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Emission of gamma radiation during the beta decay of nuclei
Physica III, no 6
Juni
1936
EMISSION OF GAMMA RADIATION DURING THE BETA DECAY OF NUCLEI by J. K. K N I P P *) and G. E. U H L E N B E C K Natuurkundig Laboratorium der Rijks Universiteit te Utrecht
Summary E x p e r i m e n t a l i n v e s t i g a t i o n s h a v e s h o w n t h a t t h e d i s i n t e g r a t i o n of R a E is a c c o m p a n i e d b y a f a i r l y w e a k y - r a d i a t i o n . T h e o r e t i c a l c a l c u l a t i o n s h a v e b e e n m a d e w h i c h s h o w t h a t a l a r g e p a r t , if n o t all of t h i s r a d i a t i o n a r i s e s f r o m t h e c r e a t i o n of t h e e l e c t r o n a n d its loss of e n e r g y as i t l e a v e s t h e n u c l e u s . T h i s p r o c e s s is a n " i n n e r " B r e m s s t r a h l u n g t o b e c o n t r a s t e d w i t h t h e " o u t e r " B r e m s s t r a h l u n g d u e t o t h e i m p i n g e m e n t o n t h e n u c l e u s of e l e c t r o n s c o m i n g f r o m t h e o u t s i d e . I t h a s t h e o r d e r of m a g n i t u d e a = 1/137 p e r o u t c o m i n g e l e c t r o n . T w o m e t h o d s of t r e a t i n g t h e p r o b l e m h a v e b e e n e m p l o y e d a n d i t is f o u n d t h a t in t h e B o r n a p p r o x i m a t i o n t h e y g i v e i d e n t i c a l r e s u l t s . I n t h e f i r s t t h e e l e c t r o n is r e p r e s e n t e d b y t h e free, o u t g o i n g , n o t - a l l o w e d , s p h e r i c a l s o l u t i o n s of t h e D i r a c e q u a t i o n a n d r a d i a + i o n transitions are calculated to standing waves. In the second method the c a l c u l a t i o n s a r e b a s e d o n t h e F e r m i t h e o r y of ~ - d e c a y . T h e e f f e c t of a q u a n t i z e d e l e c t r o m a g n e t i c r a d i a t i o n field is i n t r o d u c e d b y m e a n s of a c o u p l i n g t e r m b e t w e e n t h e e l e c t r o n s a n d t h e r a d i a t i o n field. R a d i a t i o n i n t e n s i t y c u r v e s a r e o b t a i n e d w h i c h a r e v e r y s i m i l a r t o t h o s e for o r d i n a r y Bremsstrahlung.
I. Introduction. The usual continuous electron spectrum of radioactive nuclei has superimposed on it certain homogeneous groups of electrons arising from the internal conversion of nuclear y-rays. These y-rays are monochromatic and are due to nuclear transitions taking place before or after the process of ~-decay. That there is a small amount of y-radiation accompanying the ~-disintegration of nuclei which cannot be accounted for in this way was first shown by A s t o n 1) in his measurements on RaE. This ~-radioactive element has a normal continuous electron spectrum, but there are no homogeneous groups of electrons superimposed on it and therefore no *) Sheldon Fellow from Harvard University.
425
426
j.K.
KNIPP AND G. E. UHLENBECK
monochromatic y-rays of the usual kind. A s t o n found, however, that RaE is accompanied by a fairly weak y-radiation and from his measurements concluded the average to be about 10,000 volts per electron of disintegration. More accurate measurements have been made by S. B r a m s o n 2). She obtained an absorption curve for the radiation in copper. The form of this curve suggests immediately that the radiation is continuous in nature. She has estimated the number of quanta and found roughly 1.6 per hundred electrons of disintegration. The interpretation made that a large portion of the radiation is due to K radiation from the product atoms in which the F-particles have knocked out K electrons is not correct, however, as can be seen on theoretical grounds. Such a process has a probability of taking place proportional to e 4, per outcoming electron, or the square of the fine structure constant, e = 1/137. It can therefore be expected to be of the order of magnitude of 1/10,000. Such a probability is much too small to account for the number of quanta observed. A process with a probability of the right order of magnitude is the one in which the radiation arises from the creation of the electron and its loss of energy as it leaves the nucleus. This is an ,,inner" Bremsstrahlung to be contrasted with the "outer" Bremsstrahlung due to the impingement on the nucleus of electrons coming from the outside. The "inner" Bremsstrahlung has a probability per electron proportional to e2 or e and in the first approximation is independent of the nuclear charge, Z. The "outer" Bremsstrahlung has a crosssection proportional to ~aZ2 and vanishes for zero nuclear charge *). We shall treat the theory of the "inner" Bremsstrahlung by two methods. In the first the electron is described as a particle leaving the nucleus with a definite energy. The probability that it radiate is then calculated. On integrating this probability over the energy distribution of the electrons, the probability spectrum of the gamma rays is obtained. In the second method we use the F e r m i theory of F-decay, according to which a neutron changes to a proton under the influence of the electron-neutrino field. The effect of a quantized *) T h e r e is a n a n a l o g o u s s i t u a t i o n in t h e p r o d u c t i o n of p a i r s b y y - r a y s . F o r n u c l e a r y - r a y s t h e p r o b a b i l i t y is p r o p o r t i o n a l to cx, i n d e p e n d e n t of t h e n u c l e a r c h a r g e in t h e f i r s t a p p r o x i m a t i o n , as f o u n d b y O p p e n h e i m e r and Nedelsky4).Ontheother hand the B e t h e-H e i t 1 e r calculation gives a l~robability for plane gamma waves p r o p o r t i o n a l to 00Z ~- a n d zero for zero n u c l e a r c h a r g e ~).
E M I S S I O N OF GAMMA R A D I A T I O N
427 i
electromagnetic radiation field is introduced by means of a coupling term between the electrons and the radiation field. We calculate the probability of obtaining a neutrino, an electron and a y-ray. On s u m m i n g over all the allowed states of the electron and the neutrino, the total probability for radiating a q u a n t u m of a definite energy is obtained. The first m e t h o d has the a d v a n t a g e t h a t the unobserved neutrino enters only indirectly in the description of the ~-spectrum. On the othc.r h a n d it requires the use of the not-allowed s o l u t ~ n s of the D i r a c equation, which up to the present have been excluded from q u a n t u m mechanical calculations. II. CalculationsUsing Outgoing Waves. In this section a n d the one following we shall use the Born approximation, in which ~Z/p is supposed to be small compared to unity. This approximation is only go,~d for light nuclei or particles of high energy and so will give results which describe only qualitatively the radiation from a nucleus as h e a v y as t h a t of R a E (~Z = 0.606). We require solutions of the D i r a c equation in polar coordinates which describe a free electron leaving the origin. In the following we employ h/mc as the unit of length, t~/mc2 as the unit of time and the electron mass m as the unit of mass. The solutions are readily found to be t y p e a, ]' = l + 1/2
Na +,,'at,,,=~v,
- - i Hi+,~. (pr) PI'~-t --i Ht+./. (pr) "l+lz)"+l p p],, ( l + m + 1) W - - 1 H,+,/,(pr) - - (l - - m) W P -- 1
Hi+,/: (pr) P7'+1
6(l--m)! (l+m+l)!,l>_
0,--(l+l)_
and t y p e b, .I" = l - - 1/2, --i
Nb
+,vbt,,,= r.i;
(l + m) Hi_,~, (pr) Pi"---i i (l - - m - - 1) Hi_,/.. (pr) ~--ID" +1 -- p p~,,
W-
1 H,+,/. (pr)
W- -- P 1 H~+,/, (pr) p7,+i N ~ = ( W - - I ) / 1 6 ( I + m)! (l--m--1)!,l>_ l , - - l _ < m _< l - - l ,
428
J . K . K N I P P A N D G. E . U H L E N B E C K
where W is the energy and p the m o m e n t u m of the electron. H~+l/,(pr) is a H a n k e 1 function of the first kind and half integral order. The functions P~' are surface harmonics as defined by D a r w i n 6). There are 2 (2l + 1) solutions having positive energy W and each has been normalized to represent one electron leaving a large sphere per unit time. Replacing the H a n k e 1 functions by B e s s e 1 functions would give, aside from the normalization factor, the ordinary standing spherical solutions of the D i r a c equation. These are not the solutions we want, however, since they do not describe an electron leaving the nucleus. One can easily see t h a t they do not give rise to radiation (see Note 1). In the Hamiltonian the coupling term between the electron and the radiation field is --->
L = - - x/~ (~. A),
(2)
where ~ is the D i r a c velocity m a t r i x * ) and A is the vector potential of the quantized radiation field having components given by
A = Y, qk Ak + c.c.,
A =
e
. rl,
(3)
k
the momentum of the light quantum being denoted by k. The amplitudes q~ are the "coordinates" of the field and have their matrix elements the same as those of the harmonic oscillator
(n~ [ qk [ nh + 1) ---- (nk + 1/2k)V,, (n~ ] q~ ]nk - - 1) ---- (nk/2k)v,,
(4)
where nk is the number of quanta of a particular polarization having energy k. The direction of polarization of the light wave is given by ...->
..->
the unit vector ek perpendicular to k and directed along the electric field strength. Each component of the field is normalized to 4~ quanta per unit volume. For emission of a q u a n t u m of radiation one can omit the complex conjugate and L becomes simply ->.
L
x
-->.
• rl,
~
=
---->
(s)
k
We suppose that at time t = 0 the electron is born at the nucleus with energy W~. We apply the method of variation of parameters to find the probability t h a t the system of electron and radiation field be in the final state / with total energy W t = W + k at the time t, this • ) Not to be c o n f u s e d w i t h t h e fine s t r u c t u r e c o n s t a n t .
E M I S S I O N OF GAMMA R A D I A T I O N
429
state being one in which the electron has the energy W and the radiation field has gained a quantum of energy k. A first order perturbation gives for this probability ] a ! [ 2 = I ( / ] L ] O ) 124sin 2 ( w ! - W e ) t
( W ! - We)2
(6)
The total probability per unit time that an electron, created with energy We, radiate a light quantum having energy between k and k + dk is then given b y summing over all possible final states of the electron and the polarization and direction of the light quantum and dividing b y the time interval t. We shall call this probability O(We, k) dk. If plane waves are used to represent the final states of the electron, we have
(We, k) dk = X X ]a, ]2 1 ,
(7)
k s
where oo
y, X = (1/(2~) 6) k 2 dk f dtL Sk f d~, Ss f dW Wp. ks
(8)
1
Here dt~, and dglk are the two elements of solid angle in the direction of emission of the electron in its final state and the light quantum respectively; Sk is the summation over the two directions of polarization of the light wave and S, is the summation over the two states of the electron with momentum p and positive energy. The integration over the energy of the electron gives the conservation of energy W I = We = W + k and the factor 2~t instead of the second factor of (6). To carry out the calculations we need the matrix elements .->
.=~
(/!L 10) ----(psiL I We jZm) - - - - (2~/k)'/, f dr+'~,o~k+,rei,,,,e-~(k" ')
(9)
The wave functions for the plane wave are very simple s). The typical integral which arises is discussed in Note 1. Let us consider the case l = m = 0 , which occurs only with type a wave function. O11 taking k along the z axis *), we readily obtain
SkS, l(ps[L[Wel/200)[2 = 4v~2°~(g2p2 + q2--2gp ,q,) gwpek
(lO)
where g = (We + 1)/(W + 1) and q = p + k. This expression can *) Although z is the axis of q u a n t i z a t i o n this involves no loss of generality.
430
j.
K. KNIPP
A N D .G. E . U H L E N B E C K
be resolved into partial fractions. On introducing the angle between ....>
--->
p and k, we obtain for the differential probability function ~ d~ j[(W w'2i3+ ~-~-~ ~ ) +2p--~os ww2 dO0= :cp2~_p~_ksin
( w - p 2cos 1 ~)
1 p/. (11)
This can be integrated at once to give the integral probability function (I)°= ~-p,k[
p2W-+-w
/
log (W + p ) - - 2 . .
(12)
It is shown in Note 2 that these results hold for all even l (type a or b) after averaging over m. Likewise we find for l -- 1 and all odd l d(i)l = ~ p sin ~ d~ J W'2 --i_2 W _+ _W2 2~ p,k [(W,--1) (W - - p cos ~)
P
1 (W--pcos~) 2
} 1..
}
(I)x= ---p, k |
log (W + p) - - 2 . .
(13)
It is seen that for a given W, all electrons radiate the same amount of soft radiation. To obtain the actual probability functions for some particular ~radioactive nucleus we must take the proper linear combination of the functions for the different states of the electrons emitted. Thus the actual function is an expression of the form
• = X (atO.~ + btObt), Z (at + bt) ---- 1, t
where a t and bt are weight According to the F e r m i transitions of the nucleus + 1 ) / ( W , - 1) (see Note 3) ao
(14)
t
factors for the two types for a given l. theory only a0 and b I enter for "allowed" 7) and these in the ratio ao/bl : (W,+ *). We have
= (W,+I)/2W,,
b t = (W,--1)/2W,;
(15)
and the probability functions for an allowed transition of the nucleu,.~ therefore are d(I) ~ ~ p sin 5 d~ ] W,2 + W 2 1 - - 1/ -2~p-Q~ IW,(W---~t~os ~) - - (W - - p cos ~)2 , , ~P J w`2+ W21og ( w + p ) - 2
/
• ) T h i s h o l d s also for t h e m o d i f i c a t i o n of t h e F e ' r m i t h e o r y p r o p o s e d b y pinski and Uhlertbeck, r e f e r e n c e (9).
(16) K o n o-
EMISSION
431
OF GAMMA RADIATION
The radiation probability spectrum,S(k), for the entire distribution of electrons is obtained b y multiplying byP(Wc) dW,, the probability t h a t an electron be created in the energy interval W, to Wc + dW~, and integrating. We h a v e Wa
S(k) =
f
l+k
dW~P(W~) ap (W~, k),
(17)
where Wo is the upper limit of the ~-spectrum. Here P(W,) is to be taken as the theoretical distribution given b y the original or modified F e r m i theory. If one assumes t h a t the proper linear combination will always be determined b y the density of the electrons near the nucleus, it seems even reasonable, according to the m e t h o d of this section, to use for P(W,) the experimental energy distribution. III. Calculations with the Fermi Theory. The process described in the last section m a y appear to be puzzling because if the nuclear charge is zero there is no acceleration and hence no reason classically w h y the electron should radiate *). The question is clearly connected with how the particle is created in the nucleus. In this section we shall base our calculations on the F e r m i t h e o r y of the nucleus, which affords a consistent account of the emission of electrons b y nuclei. We introduce the electromagnetic radiation field into the F e r m i theory b y means of a coupling term between the electrons and the radiation field. The H a m i l t o n i a n of the system contains the two perturbing terms H and L, where H is the F e r m i coupling term between the h e a v y particle and the electron-neutrino field and L is the coupling t e r m between the electrons and the electromagnetic field. We can use the original F e r m i notation and write for the first of these terms -..>
-..>.
....>
-.->
n = Q[Ao + (c¢hea,,y.A)] + Q*[A~ + (~h,avy. A)*].
(18)
Here Q and Q* are H e i s e n b e r g matrices which operate on the inner coordinate of the h e a v y particle. This coordinate p determines whether the particle is a n e u t r o n (0 = + I) or a proton (p = - - 1). Q corresponds to a proton to n e u t r o n transition and Q* the reverse. ~-h,,vyis the D i r a c velocity m a t r i x which acts on the spin coordi*) I t s h o u l d be p o i n t e d o u t t h a t t h e r e is a s i m i l a r l a c k of a c l a s s i c a l a n a l o g u e in t h e c a s e of t h e a n n i h i l a t i o n r a d i a t i o n a r i s i n g f r o m t h e r e c o m b i n a t i o n of a free p o s i t r o n a n d a free e l e c t r o n .
432
j.
K. K N I P P A N D G. E. U H L E N B E C K
n a t e of the he' v y particle. We shall neglect t e r m s c o n t a i n i n g this m a t r i x because t h e y are of a smaller order of m a g n i t u d e t h a n the o t h e r terms. A is the four v e c t o r " p o t e n t i a l " of the e l e c t r o n - n e u t r i n o field and is built u p out of the q u a n t i z e d wave functions + and £0 of the electron and n e u t r i n o respectively. Following F e r m i we can write for the i n t e r a c t i o n
H=G
E X IQa, b ~ ( + s ~ % ) + Q * a * b * ( + , ~ , ~ ) * ,
(19)
$
where G is the coefficient of" coupling and
:
0
0
0
0
(20) "
0--1
F o r the coupling t e r m L we can take the sum of terms of the t y p e used in the last section, the s u m m a t i o n being over the states of the electron L = F~NsL$, (21) $
where N$ is the n u m b e r of electrons (0 or l) in the state s. T h u s for emission L is simply L = - - X E ( 2 ~ ~lk) '/, N$ o~h e-Ik" "~. $
(22)
k
We t a k e the initial state to be one in which there is a n e u t r o n , and no electrons, neutrinos or light q u a n t a (p = + 1, N, = 0, M~ = 0, n , = 0 ) . T h e t o t a l e n e r g y of the initial state, W0, can be t h o u g h t of as the available energy. W e are i n t e r e s t e d in the p r o b a b i l i t y of a transition to a state in which the n e u t r o n has b e c o m e a p r o t o n and an electron, n e u t r i n o and light q u a n t u m , all of definite energy, h a v e been created (O = - - 1, N , = 1, M , = 1, n, -~ 1). We indicate the t o t a l energy of this final state b y W I . I t is clear this transition c a n n o t be m a d e directly since n e i t h e r H nor L act on more t h a n three particles. W e m u s t consider i n t e r m e d i a t e states of the t y p e (? = - - 1 , N, = 1, M~----1, n~ = 0). L e t the total e n e r g y be W~. O t h e r i n t e r m e d i a t e states are excluded to this a p p r o x i m a t i o n since L c a n n o t s t a r t the process because it does n o t h i n g to the h e a v y particle or neutrinos and all the effective N , are zero in the initial state. We h a v e the p e r t u r b a t i o n H causing the transition from the s t a t e 0 to the state l and t h e n the p e r t u r b a t i o n L causing f u r t h e r transitions to
E M I S S I O N OF
433
GAMMA R A D I A T I O N
the state/. This is a second order perturbation and the square of the probability amplitude for the final state after a time t is given b y
l al j2 = I 2 ~ (/tL Jz) (zI n Io/2 4 sin~ (Wl-- W0) t , ~-, ---ff-~o (w~-- Wo)~
(23)
The particles can be described by plane waves. The matrix elements are, making the same approximations as does F e r m i for an "allowed" nuclear transition,
(/ IL I/) = (ps ILl p' s') ---- - - (2==/k)'/, (u*=,u'), p' = p + k, (IIH IO) --- GM(u'Scr)*,
(24)
where primes indicate the intermediate state and u and . are the amplitudes of the plane wave functions for the electron and neutrino respectively and designate states with particular momentum, spin and sign of the energy. M is the matrix of the nuclear wave functions We have for the energies
W ~ = K,, + W', W t :
K,, + W + k.
(25)
Defining an initial energy of the electron b y W, = W o - Ko and substituting in the above expression, we have
lallm= GZ l M IZ (2~oc/k)
(u* ~.,u')(u'8,)* 2 4sin2(Wi--W0 )t d~
W , - - W'
(Wl - - Wo) 2
(26)
where the summation formally is over all the intermediate states of the electron of both the positive and negative continuum; however because of the conservation of momentum it is reduced to a summation over the four states of a particular momentum p'. The total probability S(k)dk that a light quantum be emitted in the energy range between k and k + dk is obtaified b y summing over all the allowed states of the electron and neutrino and the direction and polarizations of the light quantum. We have
S(k)dk----
Z
k
Z
~"
Z
21
l a/I 7 '
(27)
s
where in detail EE]~---k
O"
s
= (1/(2re) . 9)j "d p ,, p ,,K,,k2d k ./ "dfl :,S, j
"dflkS kj "dfl~S ,, '~"
(28)
1
Physica I I I
28
434
J.
K. K N I P P A N D G. E. U H L E N B E C K
W e have written the operations in the order in which we shall perform them. The last integral gives the factor 2r~twp and the energy conservation law W 1 = W0 = Ko + W + k, which defines W. The integration over the m o m e n t u m of the neutrino can be transformed into one over W, Wo--I--/~
6
W0
lr~,
If then we write
P(W,) = (G21M 12/2na) Ko p~ W, p,,
(30)
S(k) can be put in the form P
W,
s(k) = / a w , where • (W~,k)
(3])
P(W,) ¢ (w,, k), W, = W + k,
1¢,
,t p Wk
w,-
w'
(32)
(W,, k)dk can be interpreted as the probability that an electron, created with energy W,, create a fight quantum of energy k. The summation over t h e i n t e r m e d i a t e s t a t e a n d the spin of the neutrino can be effected by the usual methods. Neglecting the terms that drop out in the subsequent integration over the direction of the m o m e n t u m of the neutrino, we obtain for this summation [1/2(W~-- W'2) 2] [W:,(u*u)+2W, (u*=kH' ~z,u)+ (u*=kH'2c(,u)]
(33)
w h e r e - - H ' ~- (~. p ' ) + } . Introducing this expression and summing over the two directions of polarization of the light, we obtain
[1/(W~--W'2) 2] [W2~ + W ' 2 -
(2W, (p'p, + 1)/W)]
(34)
which can be broken up into partial fractions. Noticing t h a t W~ - - W '2 = p~ - - p,2 = 2k (W - - p cos 8) where ~ is the angle between the directions of the light quantum and the electron, we obtain finally for the differential and integral probabilities
d , _ e p sin S d8 J. W~_+__W2 2np, k [W,(W--pcos~)
P "( W , p (1) = r: p,k
1 (W - - p cos ff)2
l o g ( W + p ) - - 2 . [ . .I
) 1, ,
(3_5)
EMISSION OF GAMMA RADIATION
435
These are just the expressions obtained by the method of section II. It m a y be pointed out that if one changes the interaction "Ansatz" (19) als proposed by K o n o p i n s k i and U h l e n b e c k the final formulas (3 l) and (35) are not changed. One has only to replace P(W,) as given by (30) by
P(W,) = (G= I M 12/2x 3) K3oPerWe p,.
(36)
IV. Discussion. In Fig. 1 we give curves showing the distribution in energy of the gamma radiation for different initialelectron energies. We have plotted the intensity, that is k times the probability 1,0
Inten.sity Dtsteibution Cur've5
0,8 10-2; 0,6 •mc~
,,x~e= 10 ~e
=5
0.4 o£_
0.2
OA
0.6
K/(We-1)
1.0
Fig. 1. I n t e n s i t y d i s t r i b u t i o n c u r v e s for d i f f e r e n t i n i t i a l e n e r n i e s of t h e e l e c t r o n , We. T h e s e a r e for " a l l o w e d " n u c l e a r t r a n s i , t i o n s as g i v e n b y (35). E n e r g i e s a r e m e a s u r e d i n u n i t s of m c * or 0.51 m i l l i o n e l e c t r o n v o l t s a n d i n c l u d e t h e r e s t mass.
function, rather than the function itself since ~ behaves roughly as 1/k and as a consequence becomes very large for low energy quanta. The intensity starts with a finite value for soft radiation and decreases monotonically to zero as k increases to its upper limit, W , - 1. The curves are very similar to those for "outer" Bremsstrahlung.
436
J.
K. K N I P P AND G . E . U H L E N B E C K
Fig. 2 shows the total energy spectrum of the gamma radiation from a nucleus with W 0 = 3.6, which is approximately the upper limit of RaE.This curve is for an "allowed" nuclear transition. It was calculated using for P(W,) the expression (20) normalized to unity. In the same figure we have plotted the number of quanta of energy greater than k per electron. This number is given b y
/.
Wo--I
N(k) =
(37)
dx S(x),
where S(k) is the expression (31). With the F e r m i "Ansatz" the B o r n approximation gives three quanta having energy greater than 50,000 volts per thousand
2,5
5
1
NfKJ
10`3 20
2
mC ~
10-~"
i
i
0.5
10
10 mc ~ 1,5
2,0
1,5
t0
K
0,5
Q5
H 2,5
F i g . 2. T o t a l e n e r g y s p e c t r u m o f t h e g a m m a r a y s f o r a n " a l l o w e d " n u c l e a r t r a n s i t i o n w i t h W0 = 3.6 a n d Z = 0 a n d u s i n g t h e F e r m i d i s t r i b u t i o n c u r v e for t h e e l e c t r o n s . T h e s m a l l e r c u r v e g i v e s t h e n u m b e r of q u a n t a h a v i n g e n e r g y g r e a t e r t h a n k a s a f u n c t i o n of k.
outcoming electrons for a nucleus having a continuous ~-spectrum with upper limit of 3.6 mc 2. If for P(W,) the express in (36) as given b y the modified theory is used, considerably less hard radiation is
437
E M I S S I O N OF GAMMA R A D I A T I O N
obtained. For 1000 outcoming electrons there are about 1.5 quanta having energy greater than 50,000 volts. This is roughly a factor 1/ 10 smaller than the number measured for RaE. According to the F e r m i theory the disintegration of R a E is a "forbidden" transition of the nucleus. It is clear also that we cannot make comparisons with the experimental results of S. B r a m s o n because R a E has such a large nuclear charge (Z = 83). To get a rough idea of the influence of the nuclear charge on the gamma radiation emitted we have made calculations after the method of section II using the S c h r 6 d i n g e r approximation for the electron in the final state. This is a non-relativistic approximation (p ~ 1 and ~Z ~ 1) and is valid only when the light quantum gets almost all of the kinetic energy of the electron. It is not necessary to take the small components in the wave functions. We give only the results. For l even we obtain for the integral probability 80¢2 Zk3W *o = p, (W, + 1) (p~ - - k2)2 (1--
e -2"~'z/p)
"
(38)
To be consistent with the approximations, one should replace k by W, - - ! and W b y unity, so that 2~2 Z (We --1) D° = Pe (W, + 1) ( 1 - - e-2'~az/#) "
(39)
The corresponding expressions for l odd are obtained b y replacing (We + 1) occuring in the denominators of the above b y ( W e - 1). Using (! 5) we obtain for an "allowed" transition *) 8~ 2 Zk3W = p e W e (p2 ___ k2)2 (1-- e--2'~az/#) '
(40)
or
@=
2~x2Z ( W , - - 1) p, W, ( l - - e-2"~/*) "
(41)
These expressions give a finite limit for the intensity curves as k approaches its maximum value. The amount of hard radiation is increased and the results indicate that the theory can account for the observed measurements on the number of quanta. We hope to investigate the dependence on the nuclear charge on the basis of the *) All these expressions have the correct l i m i t i n g connections with the corresponding expressions of the B o r n approximation. See ref. (8).
j . K . KNIPP AND
438
G.. E. I#HLENBECK
F e r m i t h e o r y and to carry out a more detailed comparison with experiment. We wish to express our gratitude to Professor L. M e i t n e r of Berlin for valuable assistance in the interpretation of the experim e n t a l data. We are t h a n k f u l to Dr. A r n o 1 d N o r d s i e c k for m u c h stimulating discussion. Note 1. The General Integral Arising in Section II. All the integrals in the m a t r i x elements of section II are of the form d'r e ---i(q" r)
HI+,I, (P, r)
r -'1. P'~.
(42)
On carrying out the angular integrations there remains the integral
J,+,/, (qr) Hz+,/, (p, r) r dr,
(43)
o
This is divergent at infinity b u t has to be understood t h a t in evaluating it one introduces as usual exp. ( - - ar) and puts ~ = 0 after the integration. The integral can be split into two parts b y means of the relation
nl+,h = J~+,l, + i ( - - ) ~+a J-c~+'h). Since q :/: Po the first of these is found to give zero (as seen using Eqn. 2, p. 389, W a t s o n, Theory o/ Bessel Functions). If one represented the electron initially b y standing waves, only integrals of this k i n d would arise. F o r the second we find oo
f J,+'l. (qr) J-I+v., (p.r)
r
dr
=
2i
1
(- q),+'l,
(44)
O.
for p, > q, where ¢ = 0 , l even and ~ = 1, l odd (using Eqn. 6, p. 404, Watson, Theory o/Bessel Functions). Note 2. B o r n A p p r o x i m a t i o n for a n y (W, tim). Consider an initial state of t y p e a and l even. B y means of relations a m o n g the D a r w i n functions a n d using the integrals of Note 1, the summ a t i o n s over the polarizations and the spin of the electron in the final state can be reduced to the expression on the right side of (10) w i t h the additional factor
(IPr+xl2 + IPf+lp)/(l--m)l ( l + m +
1)!.
(45)
E M I S S I O N OF GAMMA RA D I A T I O N
439
We can use the addition theorem for these functions to sum this expression over m. We obtain in this way the factor 2 / + 2, which drops out on dividing by the number of states. On the other hand we have for l odd
Sk S, ]ps [L ] W, alm) [2 _ 4~2m [ i p ~ l [ 2 T. ,D,.+1,2~ t--l+l, I g2p2q2+p4--2gP•P*q"
(46)
which can be treated in the same way. For states of type b we find that the summations are given by identical expressions except that in the result l is replaced by l - 1. It should be mentioned that cutting off the wave functions for the electron in the initial state at a distance ~ from the origin, which is about the nuclear radius, introduces negliglible corrections of the order ~3. Note 3. Determination of the States Entering in an ,,Allowed" Nuclear Transition. In the F e r m i thecry for ~-disintegration there enters in the expression for P(W~) the density of the electrons in the various states averaged over the nuclear volume. Only those states having ~ = { are appreciably different from zero near the nucleus, however, so this factor becomes simply
S, lq, I2 -- (i/2=) (fi + ~ . + p_2 + g~=), evaluated near the origin, where [ and g are the two kinds of radial wave functions for the D i r a c electron, those with the subscript zero coming from type a, l = 0, and those with the subscript - - 2 from type b, l ---- 1. The ratio of the two types at a small distance from the origin is then found. The re ult is
/~+~ /2-2 + g~2
--
ao
w,+
bl -- W,
--
~ y
where T = ( 1 - - ,1222)'/, and in section II was put equal to unity. Received April 24, 1936. REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9)
G. H. A s t o n, Proc. Cambr. phil. Soc. 22, 935, 1927. S. B r a m s o n, Z. Phys. 66, 721, 1930. J . A . G r a y and J. F. H i n d s, Phys. Rev. 49, 477, 1936. J. R. O p p e n h e i m e r and L. N e d e l s k y , Phys. Rev. 44, 948, 1933. H. B e t h e and W. H e i t l e r , Proc. roy. Soc. London, (A) 146, 83, 1934. C.G. D a r w i n, Proc. roy. Soc. London, (A) 118, 654, [928. E. F e r m i , Z. Phys. BS, 161, 1934. M . E . R o s e a n d G . E. U h l e n b e c k , Phys. Rev. 4B, 211, 1935. E.J. Konopinski a n d G . E. U h l e n b e c k , Phys. Rev. 4 8 , 7 , 1935.
(47)
Juni
1936
EMISSION OF GAMMA RADIATION DURING THE BETA DECAY OF NUCLEI by J. K. K N I P P *) and G. E. U H L E N B E C K Natuurkundig Laboratorium der Rijks Universiteit te Utrecht
Summary E x p e r i m e n t a l i n v e s t i g a t i o n s h a v e s h o w n t h a t t h e d i s i n t e g r a t i o n of R a E is a c c o m p a n i e d b y a f a i r l y w e a k y - r a d i a t i o n . T h e o r e t i c a l c a l c u l a t i o n s h a v e b e e n m a d e w h i c h s h o w t h a t a l a r g e p a r t , if n o t all of t h i s r a d i a t i o n a r i s e s f r o m t h e c r e a t i o n of t h e e l e c t r o n a n d its loss of e n e r g y as i t l e a v e s t h e n u c l e u s . T h i s p r o c e s s is a n " i n n e r " B r e m s s t r a h l u n g t o b e c o n t r a s t e d w i t h t h e " o u t e r " B r e m s s t r a h l u n g d u e t o t h e i m p i n g e m e n t o n t h e n u c l e u s of e l e c t r o n s c o m i n g f r o m t h e o u t s i d e . I t h a s t h e o r d e r of m a g n i t u d e a = 1/137 p e r o u t c o m i n g e l e c t r o n . T w o m e t h o d s of t r e a t i n g t h e p r o b l e m h a v e b e e n e m p l o y e d a n d i t is f o u n d t h a t in t h e B o r n a p p r o x i m a t i o n t h e y g i v e i d e n t i c a l r e s u l t s . I n t h e f i r s t t h e e l e c t r o n is r e p r e s e n t e d b y t h e free, o u t g o i n g , n o t - a l l o w e d , s p h e r i c a l s o l u t i o n s of t h e D i r a c e q u a t i o n a n d r a d i a + i o n transitions are calculated to standing waves. In the second method the c a l c u l a t i o n s a r e b a s e d o n t h e F e r m i t h e o r y of ~ - d e c a y . T h e e f f e c t of a q u a n t i z e d e l e c t r o m a g n e t i c r a d i a t i o n field is i n t r o d u c e d b y m e a n s of a c o u p l i n g t e r m b e t w e e n t h e e l e c t r o n s a n d t h e r a d i a t i o n field. R a d i a t i o n i n t e n s i t y c u r v e s a r e o b t a i n e d w h i c h a r e v e r y s i m i l a r t o t h o s e for o r d i n a r y Bremsstrahlung.
I. Introduction. The usual continuous electron spectrum of radioactive nuclei has superimposed on it certain homogeneous groups of electrons arising from the internal conversion of nuclear y-rays. These y-rays are monochromatic and are due to nuclear transitions taking place before or after the process of ~-decay. That there is a small amount of y-radiation accompanying the ~-disintegration of nuclei which cannot be accounted for in this way was first shown by A s t o n 1) in his measurements on RaE. This ~-radioactive element has a normal continuous electron spectrum, but there are no homogeneous groups of electrons superimposed on it and therefore no *) Sheldon Fellow from Harvard University.
425
426
j.K.
KNIPP AND G. E. UHLENBECK
monochromatic y-rays of the usual kind. A s t o n found, however, that RaE is accompanied by a fairly weak y-radiation and from his measurements concluded the average to be about 10,000 volts per electron of disintegration. More accurate measurements have been made by S. B r a m s o n 2). She obtained an absorption curve for the radiation in copper. The form of this curve suggests immediately that the radiation is continuous in nature. She has estimated the number of quanta and found roughly 1.6 per hundred electrons of disintegration. The interpretation made that a large portion of the radiation is due to K radiation from the product atoms in which the F-particles have knocked out K electrons is not correct, however, as can be seen on theoretical grounds. Such a process has a probability of taking place proportional to e 4, per outcoming electron, or the square of the fine structure constant, e = 1/137. It can therefore be expected to be of the order of magnitude of 1/10,000. Such a probability is much too small to account for the number of quanta observed. A process with a probability of the right order of magnitude is the one in which the radiation arises from the creation of the electron and its loss of energy as it leaves the nucleus. This is an ,,inner" Bremsstrahlung to be contrasted with the "outer" Bremsstrahlung due to the impingement on the nucleus of electrons coming from the outside. The "inner" Bremsstrahlung has a probability per electron proportional to e2 or e and in the first approximation is independent of the nuclear charge, Z. The "outer" Bremsstrahlung has a crosssection proportional to ~aZ2 and vanishes for zero nuclear charge *). We shall treat the theory of the "inner" Bremsstrahlung by two methods. In the first the electron is described as a particle leaving the nucleus with a definite energy. The probability that it radiate is then calculated. On integrating this probability over the energy distribution of the electrons, the probability spectrum of the gamma rays is obtained. In the second method we use the F e r m i theory of F-decay, according to which a neutron changes to a proton under the influence of the electron-neutrino field. The effect of a quantized *) T h e r e is a n a n a l o g o u s s i t u a t i o n in t h e p r o d u c t i o n of p a i r s b y y - r a y s . F o r n u c l e a r y - r a y s t h e p r o b a b i l i t y is p r o p o r t i o n a l to cx, i n d e p e n d e n t of t h e n u c l e a r c h a r g e in t h e f i r s t a p p r o x i m a t i o n , as f o u n d b y O p p e n h e i m e r and Nedelsky4).Ontheother hand the B e t h e-H e i t 1 e r calculation gives a l~robability for plane gamma waves p r o p o r t i o n a l to 00Z ~- a n d zero for zero n u c l e a r c h a r g e ~).
E M I S S I O N OF GAMMA R A D I A T I O N
427 i
electromagnetic radiation field is introduced by means of a coupling term between the electrons and the radiation field. We calculate the probability of obtaining a neutrino, an electron and a y-ray. On s u m m i n g over all the allowed states of the electron and the neutrino, the total probability for radiating a q u a n t u m of a definite energy is obtained. The first m e t h o d has the a d v a n t a g e t h a t the unobserved neutrino enters only indirectly in the description of the ~-spectrum. On the othc.r h a n d it requires the use of the not-allowed s o l u t ~ n s of the D i r a c equation, which up to the present have been excluded from q u a n t u m mechanical calculations. II. CalculationsUsing Outgoing Waves. In this section a n d the one following we shall use the Born approximation, in which ~Z/p is supposed to be small compared to unity. This approximation is only go,~d for light nuclei or particles of high energy and so will give results which describe only qualitatively the radiation from a nucleus as h e a v y as t h a t of R a E (~Z = 0.606). We require solutions of the D i r a c equation in polar coordinates which describe a free electron leaving the origin. In the following we employ h/mc as the unit of length, t~/mc2 as the unit of time and the electron mass m as the unit of mass. The solutions are readily found to be t y p e a, ]' = l + 1/2
Na +,,'at,,,=~v,
- - i Hi+,~. (pr) PI'~-t --i Ht+./. (pr) "l+lz)"+l p p],, ( l + m + 1) W - - 1 H,+,/,(pr) - - (l - - m) W P -- 1
Hi+,/: (pr) P7'+1
6(l--m)! (l+m+l)!,l>_
0,--(l+l)_
and t y p e b, .I" = l - - 1/2, --i
Nb
+,vbt,,,= r.i;
(l + m) Hi_,~, (pr) Pi"---i i (l - - m - - 1) Hi_,/.. (pr) ~--ID" +1 -- p p~,,
W-
1 H,+,/. (pr)
W- -- P 1 H~+,/, (pr) p7,+i N ~ = ( W - - I ) / 1 6 ( I + m)! (l--m--1)!,l>_ l , - - l _ < m _< l - - l ,
428
J . K . K N I P P A N D G. E . U H L E N B E C K
where W is the energy and p the m o m e n t u m of the electron. H~+l/,(pr) is a H a n k e 1 function of the first kind and half integral order. The functions P~' are surface harmonics as defined by D a r w i n 6). There are 2 (2l + 1) solutions having positive energy W and each has been normalized to represent one electron leaving a large sphere per unit time. Replacing the H a n k e 1 functions by B e s s e 1 functions would give, aside from the normalization factor, the ordinary standing spherical solutions of the D i r a c equation. These are not the solutions we want, however, since they do not describe an electron leaving the nucleus. One can easily see t h a t they do not give rise to radiation (see Note 1). In the Hamiltonian the coupling term between the electron and the radiation field is --->
L = - - x/~ (~. A),
(2)
where ~ is the D i r a c velocity m a t r i x * ) and A is the vector potential of the quantized radiation field having components given by
A = Y, qk Ak + c.c.,
A =
e
. rl,
(3)
k
the momentum of the light quantum being denoted by k. The amplitudes q~ are the "coordinates" of the field and have their matrix elements the same as those of the harmonic oscillator
(n~ [ qk [ nh + 1) ---- (nk + 1/2k)V,, (n~ ] q~ ]nk - - 1) ---- (nk/2k)v,,
(4)
where nk is the number of quanta of a particular polarization having energy k. The direction of polarization of the light wave is given by ...->
..->
the unit vector ek perpendicular to k and directed along the electric field strength. Each component of the field is normalized to 4~ quanta per unit volume. For emission of a q u a n t u m of radiation one can omit the complex conjugate and L becomes simply ->.
L
x
-->.
• rl,
~
=
---->
(s)
k
We suppose that at time t = 0 the electron is born at the nucleus with energy W~. We apply the method of variation of parameters to find the probability t h a t the system of electron and radiation field be in the final state / with total energy W t = W + k at the time t, this • ) Not to be c o n f u s e d w i t h t h e fine s t r u c t u r e c o n s t a n t .
E M I S S I O N OF GAMMA R A D I A T I O N
429
state being one in which the electron has the energy W and the radiation field has gained a quantum of energy k. A first order perturbation gives for this probability ] a ! [ 2 = I ( / ] L ] O ) 124sin 2 ( w ! - W e ) t
( W ! - We)2
(6)
The total probability per unit time that an electron, created with energy We, radiate a light quantum having energy between k and k + dk is then given b y summing over all possible final states of the electron and the polarization and direction of the light quantum and dividing b y the time interval t. We shall call this probability O(We, k) dk. If plane waves are used to represent the final states of the electron, we have
(We, k) dk = X X ]a, ]2 1 ,
(7)
k s
where oo
y, X = (1/(2~) 6) k 2 dk f dtL Sk f d~, Ss f dW Wp. ks
(8)
1
Here dt~, and dglk are the two elements of solid angle in the direction of emission of the electron in its final state and the light quantum respectively; Sk is the summation over the two directions of polarization of the light wave and S, is the summation over the two states of the electron with momentum p and positive energy. The integration over the energy of the electron gives the conservation of energy W I = We = W + k and the factor 2~t instead of the second factor of (6). To carry out the calculations we need the matrix elements .->
.=~
(/!L 10) ----(psiL I We jZm) - - - - (2~/k)'/, f dr+'~,o~k+,rei,,,,e-~(k" ')
(9)
The wave functions for the plane wave are very simple s). The typical integral which arises is discussed in Note 1. Let us consider the case l = m = 0 , which occurs only with type a wave function. O11 taking k along the z axis *), we readily obtain
SkS, l(ps[L[Wel/200)[2 = 4v~2°~(g2p2 + q2--2gp ,q,) gwpek
(lO)
where g = (We + 1)/(W + 1) and q = p + k. This expression can *) Although z is the axis of q u a n t i z a t i o n this involves no loss of generality.
430
j.
K. KNIPP
A N D .G. E . U H L E N B E C K
be resolved into partial fractions. On introducing the angle between ....>
--->
p and k, we obtain for the differential probability function ~ d~ j[(W w'2i3+ ~-~-~ ~ ) +2p--~os ww2 dO0= :cp2~_p~_ksin
( w - p 2cos 1 ~)
1 p/. (11)
This can be integrated at once to give the integral probability function (I)°= ~-p,k[
p2W-+-w
/
log (W + p ) - - 2 . .
(12)
It is shown in Note 2 that these results hold for all even l (type a or b) after averaging over m. Likewise we find for l -- 1 and all odd l d(i)l = ~ p sin ~ d~ J W'2 --i_2 W _+ _W2 2~ p,k [(W,--1) (W - - p cos ~)
P
1 (W--pcos~) 2
} 1..
}
(I)x= ---p, k |
log (W + p) - - 2 . .
(13)
It is seen that for a given W, all electrons radiate the same amount of soft radiation. To obtain the actual probability functions for some particular ~radioactive nucleus we must take the proper linear combination of the functions for the different states of the electrons emitted. Thus the actual function is an expression of the form
• = X (atO.~ + btObt), Z (at + bt) ---- 1, t
where a t and bt are weight According to the F e r m i transitions of the nucleus + 1 ) / ( W , - 1) (see Note 3) ao
(14)
t
factors for the two types for a given l. theory only a0 and b I enter for "allowed" 7) and these in the ratio ao/bl : (W,+ *). We have
= (W,+I)/2W,,
b t = (W,--1)/2W,;
(15)
and the probability functions for an allowed transition of the nucleu,.~ therefore are d(I) ~ ~ p sin 5 d~ ] W,2 + W 2 1 - - 1/ -2~p-Q~ IW,(W---~t~os ~) - - (W - - p cos ~)2 , , ~P J w`2+ W21og ( w + p ) - 2
/
• ) T h i s h o l d s also for t h e m o d i f i c a t i o n of t h e F e ' r m i t h e o r y p r o p o s e d b y pinski and Uhlertbeck, r e f e r e n c e (9).
(16) K o n o-
EMISSION
431
OF GAMMA RADIATION
The radiation probability spectrum,S(k), for the entire distribution of electrons is obtained b y multiplying byP(Wc) dW,, the probability t h a t an electron be created in the energy interval W, to Wc + dW~, and integrating. We h a v e Wa
S(k) =
f
l+k
dW~P(W~) ap (W~, k),
(17)
where Wo is the upper limit of the ~-spectrum. Here P(W,) is to be taken as the theoretical distribution given b y the original or modified F e r m i theory. If one assumes t h a t the proper linear combination will always be determined b y the density of the electrons near the nucleus, it seems even reasonable, according to the m e t h o d of this section, to use for P(W,) the experimental energy distribution. III. Calculations with the Fermi Theory. The process described in the last section m a y appear to be puzzling because if the nuclear charge is zero there is no acceleration and hence no reason classically w h y the electron should radiate *). The question is clearly connected with how the particle is created in the nucleus. In this section we shall base our calculations on the F e r m i t h e o r y of the nucleus, which affords a consistent account of the emission of electrons b y nuclei. We introduce the electromagnetic radiation field into the F e r m i theory b y means of a coupling term between the electrons and the radiation field. The H a m i l t o n i a n of the system contains the two perturbing terms H and L, where H is the F e r m i coupling term between the h e a v y particle and the electron-neutrino field and L is the coupling t e r m between the electrons and the electromagnetic field. We can use the original F e r m i notation and write for the first of these terms -..>
-..>.
....>
-.->
n = Q[Ao + (c¢hea,,y.A)] + Q*[A~ + (~h,avy. A)*].
(18)
Here Q and Q* are H e i s e n b e r g matrices which operate on the inner coordinate of the h e a v y particle. This coordinate p determines whether the particle is a n e u t r o n (0 = + I) or a proton (p = - - 1). Q corresponds to a proton to n e u t r o n transition and Q* the reverse. ~-h,,vyis the D i r a c velocity m a t r i x which acts on the spin coordi*) I t s h o u l d be p o i n t e d o u t t h a t t h e r e is a s i m i l a r l a c k of a c l a s s i c a l a n a l o g u e in t h e c a s e of t h e a n n i h i l a t i o n r a d i a t i o n a r i s i n g f r o m t h e r e c o m b i n a t i o n of a free p o s i t r o n a n d a free e l e c t r o n .
432
j.
K. K N I P P A N D G. E. U H L E N B E C K
n a t e of the he' v y particle. We shall neglect t e r m s c o n t a i n i n g this m a t r i x because t h e y are of a smaller order of m a g n i t u d e t h a n the o t h e r terms. A is the four v e c t o r " p o t e n t i a l " of the e l e c t r o n - n e u t r i n o field and is built u p out of the q u a n t i z e d wave functions + and £0 of the electron and n e u t r i n o respectively. Following F e r m i we can write for the i n t e r a c t i o n
H=G
E X IQa, b ~ ( + s ~ % ) + Q * a * b * ( + , ~ , ~ ) * ,
(19)
$
where G is the coefficient of" coupling and
:
0
0
0
0
(20) "
0--1
F o r the coupling t e r m L we can take the sum of terms of the t y p e used in the last section, the s u m m a t i o n being over the states of the electron L = F~NsL$, (21) $
where N$ is the n u m b e r of electrons (0 or l) in the state s. T h u s for emission L is simply L = - - X E ( 2 ~ ~lk) '/, N$ o~h e-Ik" "~. $
(22)
k
We t a k e the initial state to be one in which there is a n e u t r o n , and no electrons, neutrinos or light q u a n t a (p = + 1, N, = 0, M~ = 0, n , = 0 ) . T h e t o t a l e n e r g y of the initial state, W0, can be t h o u g h t of as the available energy. W e are i n t e r e s t e d in the p r o b a b i l i t y of a transition to a state in which the n e u t r o n has b e c o m e a p r o t o n and an electron, n e u t r i n o and light q u a n t u m , all of definite energy, h a v e been created (O = - - 1, N , = 1, M , = 1, n, -~ 1). We indicate the t o t a l energy of this final state b y W I . I t is clear this transition c a n n o t be m a d e directly since n e i t h e r H nor L act on more t h a n three particles. W e m u s t consider i n t e r m e d i a t e states of the t y p e (? = - - 1 , N, = 1, M~----1, n~ = 0). L e t the total e n e r g y be W~. O t h e r i n t e r m e d i a t e states are excluded to this a p p r o x i m a t i o n since L c a n n o t s t a r t the process because it does n o t h i n g to the h e a v y particle or neutrinos and all the effective N , are zero in the initial state. We h a v e the p e r t u r b a t i o n H causing the transition from the s t a t e 0 to the state l and t h e n the p e r t u r b a t i o n L causing f u r t h e r transitions to
E M I S S I O N OF
433
GAMMA R A D I A T I O N
the state/. This is a second order perturbation and the square of the probability amplitude for the final state after a time t is given b y
l al j2 = I 2 ~ (/tL Jz) (zI n Io/2 4 sin~ (Wl-- W0) t , ~-, ---ff-~o (w~-- Wo)~
(23)
The particles can be described by plane waves. The matrix elements are, making the same approximations as does F e r m i for an "allowed" nuclear transition,
(/ IL I/) = (ps ILl p' s') ---- - - (2==/k)'/, (u*=,u'), p' = p + k, (IIH IO) --- GM(u'Scr)*,
(24)
where primes indicate the intermediate state and u and . are the amplitudes of the plane wave functions for the electron and neutrino respectively and designate states with particular momentum, spin and sign of the energy. M is the matrix of the nuclear wave functions We have for the energies
W ~ = K,, + W', W t :
K,, + W + k.
(25)
Defining an initial energy of the electron b y W, = W o - Ko and substituting in the above expression, we have
lallm= GZ l M IZ (2~oc/k)
(u* ~.,u')(u'8,)* 2 4sin2(Wi--W0 )t d~
W , - - W'
(Wl - - Wo) 2
(26)
where the summation formally is over all the intermediate states of the electron of both the positive and negative continuum; however because of the conservation of momentum it is reduced to a summation over the four states of a particular momentum p'. The total probability S(k)dk that a light quantum be emitted in the energy range between k and k + dk is obtaified b y summing over all the allowed states of the electron and neutrino and the direction and polarizations of the light quantum. We have
S(k)dk----
Z
k
Z
~"
Z
21
l a/I 7 '
(27)
s
where in detail EE]~---k
O"
s
= (1/(2re) . 9)j "d p ,, p ,,K,,k2d k ./ "dfl :,S, j
"dflkS kj "dfl~S ,, '~"
(28)
1
Physica I I I
28
434
J.
K. K N I P P A N D G. E. U H L E N B E C K
W e have written the operations in the order in which we shall perform them. The last integral gives the factor 2r~twp and the energy conservation law W 1 = W0 = Ko + W + k, which defines W. The integration over the m o m e n t u m of the neutrino can be transformed into one over W, Wo--I--/~
6
W0
lr~,
If then we write
P(W,) = (G21M 12/2na) Ko p~ W, p,,
(30)
S(k) can be put in the form P
W,
s(k) = / a w , where • (W~,k)
(3])
P(W,) ¢ (w,, k), W, = W + k,
1¢,
,t p Wk
w,-
w'
(32)
(W,, k)dk can be interpreted as the probability that an electron, created with energy W,, create a fight quantum of energy k. The summation over t h e i n t e r m e d i a t e s t a t e a n d the spin of the neutrino can be effected by the usual methods. Neglecting the terms that drop out in the subsequent integration over the direction of the m o m e n t u m of the neutrino, we obtain for this summation [1/2(W~-- W'2) 2] [W:,(u*u)+2W, (u*=kH' ~z,u)+ (u*=kH'2c(,u)]
(33)
w h e r e - - H ' ~- (~. p ' ) + } . Introducing this expression and summing over the two directions of polarization of the light, we obtain
[1/(W~--W'2) 2] [W2~ + W ' 2 -
(2W, (p'p, + 1)/W)]
(34)
which can be broken up into partial fractions. Noticing t h a t W~ - - W '2 = p~ - - p,2 = 2k (W - - p cos 8) where ~ is the angle between the directions of the light quantum and the electron, we obtain finally for the differential and integral probabilities
d , _ e p sin S d8 J. W~_+__W2 2np, k [W,(W--pcos~)
P "( W , p (1) = r: p,k
1 (W - - p cos ff)2
l o g ( W + p ) - - 2 . [ . .I
) 1, ,
(3_5)
EMISSION OF GAMMA RADIATION
435
These are just the expressions obtained by the method of section II. It m a y be pointed out that if one changes the interaction "Ansatz" (19) als proposed by K o n o p i n s k i and U h l e n b e c k the final formulas (3 l) and (35) are not changed. One has only to replace P(W,) as given by (30) by
P(W,) = (G= I M 12/2x 3) K3oPerWe p,.
(36)
IV. Discussion. In Fig. 1 we give curves showing the distribution in energy of the gamma radiation for different initialelectron energies. We have plotted the intensity, that is k times the probability 1,0
Inten.sity Dtsteibution Cur've5
0,8 10-2; 0,6 •mc~
,,x~e= 10 ~e
=5
0.4 o£_
0.2
OA
0.6
K/(We-1)
1.0
Fig. 1. I n t e n s i t y d i s t r i b u t i o n c u r v e s for d i f f e r e n t i n i t i a l e n e r n i e s of t h e e l e c t r o n , We. T h e s e a r e for " a l l o w e d " n u c l e a r t r a n s i , t i o n s as g i v e n b y (35). E n e r g i e s a r e m e a s u r e d i n u n i t s of m c * or 0.51 m i l l i o n e l e c t r o n v o l t s a n d i n c l u d e t h e r e s t mass.
function, rather than the function itself since ~ behaves roughly as 1/k and as a consequence becomes very large for low energy quanta. The intensity starts with a finite value for soft radiation and decreases monotonically to zero as k increases to its upper limit, W , - 1. The curves are very similar to those for "outer" Bremsstrahlung.
436
J.
K. K N I P P AND G . E . U H L E N B E C K
Fig. 2 shows the total energy spectrum of the gamma radiation from a nucleus with W 0 = 3.6, which is approximately the upper limit of RaE.This curve is for an "allowed" nuclear transition. It was calculated using for P(W,) the expression (20) normalized to unity. In the same figure we have plotted the number of quanta of energy greater than k per electron. This number is given b y
/.
Wo--I
N(k) =
(37)
dx S(x),
where S(k) is the expression (31). With the F e r m i "Ansatz" the B o r n approximation gives three quanta having energy greater than 50,000 volts per thousand
2,5
5
1
NfKJ
10`3 20
2
mC ~
10-~"
i
i
0.5
10
10 mc ~ 1,5
2,0
1,5
t0
K
0,5
Q5
H 2,5
F i g . 2. T o t a l e n e r g y s p e c t r u m o f t h e g a m m a r a y s f o r a n " a l l o w e d " n u c l e a r t r a n s i t i o n w i t h W0 = 3.6 a n d Z = 0 a n d u s i n g t h e F e r m i d i s t r i b u t i o n c u r v e for t h e e l e c t r o n s . T h e s m a l l e r c u r v e g i v e s t h e n u m b e r of q u a n t a h a v i n g e n e r g y g r e a t e r t h a n k a s a f u n c t i o n of k.
outcoming electrons for a nucleus having a continuous ~-spectrum with upper limit of 3.6 mc 2. If for P(W,) the express in (36) as given b y the modified theory is used, considerably less hard radiation is
437
E M I S S I O N OF GAMMA R A D I A T I O N
obtained. For 1000 outcoming electrons there are about 1.5 quanta having energy greater than 50,000 volts. This is roughly a factor 1/ 10 smaller than the number measured for RaE. According to the F e r m i theory the disintegration of R a E is a "forbidden" transition of the nucleus. It is clear also that we cannot make comparisons with the experimental results of S. B r a m s o n because R a E has such a large nuclear charge (Z = 83). To get a rough idea of the influence of the nuclear charge on the gamma radiation emitted we have made calculations after the method of section II using the S c h r 6 d i n g e r approximation for the electron in the final state. This is a non-relativistic approximation (p ~ 1 and ~Z ~ 1) and is valid only when the light quantum gets almost all of the kinetic energy of the electron. It is not necessary to take the small components in the wave functions. We give only the results. For l even we obtain for the integral probability 80¢2 Zk3W *o = p, (W, + 1) (p~ - - k2)2 (1--
e -2"~'z/p)
"
(38)
To be consistent with the approximations, one should replace k by W, - - ! and W b y unity, so that 2~2 Z (We --1) D° = Pe (W, + 1) ( 1 - - e-2'~az/#) "
(39)
The corresponding expressions for l odd are obtained b y replacing (We + 1) occuring in the denominators of the above b y ( W e - 1). Using (! 5) we obtain for an "allowed" transition *) 8~ 2 Zk3W = p e W e (p2 ___ k2)2 (1-- e--2'~az/#) '
(40)
or
@=
2~x2Z ( W , - - 1) p, W, ( l - - e-2"~/*) "
(41)
These expressions give a finite limit for the intensity curves as k approaches its maximum value. The amount of hard radiation is increased and the results indicate that the theory can account for the observed measurements on the number of quanta. We hope to investigate the dependence on the nuclear charge on the basis of the *) All these expressions have the correct l i m i t i n g connections with the corresponding expressions of the B o r n approximation. See ref. (8).
j . K . KNIPP AND
438
G.. E. I#HLENBECK
F e r m i t h e o r y and to carry out a more detailed comparison with experiment. We wish to express our gratitude to Professor L. M e i t n e r of Berlin for valuable assistance in the interpretation of the experim e n t a l data. We are t h a n k f u l to Dr. A r n o 1 d N o r d s i e c k for m u c h stimulating discussion. Note 1. The General Integral Arising in Section II. All the integrals in the m a t r i x elements of section II are of the form d'r e ---i(q" r)
HI+,I, (P, r)
r -'1. P'~.
(42)
On carrying out the angular integrations there remains the integral
J,+,/, (qr) Hz+,/, (p, r) r dr,
(43)
o
This is divergent at infinity b u t has to be understood t h a t in evaluating it one introduces as usual exp. ( - - ar) and puts ~ = 0 after the integration. The integral can be split into two parts b y means of the relation
nl+,h = J~+,l, + i ( - - ) ~+a J-c~+'h). Since q :/: Po the first of these is found to give zero (as seen using Eqn. 2, p. 389, W a t s o n, Theory o/ Bessel Functions). If one represented the electron initially b y standing waves, only integrals of this k i n d would arise. F o r the second we find oo
f J,+'l. (qr) J-I+v., (p.r)
r
dr
=
2i
1
(- q),+'l,
(44)
O.
for p, > q, where ¢ = 0 , l even and ~ = 1, l odd (using Eqn. 6, p. 404, Watson, Theory o/Bessel Functions). Note 2. B o r n A p p r o x i m a t i o n for a n y (W, tim). Consider an initial state of t y p e a and l even. B y means of relations a m o n g the D a r w i n functions a n d using the integrals of Note 1, the summ a t i o n s over the polarizations and the spin of the electron in the final state can be reduced to the expression on the right side of (10) w i t h the additional factor
(IPr+xl2 + IPf+lp)/(l--m)l ( l + m +
1)!.
(45)
E M I S S I O N OF GAMMA RA D I A T I O N
439
We can use the addition theorem for these functions to sum this expression over m. We obtain in this way the factor 2 / + 2, which drops out on dividing by the number of states. On the other hand we have for l odd
Sk S, ]ps [L ] W, alm) [2 _ 4~2m [ i p ~ l [ 2 T. ,D,.+1,2~ t--l+l, I g2p2q2+p4--2gP•P*q"
(46)
which can be treated in the same way. For states of type b we find that the summations are given by identical expressions except that in the result l is replaced by l - 1. It should be mentioned that cutting off the wave functions for the electron in the initial state at a distance ~ from the origin, which is about the nuclear radius, introduces negliglible corrections of the order ~3. Note 3. Determination of the States Entering in an ,,Allowed" Nuclear Transition. In the F e r m i thecry for ~-disintegration there enters in the expression for P(W~) the density of the electrons in the various states averaged over the nuclear volume. Only those states having ~ = { are appreciably different from zero near the nucleus, however, so this factor becomes simply
S, lq, I2 -- (i/2=) (fi + ~ . + p_2 + g~=), evaluated near the origin, where [ and g are the two kinds of radial wave functions for the D i r a c electron, those with the subscript zero coming from type a, l = 0, and those with the subscript - - 2 from type b, l ---- 1. The ratio of the two types at a small distance from the origin is then found. The re ult is
/~+~ /2-2 + g~2
--
ao
w,+
bl -- W,
--
~ y
where T = ( 1 - - ,1222)'/, and in section II was put equal to unity. Received April 24, 1936. REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9)
G. H. A s t o n, Proc. Cambr. phil. Soc. 22, 935, 1927. S. B r a m s o n, Z. Phys. 66, 721, 1930. J . A . G r a y and J. F. H i n d s, Phys. Rev. 49, 477, 1936. J. R. O p p e n h e i m e r and L. N e d e l s k y , Phys. Rev. 44, 948, 1933. H. B e t h e and W. H e i t l e r , Proc. roy. Soc. London, (A) 146, 83, 1934. C.G. D a r w i n, Proc. roy. Soc. London, (A) 118, 654, [928. E. F e r m i , Z. Phys. BS, 161, 1934. M . E . R o s e a n d G . E. U h l e n b e c k , Phys. Rev. 4B, 211, 1935. E.J. Konopinski a n d G . E. U h l e n b e c k , Phys. Rev. 4 8 , 7 , 1935.
(47)