Second Series February sgzg Pol. 33, ¹.z
THE
PHYSICAL REVIEW
QUANTUM MECHANICS AND RADIOACTIVE DISINTEGRATION'
BY R. %.GURNEY AND E.U. CQNDoN
ABSTRACT
Application of quantum mechanics to a simple model of the nucleus gives the
phenomenon of radioactive disintegration. The statistical nature of the quantum
mechanics gives directly disintegration as a chance phenomenon without any special
hypothesis. )i contains a presentation of those features of quantum mechanics which
are here used and gives a simple calculation of the disintegration constant. $2 discusses
the qualitative application of the model to the nucleus. $3 presents quantitative
calculations amounting to a theoretical interpretation of the Geiger-Nuttall relation
between the rate of disintegration and the energy of the emitted a-particle. In getting
this relation one arrives at the rather remarkable conclusion that the law of force
between emitted e-particle and the rest of the nucleus is substantially the same in all
the atoms even where the decay rates stand in the ratio 10". $4 calls attention to the
natural way in which the paradoxical results of Rutherford and Chadwick on the
scattering of fast O.-particles by uranium receive explanation with the model here used.
(5 discusses certain limitations inherent in the methods employed.
HE study of radioacitivity itself together with the application of it as a
working source of high speed helium nuclei and electrons has played a
fundamental role in the development of quantum physics. The scattering
experiments of Rutherford and his associates gave the picture of the nuclear
atom on which all of the success of modern atomic theory depends. Bohr's
formulation of quantum postulates to be applied to such a model was a great
step in the extension of knowledge of atomic structure and finally culminated
in 1925 in the discovery by Heisenberg and by Schrodinger of a reformulation
of mechanical laws which has subsequently proved extremely powerful
in handling atomic structure problems. In this development of the last
fifteen years little advance has been made on the problem of the structure of
the nucleus.
It seems, however, that the new quantum mechanics has had sufhcient
success to justify the hope that it is competent to carry out an effective
attack on the problem. The quantum mechanics has in it just those statistical
elements which would seem appropriate to an explanation of the phenomenon
' An account of this work was first published in Nature for September 22, 1928. In a
number of the Zeitschrift fur Physik (51, 204, 1928) received here two weeks ago there appears
a paper by Gamow who has arrived quite independently at the same basic idea as was presented
in our letter and which is here treated in detail. Reports of this paper were also given
at the Schenectady meeting of the National Academy of Sciences on November 20, 1928 and
at the Minneapolis meeting of the American Physical Society on December 1, 1928.
127
128 R. IV. GURNEY AND E. U. CONDON
of radioactive decay. This is the feature of the general problem with which
we are concerned in this paper. We believe that the results provide at last an
interpretation of nuclear disintegration which in its fundamental points is
very close to the truth although it is necessarily quite incomplete.
The outstanding difficulties in the way of a good theoretical treatment of
nuclear structure at present are mainly bound up with our lack of understanding
of the quantum mechanics of the magnetism of the fundamental
particles. This question has been much advanced this year by Dirac's extension
of Pauli's theory of the spinning electron' but this remains essentially
a theory of the behavior of one electron in an electromagnetic field. Not only
is it apparently still unsatisfactory as such but this limitation must necessarily
be disposed of in principle before the many body nuclear problem can
be approached. And with that done there will remain the inevitable analytical
difficulties.
Enough is known, however, to teach us that probably the magnetic interaction
is not to be handled simply by an alteration of a potential energy
function depending solely on the coordinates of the several interacting particles.
This tends to detract from the value of arguments based simply on the
use of quantum mechanics with the positional coordinates of the nuclear
constituents. Nevertheless we shall restrict ourselves to the use of such
methods in the discussion of the instability or capacity for spontaneous
disintegration of a very much simplified nuclear model. The simplification
to be made will consist in supposing that we can discuss the behavior of
any one constituent by applying the quantum mechanics to it as a single
body moving in a force field due to the rest of the nucleus.
The difference between quantum mechanics and classical mechanics which
is here made responsible for the disintegration process is easily stated. In
classical mechanics the orbit of a particle is entirely confined to those points
in space at which its potential energy is less than its total energy. This is
not true in quantum mechanics. Classically if a particle be moving in a basin
of low potential energy and have not as much total energy as the maximum
of potential energy surrounding the basin, it must cerfairlly remain there for
all time, unless it acquires the deficiency in energy somehow. But in quantum
mechanics most statements of certainty are replaced by statements of probability.
And the above statement must now be altered to read " it may
remain there for a long time but as time goes on the probability that it has
escaped, even without change in its total energy, increases toward unity. "
In $1 of this paper the detailed development of the argument leading to
the conclusion of the preceding paragraph is given. In f2 we discuss its
qualitative application to the nuclear disintegration problem. f3 is devoted
to semi-quantitative estimates of the rates of decay.
CQUPLING QF JUIQTtoNs oF EQUAL ENERGY
Consider a particle of mass p, It is sufficient to consider one degree of
freedom; let the coordinate of the particle be x and let the forces be measured
by the potential energy function V(x).
~
Dirac, Proc. Roy. Soc. A117, 610; A118, 351 {1928).
QUANTUM MECHANICS: RADIOACTIVE DISINTEGRATION 129
In classical mechanics the equations of motion possess the energy integral
p'/2@+ V(x) = W
(p =momentum) which, for values of x such that W —V(x) &0, can only be
satisfied by P pure imaginary. Therefore, classically, one had the result that
a particle could only be where W —V(x) &0. An important consequence of
this was that if there were several ranges of x for which W —V(x) &0 separated
by ranges where W —V(x) & 0, then there were several different motions
possible with the energy level 8', each of which was wholly confined to one
of these separate ranges. Thus in Fig. 1 for the energy level indicated there
mould be two distinct types of motion of the same energy 8";one is a libration
in the range I, and the other a libration in the range II.
These results are modified considerably
by the new quantum mechanics. Iil V
the first place, Fq. (1) loses its validity
and is replaced by an integral theorem,
as Born' has shown, in which there is no
longer a definite correlation between
simultaneous value of position and momentum
as (1) implies. The quantum Fig, 1.
mechanical form of (1) is, if lf (x) is
Schroedi'nger's wave function
W
I I
I
+II+ . x
f'+" h' dP df
tr= II
———+V(*)04)d* (& )
J „8x'p dx dx
The lack of a precise correlation has been much emphasized by Heisenberg
and by Bohr,4
and is a general characteristic of quantum mechanics. From
the new standpoint, one has to consider the behavior of Schrodinger's
equation for the problem
' Born, Zeits. f. Physik 38, 806 (1926).
4 Heisenberg, Zeits. f. Physik 43, 172 (1927); Bohr, Nature, April 14, 1928,
d'P Sm-'p
—+ (W —V(x))iP =0. (2)
dx' h'
As is well known, in some problems there are solutions f(W, x) for certain
values of 8' which are finite and continuous everywhere. These are the
"allowed" values of quantum theory. For the iP(W, x) which comes out of (2)
as a by-product, Born has shown that its square may be satisfactorily interpreted
as giving the probability that the particle lies between x and x+dx
when it is in the state of energy B. This is really the ground for requiring
that iP remain finite. For an energy level, such that P(W, x), does not remain
finite as x~+ 00, the probability that it is not "at infinity" is vanishingly
small, and therefore these states do not exist physically. Adopting the
probability interpretation of iP(W, x) one has at once the result that there is
a finite probability of being outside the range of the classical motion of that
energy.
130 R. lK GURNEY AND E. U. CONDON
A simple case is the lowest state of the harmonic oscillator, which has the
energy hv/2. The P(W, x) for this state is e r'&*&'&&' so P=e &*'&' where u is
the classical amplitude of motion associated with this energy. The probability
of being outside the classical range is therefore
a
=0.157
Fig. 2.
or more than 15 percent.
When one studies the behavior of f(W, x) from (2) for a V(x) somewhat
like the one in Fig. 1, he finds that, if the W is one for which P is finite everywhere,
then P approaches zero very rapidly (exponential decrease) as
x~+ ~ . In the neighborhood in which W—V(x) is small, the function takes
on appreciable values and has oscillatory character where W —V(x) (0, and
non-oscillatory character elsewhere. Cases like that of Fig. 1 have been discussed
by Hund' in connection with his studies of molecular spectra.
An important case is that in which the potential energy curve consists
of a single "obstacle" or barrier as in Fig. 2, and the motion is one of insuScient
energy, 8', to clear the obstacle. In such cases there are taro finite
solutions lf &( W, x), and tp2(W, x) associated with each
energy level, W, and so an arbitrary linear combi-
V
nation of them is also a solution of (2). Born has
shown that there is always a combination of them
which depends on x as e+' and represents a pure
left-to-right progressive wave motion as x~+ ~. Such a solution for x large
and negative can then be said to represent an incident left-to-right wave
coming from the left side and a reflected wave which is not as strong as the
incident wave. The interpretation is that the incident beam of particles is
partly reflected and partly transmitted. In the range where (W—V) (0
the de Broglie wave-length h/p becomes imaginary, and so gives rise to an
exponential behavior of P whose nearest analogue is, perhaps, in optics in
the slight penetration of a refracted ray into a rarer medium even beyond the
angle of total reHection where the refracted angle is imaginary. In this way,
one can find the probability that a particle comi'ng up from the left will
get through the wall and escape to the right. The case illustrated in Fig. 3a
for which
Fig. 3a.
V(x) =0
V(x) =V
V(x) =0
x& —a,
—a&x&0,
and for 0& 8"& V, is a simple one with which to illustrate the nature of the
calculation. For a given energy level, W, there are two g functions satisfying
~ Hund, Zeits. f. Physik 40, 742 (1927);Kentzel, Zeits. f. Physik 38, 518 (1926).
QUANTUM MECHANICS: RADIOACTIVE DISINTEGRA TION 13i
the requirements of finiteness everywhere and of continuity for the ordinates
and slopes at the discontinuities in V(x).
These are readily found to be
( cosh o,a cos o,(x+a) —(o,/o. q)sinh o,a sin uq(x+a) (x& —a)
fg(W, x) = ~ cosh o,x (—a&x&0)
COS 0 yX (o&*)
(3)'
—(o&/a2)sinh 02a cos o.z(x+a)+cosh 02a sin o.q(x+u) (x& —a)
Pg(W, x) = (o ~/u, ) sinh u2x ( —a&x&O)
Sln 0'yX (0&*)
where a&=(2s./b)(2pW)'~' and o, = (2s/h) [2p(V —W) j'~' To find the lP function
corresponding to a beam of particles incident from the left which is
partly transmitted and partly reHected, one has to add these together in
such a way that to the right of the obstacle there is only the pure left-to-right
How, i.e., one must take
Pg(lV, x)+i/2(W, x)
To the left of the obstacle, the P function represents the superposition of a
left-to-right, or incident beam
Z oy og
cosh O.~a —————sinh cr2a e' 1(*+'
2 0 2 0'y
and a reHected beam
tT y 0'2
—+—sinh o ~ae
—' 1( +')
2 0'g 0 y
The transmitted beam is simply
ego'1a
These expressions have, of course, the conservation property
(4'4') *-=(44)"I+(AW i'
(4)
The probability that a particle coming up to the wall shall get through to
the other side is simply (fP)~„'. (PP);„, which —fore'~ &&1 is clearly equal to
I' (W) =16(W/V)(1 —W/V)s "~'. (6)
The controlling factor is the exponential term except when W/V is very near
to Oor i.
For application to a theory of the pulling of electrons out of metals by
electric fields Fowler and Nordheim' have derived the probability expression
by similar methods for the curve of Fig. 3b, i.e.
V(x) =0
V(x) =C Fx—x&0
Fig. 3b.
6
Fow1er and Nordheim, Proc. Roy. Soc. A119, 1 (1928).
132 R. O'. GURNEY AND E. U. CONDON
The probability that a particle of energy S"get through the wall they find
to be
Pi, (W) =4 [(W/C) (1—W/C) ]"'exp( —4k(C —W)' "/3F)
from their Eq. (18) p. 178. The term in the exponent can be written
4k(C W) '—"/3F =4ee/3a (k' =8ir'p/k') (7)
to exhibit the similarity with the case of the square wall. Here a is the positive
value of x for which V(x) = W, and o ~ is defined as
ee = (2x/k) [2ti(C—W) ]'t'
The exponents in each of these cases can be written in the form
(4x/k) ~f [2&(V—W)'t'dx
the integration extending across the barrier, the limits being the two places
where V(x) —W=O.
Application of the method of approximate integration of Schrodinger s
wave equation which was first used in quantum mechanics by Wentze15 indicates
that such a result is quite general. The probability of getting through
the wall at a single approach is governed essentially by the factor
exp —4' h 2p V —W ' 'dx (8)
being equal to it except for a factor of the order of magnitude of unity.
We have next to consider the case of a potential energy curve of the type
shown in Fig. 4. According to classical mechanics there are two modes of
motion associated with energy levels below the maximum such as 8' in the
figure. One is a periodic motion in the range I while the other is an aperiodic
motion in the range II. By the Bohr-Sommerfeld rule the periodic motions
would give a discrete spectrum of allowed energy levels which would overlie
the continuous spectrum associated with the
aperiodic motions. On the quantum mechanics
every energy level is allowed with the
essential difference that there are no energyI
T1 levels with which two tyPes of motion are asso
Fig. 4. ciated. With each energy level there is associated
just one wave function f(W, x) whose
square gives the relative probability of being at different parts of the possible
range of x. The P(W, x) functions do show traces of the discreteness of the
energy levels which the Bohr-Sommerfeld rule associates with the perodic
motions in I, in an interesting way. The f(W, x) for every W show sinusoidal
oscillations as x~~ and also oscillate in the range I. For most energies
the amplitude of the oscillations in the range II is overwhelmingly large
compared to that in range I, the ratio being of the order of exp[(2ir/k)
f[2ti(V- +]"'dx] the integration extending across the barrier. This
situation is just reversed however for little ranges of S' values near those
given by the old quantization rules. For these the amplitude in I is large
compared to that in II in the same ratio. These then are the "allowed"
QUANT UM MECHANICS: RADIOA CTIVE DISINTEGRA TION 133
energy levels. It is not a stationary state for the particle to be in range I and
remain there. But for certain energy levels there is an extraordinarily large
probability of being in unit length of range I relative to unit length of
range II.
We have to find the mean time which a particle remains in the range I
before "leaking through" to the outer range II. This can be obtained from
the following simple consideration. When the particle is at a place of x large
and positive, V(x) =0 (Fig. 4) so the energy is all kinetic and the speed is
therefore (2W/]M)'/'. The amount of time which the particle spends in unit
length for x large is therefore (p/2 W)'/'. The time spent in a range of length
a is therefore o (]s/2W)"'. Now according to the wave-functions the probability
of being in unit length of range I for one of the quasi-discrete energy
values relative to the probability of being in unit length of range II is of the
order exp I (4m./h) f[2p( V—W) j'/'dx I . Therefore since the motion is aperiodic
and the particle escaping from range I mill in the mean only go through unit
length of II once, the time T which must be spent in range I before getting
through to range II is of the order of
1' (p/2W) ~
p (]4 /h) Jf [2l (V—W)]"'1
where a is of the order of the breadth of range I.
Like all of the results of quantum mechanics this is to be interpreted as a
probability result. So that if we start with a number of particles in the same
allowed energy level in identical regions similar to range I, the number which
leak out in time dt is governed by
dÃ= —EXdt
which gives the usual exponential law of decay X(]') =Roe "' where
X= 1/T. (9)
The expression for 1may be arrived at in a somewhat different way. One
can think of the particle as executing its classical motion in range I, but as
having at each approach to the barrier the probability of escaping to range
II given by expression (8) above. The frequency of the periodic motion in I,
which represents the number of approaches to the barrier in unit time, is of
the order a(]s/2 W)'" so the mean time of remaining in range I before excape
comes out as the quotient of these two quantities as before. The reader will
find it of interest to examine Oppenheimer's formula' for the pulling of electrons
out of hydrogen atoms by an electric field. His formula for the mean
time required for dissociation of the atom by a steady electric field splits
naturally into a factor which is the classical frequency of motion in the Bohr
orbit multiplied by an exponential probability factor of the type of expression
(8) used in this paper.
2. APPLICATION TO RADIOACTIVE DISINTEGRATION
After the exponential law in radioactive decay had been discovered in
1902, it soon became clear that the time of disintegration of an atom was
~
Oppenheimer, Proc. Nat. Acad. Sci. 14, 363 (1928).
134 R. tV. GURNEY AND E. U. CONDON
as independent of the previous history of the atom as it was of its physical
condition. One could not for example suppose that an atom at its birth begins
to lose energy by radiation and that its instability is the result of the drain
of energy from the nucleus. On such a view it would be expected that the
rate of decay would increase with the age of the atoms. When later it was
observed that the number of atoms breaking up per second showed the fIuctuations
demanded by the laws of probability it became clear that the disintegrating
depended solely on chance. This has been very puzzling so long as we
have accepted a dynamics by which the behaviour of particles is definitely
fixed by the conditions. We have had to consider the disintegration as due
to the extraordinary conjunction of scores of independent events in the orbital
motions of nuclear particles. Now, however, we throw the whole responsibility
on to the laws of quantum mechanics, recognizing that the behaviour
of particles everywhere is equally governed by probability.
From what was said in the preceding section it is clear that the property
of the nucleus which we need to know in order to apply the theory is its
potential energy curve; and this happens to be a property which we know
fairly definitely. Outside a nucleus whose net charge is given by the atomic
number we should expect to find a Coulomb inverse-square field of the approI
II I I I I I I I I I
0 l 2 0 g 8 X
F'ig. 5. &he unit of abscissas is 10 "cm. The horizontal line gives the energy of
the a-particle emitted by uranium, 0.5 &(10 ' ergs.
priate strength. And it is well known that in experiments on the scattering
of alpha particles from heavy nuclei the proper inverse-square field is found
to extend through the whole accessible region. In Fig. 5 the curve AB is a
plot of the potential energy of an alpha particle in this field against the distance
from the centre of a nucleus of atomic number Z =90. To provide the
attractive field which holds alpha particles in the nucleus it has Iong been
recognized that the potential energy curve must turn over in the way shown
in Fig. 5. And it has been shown, for example by Enskog, ' that curves of
this type may be obtained by giving the particle a magnetic moment.
In order to explain the ejection of a particle one has hitherto supposed
that the particle in the internal region received energy sufhcient to raise it
over the potential barrier. The suggestion that this energy was obtained by
absorption of some ultra-penetrating radiation from outside never received
wide acceptance. But it was necessary on classical mechanics to suppose
that the-emitted particle had received energy, if not from outside then from
the other nuclear particles. Now the potential barrier which confi. nes particles
in the nucleus, i.e. the area under the curve in Fig. 5, is a region where
' Enskog, Zeits. f. Physik 45, 852 (1927).
QUANTUM MECIIANICS: RADIOACTIVE DISINTEGRATION
the total energy would be less than the potential energy. And since the
quantum mechanics endows particles with the new property of being able
to penetrate such regions, this gives us at last a nucleus which can disintegrate
without the absorption of energy.
We see that a mere qualitative application of the principles of quantum
mechanics seems to account for the principal properties of radioactive
atoms, most of which have been familiar for nearly thirty years. We have
now to consider the question: How can nearly similar nuclei have periods
of decay of anything from a small fraction of a second to over 10' years?
It has been shown above that in coupling the possible motions of a particle
on either side of a potential barrier, the probability of transmission through
the barrier is extremely sensitive to the area of the barrier; in fact the relation
to it is exponential. In this way we shall show that we can obtain all rates
of decay up to practical stability, and that from atoms whose potential
curves are almost identical.
3. QUANTITATIVE APPLICATION
If the height of CD above OX in Fig. 5 gives the energy of the alphaparticle
emitted, we have to consider the coupling of the motion along CD
with the motion along EJ' inside the nucleus. We clearly do not have much
choice in the area of the potential barrier we may take, since both the point
C is fixed and the curve passing through C. For the purposes of numerical
calculation we will compare radium A which has a period of 4.4 minutes
(half-value period 3.05 minutes) with the extreme cases of uranium and
t
f
I
1.0— I
I
I
I
l
f
l
f
Fig. 6. The unit of ordinates is 10 ~
ergs, and the unit of abscissas 10 "cm.
radium C', which have decay periods of about six thousand million years
and a millionth of a second respectively.
An alpha-particle emitted by an element of atomic number Z escapes
through the Coulomb field corresponding to (Z-2). Hence the potential
energy for Z =82, which has been plotted in Fig. 6 is appropriate to radium
A. The three horizontal lines in Fig. 6 give the energies of the alpha-particles
9 Blackett, Proc. Roy. Soc. A10'7, 369 (1925).
R. TV. GURNEF AND E. U. CONDON
from Ra A, Ra C', and Uranium, wh'ch are 1.22X10 ', 9.55X10 ', and
6 5 X10 ' ergs respectively. It is clear that the factor (V-W), which occurs
in the expression (9), given above for the rate of decay, is simply the vertical
distance between the horizontal line for 8' and the potential energy curve
for the proper value of Z-2. In Fig. 7 is plotted a curve derived from Fig. 6
giving the value of (4m/h) [2p(U-W)]'i' for Ra A as a function of the radius.
The upper curve is for uranium and the lower for Ra C', derived from curves
for the proper atomic numbers.
From these curves we can at once find how large a barrier we have to
take in order to obtain any observed rate of decay. For the integral occurring
in the exponent of expression (9) is merely the area that we will take under
the curve in Fig. 7. Since the unit of abscissas taken is 10 "cm and the
unit of ordinates 10"cm ', each of the squares in Fig. 7 has the dimensionless
value 10; so that for an element whose potential barrier has an area of one
5'2
~1~/c
ff
I
I ~--ll~—
gl/
ill
I
pig. 7. Ordinates give the value oi (4v/h}[2p(V —W) j'I' the unit heing
10"cm '. The unit of abscissas is 10 "cm.
square on this diagram we should employ the factor e ". The broken line
in Fig. 6 has been drawn so as to give for Ra A in Fig. 7 an area of approximately
the value 53.7. For substituting this value in expression (9) together
with p'=9.55&&10 ' and a =10 ' we obtain the decay constant 8.45&(10»x
e '3 r =3.8X 10 ' sec. ', or the decay period 1/X is 4.4 minutes in agreement
with observation. In the expression for ) the precise value of the first factor
is obviously unimportant, for if it were taken five times larger or smaller
this would only alter the area of the required barrier by about 1 percent.
The general size of the potential barrier in Fig. 6 that we have had to take
seems to be a very reasonable one.
Now we reach an unexpected result. In drawing the areas for uranium
and Ra C' jn Fig. 7 the continuous lines were predetermined, and the broken
lines have been derived from the curve in Fig. 6, already used for Ra A.
The values of the two areas are found to be 34.4 and 90, though the exact
values depend on how the broken line is made to join the Coulomb potential
curves. On substituting the values 34.4 and 90 in the expression for 1we
obtain for Ra C' and uranium decay periods of the order of 10—' sec. and
10"years respectively, in agreement with observation.
It was already clear from Fig. 6 that we should obtain for all elements
some qualitative agreement with the Geiger-Nuttall relation: the higher
QUANTUM MECHANICS: RADIOA CTIVE DISINTEGRA TION
the energy of the alpha-particle the greater the rapidity of decay. But now
we have found the unexpected result that the agreement is almost quantitative;
that we do not have to choose a diferent potential energy inside
the nucleus for each alpha-particle but having taken one potential curve
for the whole series, it is the energy of the emitted alpha-particle which
determines its own rate of decay. The mere fact that the velocity of the
alpha-particle from Ra A, 1.69X10' cm per sec. is a little greater than the
1.4&&10' cm per sec. of uranium, and a little less than the 1.92 '1.0' cm per
sec. of Ra C', gives Ra A a decay period 10"times as short as that of uranium
and 10' times as long as that of Ra C', in agreement with observation.
Questions raised by this agreement with the factor of Geiger and Nuttall
will be discussed in the last section of this paper. The radius 2)&10 "cm,
at which we have taken the deviation from the inverse-square law, seems to
be of the magnitude which our knowledge of the nucleus would lead us to
expect.
Further we see at once why it is that no slow alpha-particles have been
discovered. Although particles of ranges between 2.5 and 7 cm are plentifully
distributed, no alpha-particles of energy less than 6.5X10 ' ergs have been
found. But we now see from Figs. 6 and 7 that for particles of lower energy
the area of the potential barrier increases very rapidly; so that for particles
of range 2 cm or less the exponential factor would reduce the rate of decay
of the element to a value at which its manifestation of radioactivity would
be beyond the limits of detection.
Beta-ray disintegration. —It has been customary to assign the central
core of the nucleus as the habitat of the nuclear electrons, with a potential
energy curve of the type shown in Fig. 8. The outer slope AB again represents
the Coulomb inverse-square field, as in Fig. 5. But since the charge
of the electron is —e instead of +2e the
potential energy is reversed in sign, and of
half the magnitude of that in Fig. 5.
There is nothing new in this assumed
curve, although it looks somewhat arti- A
ficial; thils type of curve for the nuclear B
electron was obtained for example by Fig. 8.
Enskog in the paper referred to above. '
What is new is the suggestion that an electron in the internal region again
has a certain chance of penetrating the barrier, and of escaping at any time
along CD with kinetic energy given by the height of CD above the axis.
If we have alpha and beta-particles both with this chance of escaping
from the nucleus, it might be thought that every radioactive element should
be found to disintegrate part with expulsion of alpha-particles and part with
beta-particles. But we would repeat that the chance of escape is extremely
sensitive to the height to which the potential energy curve rises above the
energy-level in question; and that if the size of this potential barrier be
increased by a small factor the probability of escape may be decreased more
than a million-fold. There seems then no reason why there should not be
138 R. W. GURcVBF AND E. U, CONDOR
the three types of disintegration: that in which the probability of escape is
much greater for an alpha-particle than for an electron; that in which it is
much greater for an electron than for an alpha-particle; and that in which
the probabilities of escape are comparable. The last gives the branching
type of disintegration as shown by Ra C, of which 99.97 percent emits betaparticles,
and 0.03 percent alpha-particles. By taking this view of the
disintegration process, we have raised the question: Does any radioactive
element have a unique mode of disintegration, or does it merely appear
unique in most cases because the secondary mode is a million times less
frequent and escapes detection' The present discussion certainly favours
the latter alternative. It need not surprise us then that so few cases of
branching disintegration have so far been discovered, since it is unlikely
(so far as we know) that the areas of the potential barriers will in many
nuclei happen to have just that relative size which will give for alpha and
beta-particles comparable probabilities of escape.
Arti6cial disintegration. —Blackett's cloud-chamber photographs' of
artificial disintegration in nitrogen showed that the impinging alpha-particle
was caught and retained by the nucleus. One is tempted to apply the present
theory, using again the fact that the impinging alpha-particle may penetrate
the barrier of potential, this time from the outside, instead of passing over
the top as required by classical theory. But when we do this we are at once
confronted by the fact that instead of approaching the barrier 10"times per
second, like a nuclear particle, our alpha-particles will only make one impact
apiece. So it would seem that the capture of the alpha-particle could not
be due to penetration. There is, however, another consideration; and that
is that if the impinging alpha-particle have an energy very near that of an
allowed but unoccupied nuclear energy-level, the chance of its penetrating
the barrier at a single impact approaches unity. This property has already
been referred to in section 1.
4. EXPERIMENTAL EVIDENCE FOR THE PENETRATION OF
POTENTIAL BARRIERS
The essential basis of the present theory is the assumed power of particles
to pass through regions where their total energy would by classical mechanics
be less than their total energy. For this property there is no direct experimental
evidence in physics, although it follows from the laws of quantum
mechanics. But in applying this to the nucleus we have found that we can
actually obtain direct experimental evidence. Though on classical mechanics
the passage of a particle through such a forbidden region was a manifest
absurdity, it was found in 1925 by Rutherford and Chadwick" that that is
exactly what the alpha-particles from uranium appear to do.
Consider the alpha-particIe which the uranium nucleus emits during its
disintegration. The alpha-particle will gain energy in escaping through the
repulsive Coulomb field outside the nucleus. This energy is given on classical
theory as 2Ze'/r. Even if the alpha-particle leaves its place in the nucleus
Rutherford and Chadwick, Phil. Ma@. 50, 889 (192&}.
QUANTUM MECHANICS: RADIOACTIVE DISINTEGRATION 139
with no initial veIocity, its energy cannot be less than this amount. The
energy with which the alpha-particles leave the disintegrating Uranium
atom is observed experimentally to be 6.5 X 10 ' ergs. On referring to Fig. 5,
which was drawn for Z =90, we see that this energy corresponds to r =6.3 X
10 "cm and if any of the energy was initial energy and not acquired through
falling through the repulsive field, the value of r would have to be greater
than this value.
It was concluded that the inverse-square law of repulsive field could
not possibly hold within this value of r. Consequently if we fire at the
uranium nucleus an alpha-particle having slightly more energy than the 6.5
10 ' ergs, it should penetrate its structure to where the Coulomb law no
longer holds; while still faster particles should penetrate, even when not
fired directly at the nucleus. It was therefore disconcerting when, on examining
the scattering of fast alpha-particles fired at uranium, Rutherford and
Chadwick could find no indication of any departure from the inverse-square
laws. The Coulomb field was found to hold inside the radius from which the
uranium alpha-particIe appeared to come. That is to say, the uranium
alpha-particle appeared to emerge from a region where its kinetic energy
was negative. To escape this conclusion Rutherford" supposed that the
uranium alpha-particles before ejection are electrically neutral, having been
neutralised by two electrons which they leave behind when they are ejected.
This hypothesis succeeded in circumventing the paradox. But if we abandon
classical mechanics, the paradox disappears, yielding us direct experimental
evidence in favor of the phenomenon of quantum mechanics in which we are
interested.
5. DISCUSSION OF LIMITATIONS
It must be clearly understood that although the Coulomb part of the
potential curve outside the nucleus, represented by AB in Fig. 5 is necessarily
common to all particles, the internal part is merely intended to represent
the potential energy of a particular alpha-particle. And it must not be
taken to represent a general central field common to many particles, such
as we are so accustomed to in atomic structure. There is no reason why the
internal field should be necessarily symmetrical about the center of the
nucleus as drawn in Figs. 5 and 8. In fact, Rutherford" has suggested that
the nucleus may have something analogous to a crystalline structure. If
this caution is lost sight of, difficulties are encountered.
For the atom of each radioactive element contains within its nucleus
not only the alpha-particle which it will itself emit, but also the alphaparticles
destined to be emitted by its successors in the radioactive series.
Now if the velocity of escape of the alpha-particles from each element were
always less than that of those emitted by its predecessors, there mould be
no serious difficulty; for from an atom loaded with alpha-particles in various
allowed energy levels, the particle in the highest level would have the
Rutherford, Phil. Mag. 4, 580 (1927); Proc. Phys. Soc. 39, 370 (1927).
~ Rutherford, Jour. Franklin inst. 198, 743 (1924).
R. 8". GURNEY AND E. U. CONDON
greatest probability of escape. This however is the opposite of what is
observed; and we have to account for the subsequent emission of particles
of higher energy than that emitted by the parent substance. We may do this
by supposing either (a) that the alpha-particles of higher energy have in
the parent element been confined by correspondingly high barriers; or (b)
by supposing that the alpha-particles in the nucleus are not permanently in
the high energy levels from which they emerge, but are temporarily raised
up from lower levels. The latter seems to be a retrograde step, for the
principle advantage of the present theory is that it has overed an escape
from such processes.
If, however, we accept the former supposition (a), we see that the emission
of one alpha-particle must profoundly modify the potential barrier which
confines the alpha-particle destined to be emitted next. As we have shown,
the Geiger-Nuttall relation seems to require that the barrier through which
this alpha-particle emerges be approximately the same in all elements of
the series. But until we know how this comes about, it seems inadvisable
to discuss the Geiger-Nuttall relation in greater detail. In speaking of the
energy of one particle in the nucleus, it must not be forgotten that we are
making use of the simplification mentioned in the introduction: that of
discussing one nuclear constituent alone.
PALMER PHYSICAL LABORATORY,
PRINCETON UNIVERSITY,
November 20, 1928.