Chapter 3
Interaction of radiation with matter
3.1 Types of radiation
Interaction of radiation with the matter is necessary in order to characterize the microstructures:
• photons
• electrons
• neutrons
• protons
• ions/atoms
The penetration depth or mean free path of the incident beam determines the depth and volume
of material that will be sampled. In many cases one is probing with one type of radiation but
detecting a second type. This occurs in X-ray photoelectron spectroscopy (XPS) where the incident
probe is a beam of X-ray photons but emitted electrons are detected,whereas this is reversed for
the technique of energy dispersive X-ray(EDX) analysis. Generally the particle or radiation which
has the shortest mean free path in the material will determine the volume analysed. Whatever
beam is selected we must be aware of its interaction with the material, what photons, electrons
or other particles are ejected and how they, in turn, inter-act with the material. Only in this way
can we use the emitted signals to gain an understanding of the material being examined. The
material can be damaged by this interaction:
• surface observations ⇒visible light is bombarding the surface (damage to the photographic
layer)
• for a higher magnification ⇒ electrons with energies between 10–30 keV (greater penetration
and consequently greater damage)
• high resolution and high sensitivity ⇒ high energy ions (visible damage)
It is necessary to choose the right method due to their damage characteristics.
1
2 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
3.2 Photon interactions
The electromagnetic radiation can interact with electrons or phonons. Photons are discrete quanta
of electromagnetic radiation. The photon is identified by the wavelength, λ, energy, E, and
frequency, ν , all of which are related by the equation
hν = E = hc/λ (3.1)
where h is the Planck constant and c the velocity if light.The electromagnetic spectrum spans a
vast range with wavelengths varying from 106
m down to 10−14
m. If we are to use electro-magnetic
radiation for microstructural characterisation of materials a photon wavelength is needed that is
of comparable size to the features being studied. This means that photon wavelengths greater
than 10−4
m would result in an inadequate spatial resolution and we do not require radiation less
than about 10−10
m.
The penetration of photons shows considerable and dramatic variations between different types of
material and photon energy or wavelength. It is not possible or instructive to go into any detail
regarding penetration depths over the whole of the electromagnetic spectrum, but only some
specific wavelengths that are important for interrogating the microstructure of materials. The
long wavelength infrared radiation is used to characterise materials by determining how specific
wavelengths are absorbed, visiblelight is used in a variety of instruments mainly to obtain a visual
image of the surface while at the shorter wavelength ultraviolet radiation is often used to obtain
information concerning the electron distribution in the surface atoms. Some materials are opaque
while others are transparent to this range of wavelengths. However, even the most opaque or
highly reflecting of these materials will allow the radiation to penetrate at least a fraction of a
wavelength below the surface. In the case of visible light, where the wave-length is approximately
500 nm this penetrates an average of between 50 to 300 nm into the bulk so that any analysis
performed or image obtained will average over several hundred atom layers.
3.2.1 Very short wavelengths (< 10−12
m, γ-rays)
The method that uses γ-rays to characterize materials is called Mössbauer spectroscopy.
Key to the success of the technique is the discovery of recoilless γ-ray emission and absorption,
now referred to as the Mössbauer effect, after its discoverer Rudolph Mössbauer, who first observed
the effect in 1957 and received the Nobel Prize in Physics in 1961 for his work.
Mössbauer effect
Nuclei in atoms undergo a variety of energy level transitions, often associated with the emission or
absorption of a gamma ray. These energy levels are influenced by their surrounding environment,
both electronic and magnetic, which can change or split these energy levels. These changes in the
energy levels can provide information about the atom’s local environment within a system and
ought to be observed using resonance-fluorescence. There are, however, two major obstacles in
obtaining this information:
• the ’hyperfine’ interactions between the nucleus and its environment are extremely smaller,
3.2. PHOTON INTERACTIONS 3
• the recoil of the nucleus as the gamma-ray is emitted or absorbed prevents resonance.
Just as a gun recoils when a bullet is fired, conservation of momentum requires a free nucleus
(such as in a gas) to recoil during emission or absorption of a gamma ray.
• If a nucleus at rest emits a gamma ray, the energy of the gamma ray is slightly less than the
natural energy of the transition (difference is equal to the recoil energy ER),
• but in order for a nucleus at rest to absorb a gamma ray, the gamma ray’s energy must be
slightly greater than the natural energy.
Figure 3.1: Recoil of free nuclei in emission or absorption of a gamma-rays
This means that nuclear resonance (emission and absorption of the same gamma ray by identical
nuclei) is unobservable with free nuclei, because the shift in energy is too great and the
emission and absorption spectra have no significant overlap.
4 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
Figure 3.2: Resonant overlap in free atoms. The overlap shown shaded is greatly exaggerated.
3.2. PHOTON INTERACTIONS 5
In order to achieve resonance in emission and absorption the loss of the recoil energy must be
overcome in some way.
Nuclei in a solid crystal are not free to recoil because they are bound in place in the crystal
lattice. When a nucleus in a solid emits or absorbs a gamma ray, some energy can still be lost
as recoil energy, but in this case it always occurs in discrete packets called phonons (quantized
vibrations of the crystal lattice). Any whole number of phonons can be emitted, including zero,
which is known as a "recoil-free" event. In this case conservation of momentum is satisfied by the
momentum of the crystal as a whole, so practically no energy is lost.
Mössbauer found that a significant fraction of emission and absorption events will be recoilfree,
which is quantified using the Lamb–Mössbauer factor. This fact is what makes Mössbauer
spectroscopy possible, because it means gamma rays emitted by one nucleus can be resonantly
absorbed by a sample containing nuclei of the same isotope, and this absorption can be measured.
Figure 3.3: Atoms fixed in lattice.
6 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
In its most common form, Mössbauer absorption spectroscopy,
• a solid sample is exposed to a beam of gamma radiation,
• and a detector measures the intensity of the beam transmitted through the sample.
The atoms in the source emitting the gamma rays must be of the same isotope as the atoms in
the sample absorbing them.
If the emitting and absorbing nuclei were in identical chemical environments, the nuclear
transition energies would be exactly equal and resonant absorption would be observed with both
materials at rest. The difference in chemical environments, however, causes the nuclear energy
levels to shift in a few different ways. Although these energy shifts are tiny (often less than a microelectronvolt),
the extremely narrow spectral linewidths of gamma rays for some radionuclides make
the small energy shifts correspond to large changes in absorbance. To bring the two nuclei back
into resonance it is necessary to change the energy of the gamma ray slightly, and in practice this
is always done using the Doppler effect.
During Mössbauer absorption spectroscopy, the source is accelerated through a range of velocities
using a linear motor to produce a Doppler effect and scan the gamma ray energy through a
given range. A typical range of velocities for 57Fe, for example, may be ±11 mm/s (1 mm/s =
48.075 neV).
In the resulting spectra, gamma ray intensity is plotted as a function of the source velocity.
At velocities corresponding to the resonant energy levels of the sample, a fraction of the gamma
rays are absorbed, resulting in a drop in the measured intensity and a corresponding dip in the
spectrum. The number, positions, and intensities of the dips (also called peaks; dips in transmitted
intensity are peaks in absorbance) provide information about the chemical environment of the
absorbing nuclei and can be used to characterize the sample.
Mössbauer spectroscopy is limited by the need for a suitable gamma-ray source. Usually, this
consists of a radioactive parent that decays to the desired isotope. For example,
• the source for 57Fe consists of 57Co, which decays by electron capture to an excited state
of 57Fe,
• then subsequently decays to a ground state emitting the desired gamma-ray.
• The radioactive cobalt is prepared on a foil, often of rhodium.
Ideally the parent isotope will have a sufficiently long half-life to remain useful, but will also have
a sufficient decay rate to supply the required intensity of radiation.
Mössbauer spectroscopy has an extremely fine energy resolution and can detect even subtle
changes in the nuclear environment of the relevant atoms. Typically, there are three types of
nuclear interactions that are observed
• isomer shift (or chemical shift) - is a relative measure describing a shift in the resonance
energy of a nucleus due to the transition of electrons within its s orbital. The whole spectrum
is shifted in either a positive or negative direction depending upon the s electron charge
density. This change arises due to alterations in the electrostatic response between the
non-zero probability s orbital electrons and the non-zero volume nucleus they orbit.
3.2. PHOTON INTERACTIONS 7
• quadrupole splitting - reflects the interaction between the nuclear energy levels and surrounding
electric field gradient (EFG). Nuclei in states with non-spherical charge distributions,
i.e. all those with angular quantum number (I) greater than 1/2, produce an asymmetrical
electric field which splits the nuclear energy levels. This produces a nuclear quadrupole
moment.
Mossbauer_Isomer_Shift_and_Quadrupole_Splitting_for_57Fe.png
Figure 3.4: Chemical shift and quadrupole splitting of the nuclear energy levels and corresponding
Mössbauer spectra
8 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
• hyperfine splitting (or Zeeman splitting) - is a result of the interaction between the nucleus
any surrounding magnetic field. A nucleus with spin, I, splits into 2I + 1 sub-energy levels
in the presence of magnetic field. For example, a nucleus with spin state I= 3/2 will split
into 4 non-degenerate sub-states with mI values of +3/2, +1/2, -1/2 and −3/2. Each split is
hyperfine, being in the order of 10−7
eV. The restriction rule of magnetic dipoles means that
transitions between the excited state and ground state can only occur where mI changes by
0 or 1. This gives six possible transitions for a 3/2 to 1/2 transition.
3.2. PHOTON INTERACTIONS 9
Mossbauer_Magnetting_Splitting.jpg
Figure 3.5: Magnetic splitting of the nuclear energy levels and the corresponding Mössbauer
spectrum
10 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
3.2.2 Short wavelengths (10−12
to10−9
m, including X-rays and ultravio-
let)
If photon has enough energy to excite electron to high energy levels, two effects can occur:
• internal photoeffect (e.g. photoconductivity)
• external photoeffect or the photoemission.
The kinetic energy is:
1
2
mv2
= EF ± δE − Eaf + hν − ∆E. (3.2)
At 0 K the kinetic energy is maximum
(
1
2
mv2
)max = hν − (Eaf − EF) = hν − χ = h(ν − ν0), (3.3)
where ν0 is the frequency of photoelectric effect.
In the case of photoemission induced absorption (X-ray), the photon has enough energy to
excite an electron from some inner layers of solid.
1
2
mv2
= hν − χ − EB, (3.4)
where EB is the binding energy of the inner layers. The binding energy is characteristic for each
substance, so the kinetic electrons emitted bear information related to the electronic structure
and thus, the type of material.
3.2.3 Fowler‘s theory for metal/vacuum interfaces
Fowler proposed a theory in 1931 which showed that the photoelectric current variation with the
light frequency could be accounted for by the effect of temperature in the number of electrons
available for emission, in accordance with the distribution law of Sommerfeld‘s theory for metals.
Sommerfeld‘s theory had resolved some of the problems surrounding the original models for
electron in metals. In classical Drude theory, a metal had been envisaged as a three-dimensional
potential well (box) containing a gas of freely mobile electrons. This adequately explained their
high electrical and thermal conductivities. However, because experimentally it was found that
metallic electrons do not show a gas-like heat capacity, the Boltzman distribution law is inappropriate.
A Fermi-Dirac distribution function is required, consistent with the need that the electrons
obey the Pauli exclusion principle and this distribution function has the form
P(E) =
1
1 + exp[(E − EF )/kT]
(3.5)
At ordinary temperatures, the Fermi energy EF kT. Fowler‘s law, referred to above as the
"square law", is readily tested in practice since it predicts at T=0 K
i ∝ (ν − ν0)2
(3.6)
3.2. PHOTON INTERACTIONS 11
where ν0 is the photoelectric threshold frequency and ν refers to a range of energies from threshold
to a few kT above threshold. To a good approximation, the photoelectric yield per unit light
intensity near ν0 is proportional simply to the number of electrons incident per unit time. The
energy condition can be written
(1/2m)p2
+ hν > φ (3.7)
where p is the initial momentum of the electron normal to the surface. The escaping electrons
have been termed the "available electrons".
Fowler‘s derivation for the single photon does not explicitly involve the quantum mechanical form
of current; instead, a semi-classical flux of electrons arriving at the metal surface is used. The
electron gas in the metal will obey Fermi-Dirac statistics, and the number of electrons per unit
volume having velocity components in the ranges u, u + du, ν, ν + dν and w + dw is given by the
formula
n(u, ν, w)dudνdw = 2(
m
h
)2 dudνdw
1 + exp[1
2
m(u2 + ν2 + w2) − EF ]/kT
(3.8)
where u is the velocity normal to the surface and m is the electron mass. The number of electrons
per volume, n(u)du, with their velocity component normal to the surface in the range u, u + du is
given by:
n(u)du =
4πkT
m
(
m
h
)3
log[1 + exp(EF −
1
2
mu2
)/kT]du (3.9)
The solution for the photoelectric current resulted in the form
i ∝
(hν − φ)3/2
(φ0 − hν)1/2
. (3.10)
A method for determining the threshold frequency of the illuminated surface ν0 is that which
makes use of the complete photoelectric emission. When a metal surface is exposed to the total
thermal radiation from a black body with a temperature T, the total photoelectric current is given
by:
i = A‘T2
e−
hν0
kT (3.11)
were A‘ is a constant.
3.2.4 Measurement of the photoelectric work function
• Fowler isotherm: measurement of the photocurrent depending on the frequency of incident
radiation at constant temperature, the ln if/T2
is expressed as hν/kT and compared with
the theoretical curve
ln
if
T2
= ln(αA0) + ln f(x) = B + Φ[
h
kT
(ν − ν0)]. (3.12)
From the Fowler equation, giving the relation between temperature and photoelectric emission,
ν0 can be determinate by plotting the ln f(∆) as a function of ∆. A universal curve is
obtained. Then ln i
T2 is plotted against hν
kT
. A curve us obtained of the same shape as the
universal curve, which can be made to coincide with it by a parallel shift. The vertical shift
is measure of B, the horizontal is equal to hν0
kT
.
12 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
• duBridge isochromatic method: measurements at one frequency ν but different temperatures.
For this purpose ln f(∆) is plotted against ln |∆|, and again a universal curve is
obtained. From the experimental data ln if/T2
is plotted against ln 1/T. For different frequencies
the temperature dependence of the photoemission differs
• for ν = ν0 is if = αA0 = π2
/12T2
• for x 1; ν > ν0, is f(x) = π2
/6 + x2
/2 and
if =
αA0
2
h2
(ν − ν0)2
k2
+
π2
3
, (3.13)
where the parabolic dependence is applied obly when the first term in the brackets is sufficiently
small.
• for x 1, ν < ν0, is f(x) = ex
and
if = αA0T2
exp(−
h(ν0 − ν)
kT
). (3.14)
3.2.5 Interaction of excited electrons with matter
The hot-electron problem represents the study of the deviations from the linear response regime
due to heating of charge carriers above the thermal equilibrium. Studies have been carried out in
1930s, mainly by Russian physicist Landau and Davidov. This means that electron temperature
Te which , in the presence of an external high electric field, is higher that the lattice temperature
T.
Nonlinear transport deals with problems that arise when a sufficiently strong electric field is
applied to a semiconductor sample, so that the current deviates from the linear response. This
effect is typical of semiconductors and cannot be seen on metals since, owing to their large
conductivities, Joule heating would destroy the material before deviations from the linearity
could be observed.
Considering
v(i)
(t) = v
(i)
0 + a∆t(i)
(3.15)
where v(i)
(t) is the instantaneous velocity if the i-th electron at time t, ∆t(i)
is the same time elapsed
after its last scattering event, and a = eE/m is the electron acceleration, due to a constant and
uniform applied electric field E. The region of field strengths where transport starts to deviate
from linearity is called warm-electron region.
3.2.6 Photoemission from semiconductors
There are some differences when we speak about the photoemission from metals and semiconductors.
All high quantum yield photocathodes are semiconductor ones. The absorption of light is
affected by:
• internal absorption due to their band gap
3.2. PHOTON INTERACTIONS 13
• external absorption from the external layer
The Fermi level is influenced by the surface layers and the type of semiconductor used. Electron
coming from different depths have different space charge and work function ⇒ as they would have
different work function for different photoelectrons excited at different wavelengths.
The density of stated around Fermi level increases strongly and we cannot assume that photon
absorption is independent from electron‘s energy state. The quantum yield increases with the
temperature faster than in metals.
3.2.7 Photocathode
• IR region: Ag-O-Cs, also called S-1. This was the first compound photocathode material,
developed in 1929. Sensitivity from 300 nm to 1200 nm. Since Ag-O-Cs has a higher dark
current than more modern materials photomultiplier tubes with this photocathode material
are nowadays used only in the infrared region with cooling.
• visible region: Sb-Cs has a spectral response from UV to visible and is mainly used in
reflection-mode photocathodes. and also bi-alkali (antimony-rubidium-caesium Sb-Rb-Cs,
antimony-potassium-caesium Sb-K-Cs). Spectral response range similar to the Sb-Cs photocathode,
but with higher sensitivity and lower dark current than Sb-Cs. They have sensitivity
well matched to the most common scintillator materials and so are frequently used
for ionizing radiation measurement in scintillation counters.
• UV region: metals and Cs-Te (Cs-I). These materials are sensitive to vacuum UV and UV
rays but not to visible light and are therefore referred to as solar blind. Cs-Te is insensitive
to wavelengths longer than 320 nm, and Cs-I to those longer than 200 nm.
3.2.8 Interaction of matter with very long wavelengths photons (≥
1mm) - including radio and microwaves
All nucleons, that is neutrons and protons, composing any atomic nucleus, have the intrinsic
quantum property of spin. The overall spin of the nucleus is determined by the spin quantum
number I. Some nuclei have integral spins (e. g. I = 1, 2, 3 . . .), some have fractional spins (e.g.
I = 1/2, 3/2, 5/2....), and a few have no spin, I = 0 (e.g. 12C, 16O, 32S, ....). Isotopes of particular
interest and use to organic chemists are 1H, 13C, 19F and 31P, all of which have I = 1/2.
A non-zero spin is always associated with a non-zero magnetic moment µ
µ = γS (3.16)
where γ is the gyromagnetic ratio.
It is this magnetic moment that allows the observation of nuclear magnetic resonance (NMR)
absorption spectra caused by transitions between nuclear spin levels in the presence of external
magnetic field (spin states splitting).
http://www2.chemistry.msu.edu/faculty/reusch/VirtTxtJml/Spectrpy/nmr/nmr1.htm
Electron spin resonance (ESR) or electron paramagnetic resonance (EPR) is a related technique
in which transitions between electronic spin levels are detected rather than nuclear ones. The basic
14 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
principles are similar but the instrumentation, data analysis, and detailed theory are significantly
different. Moreover, there is a much smaller number of molecules and materials with unpaired
electron spins that exhibit ESR absorption than those that have NMR absorption spectra. ESR
has much higher sensitivity than NMR does.
3.3 Electron interactions
• desorption of neutral particles
• excitation of photons (x-ray, chatodoluminescence)
• reality of material: cleaning, activation energy for chemical and physical processes, electron
lithography
• change of material
• emission of electrons: (a) without interaction, (b) elastic scattering (diffraction on passage),
(c) elastic reflection (diffraction on reflection), (d) inelastic reflection (characteristic energy
loss), (e) backscatter electron, (f) electron transmission with the possibility to emit others
and (g) electron transmission with secondary electron emission.
When a sample is bombarded with charged particles, the strongest region of the electron
energy spectrum is due to secondary electrons. The secondary electron yield depends on
many factors, and is generally higher for high atomic number targets, and at higher angles
of incidence. There is a lot of information in this secondary electron "background", but,
unlike Auger and other electron spectroscopies, it is not directly chemical or surface specific
in general.
Total electron current from a substance is the current of secondary electrons on reflection
or transmission.
3.3.1 Electron emission
The ration between this current Is and primary Ip is called secondary emission coefficient
σ = Is/Ip. By selecting the secondary electrons by their energies we can identify individual
interactions:
• the highest maximum of primary energy level ⇒ elastic reflection of e−
interaction with the atom like a whole, there is no change in energy but, the electron momentum
can be changed dramatically. For electron energy higher than (≥ 5 keV)the effect
is called Rutherford Backscattering (RBS)
dQ(θ)
dΩ
=
1
4
e2
(Z − F(θ))
4π 0 mred g2
2
1
sin4
(θ/2)
, (3.17)
where F(θ) is the screening effect of electrons. F(θ) is insignificant at high angles θ and
strong scattering can occur only in heavy atoms (used for TEM contrast). When the target
3.3. ELECTRON INTERACTIONS 15
Figure 3.6: Electron interaction with the solid state.
is a crystalline material diffraction and channelling occurs. When electron scattering
from crystal lattice is coherent, amplification in those direction occurs, which is represents
Bragg law:
nλ = 2dhkl sin θ, (3.18)
where n is a integer dhkl is is the spacing between the planes in the atomic lattice expressed
with Miller indexes. This method is used to examine the bulk crystalline structure (E ≈
10 keV) or the surface structure using low-energy electron diffraction (LEED) or very small
angle relection in reflection high-energy electron diffraction (RHEED).
• the smaller peaks to the left show the shift in maximum elastic reflection of e−
⇒ electrons
which suffer energy losses.
Electrons undergo a variety of interactions:
• adsorbed molecules are excited on vibrational state ∆E ≈ 50–500 meV;
• individual interaction with valence e−
, ∆E ≈ 3–20 eV but without a sharp peak because the
valence band has a width of several eV;
• multiple interactions with the valence e−
which is bombarded by an electron, vibrating with
frequency
ωp =
1
0
n0e2
m
(3.19)
for plasmon in volume, or
ωs =
ωp
√
1 + r
=
ω
√
2
(3.20)
for excitation of the surface plasmon. ∆E is between 5 and 60 eV.
16 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
Inelastic interactions include phonon excitations, inter and intra band transitions, plasmon
excitations, inner shell ionizations are studied by electron energy loss spectroscopy (EELS).
• smaller peaks which are not shifted by energy ⇒ Auger e−
which have for each substance
a characteristic position.
Figure 3.7: Electron energy distribution.
Primary e−
removes a core e−
(e.g. from K shell). The core state electron is removed
leaving behind a hole. As this is an unstable state, the core hole can be filled by an outer shell
electron, whereby the electron moving to the lower energy level loses an amount of energy equal
to the difference in orbital energies. The transition energy can be coupled to a second outer shell
electron which will be emitted from the atom if the transferred energy is greater than the orbital
binding energy. The Auger energy released
Ek = EK − EL − EV − Eaf. (3.21)
The EV is not equal with the original level energy EV, because the e−
on the level V moves in a
potential and the e−
on the level L is missing. The Auger requires the existence of at least two
energy levels to three electrons. That is why it can not measure H and He. The elements with
atomic number between 3 and 14 have KLL line, elements between 14 and 40 have LMM line,
between 40 and 79 are MNN line and for heavier elements NOO line exists.
3.3. ELECTRON INTERACTIONS 17
Figure 3.8: Overwiev of measuring methods using emitted electrons.
18 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
3.4 Interaction of ions with solid surface
Differences exists in the interaction of ions and electrons with matter. Ions are much heavier, have
bigger dimensions and they can have higher charges.
Interakce iontů a atomů s pevnými látkami je základem mnoha technologických procesů materiálového
inženýrství, protože může vyústit ve strukturní změny materiálu, implantaci dopadající
částice, či k rozprášení pevné látky. Také mnoho metod charakterizace povrchu materiálů je založeno
na detekci odražených nebo emitovaných částic při iontovém bombardu. Emitované částice
můžou být
• secondary electrons (secondary ion-electron emission),
• photons (characteristic x-rays and visible radiation - ionoluminescence),
• ions (ion scattering, ion-ion secondary emission),
• neutral particles (cathode sputtering),
• products of nuclear reactions
Všechny procesy způsobené dopadem atomu či iontu na pevný materiál jsou schematicky naznačeny
na obrázku Fig. 3.9.
Který proces nastane po dopadu atomu či iontu na povrch záleží na mnoha faktorech, jako je
energie dopadající částice (E0), úhel dopadu, tok dopadajících částic (γi), hmotnost a atomové
číslo dopadající částice (M1 respektive Z1), hmotnost a atomové číslo atomů terče (M2 respektive
Figure 3.9: Důsledky iontového bombardování.
3.4. INTERACTION OF IONS WITH SOLID SURFACE 19
Z2) a náboj dopadajících částic a částic terče. Za podmínek obvyklých pro opracovávání materiálů
lze interakci projektilu a terče rozdělit na dva odlišné mechanismy
• Jaderné srážky, například elastické srážky atomů
• Elektronické srážky, při nichž jsou excitovaný, nebo emitovány elektrony z materiálu.
3.4.1 Kinematika dvojné srážky
V případě jaderných srážek jsou hmotnosti interagujících částic poměrně blízké. V případě
elastické srážky tak může docházet k velké ztrátě kinetické energie dopadající částice, velkému
rozptylu dopadající částice a vzniku energetických zpětně odražených atomů. Ei,1 a Et jsou dány
vztahy
E1 = KE0 (3.22)
E2 = (1 − K)E0
kde K je kinetický faktor daný poměrem hmotností M1/M2 a úhlem rozptylu v laboratorní
soustavě θ. Pokud je hmotnost dopadajícího atomu větší, než hmotnost zasaženého atomu,
M1/M2 > 1, je kinetický faktor daný vztahem:
K =



cos θ ± M2
M1
2
− sin2
θ
1/2
M2
M1
+ 1



2
. (3.23)
Pokud M1/M2 ≤ 1 výraz se zjednoduší na tvar
K =



cos θ + M2
M1
2
− sin2
θ
1/2
M2
M1
+ 1



2
. (3.24)
Energie předaná zasaženému atomu lze vyjádřit zjednodušením rovnice (3.23) jako
E2 =
4M1M2
(M1 + M2)2
sin2 θc
2
E0, (3.25)
kde θc je rozptylový úhel v těžišťové soustavě.
3.4.2 Účinný průřez
Vystavení materiálu proudu atomů, či iontů vede k vzájemné interakci velkého množství částic.
Proto je obvyklé popisovat tento děj pomocí pravděpodobnostních veličin, založených na
pravděpodobnosti rozptylu částic. Uvažujme částici mířící do bodu ve vzdálenosti b od středu jiné
částice. Schéma odklonění této částice je na obrázku 3.10.
Veličina b je jeden ze srážkových parametrů, tzv. záměrná vzdálenost. Tok částic bude značen
Γ. Částice vstupující do srážky z diferenciální oblasti b db dφi opouštějí s určitou pravděpodobností
20 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
db
b
trajektorie
rozptylove
centrum
Figure 3.10: Schéma rozptylu lehké částice
srážku skrze diferenciální prostorový úhel Ω definovaný pomocí rozptylového úhlu θi a orientace
roviny srážky φi jako
dΩ = sin θidθidφi. (3.26)
Konstantou úměrnosti je diferenciální účinný průřez σ(θi, φi):
Γbdbdφ = σ(θi, φi) ΓdΩ. (3.27)
Protože obě strany rovnice (3.27) odpovídají množství částic rozptýlených za jednotku času,
tedy výrazu dN/dt, lze diferenciální účinný průřez považovat za množství částic rozptýlených za
jednotku času a vstupního úhlu do prostorového úhlu dΩ:
σ(θi, φi) =
dN
dt Γ dΩ
(3.28)
Diferenciální účinný průřez lze vyjádřit z rovnic (3.26) a (3.27) jako
σ(θi, φi) =
b
sin θi
db
dθi
. (3.29)
Velikost db/dθ je určena interakční silou, která vyvolává rozptyl. Absolutní hodnota je použita,
protože θ obvykle klesá se vzrůstajícím b a σ(θiaφi) jsou kladné veličiny.
Celkový účinný průřez σt lze vypočítat integrací σ přes celý prostor
σt =
Ω
σ(θi, φi)dΩ =
2π
0
π
0
σ(θi, φi)dθdφ. (3.30)
3.4. INTERACTION OF IONS WITH SOLID SURFACE 21
Veličiny σ(θi, φi) a σt závisí na velikosti vzájemné rychlosti částic. V případě centrálně působící
síly závisí vzájemný interakční potenciál pouze na vzájemné vzdálenosti částic r. Takový potenciál
je izotropní, takže výsledný diferenciální účinný průřez nezávisí na φ:
σt = 2π
π
0
σ(θi)dθ. (3.31)
Například Coulombova síla splňuje toto kritérium.
Uvažujme nepohybující se částice rozmístěné s hustotou nt. Pokud ke srážkám s dopadajícím
svazkem částic dochází poměrně zřídka, lze počet částic, které projdou srážkou během průletu
oblastí dx lze vyjádřit jako
dn = −σtotnntdx. (3.32)
Při studiu zastavení iontů v materiálu bývá užitečné znát pravděpodobnost, že projektil s energií
Ei,0 předá zasaženému atomu množství energie v rozmezí Et a Et + dEt. Tato pravděpodobnostní
funkce definuje účinný průřez pro přenos energie σE(Et), který je ve vztahu s účinným
průřezem σ(θi, φi) jako
σE(Et)dEt = σ(θi, φi)dθidφi. (3.33)
3.4.3 Dynamika elastické srážky
Aby bylo možno určit diferenciální účinný průřez z rovnice (3.29) je nutné najít závislost mezi
parametrem rozptylu b a úhlem rozptylu θi. Závislost lze stanovit i bez konkrétní znalosti přesné
závislosti interakční síly působící mezi částicemi. Za předpoklady centrálně působící síly F(r) =
F(r)ˆr lze využít potenciální energie U(r) díky vztahu
F(r) = − U(r) = −
dU(r)
dr
ˆr. (3.34)
Úhel rozptylu θi pak lze zapsat ve tvaru
θi(b, v) = π − 2
∞
rmin
b
r2
1 −
b2
r2
−
2U(r)
µv2
−1/2
dr (3.35)
kde v je relativní rychlost částic a rmin je nejmenší vzájemná vzdálenost, která je dána vztahem
rmin = b 1 −
2U(rmin)
µv2
−1/2
. (3.36)
3.4.4 Meziatomový potenciál
Stínící potenciál
Interakce dvou atomových jader je dána coulombovskou potenciální energií
UC(r) =
1
4πε0
ZiZte2
r
. (3.37)
22 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
Interakce dvou atomů je však složitější, neboť zahrnuje i vliv elektronového obalu a jeho závislost
na vzdálenosti Existují dvě užitečné veličiny, které ohraničují problém a to Bohrův poloměr a0
a0 =
4π 0¯h2
mee2
, (3.38)
a meziatomová vzdálenosti v krystalu r0. Bohrův poloměr a0 = 0.053 nm, označuje dosah elektronové
slupky atomu. Meziatomová vzdálenost, typicky r0 = 0.25 nm, je vzdálenost mezi mezi
dvěma vázanými atomy daná minimem potenciální energie krystalu. Náboj jádra je na vzdálenost
r r0 velmi dobře odstíněn elektrony z elektronového obalu. Jak se atomy k sobě přibližují, začnou
se elektronové obaly překrývat a může dojít k přitahování atomů a vzniku vazby. V extrémním
případě r a0 se jádra stanou vzájemně nejblíže položenými nabitými částicemi v systému. V
takovém případě jejich Coulombovská interakce dominuje a potenciální energie je velmi dobře
popsána rovnicí (3.37).
Ve střední vzdálenosti a0 < r ≤ r0, má kladná interakční energie, vedoucí na odpudivou sílu,
dvě složky (i) elektrostatická odpudivá interakce mezi dvěma jádry (ii) zvýšení energie na základě
Pauliho vylučovacího principu.
Ačkoliv přesný popis meziatomové interakce zahrnuje komplikované efekty elektronového obalu,
předchozí diskuze ukazuje, že stačí uvažovat rovnici (3.37) upravenou vhodnou limitující funkcí.
Ve výsledku je stíněný Coulombovský potenciál popisován tvarem
U(r) =
1
4πε0
ZiZte2
r
χ(r) (3.39)
kde χ je stínící funkce. Za ideálních podmínek by χ(r) mělo jít k nule pro velké vzdálenosti a k
jedničce pro malé vzdálenosti.
V podstatě existují dva způsoby vyjádření stínící funkce, (i) jednoduchý statistický a (ii)
kvantově-mechanický Hartee-Fockův atomový model.
Thomas-Fermiho statistický model
Statistický Thomas-Fermi (TF) popis předpokládá, že se elektrony chovají jako ideální plyn složený
z částic o energii E. Elektrony podléhají Fermi-Diracově statistice a vyplňují potenciálovou jámu
v okolí pozitivně nabitého jádra. Tento model evidentně nebere v potaz různé elektronové hladiny.
Přesné řešení TF modelu stínící funkce je obvykle získáno numericky.Avšak pro mnoho aplikací
je výhodné mít k dispozici analytické řešení, které přibližně odpovídá TF rovnici. Nejstarší a
nejznámější je Sommerfeldův asymptotický výraz:
χ(x) = 1 +
x
a
λ −c
(3.40)
kde konstanty a, λ a c jsou voleny následujícím způsobem: a = 122/3
a cλ = 3. Normalizovaná
meziatomová vzdálenost x = r/aTF je následně modifikována pomocí TF stínícím poloměrem pro
srážky mezi atomy
aTF =
1
2
3π
4
3/2
a0
Z
1/3
eff
(3.41)
3.4. INTERACTION OF IONS WITH SOLID SURFACE 23
kde Zeff je efektivní náboj při interakci dvou rozdílných atomů
Zeff = (Z
1/2
i + Z
1/2
t )2
. (3.42)
Sommerfield zjistil, že pro velké x jsou přibližné hodnoty λ a c λ = 0.772 a c = 3.886. Tím dává
konečný tvar rovnice (3.43) jako
χ(x) = 1 +
x
122/3
0.772 −3.886
(3.43)
Další častou používanou aproximací stínící funkce je tvar odvozený Molierem ve formě tří
exponenciál:
χ(x) = 7p exp(−qx) + 11p exp(−4qx) + 2p exp(20qx) (3.44)
kde p = 0.05 a q = 0.3. Matematicky jednoduché analytické řešení stínící funkce lze vytvořit při
využití inverzní x = r/aTF s exponentem pro různé rozsahy r/aTF:
χ(r) =
ks
s
aTF
r
s−1
(3.45)
kde s = 1, 2 . . . a ks je numerická konstanta.
Univerzální meziatomový potenciál - kvantově mechanické odvození
Stínící funkce odvozená za použití kvantové mechaniky vytváří tvar, který je obvykle nazýván
jako univerzální meziatomový potenciál. Díky práci Zieglera, Biersacka a Littmarka byla odvozena
stínicí funkce ve tvaru:
χU = 0.1818 exp(−3.2x) + 0.5099 exp(−0.9423x) + 0.2802 exp(−0.4028x) + 0.02817 exp(−0.2016)
(3.46)
kde je redukovaná dálka x dána jako
x =
r
aU
(3.47)
a aU, univerzální stínící délka, je definována jako
aU =
0.8854a0
Z0.23
i + Z0.23
t
. (3.48)
Aplikujeme-li meziatomový potenciál se stínící funkcí Eq. (3.45) na rozptylový proces, získáme
diferenciální účinný průřez pro rozptyl ve tvaru
σE(Ei) =
Cm
Em
i E1+m
t
, (3.49)
kde m = 1/s v rovnici (3.45) a konstanta Cm je dána jako
Cm =
π
2
λma2
TF
ZiZte2
2πε0aTF
2m
M1
M2
2
, (3.50)
kde λm je definováno
λ1/3 = 1.309 λ1/2 = 0.327 λ0.5 = 0.5. (3.51)
24 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
3.4.5 Brždění iontů
Because mion ∼ mtarget, it leads to considerable energy loss due to elastic collisions. Energy loss
for penetration by the distance dx
S = −
dE
dx
(3.52)
is called stopping power or specific energy loss. The stopping cross section can be defined
as:
=
1
N
dE
dx
or =
1
ρ
dE
dx
, (3.53)
where N represent [atoms/m3
] and ρ is the density [kg/m3
]. Sometimes also represents stopping
power.
Obecně lze celkovou ztrátu energie projektilu pohybujícího se v materiálu získat jako součet
atomového a elektronového příspěvku:
S = Sn + Se. (3.54)
The relative importance of interactions between ions and matter depends on the ion speed,
their charge and the atoms of the target
• for vion v0, where v0 is the Bohr velocity of electrons in the atom. Ions will carry their
electrons and they tend to be neutralized by electron capture. At these speeds, the elastic
collisions dominates and the losses are callednuclear energy losses.
• for higher velocities, the importance of nuclear energy losses decreases with a factor of 1/E
and the inelastic collisions with electrons because important. The collision can result in
excitation of electrons of the material and/or excitation of electron in ion electron shells.
Above 200 keV/amu, the contribution of nuclear energy losses is typically less than 1 % than
electronic one. Between 0.1v0 ≤ vion ≥ Z
2/3
1 v0 electron collisions are proportional with the
velocity (E1/2
).
• for v v0 the ion charge increases till it completely losses all electrons. The Bethe-Bloch
Formula for the electron energy is
dE
dx
= NZ2(Z1e2
)2
f(E/M1), (3.55)
where Z1 is the ion atomic number.
Jaderné brzdění
Průměrná ztráta energie částice při pohybu o vzdálenost dx lze získat pomocí diferenciálního
účinného průřezu pro přenos energie σE. Ten je definován z rovnice (3.33) jako
dE
dx
= n
Et,max
Et,min
EtσEdEt, (3.56)
3.4. INTERACTION OF IONS WITH SOLID SURFACE 25
kde Et,min a Et,max jsou minimální, respektive maximální energie předané zasaženému atomu.
Aplikujeme-li diferenciální účinný průřez ze vtahu (3.49), bude účinný průřez pro nukleární
zastavování dán
Sn(E) =
CmE1−2m
1 − m
4MiMt
(Mi + Mt)2
1−m
, (3.57)
kde Cm je definováno v rovnici (3.50).
Účinný průřez pro jaderné brzdění roste pro nízké hodnoty energie a dosahuje maxima pro
jednotky keV pro lehké ionty a stovky keV pro těžké ionty. Lze ho spočítat pro iont s energií E
na základě vztahu odvozeného Zieglerem a kolektivemjako
Sn =
8.462ZiZtSn(Er)
(Mi + Mt) + (Z0.23
i + Z0.23
t )
eVcm2
/1015
atomů, (3.58)
kde Er je redukovaná energie vyjádřená jako
Er =
32.53MtE
ZiZt(Mi + Mt)Z0.23
i + Z0.23
t )
(3.59)
a Sn(Er) je redukované nukleární brzdění definované jako
Sn(Er) =
ln(1 + 1.1383Er)
2(Er + 0.01321E0.21226
r + 0.19593E0.5
r )
pro Er ≤ 30 keV (3.60)
nebo
Sn(Er) =
ln Er
2Er
pro Er > 30 keV. (3.61)
Nad 200 keV/amu je příspěvek nukleárního brzděný malý, typicky pod 1 % elektronového brzdění.
Elektronové brzdění
Množství srážek s elektrony, které podstoupí iont při průletu pevnou látkou je obrovské. Zároveň
může velmi často docházet ke změně náboje iontu. Je proto velice obtížné popsat všechny možné
interakce pro všechny možné stavy iontu. Místo toho se brzdění obvykle vyjadřuje jako průměrná
ztráta energie pro různé stavy iontu. Tímto přístupem lze teoreticky určit výsledek s chybou
odpovídající několika málo procentům při energiích kolem stovky keV. Nejpřesnější je Betheho
formule:
S =
4π
mec2
nz2
β2
e2
4πε0
2
ln(
2mec2
β2
I(1 − β2)
) − β2
, (3.62)
kde v rychlost iontu, c je rychlost světla, β = v/c, ze je náboj iontu, me je klidová hmotnost
elektronu, n = NAZρ/A je elektronová hustota terče a I excitační potenciál terče.
Pro nižší energie pod přibližně 100 keV na nukleon, bývá obtížnější určit brzdný účinek z teorie
Pro rychlosti iontu v rozsahu ≈ 0.1v0 to Z
2/3
i v0 je ztráta energie díky elektronům přibližně
úměrná E1/2
, jak odvodil Lindhard a kolektiv.Při vyšších rychlostech vi,0 v0 dochází k postupné
ztrátě elektronů z obalu iontu až je nakonec zcela ztratí. V takovém případě je ztráta energie
úměrná druhé mocnině náboje iontu.
26 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
3.4.6 Dolet iontů
Jeden z nejdůležitějších parametrů jakékoliv interakce mezi ionty a pevnou látkou je rozložení
hloubky (vzdálenosti), ve které se ionty zastaví. Obvyklé rozložení v amorfní látce iontů majících
stejnou energii je přibližně gaussovské. Může proto být obvykle charakterizováno projekcí dráhy
Rp a šířkou rozptylu ∆Rp kolem této střední hodnoty, jak je zobrazeno na obrázku 3.11.
Figure 3.11: Iont dopadající na pevnou látky se v ní pohybuje po dráze R, která vytváří projekci
dráhy Rp do původního směru přilétajícího atomu.
3.4.7 Radiační poškození
Přilétající ion s energií odpovídající 100 keV, se v pevné látce zastaví za čas řádově 10−13
s díky
elektronovému a nukleárnímu brzdění. Během své dráhy v pevné látce vykoná dopadající iont
mnoho srážek s atomy pevné látky. Když se jedná o krystalickou látku, může být energie předaná
při srážce dostatečná k vyražení atomu z jeho polohy v mřížce.
Takto primárně vyražený atom může následně vyrážet další atomy (sekundární, terciární), čímž
vytvoří kaskády atomových srážek. Takovéto srážky vedou k vytvoření prázdných míst, uvěznění
atomu mimo mřížku a dalším typům krystalových poruch podél dráhy dopadajícího iontu. Tato
sekvence mnoha srážek je obvykle nazývána srážkovou kaskádou.
Teorie popisující radiační poškození v pevných látkách je založena na předpokladu, že atom
vyražený ze svého místa v krystalu buď iontem, nebo jiným odskočeným atomem musí během
srážky získat určitou minimální energii. Dislokační energie Ed je energie kterou musí atom získat,
aby byl vyražen z mřížky.
3.4.8 Termální ohřev
Během srážkové kaskády postupně nastane situace, kdy vyražené atomy vyšších řádů již nebudou
mít dostatek energie, aby vyrazily další atomy z jejich poloh v mřížce. Další srážky s atomy pevné
látky tak pouze způsobí rozvibrování zasažených atomů s velkou amplitudou výchylky. Tyto
vibrace se postupně přenáší na další atomy a energie je rozložena do vibrace mřížky, tedy tepla.
3.4. INTERACTION OF IONS WITH SOLID SURFACE 27
Po přibližně 10−12
s je dosažen stav termodynamické rovnováhy, kdy se rozdělení vibračních
energie začne blížit Maxwell-Boltzmanově rozdělení. Tato fáze srážkové kaskády se nazývá termální
ohřev a může trvat i několik pikosekund, než dojde k poklesu na původní teplotu.
3.4.9 Rozprašování
Figure 3.12: Rozprašování
Výtěžnost Y popisuje, kolik atomů je vyraženo během rozprašovací události. Celková výtěžnost
je pak definována jako průměrný počet rozprášených atomů připadající na jeden dopadající atom.
Celkové chování atomů terče po dopadu částice lze rozdělit do pěti rozdílných skupin podle
Ei, Zi a Mi, Zt a Mt. V lineární kaskádní teorii předává počáteční iont energii atomům v mřížce,
které jsou v klidu. Produkuje tak velké množství rychlých atomů, které následně produkují další
pomalejší atomy. Vzniká tak izotropická kaskáda. Přibližně za 1–5×1013
s po dopadu iontu již
energie na hranici kaskády klesne pod limitní energii nutnou k vyražení dalšího atomu (přibližně
10 eV). Kaskáda tak zanikne rozptýlením energie do vibrací mřížky. Použijeme-li tuto teorii, lze
výtěžnost spočítat pomocí analytického vztahu.
Y (Ei, θi) =
Kit
U0
Sn(Ei/Eit) f(θi), (3.63)
kde U0 je velikost potenciálového valu na povrchu (v eV), Kit a Eit jsou škálovací konstanty závislé
na chemickém složení terče a dopadajícím iontu, Sn(Ei/Eit) je redukovaný účinný průřez pro
jaderné brzdění, ( = Ei/Eit je redukovaná energie) a f(θi) je funkce popisující sklon, pod nímž
došlo k nárazu.
3.4.10 Secondary ion-electron emission
According to the adiabatic principle, heavy ions are not efficient for transferring energy to light
electrons to provide ionization. This general statement can be referred to the direct electron
28 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
emission from solid surfaces induced by ion impact.
The secondary electron emission coefficient γ (electron yield per ion) becomes relatively high only
at very high ion energies exceeding 1 keV. Although the secondary electron emission coefficient γ
is low at lower ion energies, it is not negligible and remains almost constant at ion energies below
the kilovolt range.
γ =
ie
ii
. (3.64)
Total γ coefficient is the sum of
γ = γp + γk, (3.65)
where γp is the potential emission and γk for the kinetic one.
Iont blížící se k povrchu je reprezentován potenc. jámou ⇒ dojde k tunelovému přechodu e−
z
kovu na prázdnou valenční hladinu v iontu ⇒ e−
přejde na nižší hladinu a přebytek energie předá
dalšímu e−
, který je emitován. Tento jev je základem tzv. neutralizační spektroskopie.
The Penning mechanism of secondary ion-electron emission, also called the potential
mechanism: the ion approaching the surface represents a potential well ⇒ dojde k tunelovému
přechodu e−
z kovu na prázdnou valenční hladinu v iontu ⇒ e−
přejde na nižší hladinu a
přebytek energie předá dalšímu e−
, který je emitován. Tento jev je základem tzv. neutralizační
spektroskopie.
The electron is extracted because the ionization potential Φ exceeds the work function φ.
The energy Φ − φ is usually large enough to enable the escape of more than one electron from
the surface. The secondary ion-electron emission coefficient γ can be estimated using the
empirical formula:
Φ ≥ 2φ, (3.66)
⇒ potential emissions are observed especially when bombarding the surface with ions of inert
gases that have high Φ. γp is higher for higher ion charge.
Ion-neutralization spectroscopy (INS) is an emission electron spectroscopy. In INS the
externally applied agent is a slowly moving positive ion, such as He+
, presented to the solid
surface. Near the surface a non radiative, Auger type electronic transition process occurs which
simultaneously neutralizes the ion to the ground state of the parent atom and excites a second
electron, which may be ejected into vacuum. The kinetic-energy distribution of these ejected
electrons contains spectroscopic information concerning the electronic structure in the surface
region of the solid.
• When the incoming ions is just outside the metal surface, two electrons in the filled valence
band of the metal interact, exchanging energy and momentum.
• One electron, the neutralizing electron, tunnels through the potential barrier into the potential
well presented by the ion, and drops to the vacant atomic ground level.
• The energy released in this transition is taken up by the second interacting election which
now may have sufficient energy to escape from the metal
• These Auger type transition can take place anywhere within the filled valence band so that
the ejected electrons have a range of energies rather than one specific energy. Outside the
metal surface the electron energy distribution can be measured quite straightforwardly.
3.5. PENETRATION DEPTH 29
3.5 Penetration depth
Penetration depth is a measure of how deep light or any electromagnetic radiation can penetrate
into a material. It is defined as the depth at which the intensity of the radiation inside the material
falls to 1/e of its original value at (or more properly, just beneath) the surface.
When electromagnetic radiation is incident on the surface of a material, it may be (partly) reflected
from that surface and there will be a field containing energy transmitted into the material. This
electromagnetic field interacts with the atoms and electrons inside the material. Depending on
the nature of the material, the electromagnetic field might travel very far into the material, or
may die out very quickly. For a given material, penetration depth will generally be a function of
wavelength.
• photons
The huge electromagnetic spectrum covers wavelengths between 106
m and 10−14
m. If we
use the photons for microstructure analysis it is necessary to select a comparable length of
radiation comparable with the structure to be analysed ⇒ between 10−4
–10−10
m. Depth of
penetration changes with photon energy and the type of material use for analysis:
– IR radiation: characterization of materials, which is depending of the absorbed wave-
lengths;
– VIS radiation: used often to see the surface. Some material are opaque while others
are transparent. Even for high reflectivity opaque materials the penetration depth is
in order of 50–300 nm.
– UV radiation: reveals information about the electron distribution for surface atoms.
Most substances absorb in this region.
– X-ray radiation: X-ray penetration is easier to predict than visible radiation. The
penetration depth is typically µm. The intensity of radiation which is transmitted
through sample thickness d is
I = I0 exp(−αd), (3.67)
where I0 is the intensity of incident radiation and α, the absorbance, increases with
the atomic number.
– Gamma (γ) radiation: has energy between 50 keV–50 MeV ≈ 10−2
nm. Absorbance is
governed by the same relation for X-rays, but here α is inversely proportional with the
atomic number. Gamma radiation penetrate almost all the laboratory samples.
• electrons
Penetration depth changes significantly with the electron energy and the atomic number of
the material:
– for stainless steel: around one tenth of a µm for energy 10 keV and 2 µm for 30 keV,
– for energies higher than 10 keV: elements with atomic number under 20 have a penetration
depth up to 10 µm while the elements with Z higher than 40 below 2 µm
– for energies between 0–2 keV: the penetration depth is reduces to 0.4–300 nm
30 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
Figure 3.13: Penetration depth for different particles.
• neutrons
Although neutrons are thousand times bigger than electrons, they still have a wave-particle
duality. Because they don‘t affect the electron clouds and interact only with the nuclei
the penetration depths are larger than those of electrons or X-rays and are in order of few
millimetres.
• protons
Protons interactions are similar to electron ones, but some peculiarities can be observed.
Being 1836 times the mass of electron, has a higher angular momentum and loses only a
small portion of eat at each collision ⇒ penetration depth is greater than for electrons. For
2.5 MeV proton, the penetration depth is around 55 µm for carbon and 28 µm for silver; it
decreases with the the proton energy and Z of the material.
• ions/atoms
Penetration depth depends on the number of protons in the ion, its energy and the target. For
low energies (several eVs) it is reflected from the target and part of its energy is transferred
to the surface atom, causing ejection of atoms, ions or clusters. It can also create cascade
collisions.
The range R - the total distance that the ion (projectile) travels in coming to rest, is longer
3.6. MATERIAL DAMAGE 31
than the penetration depth x. The projected range Rp is defined as the total path length
of the projectile with energies 0.002 ≤ ≤ 0.1 keV measured along the direction of incidence.
Rp = C1(µ)M2




Z
2/3
1 + Z
2/3
2
Z1Z2

 E


2/3
, (3.68)
where M2 is the atomic mass of the target , Z is the atomic number while C1(µ) is experimentally
determined. for projectiles with energies 0.5 ≤ ≤ 10 keV is
Rp = C1(µ)M2




(Z
2/3
1 + Z
2/3
2 )1/2
Z1Z2

 E


2/3
. (3.69)
Ion channelling is possible along crystallographic axes with small critical angles. In this
case the penetration depth is substantially increased.
Figure 3.14: Calculation of total distance of ion.
3.6 Material damage
• photons
Regarded in general as the least harmful, the photons will damage the target nevertheless.
Damage can occur by heating the target, which depends on the radiant energy, photon flux
and the depth of penetration. Moreover X-rays can induce surface oxidation and lasers can
burn holes into material. In general the most photon radiation causes very little damage ⇒
it is used at the beginning of target characterization.
32 CHAPTER 3. INTERACTION OF RADIATION WITH MATTER
• electrons
Although electrons have a dual nature, their weight leads to a relatively large transfer angular
momentum if they are accelerated to several hundreds keVs. The resulted damage depends
on the heat transferred and the thermal conductivity of the material. For metals and alloys
the damage is minimum, but in polymers and oxides the damage is worse. It is possible to
cover the surface with a conductive material (usually gold) but the disadvantage is that the
composition can not longer be investigated.
• ions and atoms
When atoms and ions penetrate the target, they can react with it or not cause any damage
at all ( e.g. Ion Scattering Spectroscopy, where ions interact elastically with the target) or
they cause significant damage:
– ejection of atoms form their normal lattice position
– breaking the bonds (requires 10x more energy)
For low ion fluxes the damage areas are isolated (the amorphous material is surrounded by
intact one) but for high fluxes, the ions create an amorphous layer.
3.7 Resolution
Mean free path determines the depth resolution, which has influence in the spatial one (defined
as the perpendicular to the direction of the incident beam). Generally, the spatial resolution is
influenced by the beam size and the wavelength.
The image can be obtain by:
• illumination of the sample and use of lens for focusing the reflected or emitted radiation.
The spatial resolution depends on the lens system used and the wavelength of emitted or
reflected radiation. Optical, X-ray and some ion microscopes are using this method.
• a narrow beam is directed on the sample and absorbed or reflected radiation is detected. The
sample surface is scanned by a narrow beam. In this case the spatial resolution depends in
the wavelength, beam size and the scattered radiation in the sample. Most of the equipment
are using electrons and ions.
Nowadays, laser microscopes are used for sample characterization.