOÀ
3 (aq)]. The latter increase is achieved by adding a
salting agent such as HNO3 or Al(NO3)3. These extraction equilibria are the basis
of the PUREX process, used almost exclusively in all modern reprocessing of
spent nuclear fuel.
A third group of extractants (the cationic extractants) are the amines, especially
the tertiary or quarternary amines. These strong bases form complexes with actinide
metal cations. The efficiency of the extraction is improved when the alkyl groups
have long carbon chains, such as trioctylamine or triisooctylamine. The extraction
is conventionally thought of as a “liquid anion exchange” in that the reaction for
metal extraction can be written as an anion exchange, that is,
xRBþ
LÀ
(o) þ MLÀx
n $ (RBþ
)xMLx
n(o) þ xLÀ
where MLÀx
n is the metal anion complex being extracted and RBþ
is the ammonium
salt of the amine. Hexavalent and tetravalent actinides are efficiently extracted using
this technique while trivalent actinides are not well extracted under ordinary
conditions.
As a practical matter, the distribution ratio D is defined as
D ¼ ½MŠorg=½MŠaq
where [M]i is the metal ion concentration in the ith phase. The relevant equilibria,
for example, to describe the extraction in systems of lipophilic acidic chelating
598 RADIOCHEMICAL TECHNIQUES
agents are
HLorg
À!À HLaq
HLaq À!À Hþ
þ LÀ
aq
M3þ
aq þ 3HLorg À!À (ML3)org þ 3Hþ
aq
where Ke is the equilibrium constant for the last reaction. The distribution coefficient
D is given as:
D ¼ ½ML3Šorg=½M3þ
Šaq ¼ Ke½HLŠ3
org=½Hþ
Š3
aq
If one introduces a water-soluble complexing agent into the system, the [M3þ
]aq
becomes [M3þ
] þ [MX2þ
] þ [MX2
þ
] þ . . . and the measured distribution ratio will
include these species as well. The separation factor between two ions, S, is given
by the ratio of their distribution coefficients:
S ¼ DA=DB
Thus, the most effective separations will involve cases where the target ion interacts
strongly with the extractant but is less strongly complexed by the aqueous ligand
X. The percent extraction is given by:
% extraction ¼
100D
D þ
Vaq
Vorg
 
where Vi is the volume of the ith phase.
19.6.3 Ion Exchange
Ion exchange is one of the most popular radiochemical separation techniques due to
its high selectivity and the ability to perform separations rapidly. In ion exchange, a
solution containing the ions to be separated is brought into contact with a synthetic
organic resin containing specific functional groups that selectively bind the ions in
question. In a later step the ions of interest can be removed from the resin by elution
with another suitable solution that differs from the initial solution. Typically, the solution
containing the ions is run through a column packed with resin beads. The
resins are typically crosslinked polystyrenes with attached functional groups.
Most cation exchangers (such as Dowex 50) contain free sulfonic acid groups,
SO3H, where the cation displaces the hydrogen ion. Anion exchangers (such as
Dowex 1) contain quaternary amine groups, such as CH2N(CH3)3Cl where the
anion replaces the chloride ion. The resin particles have diameters of 0.08–
0.16 mm and exchange capacities of 3–5 meq per gram of dry resin.
19.6 RADIOCHEMICAL SEPARATION TECHNIQUES 599
It is common to absorb a group of ions on the column material and then selectively
elute them. Complexing agents, which form complexes of varying solubility
with the absorbed ions, are used as eluants. A competition occurs between the complexing
agent and the resin for each ion, and each ion will be exchanged between the
resin and the complexing agent several times as it moves down the column. This is
akin to a distillation process. The rates at which the different ions move down the
column vary, causing a spatial separation between “bands” of different ions. The
ions can be collected separately in successive eluant fractions (see Fig. 19.6).
The most widely cited application of ion exchange techniques is the separation of
the rare earths or actinides from one another. This is done with cation exchange
using a complexing agent of a-hydroxyisobutyric acid (“a-but”). The order of
elution of the ions from a cation exchange column is generally in order of the
radii of the hydrated ions with the largest hydrated ions leaving first; thus lawrencium
elutes first and americium last among the tri-positive actinide ions (see
Fig. 19.6). In the case of the data of Figure 19.6, the separation between adjacent
cations and the order of elution is derived from the comparative stability of the
aqueous actinide or lanthanide complexes with a-hydroxyisobutyrate. As shown
in Figure 19.6, there is a strikingly analogous behavior in the elution of the actinides
Figure 19.6 Elution of tripositive lanthanide and actinide ions on Dowex-50.
600 RADIOCHEMICAL TECHNIQUES
and lanthanides that allowed chemists to prove the identity of new elements in the
discovery of elements 97–102 (Bk–No). For cation exchange, the strength of
absorption goes as M4þ
. M3þ
. MO2
2þ
. M2þ
. MO2
þ
.
The anion exchange behavior of various elements has been extensively studied.
For example, consider the system of Dowex 1 resin and an HCl eluant. Typical distribution
ratios for various elements as a function of [Cl2
] are shown in Figure 19.7.
Note that groups 1, 2, and 3 are not absorbed on the column. One usually sees a rise
in the distribution coefficient D until a maximum is reached, and then D decreases
gradually with further increases in [Cl2
]. The maximum occurs when the number of
ligands bonding to the metal atom equals the initial charge on the ion. The decrease
in D with further increases in eluant concentration is due to free anions from the
eluant competing with the metal complexes for ion exchange resin sites.
Figure 19.7 or similar data can be used to plan separations. For example, to separate
Ni(II) and Co(II), one needs simply to pass a 12 M HCl solution of the elements
through a Dowex 1 column. The Co(II) sticks to the column while the Ni(II) is
not absorbed. A mixture of Mn(II), Co(II), Cu(II), Fe(III), and Zn(II) can be separated
by being placed on a Dowex 1 column using 12 M HCl, followed by elutions
with 6 M HCl (Mn), 4 M HCl (Co), 2.5 M HCl (Cu), 0.5 M HCl (Fe), and 0.005 M
HCl (Zn).
In addition to the organic ion exchange resins, some inorganic ion exchanges,
such as the zeolites, have been used. Inorganic ion exchangers are used in situations
where heat and radiation might preclude the use of organic resins, although the
establishment of equlibria may be slow.
Figure 19.7 Elution of elements from anion exchange resin. (From Kraus and Nelson,
1956.)
19.6 RADIOCHEMICAL SEPARATION TECHNIQUES 601
Newer developments have emphasized the preparation of more selective resins.
Among these are the chelating resins (such as Chelex-100) that contain functional
groups that chelate metal ions. Typical functional groups include iminodiacetic
acids, 8-hydroxyquinoline, or macrocyclic units such as the crown ethers,
calixarenes, or cryptands. The bifunctional chelating ion exchange material, Diphonix
resin—a substituted diphosphonic acid resin, shows promise in treating radioactive
waste. Important newer resins include those with immobilized phosphorus
ligands (Bond et al., 1999).
19.6.4 Extraction Chromatography
Extraction chromatography is an analytical separation technique that is closely
related to solvent extraction. Extraction chromatography is a form of solvent extraction
where one of the liquid phases is made stationary by adsorption on a solid
support. The other liquid phase is mobile. Either the aqueous or the organic phase
can be made stationary. Extraction chromatography has the selectivity of solvent
extraction and the multistage character of a chromatographic process. It is generally
used for laboratory-scale experiments, although some attempts have been made to
use it in larger scale operations. The common applications involve the adsorption
of an organic extractant onto a variety of inorganic substrates such as silica or
alumina or organic substrates such as cellulose or styrene-divinyl benzene copolymers.
When the stationary phase is organic, the technique is referred to as
reversed-phase high-performance liquid chromatography. The stationary phase is
used in a column just as in ion exchange chromatography. High-pressure pumps
are usually used to force the liquid phase through these columns, just as in conventional
high-performance liquid chromatography.
The same extracting agents as used in solvent extraction can be used in extraction
chromatography. Early applications of extraction chromatography have employed
various traditional extractants such as the acidic organophosphorus compounds
[di-(2-ethylhexyl) phosphoric acid, HDEHP] or TBP as extractants for the actinide
elements. Recent advances have led to a variety of new solvent exchange extractants
such as the crown ethers, cryptands, or bifunctional organophosphorus compounds.
A particularly successful application is the selective sorption of actinides on TRU
resins, involving solutions of carbamolymethyl-phosphoryl (CMPO) compounds
in TBP sorbed on Amberlite XAD-7. This resin has found a number of applications
in the isolation and subsequent determination of the actinides in complex matrices.
19.6.5 Rapid Radiochemical Separations
Many of the separation techniques we have described take hours to perform. Many
interesting nuclei, such as the heavier actinides, the transactinides, or the light nuclei
used in PET studies, have much shorter half-lives. Thus, we will briefly review the
principles of rapid radiochemical separations (procedures that take seconds to
minutes) and refer the reader to Herrman and Trautman (1982), Meyer and Henry
(1979), Scha¨del et al. (1988), and Trautman (1995) for details.
602 RADIOCHEMICAL TECHNIQUES
In most chemical separation procedures, the goal is to selectively transfer the
species of interest from one phase to another, leaving behind any unwanted species.
The phase-to-phase transfer is rapid, but the procedures to place the species in the
proper form for transfer to occur are slow. The goal of rapid radiochemical separations
is to speed up existing chemical procedures or to use new, very fast chemical
transformations.
Two procedures are commonly used for rapid radiochemical separations, the
batch approach and the continuous approach. In the batch approach, the desired
activities are produced in a short irradiation, separated and counted with the procedure
being repeated many times to reduce the statistical uncertainty in the data.
In the continuous approach, the production of the active species is carried out continuously,
and the species is isolated and counted as produced.
One of the most widely used techniques for rapid chemical separation is that of
gas chromatography, which has been developed for use with the transuranium
elements by Zvara and co-workers (1972). In gas chromatography, volatile elements
or compounds are separated from one another by their differences in distribution
between a mobile gas phase and a stationary solid phase. Thermochromatography
involves passing a gas through a column whose temperature decreases continuously
with distance from the entrance. Thus, the less volatile species condense on the
column walls first with the more volatile species depositing last. Measurement of
the migration times, the deposition temperature, the temperature gradient in the
column, and so on can allow one to deduce the molar enthalpy of absorption of
the compound on the column material. This physical quantity can be compared to
quantum chemical calculations of this quantity to gain insight into the bonding properties
of the element in question. This technique was used to show the chemical
properties (Scha¨del, 1997; Eichler, 2000; Du¨llmann et al., 2002) of the transactinides
Rf–Hs and their behavior relative to their chemical congeners.
Another rapid chemical separation technique is separation by volatilization.
There are a variety of volatile compounds that can be released from an irradiated
material upon dissolution that can, with proper conditions, serve to rapidly chemically
separate the elements involved. Examples of such volatile species include
I2, At, GeCl4, AsCl3, SeCl4, OsO4, RuO4, Re2O7, Tc2O7, and so on. Separation by
volatilization has largely been used for the elements forming volatile hydrides,
As, Se, Sn, Sb, and Te.
19.7 LOW-LEVEL MEASUREMENT TECHNIQUES
19.7.1 Introduction
One of the areas in which the skills of radiochemists are used is the area of low-level
chemistry and low-level counting. Areas as diverse as the detection of solar neutrinos
or the study of environmental radioactivity involve low-level techniques. For
example, despite concentration of the radiotracers of interest during sampling procedures
in environmental studies, quite often one is left with a sample containing
19.7 LOW-LEVEL MEASUREMENT TECHNIQUES 603
a small quantity ,10 ppm of radioactivity that must be assayed. Such essays are
referred to as low-level techniques. Let us begin our discussion of low-level techniques
by considering any chemical manipulations of the sample that must be
made prior to counting it. Understandably, the fact of having activity levels
,10 dpm puts severe restrictions on the nature of low-level chemistry. Among the
requirements for low-level chemistry are a small constant blank, high chemical
yields for all procedures, high radiochemical purity for all reagents employed,
and the ability to place the sample in suitable chemical form for counting.
19.7.2 Blanks
The blank in low-level chemistry is defined as the contribution of the added reagents
and other sample constituents to the activity being measured. The blank is determined
by performing the chemical procedures without the radioactive sample
being analyzed. Care must be taken to ensure that the blank is properly measured
and includes all possible contributions to the activity that would be encountered
in a real system. For example, in the determination of fission product 144
Ce in seawater,
the blank must be determined for each new bottle of reagents used due to the
high variability of the 144
Ce content in chemical reagents.
Clearly, one of the most effective ways of dealing with a blank correction is to
reduce it to its lowest level. Among the factors contributing to the blank correction
that can be reduced with care is radiochemical contamination of analytical reagents
used in chemical procedures. DeVoe (1961) and Sugihara (1961) have written
extensive review articles on this subject, and their work should be consulted for
detailed information. Typical contamination of most reagents is in the range of
10–100 ppm/g reagent, although individual reagents may contain activity
levels of .10,000 ppm/g. Some especially troublesome reagents are rare earths
(Ce salts in particular), chlorine or sulfur-containing reagents that may contain
32
P contamination, cesium salts (which may contain 40
K or 87
Rb), and potassium
salts and other obvious offenders. Precipitating agents, such as tetraphenylborates
and chloroplatinates are also particularly pernicious with regards to contamination
problems.
Airborne contamination is another possible contribution to the blank correction.
Here one is chiefly concerned with sample contamination with the daughters of
222
Rn, which have half-lives in the 30- to 40-min range. Steps that can be taken
to avoid this problem include eschewing the use of suction filtration in chemical procedures,
prefiltering of room air, and use of radon traps.
Further lowering of the blank correction occurs when nonisotopic carriers
in chemical procedures are used to replace inert carriers of the element of
interest when it is difficult to obtain the inert carrier in a contamination-free
condition. Obviously, only clean glassware should be used, reagents should not
be reused, and the laboratory should be kept in an immaculate condition. Separations
that have high chemical yields and high radiochemical purity reduce the
blanks.
604 RADIOCHEMICAL TECHNIQUES
19.7.3 Low-Level Counting—General Principles
Once the low-level radioactive sample has been collected and any chemical procedures
performed prior to counting, it is ready for counting. Because of the extremely
small disintegration rates encountered, special techniques, called low-level
counting, must be used to assay the sample. We shall survey some of the highlights
of this area, which has been the subject of many articles and monographs (Knoll,
2000).
What are the most important characteristics the detector must possess for lowlevel
counting? The first general characteristic is one of stability. Low-level counting
frequently requires counting periods of long duration; hence counter stability is
quite important. If the sample count rate S (source þ background) is only slightly
larger than the background rate B, one’s detector should be picked to maximize
the ratio S2
/B—that is, low background and high efficiency. If the sample count
rate is large with respect to background, one need only maximize S—that is, one
chooses a high-efficiency detector.
19.7.4 Low-Level Counting—Details
For low-level a-particle counting, the choice is generally between the gas-filled ionization
chamber and a semiconductor detector system. The former can have a counting
efficiency of approximately 50% and a background of approximately 3–4 cpm;
the latter has a background rate of approximately 0.5 cph and a detection efficiency
that approaches 50%. The semiconductor detector is usually the detector of choice,
although large sample sizes may be better assayed with gas-filled ionization
chambers. Background radiation is primarily due to a-particle emitting impurities
in the counter, counter support material, and so on, plus the occurrence of cosmicray-induced
(n,a) reactions. Because of its Ra content, Al is not used in constructing
a spectrometers.
Low-level “soft” radiation counting has its own techniques. The term soft radiation
counting refers to detecting EC and low-energy b2
emitters where the selfabsorption
of the radiation in the sample is important. To solve this problem, one
typically tries to incorporate the radionuclide to be counted into the detector
itself. One typical method of assay is liquid scintillation counting, which is used
to assay samples whose activity is greater than 10 pCi. Typical liquid scintillation
counter backgrounds can be as high as approximately 100 cpm, whereas special
counters have been built with background rates of approximately 10 cpm or less.
Liquid scintillation counting is a speedy, simple method of low-level counting.
Another technique that has been used to count low-level soft radiation samples
involves the use of gas-filled proportional counter. The sample to be counted is
converted to gaseous form and added to the counter gas at a concentration of
approximately 0.05 mol % or less. This method of low-level counting, although
tedious and time-consuming, allows one to assay samples whose activity is less
than 0.5 pCi. Typical counter backgrounds are 1–2 cpm with 100% counting
efficiency for energies as low as 10 eV.
19.7 LOW-LEVEL MEASUREMENT TECHNIQUES 605
The counting of tritium in water is a special problem about which much has been
written. Current methods for assay of tritium in water have a range of 0.1–5000 TU,
where a tritium unit (TU) has the value of 7.2 dpm/L. The most desirable feature of
a tritium measuring system is that it be capable of measuring a large number of
samples rapidly, simply, and cheaply as possible with an uncertainty of +10%
or better. It is generally more important to assay 100 samples with an uncertainty
of +10% than to assay 10 samples with an uncertainty of +3%.
The basic feature of low-level b2
counting that distinguishes it from ordinary b2
counting is the use of an anticoincidence shield around the main b2
counters. An
anticoincidence shield is a single detector, or array of detectors, that surrounds
the primary detector. The output of the anticoincidence detector is fed to an anticoincidence
circuit along with the output of the primary detector. When nuclear radiation
passes through both detectors simultaneously, as in the case of a highly
penetrating cosmic ray striking both detectors, no output results from the anticoincidence
circuit. When the anticoincidence circuit receives a signal only from the
primary detector, an output signal results. The net effect is that the anticoincidence
shield detector “guards” or shields against exterior radiation background radiation
entering the primary detector. Typical ring assemblies reduce the background rate
in the primary counter by a factor of 50. A well-designed guard ring will allow
several different types of central counter to be inserted into it. Low background
b2
counters constructed of especially pure materials with anticoincidence shields
have exhibited background rates of 1 cph with efficiencies of approximately 50%.
Low-level counting of g-ray emitters using solid scintillation counters is an extensively
used technique. The most important aspect of low-level solid scintillation
counting is to decrease the counter background. Typical contributions to a solid scintillation
counter’s background rate from various sources are shown in Table 19.3.
Here four factors are seen as the major contributors to the detector background
rate. They are (a) the cosmic ray shield, (b) the atmosphere surrounding the detector,
(c) the detector itself, and (d) the cosmic rays. For the cosmic ray shield about the
detector, it is advisable to use old or virgin lead, that is, lead that was purified
over 100 years ago, thus allowing any 210
Pb present to decay. One should expect
1 cpm/g shield material. Iron can also be used in constructing the detector
TABLE 19.3 Components of a NI(Tl) Scintillation
Counter Background
Outside shield 29.200 cpm
Inside shield
Cosmic ray mesons 116.4 cpm
Cosmic ray neutrons 19.4 cpm
222
Rn daughters 25.9 cpm
40
K 8.6 cpm
Remaining background 33.1 cpm
Total 203.4 cpm
Source: Stenberg and Olsson (1968).
606 RADIOCHEMICAL TECHNIQUES
shield, but care must be taken to ensure that the iron or steel is pre-1945 in origin.
(Iron processed in the post-1945 period has 60
Co contamination due to the use of
60
Co in the blast furnace operation.) Mercury is a very good, easily purified
shield material but it is quite expensive. “Graded” shields consisting of an outer
thick layer of Pb lined with Cd, which in turn is coated with Cu, are used to
reduce the production of X-rays in the Pb shield material. The main portion of
the atmospheric contribution to the detector background is due to radon and its
daughters. Particularly troublesome in this regard is the fact that atmospheric
radon concentrations can fluctuate by a factor of 40 during the course of a day.
Once again the problem is best handled by filtering the room air, rapid air turnover,
and the use of inert atmospheres (N2 from evaporation of liquid nitrogen) inside
counting assemblies. A NaI detector will contain some 40
K impurity, which will
contribute to the background. The detector housing is also a potential contributor
to the background. Copper appears to be the best material for detector housing,
with aluminum being the least preferred, for it can be expected to contain
10213
Ci Ra/g Al. In reducing the cosmic ray component of the background,
one tries to stop the “soft” cosmic rays (electrons, X, and g rays) in the detector
shield while using an anticoincidence system to stop the “hard’ component of the
cosmic radiation (mesons, etc.).
At first one might think that germanium detectors with their low detection efficiencies
would have little use in low-level counting. However, such is not the
case because of the complexity of a radionuclide mixture found in environmental
samples, the virtual impossibility of drawing significant conclusions from NaI
detector spectra due to poor energy resolution. Clearly, there are many cases in
which the very good energy resolution of the germanium detector is a necessity.
Furthermore, recent developments in detector fabrication techniques allow the production
of germanium detectors with detector efficiencies equal to that of standard
(3 in. Â 3 in.) NaI detectors.
A number of special techniques have evolved to increase the detection sensitivity
in g-ray counting. One of the most important is the suppression of the Compton scattering
events in the g-ray spectrum by the use of anticoincidence annulus around the
central g-ray detector. The idea behind a Compton suppression spectrometer is that
most events in which the incident photon undergoes one or more Compton scattering
events in the central detector will result in partial energy deposition in the detector
with a low-energy photon escaping the detector.
Suppression is accomplished by setting up an anticoincidence between the central
detector signal and any signal coming from the annulus. A photon that is Compton
scattered from the central detector will probably give rise to a signal from the
annulus. Thus, such events will not be accepted. Other events, such as photoelectric
events in the central detector will not produce signals from the annulus and will be
counted. This reduction in number of Compton scattering events in the g-ray
spectrum leads to a more easily interpreted spectrum since the peak to Compton
ratios are much higher. Typically, the use of such annuli reduces the number of
Compton events 10-fold. More sophisticated designs have been used to further
improve the rejection of Compton scattering events.
19.7 LOW-LEVEL MEASUREMENT TECHNIQUES 607
19.7.5 Limits of Detection
Suppose you have performed a low-level experiment and you wish to state your
results in a statistically meaningful manner. You wish to answer such questions as
“Is there a result/signal/event? What is the chance it will be detected with my
apparatus? How big is it?” Currie (1968) has provided answers to these questions
by defining three different limits of detection:
. The critical level LC, the signal level above which an observed instrument
response may be reliably recognized as “detected.”
. The detection limit LD, the true net signal that may be expected a priori to lead
to detection.
. The determination limit LQ, the signal level above which a quantitative
measurement can be performed with a stated relative uncertainty.
Operationally, the recipes for calculating these limits are as follows:
LC LD LQ
Paired observations 2.33sB 4.65sB 14.1sB
“Well-known” blank 1.64sB 3.29sB 10sB
5000
2000
1000
500
200
100
50
20
10
0
1 2 3 4 5 6 7 8
Detectionlimit(pci)
Definitions:
1 – Background standard deviation (σB)
2 – 10% of the background
3 – 2σB
4 – 3σB
5 – 3σB + 3σD (σD = sample standard deviation)
6 – Twice the background
7 – 1000 dpm
8 – 100 dps
Definition
(LQ)
(LD)
(LC)
Figure 19.8 Various measures of detection limits for low-level counting. (From Currie,
1968.)
608 RADIOCHEMICAL TECHNIQUES
For example, if the background under a photopeak of interest in a g-ray spectrum
was 100 cpm, then sB ¼
ffiffiffiffiffiffiffiffi
100
p
¼ 10 cpm. According to the recipes given above,
one would need to detect a photopeak area of 23 cpm to say that a nuclide was
present and would need to detect at least 141 cpm to measure the amount of
nuclide present. One would need a count rate of 47 cpm to ensure, before making
the measurement, that the nuclide in question could be detected. The relationship
of these limits to other measures used to describe low-level counting is shown in
Figure 19.8.
PROBLEMS
1. A beam of 1 particle-mA of 48
Ca10þ
ions is incident on an Al foil that is 5 mg/
cm2
thick. (a) Estimate the energy deposit/s in the foil. (b) If the foil has an area
of 4 cm2
and it is mounted in a vacuum with no cooling, how long will it take
until the foil reaches the melting point of Al (6608C)? Assume the specific
heat of Al is independent of temperature and is 0.25 cal/deg/g.
2. Au foils are to be used as flux monitors in a nuclear reactor. What is the
maximum thickness that can be used if the self-shielding corrections are to be
less than 10%?
3. The reaction 27
Al(p,3pn)24
Na is to be used to measure the proton flux in an
irradiation with 300-GeV protons. The cross section for this reaction is
known to be 10.1 mbarn. The flux monitor is 5.0 mg/cm2
. One measures a
24
Na counting rate (background corrected) of 10,000 cpm (in a 2% efficient
detector) 24 h after a 0.5-h irradiation. What was the average proton flux
during the irradiation?
4. Imagine you have for your use a nuclear reactor with a flux of 1013
n/cm2
/s in its
irradiation facility and a cyclotron with a beam of 10.5-MeV protons, 21-MeV
deuterons, and 42-MeV a particles. What would be the best way to prepare:
(a) 140
Ba, (b) carrier-free 99
Tcm, (c) carrier-free 144
Ce, (d) 237
Np, (e) 254
Es,
and (f) carrier free 18
F for incorporation into glucose. Outline the target and reaction
to be used, how the target would be prepared and any chemical separations to
be done following irradiation.
5. What value of Br would you use in a gas-filled separator if you wanted to separate
254
No produced in the reaction of 215 MeV 48
Ca with 208
Pb. You may
assume qave for 254
No is 1 torr He gas is 17þ.
6. As a radiochemist, you have been asked to prepare the following accelerator
targets for your research group. They are: (a) 226
Ra, (b) 208
Pb, (c) 238
U, (d)
244
Pu, (e) 90
Zr, and (f) 124
Sn. Assume all targets are to be 0.5 mg/cm2
. For
each target, outline the best method of target preparation, the backing foil
used, if any, and the reasons for your choice. Describe any anticipated problems
in each procedure.
PROBLEMS 609
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Ehmann, W. D., and D. E. Vance. Radiochemistry and Nuclear Methods of Analysis, Wiley,
New York, 1991.
Friedlander, G., J. W. Kennedy, J. M. Miller, and E. S. Macias. Nuclear and Radiochemistry,
Wiley, New York, 1981.
Lieser, K. Nuclear and Radiochemistry: Fundamentals and Applications, VCH, New York,
1997.
Seaborg, G. T. and W. Loveland. The Elements Beyond Uranium, Wiley, New York, 1990.
BIBLIOGRAPHY 611
APPENDIX A
FUNDAMENTAL CONSTANTS AND
CONVERSION FACTORS
TABLE A.1 Fundamental Constants
Quantity
Symbol
(Expression) Value SI Units cgs Units
Speed of light
in vacuum
c 2.99792458 108
m s21
1010
cm s21
Elementary
charge
e 4.8032068 10210
esu
1.60217733 10219
C 10220
emu
Planck
constant
(reduced)
h
h
6.6260755
1.05457266
10234
J s
10234
J s
10227
ergs
10227
ergs
Boltzmann
constant
k 1.380658 10223
J K21
10216
erg K21
Avogadro’s
number
NA 6.0221367 1023
mol21
1023
mol21
Molar gas
constant
R 8.314510 J mol21
K21
107
erg
mol21
K21
Rydberg
constant
R1 ¼ mec a2
/2h 1.0973731534 107
m21
105
cm21
(continued)
613
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
TABLE A.1 Continued
Quantity
Symbol
(Expression) Value SI Units cgs Units
Bohr
magneton
mB ¼ eh[c]/2mec 9.2740154 10224
J T21
10221
erg G21
Nuclear
magneton
mN ¼ eh[c]/2mpc 5.0507866 10227
J T21
10224
erg G21
Fine structure
constant
a ¼ [4p10]21
e2
/hc
7.29735308
a21
137.0359895
Permittivity,
free space
10 8.854187817 10212
C2
N21
m22
Atomic mass
unit
u 1.6605402 10227
kg 10224
g
Electron rest
mass
me 9.10938974 10231
kg 10228
g
Proton rest
mass
mp 1.6726231 10227
kg 10224
g
Neutron rest
mass
mn 1.674929 10227
kg 10224
g
Muon rest
mass
mm 1.8835327 10228
kg 10225
g
Pion rest
mass
mp
+
2.4880187 10228
kg 10225
g
mp
0
2.406120 10228
kg 10225
g
Bohr radius a0 ¼ re/a2
5.29177249 10211
m 1029
cm
Compton
wavelength
Electron lc,e ¼ h/mec 2.42031058 10212
m 10210
cm
Proton lc,p ¼ h/mpc 1.32141002 10215
m 10213
cm
Neutron lc,n ¼ h/mnc 1.31959110 10215
m 10213
cm
Classical
electron
radius
re ¼ ah/mec 2.81794092 10215
m 10213
cm
Magnetic
dipole
moment
Electron me 1.001159652193 mB
Proton mp 2.792847386 mN
Neutron mn 21.91304275 mN
Proton
gyromagnetic
ratio
gp 2.67522128 108
s21
T21
104
s21
G21
614 APPENDIX A
TABLE A.2 Conversion Factors and Handy Units
Quantity Symbol Value
Atomic mass unit u 931.494 MeV
Electron mass me 0.510999 MeV
Proton mass mp 938.272 MeV
Neutron mass mn 939.566 MeV
Electron volt 1 eV 1.602177 Â 10219
J
Electron volt/particle 1 eV/k 11604.45 K
Planck constant h 6.582122 Â 10222
MeV . s
hc 197.327053 MeV . fm
(hc)2
0.389380 GeV2 . mb
Rydberg constant R1hc 13.605698 eV
Gas constant R 1.987216 cal/mol
1 degree 8 1.7453 Â 1022
rad
1 calorie cal 4.184 J
1 British thermal unit Btu 1054.4 J
1 erg 1027
J
1 ton (equivalent of TNT) 4.184 Â 109
J
1 electron radius re 2.8179 Â 10215
m
1 fermi 10215
m
1 light year ly 9.4605 Â 1015
m
1 parsec pc 3.0857 Â 1016
m
1 atmosphere atm 101325 Pa
1 torr (mm Hg, 08C) 133.32 Pa
1 day d 86400 s
1 year (365.25636 d) y 3.1558150 Â 107
s
1 Curie (Ci) 3.700 Â 1010
Bq
1 rad 1.000 Â 1022
Gy
1 rem 1.000 Â 1022
Sv
1 Roentgen (R) 2.580 Â 1024
C/kg
APPENDIX A 615
APPENDIX B
NUCLEAR WALLET CARDS
617
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
What follows is the 2005 version of the Nuclear Wallet Cards
618 APPENDIX B
APPENDIX B 619
620 APPENDIX B
APPENDIX B 621
622 APPENDIX B
APPENDIX B 623
624 APPENDIX B
APPENDIX B 625
626 APPENDIX B
APPENDIX B 627
628 APPENDIX B
APPENDIX B 629
630 APPENDIX B
APPENDIX B 631
632 APPENDIX B
APPENDIX B 633
634 APPENDIX B
APPENDIX B 635
636 APPENDIX B
APPENDIX B 637
638 APPENDIX B
APPENDIX B 639
APPENDIX C
PERIODIC TABLE OF ELEMENTS
641
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
642
APPENDIX D
LIST OF ELEMENTS
Name Sym Z Name Sym Z
Actinium Ac 89
Aluminum Al 13
Americium Am 95
Antimony Sb 51
Argon Ar 18
Arsenic As 33
Astatine At 85
Barium Ba 56
Berkelium Bk 97
Beryllium Be 4
Bismuth Bi 83
Bohrium Bh 107
Boron B 5
Bromine Br 35
Cadmium Cd 48
Cesium Cs 55
Calcium Ca 20
Californium Cf 98
Carbon C 6
Cerium Ce 58
Chlorine Cl 17
Chromium Cr 24
Cobalt Co 27
Copper Cu 29
Curium Cm 96
Darmstadtium Ds 110
Dubnium Db 105
Dysprosium Dy 66
Einsteinium Es 99
Erbium Er 68
Europium Eu 63
Fermium Fm 100
Fluorine F 9
Francium Fr 87
Gadolinium Gd 64
Gallium Ga 31
Germanium Ge 32
Gold Au 79
Hafnium Hf 72
Hassium Hs 108
643
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
Name Sym Z Name Sym Z
Helium He 2
Holmium Ho 67
Hydrogen H 1
Indium In 49
Iodine I 53
Iridium Ir 77
Iron Fe 26
Krypton Kr 36
Lanthanum La 57
Lawrencium Lr 103
Lead Pb 82
Lithium Li 3
Lutetium Lu 71
Magnesium Mg 12
Manganese Mn 25
Meitnerium Mt 109
Mendelevium Md 101
Mercury Hg 80
Molybdenum Mo 42
Neodymium Nd 60
Neon Ne 10
Neptunium Np 93
Nickel Ni 28
Niobium Nb 41
Nitrogen N 7
Nobelium No 102
Osmium Os 76
Oxygen O 8
Palladium Pd 46
Phosphorus P 15
Platinum Pt 78
Plutonium Pu 94
Polonium Po 84
Potassium K 19
Praseodymium Pr 59
Promethium Pm 61
Protactinium Pa 91
Radium Ra 88
Radon Rn 86
Rhenium Re 75
Rhodium Rh 45
Roentgenium Rg 111
Rubidium Rb 37
Ruthenium Ru 44
Rutherfordium Rf 104
Samarium Sm 62
Scandium Sc 21
Seaborgium Sg 106
Selenium Se 34
Silicon Si 14
Silver Ag 47
Sodium Na 11
Strontium Sr 38
Sulfur S 16
Tantalum Ta 73
Technetium Tc 43
Tellurium Te 52
Terbium Tb 65
Thallium Tl 81
Thorium Th 90
Thulium Tm 69
Tin Sn 50
Titanium Ti 22
Tungsten W 74
Uranium U 92
Vanadium V 23
Xenon Xe 54
Ytterbium Yb 70
Yttrium Y 39
Zinc Zn 30
Zirconium Zr 40
644 APPENDIX D
APPENDIX E
ELEMENTS OF QUANTUM
MECHANICS
Quantum mechanics provides a correct description of phenomena on the atomic or
subatomic scale, where the ideas of classical mechanics are not generally applicable.
As we describe nuclear phenomena, we will use many results and concepts from
quantum mechanics. Although it is our goal not to have the reader, in general,
perform detailed quantum mechanical calculation, it is important that the reader
understand the basis for many of the descriptive statements made in the text. Therefore,
we present, in this Appendix, a brief summary of the essential features of
quantum mechanics that we shall use. For more detailed discussion of these features,
we refer the reader to the references at the end of this Appendix.
E.1 WAVE FUNCTIONS
All the knowable information about a physical system (i.e., energy, angular momentum,
etc.) is contained in the wave function of the system. We shall restrict our discussion
to one-body systems for the present. (We could easily generalize to many
body systems.) The wave function can be expressed in terms of space coordinates
and time or momenta and time. In the former notation we write,
c (x, y, z, t) or just c (E.1)
645
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
These wave functions, must be “well-behaved,” that is, they (and their derivatives
with respect to the space coordinates) must be continuous, finite, and single
valued. The functions c are solutions to a second-order differential equation
called the Schro¨dinger equation (see below).
The probability of finding a particle within a volume element dx dy dz, W dx dy
dz, is given by
W dx dy dz ¼ cÃ
c dx dy dz (E.2)
where cÃ
is the complex conjugate of c. [To form the complex conjugate of any
complex number, replace all occurrences of i (where i ¼
ffiffiffiffiffiffiffi
À1
p
) with 2i. Real
numbers are their own complex conjugates. 6 2 5i is the complex conjugate of
6 þ 5i. So (a þ ib)Ã
(a þ ib) ¼ (a 2 ib)(a þ ib) ¼ a2
þ b2
.] The probability per
unit volume (the probability density) is W ¼ cÃ
c. If we look everywhere in the
system, we must find the particle so that
ð
cÃ
c dt ¼ 1 (E.3)
where dt is a volume element dx dy dz. Wave functions possessing this numerical
property are said to be normalized. If the value of some physical quantity P is a
function of the position coordinates, the average or expectation value of P is
given by
kPl ¼
ð
cÃ
Pc dt (E.4)
This expectation value represents the average outcome of a large number of
measurements.
E.2 OPERATORS
Often we must compute values of quantities that are not simple functions of the
space coordinates, such as the y component of the momentum, py, where Equation
(E.4) is not applicable. To get around this, we say that corresponding to every
classical variable, there is a quantum mechanical operator. An operator is a
symbol that directs us to do some mathematical operation. For example, the momentum
operators are
^px ¼ ÀihÀ @
@x
^py ¼ ÀihÀ @
@y
(E.5)
^pz ¼ ÀihÀ @
@z
646 APPENDIX E
while the total energy operator ^E is given as
^E ¼ ihÀ @
@t
(E.6)
Thus, to calculate the expectation value of the x component of the momentum, px,
we write
kpxl ¼
ð
cÃ
ÀihÀ @
@x
 
c dt
¼ ÀihÀ
ð
cà @c
@x
dt (E.7)
Similarly, the classical expression for the kinetic energy is
T ¼ p2
=2m (E.8)
which, translated to quantum mechanics terms, means the kinetic energy operator,
^T, is, in Cartesian coordinates,
^T ¼
ÀhÀ 2
2m
@2
@x2
þ
@2
@y2
þ
@2
@z2
 
(E.9)
or, using the Laplacian operator, r2
^T ¼
ÀhÀ 2
2m
r2
(E.10)
where
r2
¼
@2
@x2
þ
@2
@y2
þ
@2
@z2
(E.11)
E.3 SCHRO¨ DINGER EQUATION
In 1926, Schro¨dinger found that behavior on the atomic or subatomic scale was
correctly described by a differential equation of the form
ÀhÀ 2
2m
r2
c þ Vc ¼ ihÀ @c
@t
(E.12)
where V represents the potential energy and c the wave function of the system.
Substituting from Equation (E.6), we can write
ÀhÀ 2
2m
r2
c þ Vc ¼ Ec (E.13a)
APPENDIX E 647
This equation is an example of a general class of equations called eigenvalue
equations of the form Vc ¼ vc where V is an operator and v is the value of an
observable corresponding to that operator. (The mathematical expression c is
referred to as an eigenfunction of the operator V).
To use the Schro¨dinger equation to gain information about a physical system, we
must perform a set of steps that are as follows:
1. Specify the potential energy function of the system, that is, specify the forces
acting (Section 1.6.1).
2. Find a mathematical function, c, which is a solution to the differential
equation, the Schro¨dinger equation.
3. Of the many functions that satisfy the equation, reject those that do not
conform to certain physical constraints on the system, known as boundary
conditions.
Before illustrating this procedure for several cases of interest to nuclear chemists,
we can point out another important property of the Schro¨dinger equation. If
the potential energy V is independent of time, we can separate the space and time
variables in the Schro¨dinger equation by setting
C (x, y, z, t) ¼ c (x, y, z) t(t) (E.13b)
Substituting this expression into Equation (E.13), and simplifying, we have
1
C
ÀhÀ 2
2m
r2
C þ V ¼
ihÀ
t
@t
@t
(E.14)
The only way this equation can be true is for both sides to equal a constant. If we call
this “separation constant” E, we can write
ÀhÀ 2
2m
r2
c þ Vc ¼ Ec (E.15)
and
ihÀ dt
dt
¼ Et (E.16)
Equation (E.15) is the time-independent Schro¨dinger equation. The solution to
Equation (E.16) is
t(t) ¼ eÀi(E=hÀ
)t
(E.17)
Using the Euler relation (eiu
¼ cos u þ i sin u), we can write
t(t) ¼ cos vt À i sin vt (E.18)
648 APPENDIX E
where t(t) is a periodic function with angular frequency v ¼ E/h. The separation
constant E can be shown to be the total energy, that is, the sum of the kinetic and
potential energies, T þ V.
E.4 THE FREE PARTICLE
To illustrate how the Schro¨dinger equation might be applied to a familiar situation,
consider the case of a “free” particle, that is, a particle moving along at a constant
velocity with no force acting on the particle (V ¼ 0) (Fig. E.1). For simplicity, let us
consider motion in one dimension, the x direction. For the time-independent
Schro¨dinger equation, we have
ÀhÀ 2
2m
d2
c
dx2
¼ Ec (E.19)
or
d2
c
dx2
¼ k2
c (E.20)
where the constant k is given by
k2
¼
2mE
hÀ 2
(E.21)
The allowed values of the energy E are [Equation (E.21)]
E ¼
hÀ 2
k2
2m
(E.22)
where k can assume any value (E is not quantized). Since V ¼ 0, E is the kinetic
energy of a particle with momentum p ¼ hk. From de Broglie, we know that
lÀ
¼
hÀ
p
(E.23)
so that we can make the association that
k ¼
1
lÀ
Figure E.1 Free-particle problem.
APPENDIX E 649
The solution for the Schro¨dinger equation, including the time-dependent part is
C(x, t) ¼ A exp½ikx À ivtŠ þ B exp½Àikx À ivtŠ (E.24)
where k and v are given (E.21) as
k ¼
ffiffiffiffiffiffiffiffiffi
2mE
p
hÀ
(E.25)
v ¼
E
hÀ
(E.26)
This solution is the equation for a wave traveling to the right (þx direction, the first
term) and to the left (2x direction, second term). We can impose a boundary
condition, namely, we can specify the particle is traveling in the þx direction.
Then we have
C(x, t) ¼ A exp½ikxŠ exp½ÀivtŠ (E.27)
We can now calculate the values of any observable. For example, to calculate the
value of the momentum p, we write [see Equation (E.7)]
k pl ¼
ð1
À1
cÃ
À ihÀ @
@x
 
c dx ¼
ffiffiffiffiffiffiffiffiffi
2mE
p
(E.28)
which agrees, of course, with the classical result.
E.5 PARTICLE IN A BOX (ONE DIMENSION)
Continuing our survey of some simple applications of wave mechanics to problems
of interest to the nuclear chemist, let us consider the problem of a particle confined to
a one-dimensional box (Fig. E.2). This potential is flat across the bottom of the box
and then rises at the walls. This can be expressed as:
V(x) ¼ 0 0 x L (E.29)
V(x) ¼ 1 x , 0, x . L
The particle moves freely between 0 and L but is excluded from x , 0 and x . L.
Inside the box, the Schro¨dinger equation has the form of Equation (E.19) (the free
particle). The time-independent solution can be written
c(x) ¼ A sin kx þ B cos kx (E.30)
650 APPENDIX E
But we know that c(x) ¼ 0 at x ¼ 0 and x ¼ L. Thus B must be 0 and
A sin kL ¼ 0 (E.31)
To have sin kL ¼ 0, we must have
kL ¼ np n ¼ 1, 2, 3 (E.32)
and, using the result (E.22), we have
En ¼
hÀ 2
k2
2m
¼
hÀ 2
p2
2mL2
n2
(E.33)
In this case, the energy is quantized. Only certain values of the energy are allowed.
One can show the normalization condition is satisfied if
cn(x) ¼
ffiffiffi
2
L
r
sin
npx
L
(E.34)
The allowed energy levels, the probability densities, and the wave functions are
shown for the first few levels of this potential in Figure E.3.
Sample Problem Suppose a neutron is confined to a one-dimensional box that
is the size of a nucleus, 10214
m. (a) What is the energy of the first excited state?
(b) What is the probability of finding the neutron within a region corresponding to
Figure E.2 Schematic diagram of a particle in a one-dimensional box. The particle is free to
move between x ¼ 0 and x ¼ L, but not allowed to have x , 0 or x . L.
APPENDIX E 651
20% of the width of the box, that is, between 0.4 Â 10214
m and 0.6 Â 10214
m
in the fourth excited state?
Solution
1: E0 (the energy of the ground state) ¼
hÀ 2
p2
(n)2
2mL2
¼
(1:05 Â 10À34
Js)2
(3:14)2
(1)2
2(1:66 Â 10À27 kg)(1014 m2)
¼ 3:3 Â 10À13
J ¼ 2:0 MeV
The energy of the first excited state, n ¼ 2, will be 4E0, and the energy spacing
between the first excited state and the ground state will be 3E0 ¼ 6 MeV.
n = 5 E = 25E0
n = 4 E = 16E0
n = 3 E = 9E0
n = 2 E = 4E0
n = 1
x = 0
Excited
states
Ground
state
x = L
E = E0
Figure E.3 Allowed energy levels of a particle in a one-dimensional box. The wave function
is shown as a solid line for each level while the shaded area gives the probability density.
652 APPENDIX E
2: Probability ¼
ðx2
x1
c2
(x) dx ¼
2
L
ðx2
x1
sin2
5px
L
dx ¼
x
L
À
1
5p
sin
5px
L
 x2
x1
¼ 0:20
which is the result obtained by inspection of the c2
curve in Figure E.3.
E.6 LINEAR HARMONIC OSCILLATOR (ONE DIMENSION)
One of the classic problems of quantum mechanics that is very important for our
study of nuclei is the harmonic oscillator. For a simple harmonic oscillator, the
restoring force is proportional to the distance from the center, that is, F ¼ 2kx,
so that V(x) ¼ kx2
/2. The Schro¨dinger equation is
ÀhÀ 2
2m
d2
c
dx2
þ
1
2
kx2
c ¼ Ec (E.35)
The solution of this equation is mathematically complicated and leads to wave
functions of the form
cn(x) ¼ Nneb2
=2
Hn(b) (E.36)
where
a ;
2pm
hÀ
n0
b ;
ffiffiffi
a
p
x (E.37)
n0 ;
1
2p
ffiffiffiffi
k
m
r
(the oscillator frequency)
with a normalization constant of
Nn ¼
ffiffiffiffi
a
p
r
1
2nn!
 1=2
(E.38)
The expression Hn (b) is the nth Hermite polynomial (which can be found in handbooks
of mathematical functions). The energy eigenvalues can be shown to be
En ¼ n þ 1
2
À Á
hn0 (E.39)
where m ¼ 0, 1, 2, 3 . . . .
Thus, the energy levels are equally spaced starting with the zero-point energy hn0
(Fig. E.4). Note the solutions have the property that there is some probability of
finding the particle in classically forbidden regions, that is, the particle penetrates
into the walls.
APPENDIX E 653
E.7 BARRIER PENETRATION (ONE DIMENSION)
Another important quantum mechanical problem of interest to nuclear chemists is
the penetration of a one-dimensional potential barrier by a beam of particles. The
results of solving this problem (and more complicated variations of the problem)
will be used in our study of nuclear a decay and nuclear reactions. The situation
is shown in Figure E.5. A beam of particles originating at 21 is incident on a
barrier of thickness L and height V0 that extends from x ¼ 0 to x ¼ L. Each particle
has a total energy E. (Classically, we would expect if E , V0, the particles would
bounce off the barrier, whereas if E . V0, the particles would pass by the barrier
Figure E.4 Low-lying levels and associated probability densities for the harmonic oscillator.
Figure E.5 Schematic diagram of a particle of energy E incident on a barrier of height V0 and
thickness L. The wave function c is also shown.
654 APPENDIX E
with no change in their properties. Both conclusions are altered significantly in
quantum mechanics.)
It is conventional to divide the space into three regions I, II, and III, shown in
Figure E.5. In regions I and III, we have the “free-particle” problem treated
in Section E.4. In region I, we have particles moving to the left (the incident
particles) and particles moving to the right (reflected particles). So we expect a
wave function of the form of Equation (E.24), whose time-independent part can
be written
cI ¼ aIeik1x
þ bIeÀik1x
¼ cI! þ cI (E.40)
where kI ¼
ffiffiffiffiffiffiffiffiffi
2mE
p
=hÀ
. In region III, we have no particles incident from þ1, so, at
best, we can only have particles moving in the þx direction (b ¼ 0). Thus
cIII ¼ aIIIeÀk1x
¼ cIII! (E.41)
In region II, the time-independent Schro¨dinger equation is
d2
cII
dx2
¼ k2cII (E.42)
where k2 ¼ ½2m(V0 À E)Š1=2
=hÀ
, assuming V0 . E. The solution is
cII ¼ aIIek2x
þ bIIeÀk2x
(E.43)
Notice that the wavelength l is the same in regions I and III, but the amplitude of the
wave beyond the barrier is much less than in front of the barrier. It can be shown
that the probability of transmitting particles through the barrier is
T ¼
jcIII!j2
V
jcI!j2
V
¼
jaIIIj2
jaIj2
(E.44)
where V is the particle speed. To determine the value of aIII=aI, we eliminate
the other constants bI, aII, bII by applying the conditions that c and dc/dx must
be continuous through all space. After much algebra (see, e.g., the textbook by
Evans), we have
T ¼ 1 þ
V2
0
4E(V0 À E)
sinh2
K2L
!À1
(E.45)
For nuclear applications, the barriers are quite thick (k2L ) 1), in which case,
sinh2
k2L % 1
4 e2k2L
, thus
T % 16
E
V0
1 À
E
V0
 
eÀ2k2L
(E.46)
APPENDIX E 655
The dominant term in this expression is the exponential. For a 6-MeV a particle,
V0 ¼ 20 MeV, L ¼ 10À14
m, we have
k2 %
½2 Â 4 Â 1:6 Â 10À27
 (20 À 6)  1:6  10À12
Š1=2
1:05 Â 10À34
% 5:1 Â 1015
mÀ1
Thus
eÀ2k2L
¼ eÀ102
¼ 5:1 Â 10À45
and
T ¼ 16 Â 8
20 Â 1 À 8
20
À Á
(5:1 Â 10À45
) ¼ 1:9 Â 10À44
So we ignore the preexponential term, and write
T % eÀ2G
(E.47)
where 2G ¼ 2k2L ¼ 2[2 m(V0 2 E)]1/2
/hÀ
. For an arbitrarily shaped potential that
would be more pertinent to nuclear a decay, one can show
2G ¼
2
hÀ
ðx2
x1
½2 m(V(x) À E)Š1=2
dx (E.48)
where x1 and x2 are the points where E ¼ V(x).
What about the case where E . V0. In regions I and III, the situation is the same.
In region II, the wave functions will be given as
cII ¼ aIIeik2x
þ bIIeÀik2x
(E.49)
where
k2 ¼
½2m(E À V0)Š1=2
hÀ
(E.50)
Since the wavelength l2 is 1/k2, we can note by comparing equations that l2 . l1,
and the momentum ½p(¼(2mk2)1=2
)Š becomes less. In other words, the particle is
scattered.
656 APPENDIX E
E.8 SCHRO¨ DINGER EQUATION IN SPHERICAL COORDINATES
Many problems in nuclear physics and chemistry involve potentials, such as the
Coulomb potential, that are spherically symmetric. In these cases, it is advantageous
to express the time-independent Schro¨dinger equation in spherical coordinates
(Fig. E.6). The familiar transformations from a Cartesian coordinate system
(x, y, z) to spherical coordinates (r, u, w) are (Fig. E.6)
x ¼ r sin u cos f y ¼ r sin u sin f z ¼ r cos u (E.51)
The time-independent Schro¨dinger equation becomes
ÀhÀ 2
2m
1
r2
@
@r
r2 @c
@r
 
þ
1
r2 sin2
u
@
@u
sin u
@c
@u
 
þ
1
r2 sin2
u
@2
c
@f2
!
þ vc ¼ Ec (E.52)
When the potential is spherically symmetric, v ¼ v(r), then the wave function can be
written as
c(r, u, f) ¼ R(r)Ylm(u, f)
where Ylm are the spherical harmonic functions.
If we substitute this wave function in Equation (E.52) and collect terms, we find
that all functions of r can be separated from the functions of u and w:
1
R
d
dr
r2 dR
dr
 
þ
2mr2
hÀ 2
(E À V) ¼
1
Ylm
1
sin u
@
@u
sin u
@Y
@u
 
þ
1
sin2
u
@2
Y
@f2
!
(E.53)
Setting both sides of the equation equal to a separation constant, l (lþ 1), where l ¼
0, 1, 2. . . , we have
1
R
d
dr
r2 dR
dr
 
þ
2mr2
hÀ 2
(E À V) ¼ l(l þ 1) (E.54)
Figure E.6 Spherical polar coordinates.
APPENDIX E 657
and
1
sin u
@
@u
sin u
@Y
@u
 
þ
1
sin2
u
@2
Y
@f2
¼ l(l þ 1)Y (E.55)
Working on Equation (E.54), it is convenient to change variables
R(r) ¼
u(r)
r
(E.56)
d2
u
dr2
þ
2m
hÀ 2
E À V(r) À
l(l þ 1)hÀ 2
2mr2
 !
u ¼ 0 (E.57)
This is called the radial wave equation. Apart from the term involving l, it is the
same as the one-dimensional time-independent Schro¨dinger equation, a fact that
will be useful in its solution. The last term is referred to as the centrifugal potential,
that is, a potential whose first derivative with respect to r gives the centrifugal force.
It is important to note that Equation (E.55) does not contain the potential energy
term, and thus once we have solved it, the solutions will supply to all cases where V
does not depend on Q and w, that is, all so-called central potentials. The wave functions
Ylm (u, w) are known as the spherical harmonic functions and are tabulated in
many references. The indices l and m are related to the orbital angular momentum, L,
of the particle relative to the origin. The magnitude of L is [l (l þ 1)]1/2
h and its
2lþ1 possible projections on the z axis are equal to mhÀ
(m ¼ 0, +1,
+2 . . . + l).1
The term l is called the orbital angular momentum quantum number
while m is the magnetic quantum number, in reference to the different energies of
the m states in a magnetic field (the Zeeman effect). It follows, therefore, that the
specification of a particular spherical harmonic function (as a solution to the
angular equation) uniquely specifies the particle’s orbital angular momentum and
its z component.
E.9 INFINITE SPHERICAL WELL
As an application of the Schro¨dinger equation, expressed in spherical coordinates, to
a problem of interest in nuclear chemistry, let us consider the problem of a particle
in an infinite spherical well (Fig. E.7). This potential can be defined as
V(r) ¼ 0 r , a
V(r) ¼ 1 r . a
(E.58)
Following our discussion in Section E.8, we expect the solution of the Schro¨dinger
equation to be
c ¼ Rl(r)Ylm(u, f) (E.59)
1
In more formal language, kl2
l ¼ hÀ 2
l(l þ 1) and klzl ¼ mhÀ
.
658 APPENDIX E
where the radial wave function Rl(r) is a solution to the equation
1
r2
d
dr
r2 dRl
dr
 
þ
2m
hÀ 2
E À
l(l þ 1)hÀ 2
2mr2
!
Rl ¼ 0 (E.60)
inside the well. The solutions of this equation are the spherical Bessel functions
RlðrÞ ¼ jl(kr) À
r
k
 l 1
r
d
dr
 l
sin kr
kr
 
(E:61)
where k ¼
ffiffiffiffiffiffiffiffiffi
2mE
p
=hÀ
. The boundary conditions require c ¼ 0 at r ¼ 0, and r ¼ a,
which will occur for values of ka that make the Bessel functions have a value of
0 (the “zeros” of these functions). (Each l value will have its own set of zeros.)
These resulting values of k can be used to calculate the allowed energy levels
(Fig. E.8). Each level is labeled with a number (1, 2, 3 . . .) and a letter (s, p, d, e,
etc.). The letter follows the usual spectroscopic notation of l (l ¼ 0, s; l ¼ 1, p,
etc.) while the number designates how many times that letter has occurred (the
first d level is 1d; the second 2d, etc.).
E.10 ANGULAR MOMENTUM
Classically, the angular momentum of a particle can be written as l ¼ r  p.
(Section 1.6.2). From this classical expression, we can write down the classical
components of the vector l:
lx ¼ ypz À zpx
ly ¼ zpx À xpz
lz ¼ xpy À ypx
(E.62)
Figure E.7 Schematic diagram of the infinite square well potential.
APPENDIX E 659
These classical expressions can be converted to the operator language of quantum
mechanics by substitutions [such as x ! x, px ! ihÀ
(@/@x), etc.]
lx ¼ ÀihÀ
y
@
@z
À z
@
@y
 
¼ ÀihÀ
À sin f
@
@u
À cot u cos f
@
@f
 
ly ¼ ÀihÀ
z
@
@x
À x
@
@z
 
¼ ÀihÀ
cos f
@
@u
À cot u sin f
@
@f
 
lz ¼ ÀihÀ
x
@
@y
À y
@
@x
 
¼ ÀihÀ @
@f
(E.63)
As remarked earlier (Section E.9), the expectation values of klzl and kl2
l for a central
potential are
klzl ¼ mlhÀ
ml ¼ 0, +1, +2, . . . ,+1 (E.64)
Figure E.8 (a) Energy levels of an infinitely deep spherical square well. The radical
probability density functions r2
R2
n, l(r) are shown for different values of l. (b) The three
dimensional probability densities,
Ð
n, l, m (r, u) for an infinitely deep three-dimensional
square well.
660 APPENDIX E
and
kl2
l ¼ l(l þ 1)hÀ 2
(E.65)
We can give these results a pictorial interpretation that is worth noting.
Consider a state of definite orbital angular momentum l. Then
l ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l(l þ 1)hÀ
p
The z component of l may have any value up to +lhÀ
. The possible values of lz
can be represented as the projection of a vector of length l on the z axis
(Fig. E.9). This situation is referred to as spatial quantization. Only certain
values of l2 are allowed. Due to the uncertainty principle, the values of lx and
ly are completely uncertain. In the language of Figure E.9, the vector representing
l is rotating about the z axis, so that l and lz are fixed, but lx and ly are continuously
changing.
In chemistry, we found that to describe the complete quantum state of an electron
in an atom, we had to introduce another quantum number, the intrinsic angular
momentum or spin. This quantum number is designated as s. By analogy to the
Figure E.8 Continued.
APPENDIX E 661
orbital angular momentum quantum number l, we have
ks2
l ¼ s(s þ 1)hÀ 2
ks2l ¼ mshÀ
ms ¼ + 1
2 (E.66)
Nucleons also have values of the spin quantum number of s ¼ 1
2, like electrons. The
total angular momentum of a nucleon j can be written as
j ¼ l þ s (E.67)
The usual quantum mechanical rules apply to j, that is,
k jl ¼ j( j þ 1)hÀ 2
k jzl ¼ mjhÀ
¼ klz þ szl
where mj ¼ Àj, Àj þ 1, . . . , j À 1, j. Thus we have
mj ¼ ml þ ms ¼ ml + 1
2 (E.68)
Figure E.9 Spatial orientation and z components of a vector with l ¼ 2.
662 APPENDIX E
Since ml is always an integer, then mj must have a half integer and j must be a half
integer, either j ¼ l À 1
2 or j ¼ l þ 1
2. Alternatively, for a given l value, we have two
possible values of j, j ¼ l À 1
2 or j ¼ l þ 1
2. For example, for l ¼ 1 (p state), we have
j ¼ l À 1
2 ¼ 1
2 or j ¼ l þ 1
2 ¼ 3
2. We designate these states as p1/2 and p3/2,
respectively.
E.11 PARITY
A wave function has positive (or even) parity if it does not change sign by reflection
through the origin.
c(Àx, Ày, z) ¼ c(x, y, z) positive parity, p ¼ þ (E.69)
Alternatively, if reflection through the origin produces a change of sign, the parity of
the wave function is negative (2).
c(Àx, Ày, z) ¼ Àc(x, y, z) negative parity, p ¼ À (E.70)
When c is expressed in spherical coordinates as c(r, u, w), then “reflection through
the origin” is accomplished by replacing u and w by (p 2 u) and (p þ w), respectively.
(r cannot change sign as it is just a distance.) In other words, the parity of the
wave function is determined only by its angular part. For spherically symmetric
potentials, the value of l uniquely determines the parity as
p ¼ (À1)l
(E.71)
A corollary of this is that for a system of particles the parity is even if the sum of the
individual orbital angular momentum quantum numbers Sli is even; the parity is odd
if Sli is odd. Thus, the parity of each level depends on its wave function. An excited
state of a nucleus need not have the same parity as the ground state.
Parity will be valuable to us in our discussion of nuclei because it is not conserved
in b decay, which will tell us that a different force, the weak interaction, is acting in
b decay compared to nuclear reactions. Also the rates of the g-ray transitions
between nuclear excited states depend on the changes in parity and can be used to
determine the parity of nuclear states.
E.12 QUANTUM STATISTICS
The parity of a system is related to the symmetry properties of the spatial portion of
the wave function. Another important quantum mechanical property of a system of
two or more identical particles is the effect on the wave function of exchanging the
coordinates of two particles. If no change in the wave function occurs when the
spatial and spin coordinates are exchanged, we say the wave function is symmetric
APPENDIX E 663
and the particles obey Bose–Einstein statistics. If upon exchange of the spatial and
spin coordinates of the two particles the wave function changes sign, the wave function
is said to be antisymmetric and the particles obey Fermi–Dirac statistics. The
“statistics” these particles follow profoundly affects the property of an assembly of
such particles. Particles with half-integer spins, such as neutrons, protons, and electrons,
are fermions, obey Fermi–Dirac statistics, have antisymmetric wave functions,
and as a consequence, obey the Pauli principle. (No two particles can have
identical values of the quantum numbers, m, l, ml, s, and ms.) Photons, or other
particles with integer spins, such as the p meson, are bosons, obey Bose–Einstein
statistics, have symmetric wave functions, and do not obey the Pauli principle.
This difference between fermions and bosons is reflected in how they occupy a
set of states, especially as a function of temperature. Consider the system shown
in Figure E.10. At zero temperature (T ¼ 0), the bosons will try to occupy the
lowest energy state (a Bose–Einstein condensate) while for the fermions the
occupancy will be one per quantum state. At high temperatures the distributions
are similar and approach the Maxwell Boltzman distribution.
The Fermi–Dirac distribution can be described by the equation
fFD(E) ¼
1
eÀ(EÀEF)kT þ 1
(E.72)
Figure E.10 (a) Bose–Einstein distribution function, (b) Fermi–Dirac distribution function,
and (c) filling of levels by fermions at T ¼ 0 and T ¼ T1 . 0. The dashed line indicates the
Fermi energies EF.
664 APPENDIX E
where fFD is the number of particles per quantum state, k is Boltzman’s constant, and
EF is the Fermi energy. At T ¼ 0, all energy levels up to EF are occupied ( fFD ¼ 1)
and all energy levels above EF are empty ( fFD ¼ 0). As T increases, some levels
above EF become occupied at the expense of levels below EF.
BIBLIOGRAPHY
Cohen-Tannoudji, C., B. Diu, and F. Laloe. Quantum Mechanics, Wiley, New York, 1977.
An encyclopedic treatment.
Eisberg, R. M. Fundamentals of Modern Physics, Wiley, New York, 1961. A comprehensive
treatment of modern physics.
Krane, K. S. Modern Physics, Wiley, New York, 1983. A well-written introductory treatment
of quantum physics.
Merzbacher, E. Quantum Mechanics, Wiley, New York, 1961. Another treatment with several
nuclear physics applications.
Scharff, M. Elementary Quantum Mechanics, Wiley, London, 1969. A very lucid, elementary
treatment of quantum mechanics, emphasizing physical insight rather than formal theory.
Schiff, L. I. Quantum Mechanics, McGraw-Hill, New York, 1955. An old classic treatment
that contains several applications of interest.
APPENDIX E 665
INDEX
Abnormal data, 574
Absorbed dose, 531
Accelerator mass spectrometry (AMS), 84
Achromatic separation, 419
Actinide contraction, 452
Activation analysis, 366
Activity, 10
Allowed beta decay, 212
Alpha decay
equation, 178
recoil, 180
coulomb energy, 182
Gamow factor, 186
hindrance factors, 192
mass parabola, 183
preformation factor, 186, 188
Q-value, 178
tunneling, 184
Alvarez accelerator, 405
Angular momentum
beta decay, 200
definition, 12
intrinsic, 46
Anionic extractant, 597
Antineutrino, 202
Areal density, 498
Astrophysical S-factor, 343
Atomic mass unit, 19
Attenuation length, 526
Autoradiography, 107
B(E2), 231, 227
Background radiation, 606
Barium fluoride, 562
Barn (unit), 52
Baryons, 23
Bateman Equations, 74
Becquerel (unit), 63
Beer–Lambert Law, 256, 518
Beta decay
angular momentum, 200
equation, 202
Fermi function, 207
Fermi integral, 209
parity, 214
Q-value, 203
selection rules, 211
Beta stability, 40
Bethe–Bloch stopping
power, 502
667
Modern Nuclear Chemistry, by W.D. Loveland, D.J. Morrissey, and G.T. Seaborg
Copyright # 2006 John Wiley & Sons, Inc.
Binding energy
Electron Thomas–Fermi, 30, 178
nuclear, 32
Binomial distribution, 567
Bohr independence hypothesis, 272
Bohr magneton, 48
Bohr stopping power, 502
Bohr velocity, 505
Boiling water reactor (BWR), 391, 465
Boson, 20
Bragg counter, 542
Bragg peak, 506
Bragg’s rule, 506
Branching ratio, 76
Breit–Wigner function, 274, 344
Bremsstrahlung, 514, 517
Brennan–Bernstein rules, 146
Cadmium ratio, 370
CAMAC, 565
Carrier, 94
Carrier free, 64, 583
Cationic extractant, 598
Center of mass, 528
Centrifugal potential, 190, 263
Cerenkov radiation, 518, 539
Charge distribution, 45
Charged particle activation analysis
(CPAA), 370
Chemical blank, 604
Chernobyl reactor, 394, 457
CNO cycle, 347
Cockcroft–Walton accelerator, 400
Cold fusion, 435
Compound nucleus, 254, 272
Compton edge, 523
Compton scattering, 522
Conversion factors, 613
Coulomb barrier, 261, 397
Coulomb excitation, 280
Critical mass, 423
Critical state, 388
Cross section, 254
Cumulative yield, 321
Curie (unit), 64
Cyclotron, 406
Data rejection, 574
Daughter activity, 68
De Broglie wavelength, 17, 386, 641
Dead time, 545, 548
Decay constant, 58
Decay law, 59
Deep inelastic reaction, 280, 286
Delayed neutrons, 390
Detection limit, 608
Deuterium, characteristics, 131,138
Differential cross section, 256
Diffractive scattering, 268
Direct reaction, 254
DNA analysis, 108
Double beta decay, 41, 217
Effective length, 415
Electric moments, 50
Electrodeposition, 587
Electromagnetic radiation
angular correlation, 236
lifetime, 242
multipolarity, 224
selection rules, 225
stretched transition, 224
transition rate, 226
Electron cyclotron resonance
(ECR), 398
Electronic stopping, 500, 514
Element synthesis, 434
Energy straggling, 508
Energy width, 63, 76
Entropy, 167
Epithermal neutrons, 370
Euler relation, 640
Evaporation residue, 434, 591
Exchange capacity, 581
Excitation function, 277
Faraday cup, 589
Fermi
distribution, 43
energy, 165
function, 207
gas, 163
golden rule, 204
integral, 209
Fermi decay, 201, 204
Fermion, 20
Fission
barrier, 170, 304
charge distribution, 318
isomer, 308
mass distribution, 316
TKE, 316
width, 311
Fissionability parameter, 302
Forensic activation analysis, 372
ft value, 209
Fuel cycle, 467
Fundamental constants, 613
668 INDEX
Gamow factor, 186, 343
Gamow peak, 344
Gamow–Teller decay, 201
Gas amplification, 543
Gas flow detector, 546
Gas quenching, 548
Gaseous diffusion, 476
Gaussian distribution, 570
Geiger–Mu¨ller detector, 544, 547
Geiger–Nuttall Law, 185
Geochronometer, 83
Germanium detector, 557
Giant dipole resonance, 278
Gray (unit), 532
G-value, 582
Gyromagnetic ratio, 49
Hahn and Strassmann, 299
Half life, 10, 60
Halo nucleus, 44
Heavy cluster emission, 193
Heisenberg Uncertainty
Relationship, 19, 63, 129
Hertzsprung–Russell diagram, 340
Hill–Wheeler formula, 306
Hindrance factors, 192
Hydrogen burning, 345
Impact parameter, 12, 259
Independent yield, 320
In-flight separation, 419
Interaction barrier, 264, 282
Interval distribution, 571
Intranuclear cascade, 290
Ion chamber, 540
Ion exchange, 599
Ion pair, 540
Ionization potential, 502
ISOL, 287, 417
Isomeric states, shell model, 149
Isomers, 221
Isospin, 133
Isotope dilution analysis, 122
Isotope effect, 92, 104
Isotopic abundances, 332
Kerma, 532
Kinetic isotope effect, 106
Kurie plot, 207
Lambert–Beers Law, 256, 518
Laplacian, 639
Leptons, 20
Level density, 168, 275
Level width, 273
Linear absorption coefficient, 518
Linear energy transfer (LET), 531
Liquid scintillation, 560
Lorentz force, 407
Lorentz transformations, 12
Low-level counting, 605
Low-level waste, 489
Magnetic moment, 48, 147
Magnetic rigidity, 420
Magnetic sector, 413
Mass equation
Myers–Swiatecki, 39
Semi-empirical, 37
Weizsacher, 36
Mass excess, 32
Mass parabola, 183
Mass stopping power, 504
Mean free path, 289, 387
Mean life, 62
Mesons, 23
Mirror nuclei (table), 133
Mirror nuclei, coulomb, 150
Moment of inertia, 156
Monitor reaction, 589
Monopole moment, 51
Mosely, 5
Mossbauer effect, 241
Natural decay chain, 73, 77
Natural radioactivity, 77
Natural reactor, 395
Neptunium, 439
Neutral extractant, 597
Neutrino, 200, 215
detector, 357
oscillation, 359
solar, 355
Neutron activation analysis
(NAA), 370
Neutron moderator, 388
Neutron scattering, 528
Nilsson model, 160
Nuclear decay
characteristics, 9
equilibria, 70
general, 8
rate, 10
Nuclear density, 7, 43
Nuclear fallout, 80
Nuclear force, 11
Nuclear instrumentation module
(NIM), 565
INDEX 669
Nuclear magneton, 48
Nuclear mass, 30
Nuclear potential, 139
Nuclear radii, 42
Nuclear reaction Q-value, 251
Nuclear shapes, 155
Nuclear skin, 43
Nuclear surface, 154
Nuclear temperature, 167
Nuclear waste, 484
Oklo natural reactor, 395
Optical model, 269
Packing fraction, 32
Pair production, 524
Parity, 47, 214, 655
Penning ion guage (PIG), 398
pep reactions, 345
Photoelectric absorption, 520
Photomultiplier, 562
Pickup reaction, 270
PIN diode, 555
PIXE, 373
Plutonium, 439
Poisson distribution, 569
Positron emission tomography (PET), 117
Potential Energy, Coulomb, 12
pp1 reaction, 346
pp2 reaction, 346
p-process, 353
Preformation factor, 186, 188
Pressurized water reactor (PWR),
391, 465
Primordial nucleosynthesis, 337
Proton activation analysis
(PAA), 370
Proton emission, 195
Quadrupole moment, 50
Quarks, 23
Q-value
alpha decay, 178
beta decay, 203
equation, 252
nuclear reaction, 31, 251
Rad, 531
Radiation exposure, 531
Radiative stopping, 514
Radioactive decay law, 59
Radiocarbon, 79
Radioimmunoassay, 108
Radiolysis, 101, 493
Radiopharmaceuticals (table), 116
Radiotracer label, 99
Radiotracers (table), 98, 111
Range, 513
Range straggling, 509
Reactor poisons, 390
Relativistic mass, 14
REM (unit), 533
Residual interaction, 153
Resonance, 274
Roentgen (unit), 531
Rotational energy, 155
r-process, 352
rp process, 353
Rutherford backscattering, 376
Rutherford scattering, 265
Saddle point, 300
Saturation activity, 258
Scavengers, 583
Schmidt limit, 148
Scintillation, 559
Scission point, 300
Secular equilibrium, 72
Segre diagram, 36
Selection rules, 225
Semiconductor detector, 549
Semi-empirical mass, 37
Separation energy, 32
Shell model, 141
Sievert (unit), 533
Silicon surface detector, 554
Sodium iodide, 561
Solar
abundances, 335
neutrinos, 355
reactions, 346
Spallation, 288
Specific activity, 64
Spent fuel, 479, 484, 488
Spherical coordinates, 649
Spontaneous fission, 306
s-process, 351
Stable nuclides, 36
Standard model, 20
Statistical equilibrium, 169
Stellar populations, 340
Stopping power, 500
Straggling, 508
Stripping reaction, 270
Strong force, 130
Strutinsky method, 305
Subcritical state, 390
Superallowed beta decay, 212
670 INDEX
Supercritical mass, 423
Superdeformed nuclei, 157, 161
Superheavy nuclei, 447
Synchrocyclotron, 410
Synchrotron, 410
Tandem accelerator, 402
Thermonuclear reactions, 342, 424
Three-Mile Island reactor, 394, 457
Threshold energy, 253
Time projection chamber (TPC), 542
Townsend avalanche, 543
Track detectors, 499, 564
Transient equilibrium, 70
Transmutation, 491
Triple alpha reaction, 348
Tunneling, 184, 646
Uncertainties, 573
Uranium enrichment, 475
Uranium metal, 470
Van de Graff accelerator, 400
Vibrational motion, 158
Virtual photon, 130
Water dilution volume, 485
Watt energy spectrum, 324, 387
Wave function, 637
Weisskopf transition rate, 227, 229
Weizsacher mass equation, 36
Wheeler angular distribution, 326
Wideroe accelerator, 405
Wilzbach reaction, 94, 101
Woods–Saxon potential, 132, 153, 263
X-ray, 4
Yellowcake, 474
Yrast states, 286
Yucca mountain, 487
Yukawa potential, 132
Zeeman splitting, 236
INDEX 671
COLOR PLATES
Figure 2.10 Nuclear density distribution: (a) in a schematic view and (b) in an artist’s
conception from R. Mackinfosh, J. Al-Khalili, B. Jonson and T. Pena, Nucleus: A Trip to
the Heart of Matter. Copyright # 2001 by The Johns Hopkins University Press, 2001;
reprinted by permission of Johns Hopkins.
Figure 1.4 Two artist’s conceptions of the standard model. (a) From New York Times,
22 September, 1998. Reprinted by permission of the New York Times. (b) From “Nuclear
Science”, Contemporary Physics Education Project (CPEP), LBNL.
Figure 2.12 Schematic representation of the relative sizes of the halo nucleus 11
Li and 208
Pb.
Figure 4.2 Schematic view of DNA fingerprinting.
Figure 4.6 PET pictures of the heart of a patient with acute myocardial infarction treated
with a thrombolytic agent. Top row shows scans after administration of water containing
15
O to trace blood flow. Bottom row shows tomograms obtained after administration of
acetate containing 11
C to trace the heart’s metabolism, that is, its rate of oxygen use. The
defects are clearly visible on day 1, both in the impaired blood flow (top left) and the
impaired metabolic use of oxygen (bottom left). Recovery of blood circulation has taken
place on day 2 and is maintained.
5+ 0
5.2 yr.
60
27Co
60
28Ni
94
41Nb
94
42Mo
Q = 2823.9
Q = 2045.1
99.025% 4+ 2505.77
2+ 1332.52
stable
stable
.871
.702
1332
1173
98.1% 4+ 1573.72
2+ 871.10
0+ 0
2.0x104 yr.
0+ 0
(6)+ 0
0.057%
Figure 6.8 (a) Energy level diagram showing the first (lowest energy) 2þ
and 4þ
states in
60
Ni. The high-spin ground state, 5þ
, of 60
Co b decays primarily to the 4þ
state and initiates a
well-known g-ray cascade to the 2þ
state and then the 0þ
ground state. (b) For comparison, the
energy level diagram showing the first (lowest energy) 2þ
and 4þ
states in 94
Mo. The highspin
ground state, 6þ
, of 94
Nb also primarily feeds the 4þ
state initiating a g-ray cascade.
Figure 7.5 A (reasonably accurate) one-dimensional potential energy diagram for 238
U
indicating the energy and calculated distances for a decay into 234
Th. Fermi energy
%30 MeV, Coulomb barrier %28 MeV at 9.3 fm, Qa 4.2 MeV, distance of closest approach
62 fm.
6+ (254 W.u.) 18.6 ps
4+ (358 W.u.) 103 ps
2+ (1098 W.u.) 2026 ps
160
Dy
0+
Figure 9.4 Schematic diagram of the ground-state rotational band transitions for 160
Dy.
Fission Isomers
144
Fm 100
Es 99
Cf 98
Bk 97
Cm 96
Am 95
Pu 94
Np 93
U 92
Pa 91
Th 90
140
148
152
Figure 11.6 Position of the known spontaneously fissioning isomers in the nuclide chart.
Figure 9.10 Energy level diagram of two members of the A ¼ 57 mass chain. 57
Co decays
to excited states of 57
Fe, which result in the M1 transition from the 3
2
À
state at 14.41 keV to the
1
2
À
ground state.
Figure 12.1 Abundances of the first 40 elements as a percentage by mass of Earth’s crust
(filled squares) and in the solar system (open circles). Data from the CRC Handbook of
Chemistry and Physics, 75th ed., 1994.
0.25
0.24
0.23
0.22
10
–4
10
–5
10
–9
10
–10
1 2 5
Baryon density (10
–31
g cm
–3
)
LI
He
D
7
3
0.01 0.02 0.05
Fraction of critical density
4
HeMassfractionNumberrelativetoH
Figure 12.6 Variation of the relative abundances of the Big Bang nuclei and the 4
He mass
fraction versus the baryon density. The boxes indicate the measured data and an estimate of
the uncertainty. The curves indicate the dependence of the yield on the baryon density in the
Big Bang; the vertical bar indicates the region of concordence.
Figure 12.19 Current status of the comparison between standard solar model predictions
and experimental measurements. [From Bahcall (from Bahcall website http://www.
sns.ias.edu/jnb).]
ISOL
Heavy lon
accelarator Fragment separator
Projectile Fragmentation
Radioactive lon beam
Thin production
target
Production
accelerator
Transfer tube
Production beam
Thick, hot target
Radioactive lon beam
Ion source
Isotope / Isobar
separator
Postaccelerator
Experiment
Experiment
Figure 14.13 SchematicviewofthePFandISOLtechniquesforgeneratingradioactivebeams.
Figure 15.1 Modern periodic table showing the transuranium elements.
Figure 15.2 (a) Half-lives of the known transuranium nuclei plotted as a function of Z and
N. (b) Dominant decay modes of the nuclei shown in (a).
Figure 15.3 Predicted (Mo¨ller et al., 1997) half-lives of the transuranium nuclei with
Z 112.
Figure 15.7 Velocity filter SHIP
Figure 15.9 Glenn Seaborg points out seaborgium in the periodic table.
1024
Longest lived isotopes
of transuranium elements
1 000 000 years
1 000 years
1 year
1 day
1 hour
1 minute
1 second
Decay by losing
alpha particles
1021
1018
1015
1012
109
106
103
101
10–3
10–6
10–9
10–12
94 98 102
Atomic number
Halflife(seconds)
106 110
Decay by
spontaneous
fission
Figure 15.10 Half-lives of the longest-lived isotope of each element versus atomic number
Z circa 1970.
70
60
50
40
30
20
10
0
1 3 5 7 9 11
0
–40
–80
–120
–160
–200
temperature[°C]
relativeyield[%]
HsO4
OsO4
detector
Figure 15.17 Thermochromatogram of HsO4 and OsO4. From GSI Nachrichten 1/02.
Figure 16.4 Ammonium diuranate (yellowcake) after solvent extraction. (Photo courtesy of
UIC.)
Figure 16.13 Schematic diagram of the final steps in putting vitrified waste into a geologic
repository.
Spent
Nuclear
Fuel
Chop-
Leach
Process
UREX Solvent
Extraction
Process
Conversion
to Oxide
Oxide
Reduction
to Metal
Electrometallurgical
Separation Process
Waste
Treatment
ATW Fuel
Assembly
Fabrication
Disposal as LLW
or
Storage
ATW
Transmutation
Target
Fabrication
Hull
Cleanup
Nitrate
Solution
Iodine
Tc
Tc
TRU
Uranium
Np
Raffinate
(TRU, FP)
TRU, FP
Oxides
TRU, FP
FP
FP
TRU
Figure 16.15 Schematic representation of the new schemes for reprocessing of SNF.