Continuous gamma radiation accompanying nuclear beta decay

I I

3.A

I

Nuclear Physics 79 (1966) 113--134; (~) North-Holland Publishing Co., Amsterdam

i

Not to be reprodueed by photoprint or mlerofilm without written permission from the publisher

CONTINUOUS ACCOMPANYING

GAMMA NUCLEAR

RADIATION BETA DECAY

R. E. STRUZYNSKI ? Stevens Institute of Technology, Hoboken, New Jersey, USA and F. POLLOCK Stevens Institute of Technology, Hoboken, New Jersey, USA and Hudson Laboratories of Columbia University, Dobbs Ferry, New York, USA Received 2 August 1965

Abstract: The photon spectrum associated with the internal bremsstrahlung of the nucleus 3sS is computed. Relativistic Coulomb wave functions are used for the final states of the negaton. The Green function for the Dirac equation is computed using a potential based on a finite size nucleus of uniform charge density. The equivalence between the two methods of calculation originally proposed by Knipp and Uhlenbeck is proven. The results of our calculation agree with those of Mau. Since the present calculation includes Coulomb effects completely, the discrepancy between theory and experiment may now be attributed to either the neglecting of screening effects in the calculations or to experimental procedures.

1. Introduction T h e c o n t i n u o u s g a m m a r a d i a t i o n a c c o m p a n y i n g n u c l e a r b e t a d e c a y c a n b e exp l a i n e d b y a p r o c e s s , w h i c h m a y be r e p r e s e n t e d by t h e f o l l o w i n g n u c l e a r r e a c t i o n a n d its a s s o c i a t e d F e y n m a n d i a g r a m (fig. 1): n ~ p+e-'+~

~ p+e=+~+7.

p

/

Fig. 1. Feynman diagram. t Present address: Brooklyn College, New York City, New York, USA. 113

(1)

114

1~. :E. S T R U Z Y N S K I

AND

F. P O L L O C K

The above process represents the transformation of a neutron n, in a nucleus into a proton p with the simultaneous ejection of a negaton e - ' in an intermediate state and an anti-neutrino ~. The negaton emits a photon 7 and goes into a final state e - . This process, which is called "inner" bremsstrahlung, has a probability per negaton emission of approximately e, the fine structure constant. The filst theoretical calculations on this process were performed independently by Knipp and Uhlenbeck ~) and by Bloch 2). They computed the probability of emission of a light quantum of energy k, by using second-order perturbation theory. In this calculation free particle, plane wave states, were used for the initial and final states of the negaton. Knipp and Uhlenbeck introduced an alternative method for computing the photon spectrum. This method was based on the assumption that the probability for the overall process may be written simply as the product of the probability of a beta decay times the probability that a given beta negaton would radiate. This assumption essentially states that in the process of inner bremsstrahlung, the beta decay process and the radiation process are independent. There is no obvious reason why this assumption is valid. Nevertheless, Knipp and Uhlenbeck proceeded to compute the probability, denoted by q~(We, k), that a given beta negaton e - ' with energy We would radiate to form a photon 7 with energy k and a negaton e- with energy E. The nucleus was treated as a source of negatons and the initial negaton states were represented by free particle, outgoing spherical wave solutions to the Dirac equation. The final negaton states were represented by free particle, plane wave solutions to the Dirac equation. Under the assumption of the independence of the two processes, the probability for the emission of a gamma quantum with energy k, was expressed as,

s(k) =

(wo)¢(Wo, k)dWo,

(2)

where P ( W ~ ) represents the probability of a beta decay. The result obtained by this method for allowed beta transitions turns out to be exactly the same as that obtained from the second order perturbation method. The calculations of Knipp and Uhlenbeck and of Bloch seemed to be in agreement with the first experimental results, for example those of Wu 3), Novey 4), Bolgiano 5); but more recent experiments, for example, those of Starfelt and Svantesson 6) and Langevin-Joliot 7) appear to be in complete disagreement with the theory of Knipp and Uhlenbeck and Bloch. The number of photons per beta disintegration per m c z is observed to be much greateI than that predicted by the calculations of Knipp and Uhlenbeck and Bloch. Attempts were therefore made to improve the theory by taking into account the Coulomb interaction between the beta negaton and the residual nucleus. The first formal published calculation of the net effect of the Coulomb field is that

CONTINUOUS GAMMA RADIATION

115

due to Lewis and Ford 8). Utilizing the Born approximation, they computed the first-order correction terms due to the Coulomb field. A second calculation was published by Spruch and Gold 9), who performed a nonrelativistic calculation assuming Zc~ small compared to 1, where Z is the charge of the residual nucleus. These two calculations, although closing slightly the gap between theory and experiment, still disagree with the experimental data. A third calculation was performed by Mau 1o), who avoided all the previous approximations by representing the intermediate and final states of the negaton by relativistic Coulomb wave functions. Due to the complexity of these wave functions, the calculation had to be performed numerically. Surprisingly, the results of Mau's calculations, although better than previous calculations, still deviate significantly from the experimental results. Recently, Felsner 1~) performed a calculation, taking into account Coulomb effects by using approximate Coulomb wave functions. These wave functions resemble plane wave functions with amplitudes as given by Jackson, Treiman and Wyld 12). The Coulomb corrections as calculated by Felsner increased the bremsstrahlung yield by approximately 50 ~ independent of photon energy. This result is indeed astonishing, since it is in disagreement with experiments for small photon energies, where Coulomb effects are not expected to be significant. Recent experiments were performed by Persson 13). The results of these experiments indicate that the total yield of bremsstrahlung predicted by Felsner is much too large. The present calculation is a relativistic calculation, utilizing relativistic Coulomb wave functions for the intermediate and final states of the negaton. It differs from the calculation of Mau in the following respects. In computing the Green function for the problem, which represents the intermediate states of the negaton, rather than projecting out the solution for the Dirac equation for a point charge nucleus from the solution to the second-order Dirac equation, we have directly solved for the Green function for the Dirac equation for a potential, determined by a finite size nucleus of uniform charge density. The resulting Green function being expressed as a product of spherical outgoing waves and standing waves, has allowed us to prove the equivalence of the two methods of calculation proposed by Knipp and Uhlenbeck. One can now see that if the seond-order calculation is appropriately formulated the alternative method of calculation of Knipp and Uhlenbeck logically follows from it. For the final state wave function we have used the expression developed by Johnson and Deck 14). This expression is more convenient than the final state wave function used by Mau in as much as it is written as the sum of three terms, the first of which is of the order unity in Zc~, the second of order Zc~ and the third of order (Zc0 2. This immediately allows one to estimate the relative order of magnitude of the various terms in the matrix element. The wave function of Johnson and Deck has the added advantage that it has been shown that the first two terms in the wave function reduce

116

R.E.

S T R U Z Y N S K I A N D F. P O L L O C K

to the Sommerfeld-Maue ~s) wave function in the proper limit, the third term being identically zero in this limit. The radial integrals resulting from the relativistic Coulomb wave functions were performed numerically utilizing the IBM 1620 and Univac 1106 computers. 2. Matrix Element

The Hamiltonian density of the beta interaction may be written as,

Hp(x) = g~p(x)O~On(x)~o(x)OpOv(x)

(3)

+ hermitian conjugate, where g is the beta decay coupling constant, 0p(X) the field operator for proton, 0.(x) the field operator for neutron, 0e(x) the field operator for negaton, @v(x) the field operator for neutrino and Op - ½7~(1+75) for V-A interaction. The 7, are related to the four Dirac matrices by the following relationships: ?k = --ifiak, ( k = 1,2,3); 74 = ft. Also, 7s = YI?2Y3]/4 • The 7, are hermitian and obey the following anticommutation relationships: Yu7,+7,7, = 26u~ with y, v = 1, 2, 3, 4. The quantity ~(x) is defined as ~(x) = ~*(x)74, where Or(x) is the hermitian conjugate of O(x). Since we are only interested in beta decay we discard the hermitian conjugate part of H~(x). Each of the above field operators may be expanded in terms of appropriate creation and annihilation operators. For example:

t)e(x) = Z a,e+(x)+ Z b*,e_(x). E.

(4)

E-

The summation over E+ and E_ are summations over positive and negative energy states. The wave functions e+ (x) and e_ (x) are solutions to the Dirac equation with Coulomb potential and hence Oe(x) is in the Furry representation. The operators a,, b*, respectively, destroy a negaton in a Coulomb potential and create a positon in a Coulomb potential. The Hamiltonian density for the gamma emission is given by

H,(x) = eN[~(x)A(x)@~(x)],

(5)

where N is the normal product operator, e the charge of the negaton, A(x) = A"(x)7,; # = 1, 2, 3, 4 with AU(x) representing the photon field and x the four vector containing spatial and temporal coordinates. The system of units used in the calculation is the rational relativistic system of units. In these units fi = e = me = 1. Then e 2 - - 0¢ --= 13 7 (the fine structure constant). The total interaction part of our Hamiltonian is given by

Hx(x) = H~(x) + H~(x).

(6)

CONTINUOUS

117

GAMMA RADIATION

Since we are considering a second-order process, the term in the S matrix we are interested in is S (2)

----- - -

½f_° d *xt f °~ d 4xz T[H (

I X1

o9

-at)

)H(

I X2

)3 ,

(7)

where T is the time ordering operator. Utilizing Wick's theorem, retaining only those terms which represent the transition of a nucleus with Z protons and A - Z neutrons, where A is the atomic number, to a state with Z + 1 protons, A - Z - 1 neutrons, one negaton, an antineutrino and a gamma ray, we obtain the expression S (2) = ½eg

d4xl -co

,v - o o

d4x2[~P~(xl)A(x,)SF(xl--x2)Ofll/Iv(X2)~lp(X2)Op~ln(X2)], (8)

where SF(X~- X2) is the Feynman Green function. We now expand the field operator for the photon field.

A,(xl) = x/~V-~-(2k') -~[ E a~'ezelk'~+ E akt'e"*e-'k'x~] • k

(9)

k

In the above expression, V is the normalization volume which we choose to be unity, i.e. V = i; k' is the momentum of the photon; e" are unit polarization vectors associated with the electromagnetic field; k is the four-momentum of the photon; a,, and a~, are photon annihilation and creation operators. Taking the matrix element between the initial and final states, we obtain S(2)= ½x/~

eg

(oo

~oo

× S~(x, - x 2)o~ ~,_ (x~)F + (x2) op n + (x~)],

(lO)

where e(xl) , v(x2) , p(x2) , n(x2) are wave functions for the negaton, antineutrino, proton and neutron. The plus and minus signs indicate that the wave functions came from either the positive or negative energy parts of the expansions for the field operators. We now proceed to separate the spatial and time dependence of our wave functions

e+(xl) = e(xl)e -~°'~,

e-ik" x~ = e-i(k" x~-E~tD ~_(x~) -- ~(x~)¢~%

n+(x2) = n(x~)e -~E~'2

(11)

118

R.E.

STRUZYNSKI AND F. POLLOCK

The E refer to the energies of the particles, which are appropriately designated by subscripts. The Feynman Green function is given by,

( ~ u.(xl)~.(xz)e -'~"("-t~),

~En > 0

-~s~(~l-x~) = [ - Z u.(x~)~.(xz)e-'~°<'~-~>

t1 >

t2

t2 >

tI .

(12)

E~<0

The u, are solutions to the equation in a Coulomb field. The E, are the energies of the intermediate states of the negaton and the summations are over either positive or negative energies. Substituting the above expressions into the expression for S~2) and performing the time integrations, we get S(2) if = 2rcit(Eo _ E~- Ee- E~)M,

(13)

with

M - ~ e/ ; ~ r,~ d3x~/r.oo d% LFe(xl)e-lk" ~/2k

J - 0o

d-

x Z

Oa V(xz)P(x2)Op n(x2)

(14)

En ~

where E 0 is the available energy for the beta decay; ~ . is now over both positive and negative energies and M is the matrix element for the process. 3. Evaluation of the Coulomb Green Function The Green function for the Dirac equation with a Coulomb potential may be written as

G(We; x2, xi)

Z Hn(x2)~ln(xl) ,

.

(15)

E.--We

where We is defined as

Wo-E+k', with E the final negaton energy, U the final photon energy (or momentum in the system of units we are using) G(Wo; x2, xl) is a solution of the differential equation.

[~ " p 2 - ( W o - V(x2)) + N 6 ( w o ; x2, xl) = p~(x2- xO.

(16)

In the above expression c~and fl are the Dirac matrices, g(x2) the Coulomb potential, P2 is the momentum operator acting on coordinate x2 and 6(x2-xl) the Dirac delta function.

CONTINUOUS GAMMA RADIATION

119

We define Gt(W'e; x2, xj_) = G(~'Ve; x2, xl)]~.

(17)

We then have

[~ • ~ - ( w o -

V(x~))+ ~IG'(Wo; x~, xO = 6(x~- x,).

(18)

We may also express the Green function in the following form:

G'(W~; x2, x,) = i= Z [O~(Xz)]out[O*~U(xa)]•

(19)

K, I*

I f we explicitly show the separation of the wave functions into upper and lower components, we have

0 =

0~,

where (o'" l + 1)0 U = . ~c0 u,

( . . t + 0 0 L = ~ 0 L, j 20" = j ( j + 1)0",

where n = U or L,

J=O" = #0",

a = ax i + % j + a~k, where ax, Cry, a~ are the Pauli spin operators, l the orbital angular m o m e n t u m operator, j the total angular m o m e n t u m operator, j~ the z-component of the total angular m o m e n t u m operator, ~ , j ( j + 1), # eigenvalues of their corresponding operators, [0~(x)]o,t outgoing continuum solutions of the Dirac equation and 0~(x) are standing wave solutions of the Dirac equation. Since we are restricting ourselves to allowed beta decay, the summation over the states of t¢ and # are restricted to those states corresponding to a total angular momentum o f j = 1. These states are j=l, l=0, ~c= - 1 , j=l,

l=1,

t o = 1.

Thus, the summation in the Green function is merely over values of tc = __ 1 and # = _ 1 We may also evaluate the coordinate at the origin for allowed beta decay. Therefore, we have

G'(wo; r, 0) = i~ Z [0~(r)Jou,0~*(0).

(20)

K=±I /z= ±{T h e correct normalization for the wave function in the Green function must now be determined.

120

R.E.

STRUZYNSKI

AND

F.

POLLOCK

We consider the following expression: !

( - i ~ . v +8+ V(r)- w o ) c (wo, r, 0) = ~(r).

(21)

The adjoint equation is given by G ,t (We; r, O)(i~. V + fl + V ( r ) - We) = 5(r).

(22)

Therefore, we have the following set of equations: -->

G t t (We;r, 0)(-i~t

G t t (We;r,O)(i~

,

•

t V+fl+V(r)-We)(G(W e;r,O)=

G '+(We; r,

O)6(r),

t ! V+fl+V(r)-We)G(W e ; r , O ) = G(We;r,O)5(r) •

(23)

Subtracting, we get -+ • [G'* (We,• r, 0 ) ~ 6 ' ( W e ; r, 0)] = [G'(We; r, 0 ) - V '* (We; r, 0)]5(r). iv

(24)

Substituting our expression for the Green function, we get

-->

~(o)[<(~)]oo,~ Z [<(~)]oo,<"(o)} = { Z ~[<(O]ou,~,~t"(o)+ Z 4,~(o)[~(,)]ou,}a(~).~ , ~

(25)

We now recall that the solutions to the Dirac equation in polar coordinates may be written as O~(r) = ( g(r)x: ] \if(r)z~_J '

(26)

where g(r) andf(r) are radial functions which will, in general, depend on ~; Z~ and Z~_~ are the angular parts of the solution. They may be expressed as

Zu~ = Z C(l, ½,j; p - m , m)Yf-"Z m,

(27)

nl

where Y f - " are spherical harmonics, Z" Pauli spinors with Z* = (~) and Z -~ = (o), C(l, ½,j; p - m , m) Clebsch-Gordan coefficients, l is the orbital angular momentum of the negaton of spin ½ and z-component of spin m = _+½, j and/~ gave been defined previously. When one solves the Dirac equation for a Coulomb potential modified by a finite size nucleus, one finds

¢,~(o)=

(0 ) g~(o)z~

,

(29)

=

\

0

/

CONTINUOUS GAMMA RADIATION

121

1 Otf(O)O~;(O) = 4~ [9*- ~(0)9 - a(O)O(- tc) +f*(O)f~(O)O(~c)]6,,, ,,, 6~,,u',

(3o)

We then find that

where O ( x ) is the step function, defined as O ( x ) = O,

x < 1,

O ( x ) = 1,

x >= 1.

We now multiply eq. (25) on the left by the expression O~u(0) and on the right by the expression O2(0). After integrating over a large sphere of radius R and volume V, eq. (25) may be expressed as

RZ~[g * 1(0)9-1(0)0(--

tc) +f*(O)fl(O)O(t¢)] Im [O,,* o.t(R)f~ o.t(R)]

= O(tc) Re [ f * ( 0 ) f l (0)] + 0 ( - ~ ) Re [9" a(0)9_ ~(0)].

(30

out

Out

The subscripts refer to the states ~: = __ 1, the asterisk indicates complex conjugation, Re and Im denote the real and imaginary part of a function. We shall now adopt the following procedure for obtaining the correctly normalized /t Coulomb Green function. The solutions [~(r)]ou t shall be written in the two regions r > ro and r < to, where r o is the radius of the nucleus. The appropriate asymptotic g form for [O~(r)]ou t for large R will be chosen determining the normalization for these functions in the region r > r0. The solutions will be matched at r = r o determining the normalization in the region r < r o. Utilizing eq. (31), we shall determine what the normalization is for the standing wave solutions for r < ro. Matching the standing wave solutions at r = r o determines their normalization for r > r o. u

Step 1. Construction of [OK(r)]out

We must find continuum solutions of the equation

aO= Ii,sO-r

K)+V(r)+fll 0 =

(32)

where a, is the r component of the Pauli spin matrices; K = fi(a • l + 1) and We = 1. We wish solutions which asymptotically represent outgoing waves. We know that

( o(r)x" ] '¢' =

=

if(r)z"_J

(26) "

Let Ul = rg(r),

u2 = rJ(r).

(33)

122

R . E . STRUZYNSKI AND F, POLLOCK

Substituting eqs. (26) and (33) into eq. (32), we get du 1

/~

-

dr

du2 dr

ul-t-(W~+l-V(r))u2,

r

(We- 1 - V(r))u~ + tc - U 2 r

--

(34)

"

We shall consider a finite size nucleus, of uniform charge density. The potential is given by the following expressions: V(/')

Z~ -- --, /.

~

V(r) = -

F ~

Z_~ 3 -

,

/'o,

r < r o.

(35)

2ro (i) Consider the solutions for r > r o. We make the substitutions

ul = (W~+ 1)~ (q51+~b2), •

1

(36)

u2 = l ( W o - 1)~(~bi-¢z).

We also define the quantity Pe, which is the magnitude of the momentum at r = oo Po = (We2 - 1)½.

(37)

The solutions may be written as

(38) @51 -- q52) = C(q~l - qSz)~,g +D($1

- #~2)~ing,

where A, B, C and D are constants to be determined, (~b1 +qbz)reg solutions which are regular at the origin and (qbl-t-4)a)sing are solutions which are singular at the origin. Substituting these expressions into our differential equation leads to the solutions

P~ /

P~

where y = [ ( / ¢ ) 2 - ( Z ~ ) 2 ] ½ , and ~/ is determined so that hypergeometric function F(a, b; x ) is defined by F ( a , b; x) = 1 + -a x + a ( a + l ) x z

b

b ( b + l ) 2[

+

The confluent

~b2 = ~ .

a(a+l)(a+2) b(b+l)(b+2)

xa 3[

+

....

(40)

123

CONTINUOUS GAMMA RADIATION

The defining equation for t/is

(to- iZ~/P~) (7 + iZc~W~/P~)

e 2it/

(4t)

Therefore (\7+iZc~W~e'~(2P~r)~e-~P°'F(l+7+ /

i P~ , l + 2y; 2iP~r) (42)

_+ complex conjugate. The singular solutions are (\-7+

(¢1 +¢2)s~o~

i ZaWe]/ ei~'(2P,r)-~ e-iP°~F (1-1 ~+ i . p ~, 1-2?; 2iP~r) __ complex conjugate,

(43)

with

eZiq, = _

to- iZa/P~ .

(44)

- r + iz~ wJPo

We now investigate the asymptotic forms of our solutions. To do this we use the asymptotic behaviour of the confluent hypergeometric functions. As r --* oo

F(a, b; x) --* r(b)

r(b-a)

e~i~ax_.+ r(b) eXx._b,

(45)

r(a)

where =

1 if

0 < arg x < re,

e = - 1 if -re < a r g x < 0, and F(y) is the gamma function of y. By investigating the asymptotic form of our solutions and insisting that they be outgoing waves which reduce to the asymptotic behaviour of free particle solutions Jn the limit Za -* 0, we completely determine the constants A, B, C and D. (ii) "Let us now consider the solutions for r < r 0. The solutions will be expressed as series expansions in powers of (r/ro). For the substance 35S, it is sufficient to retain two terms in the series expansion; this yields an error in the wave functions of less than one percent. The solutions are given by the following equations for tc = 1 : r

2

r

2

(~o)+...1

-

+ (~o) ld° I c°+q \ro/(r)2+""l '

ul=

(~o)ao Ii+at

u2=

(~o)ao I b°+bl \ro/(r)Z+'" "1 +do

(46)

r

[ l + d a \r0/(r)2+'" "1"

(47)

124

R.

E. STRUZYNSKY

AND

F. POLLOCK

For the states corresponding to K = - 1 , Ul =

(r) Iao bo+bl,r,2+ {]\ro/ /,

2

r

u2= (~o)ao Ii-al

" " "

2

1

+do

I

1+dr

( r~)Z ~0

+ .... 1

+ \ro/(r)1doICo+Cl( r f ~ ) 2 + . . . l .

(48)

-

(~o)+-..1

(49,

In each case, the constants at, Co, et, bo, bl and d~ are specified by appropriate recurrence formulae. The constants ao and do are determined by matching solutions at r = ro. To construct a Green function we now need an appropriately normalized standing wave solution.

Step 2. Construction of ~ ( 0 ) For r < r o we have the following solutions: For s: = 1, we have

ul= (~o)2aoIl+al(~o)z+'" U 2

1 '

(50)

=

For ~ = - 1 , we have

1

U t

"2 -- (~o)2a° El+al (~12+ ,roJ " "1 "

(53),

The constants al, bo and b t are specified by appropriate recurrence formulae., The normalization constant ao for the standing wave solutions is to be determined from eq. (31). Substituting our solutions into eq. (31) imposing the restrictions that Re [f* (0)fl (0) ] and Re[g*a(0)g-l(0)] be finite leads to a complete specification of ao.

out

out

4. Final State Wave Function

We shall, for the final state of the negaton, use the form of the continuum-state relativistic Coulomb wave function developed by Johnson and Deck 14). This wave function expresses the spin dependence in terms of the Dirac plane wave spinors which simplifies calculations by making it possible to take traces in the usual manner.

CONTINUOUS GAMMA RADIATION

125

The adjoint of the function ~,(x) is written as ~ ( x ) = ~[N+iZ~zM'o~. ( p + ? ) + L ~ . p~- (p+P)],

(54)

where oo

N = 2Z

F(a) eV½nxY_le_½XF(a, fl; x)Ep~_ 1 ~_pik]'

(55)

~=1 r(fi)

ao M = Z F(a) e,~xe_le_½,F(a, fl; x)[p~_ l _ P ; ] , ~=, r ( ~ )

(56)

t = ~ r(~_)) e~½~x~_l e- ½~{[(y- k)F(~, ~; x)- 2yF(~, ~ - 1" x)]P~_, ~=1 r ( ~ )

-[(y+k)F(c~, fl; x ) - 2 y F ( G f l - 1 ; x)]P~},

ek=Pk(p" ~).

(57)

(58)

In the above equations, x = -2ipr,

y = (k2-zZc~2)~;

c~ = y - i v ,

fl = 27+1,

k = ]K] = j+½,

v = Z- -a-E P

(59)

The primes denote differentiations with respect to the arguments of the Legendre polynomials Pk, F(a, b; x) are confluent hypergeometric functions and u the four component Dirac plane wave spinor. For large r, ~ has the asymptotic character associated with a final negaton state, namely, a plane wave plus incoming spherical waves. Considered in terms of an expansion in the Coulomb parameter (Ze), the coefficients N and M are of order unity while the coefficient L is of order (Za) 2. Thus, the three terms on the right side of eq. (54) are, respectively, of order unity, (ZcQ and

(Z.) 2.

Johnson and Deck have shown that the Sommerfeld-Maue wave function follows directly from ~(x) as the result of approximating y by k in the latter function. This approximation is precisely stated by the equation

7 = (k2-Z~2) ~ = k 1-

Z2~2~

-Z!

~ k.

(60)

It has been shown in this approximation, the coefficient L vanishes and that the first two terms in eq. (54) reduce to the Sommerfeld-Maue wave function.

126

R.E.

S T R U Z Y ' N S K I A N D F. P O L L O C K

5. Computation of Photon Spectrum The transition probability per unit time for the process is given by

(dBk'dapdaqb(Eo-E,-E-Er)

N = (2~) -s

~

IMI 2,

(61)

~kp, ae~ ~ w ~1

where k' is the momentum of the photon, P the momentum of the final negaton, q the momentum of the anti-neutrino, E 0 the available energy for beta decay, Ev the energy of the anti-neutrino (E~ = q), E the energy of the final negaton and Ee the energy of the photon (E~ = k'). The summation covers the spin states of the negaton and antineutrino, the polarizations of the photon and the initial and final nuclear states. Designating the summation over the spin states by S, we may write,

N = (2~z)-Sfk'Zdk'(q2dq(pZdpfdg2k, Sk, fdf2qS~6(Eo-E~-E-E~)~,, ]M[Z. (62) ,

Since

1

,

)

,

3

,

)

.

3

n

PdP = EdE, we may perform the energy integration and get N=(2n)-sfk'2dk'fq2dqfdOk,

sk, fdf2qS~fdOoSePE~lMi 2,

(63)

with P = ( E 2 - 1 ) ~, We now define the photon spectrum

E =

E o - E , - E ~.

S(k') by the relation

N = fS(u)du.

(64)

We then have

S(k')=(2n)-sk'afq2dqfdOk, sk, fdQ~SqfdOoSoPE~lM[ z.

(65)

With the definition We = E o - E ~ = E+E~ and Pc = (W~2 - 1 ) ~, the integration over the momentum of the anti-neutrino c a n be transformed into one over We

We = E o - E ~ = Eo--q, dWo = -dq,

(66)

therefore, ~o-l-E~dq =

~, +1 f~o JEo dW~ = ~+t dW,.

(67)

,)0

We then have

S(k') -- (2TO-Sk'2 with

f

f

f

qa dWePE df2k,S k, df2qXq d~2eSe~,]Ml 2, k' + 1

,J

q = Eo-Wo.

d

,d

n

(68)

127

CONTINUOUS GAMMA RADIATION

The matrix element M is given by eq. (14). Considering the nuclear contributions to the matrix element, we have (69)

M(Op) = [PT~½(l + ?s)n] -= M(7~) + M(?~?5). We recall that our representation for 7~ is 7, = (-iflO~k, fl). In the non-relativistic approximation for the nucleons,

M(V~) > M(v0, M(v~v~) > M ( ~ ) .

(70)

M(1) = M(74), M~(~) = M ( ~ , ) .

(71)

We define

We now sum the square of the matrix element over the spin states of the antineutrino, integrate over the directions of the anti-neutrino momentum and average over all possible nuclear states for unoriented nuclei. From our analysis of the Green function, we know that

/ ° }

ff~(0) =

i(a°--b°)lff_l

'

~._1}

F0

f(aobo)-~ ~"-1(o) = /

roo

(72)

"

Utilizing eq. (72) we find that

S(k') - (eg)2k' 16(2~) s X

ff %

×

f

f

f[

b°)~=l q2dW~PE dg2k,S k, dg2eSe

'+1

t "2

d3"'l-~(/")e-~"'~*% E

" , [t~(r)]o.t[#q(r

d3r

,)]outYue r . e ,k,. Wile(~)]

#

(eg)2k' + 16(2rc)----q

x

(a°b°)~=-aq2dWePE '+1

×

fo

/"O 2

d3/[~(,.)e-'~"-~*%2 it

f dY2k,S k,f dY2~S~

d3/"

, /")30~,[¢,-~( , /",)3o.,~,~~,e,~'. l*'j~e(/"t)] [,/,-~(

.

(73)

128

I t . E. S T R U Z Y N S K I

AND

F, P O L L O C K

Upon examining expression (73), we notice that the photon spectrum S(k') reduces to the sum of two terms. In the first term, the expression which is integrated over the spatial coordinates r and r' represents the probability that a negaton, initially in the orbital angular momentum state l = 1, radiates a photon of momentum k'. In the second term, the expression which is integrated over the spatial coordinates r and r' represents the probability that a negaton, initially in the orbital angular momentum state l = 0, radiates a photon of momentum U. The factors multiplying these expressions represent the probabilities of beta decay, appropriately weighted to yield the proper contributions of the l = 0 and l = 1 states to the photon spectrum. Thus, we have shown that the probability for the entire process may be expressed as the sum of two terms, each term being the product of the probability of a beta decay, with a beta particle in a particular angular momentum state multiplied by the probability that this beta negaton would radiate a photon of momentum k'. Since the expression (73) for the photon spectrum was derived directly from the S matrix for the process, we have proven the equivalence of the two methods of calculation of Knipp and Uhlenbeck. This proof depends on the fact that the wave functions, eq. (72), of the negaton, when evaluated at the origin, reduce from four component spinors to two component spinors. Thus, products such as ~ ( 0 ) ~ ( 0 ) and i'~ - ~(0)0-1 (0) are the only nonzero expressions contributing to eq. (73). Since the proof of the equivalence of the two methods of calculation of Knipp and Uhlenbeck depends on the evaluation of the negaton wave function at the origin, it is only exactly true for allowed beta decays. We now substitute into the matrix element the expression for the final wave function of the negaton. In as much as the term containing L is of the order (Za) z, we immediately drop terms in the square of the matrix element containing powers of L. After substituting the final state wave function, one may now evaluate the appropriate traces of the matrix element. In order to evaluate these traces, the Green function may be expanded in terms of the Dirac matrices. We shall choose the gauge eo = 0. The direction of polarization of the light wave is given by the unit vector perpendicular to k'. We select k' to be along the z-axis. Thus Sk'?u eu -- 7 " 5+ + 7 " e - , (74) Sk'7.e *~ = 7" ~ - + 7 " ~+, where ~+

-

[11x q- i~y]

(75)

,/2

With k' being chosen along the z-axis, we may also write

e-~k,., __ ~ (_ i)L[4u(2L+ 1)]-~jL(k,r) L=0

y~,(?).

(76)

CONTINUOUS

129

GAMMA RADIATION

A discussion of the evaluation of the angular integrals appears in appendix 1. The radial integrations were performed numerically on the IBM 1620 and Univac 1106 computers. A discussion of the radial integrations appears in appendix 2. As in the calculation of Mau, only photon states corresponding to L = 0, l were retained, since the radial integrals were rapidly decreasing functions of L. Thus, due to conservation of angular momentum, the final negaton must be characterized by the states corresponding to tc = __+1, ___2for L = 0 and ~c = __ 1, +2, + 3 for L = 1. The contribution of the terms corresponding to the state L = 1, lc = _ 3 is small, of the order of 0.5 percent, and hence was neglected in the calculation. To find the number of photons emitted with energy k' per beta decay we must divide by the probability of a beta decay. Thus we compute R(k') -

S(U),

(77)

N, where N, is the probability of a beta decay corrected for Coulomb corrections by the Fermi Coulomb function.

6. Knipp and Uhlenbeck Approximation In order to give confidence in our result, at least to this point, we shall examine our formulae in the limit Z~ ~ 0. An approach to the Knipp and Uhlenbeck result will indicate that the chance of error in our analytical development is minimal. In the limit Z~ ~ 0, the only non-zero term in the final state wave function is given by ~(x) = ~N. (78) In the approximation we are using N = e-lp'L

(79)

Therefore, in the limit Z~ ~ 0,

=

",

(80)

which represents a plane wave solution to the Dirac equation. Investigating the behaviour of the Green function in the limit of Za approaching zero, we find

(

hl(P~r))~ iw +l

log--l(r)]oat

=

I P ~ ( ~ + 1)1+ (

--i

h°(P~r)z~-l} P~ hl(P~r)z~" Wo+I

"

(82)

130

R.E.

S T R U Z Y N S K I A N D 1;'. P O L L O C K

The constants ao and bo when evaluated for Za = 0 yield the following relationship s: [(ao bo)~. 1]z - 0 - Pe(W~- 1) r°2 '

(83)

7~

[(aobo)2=_l]z=o _ P~(We+ 1) r°2"

(84)

7~

Substituting the above formulae into our expression for the photon spectrum and performing the radial integrations, we obtain

S(k') = ,11+I~°k,P(We)~(We, k')dW¢,

(85)

where the probability of beta decay P(W¢) is given by

P(We) = 292 P~ Weq2(M(l)ut(1)+ M(a)" Mr(a)),

(86)

1 dp,=a(W~, k') -- - - -pl~. - ! san 0 F (w)+e2-2~) - 1 1 dO, 2nk' P¢3o [_(W¢-I)(E-P cos O) ( E - P cos O) 2 J (87) ¢l=o(W¢,k')=

a

_e _l " san6} r

2nk' Pe30

(w: + E ~-

[.(w-f+~~os

2~) 6))

-

1 ( E - P cos 6))2

-1

1 dO. (88)

In the above expressions 6) is the angle between P and the z-azis. The above formulae are exactly the expressions obtained by Knipp and Uhlenbeck.

7. Results and Discussion

The photon spectrum per beta decay per mc 2 was computed for various photon energies for the substance 3sS. Table 1 lists these results in addition to various other theoretical and experimental results. Our calculation agrees with the calculation of Vinh-Mau but disagrees with the more recent calculation of Felsner. However, Felsner predicts an increase in the, bremsstrahlung yield of approximately fifty percent for low proton energy, where Coulomb effects are not expected to be significant. This result is in disagreement with all the previous theoretical predictions and with the experimental results of Starfelt and Svantesson. The results of Felsner, however, seem to agree with the experimental results of Langevin-Joliot for low photon energies. Thus, there seems to exist a discrepancy among the experimental results.

CONTINUOUS GAMMA RADIATION

131

Recently, an experiment was performed by Persson. Unfortunately, no experimental results were obtained for asS but data were obtained for other substances such as S2p which enabled comparison with other experimental results. The results of Persson disagree with the theoretical calculations of Felsner and those experiments which recorded an increase in bremsstrahlung yield. The following remarks, extracted from Persson's paper, comment on the existing discrepancies between theory and experiment. TABLE 1 Theoretical a n d experimental results for R ( k ' ) k" = 0.05

k ' = 0.20

k ' = 0.25

K n i p p , U h l e n b e c k a n d B l o c h a)

3 2 6 x 10 -5

4.1 x 10 -5

0 . 5 6 x 10 -b

L e w i s a n d F o r d a)

3 2 4 X 10 -5

5 . 2 x 10 -5

0 . 8 2 x 10 -5

V i n h - M a u a)

3 2 4 x 10 -5

6.5 x 10 -5

1 . 1 2 x 10 -5

F e l s n e r a)

600 x 10 -5

8.8 x 10 -5

1.3 x 10 -5

G o l d a n d S p r u c h a)

1 . 0 6 x 1 0 -5

S t r u z y n s k i a n d P o l l o c k a)

326 X 10 -n

6.4 x 10 -5

1.14 x 10 -5

S t a r f e l t a n d S v a n t e s s o n b)

325 x 10 -5

8.1 x 10 - s

1 . 2 0 × 10 -~

L a n g e v i n - J o l i o t b)

516 × 10 -5

12.2 x 10 -5

~) T h e o r e t i c a l results,

b) E x p e r i m e n t a l results.

"It is difficult to understand the cause for the different results found in some other experiments in the literature. Disturbances present in these determinations of the total bremsstrahlung spectrum, however, may possibly not show up in the present coincidence experiment• It should be remarked that most of the disturbances encountered in an experimental set-up give an increased apparent yield of internal bremsstrahhing, due to e.g. external bremsstrahhing production, penetration of collimator walls, and to scattering effects. Therefore it does not seem unlikely that in some of the experiments, which show an excess over the theory, such disturbances might have been present. It is astonishing that the Coulomb correction according to Felsner increases the bremsstrahlung yield by approximately 50 per cent independent of photon energy. this is in disagreement with a number of absolute measurements for small photon energies, which are in reasonably good agreement with the Knipp and Uhlenbeck theory." The discrepancies between the present calculation and the experiments of Starfelt and Svantesson, which occur for high photon energies may be accounted for by the fact that screening effects have been neglected. A calculation which takes into account the screening of the residual nucleus by the orbital negatons would therefore be of value in determining whether screening would lessen this discrepancy. An extension of the present calculation for first-order forbidden beta decays of such elements as 89S, 91y and 9 0 y would also be of value. Because of the fairly high Z for these elements' Coulomb effects are expected to be significant. •

.

.

132

a . E. STRUZYNSKI AND F. POLLOCK

Appendix 1 ANGULAR INTEGRATIONS

The angular integrations consist of integrals of products of two, three and four spherical harmonics. These may be evaluated in the following manner 16). The spherical harmonic functions are orthonormal, hence f

ml*

m2* A

The spherical harmonic functions also obey the following coupling rule:

r

V(> L 47r(21+ 1)

A <,,,

,; x

C(ll, 12, l; O, O)ylm'+m2(F),

(90)

where the C are Clebsch-Gordan coefficients. The 11, 12 and l are angular momenta and ml and m2 the z-components of l t and 12. The summation over l extends over the range l = Ill-lzl l = I t + l 2. This expansion permits an easy evaluation of the integral containing three spherical harmonics. Multiplication of eq. (90) by Yt~'3*(f) and integration over the full solid angle gives

to

f

[(21' + 1)(212+ 1)~~ × C(ll,

12, l; 0, 0)j'dOr~?3*(~)~'+'2(~).

(91)

Utilizing the orthonormality condition (89), we get f

dO~YI:~*(#)Y~7~(~)Yt]"(~)-- ~(21, + 1)(212 + 1)~"J~ C(1,, L ~-~(2~a+i)

lz ,

13; m,, m2) x C(ll,

12,13; 0, 0).

(92)

We may now also evaluate integrals of four spherical harmonics m3 A ~: ml fdf2,[g/,m4 (~ )Y/~ (r)] [Y~m2 (P)Y~, (~)] -- [~[(214+1)(213+1-)] ~C(l,, L -4~(2~

x

]

13, I; rrl,, m3)C(13,13, I; O, O)

[~F(21z+l)(2/1-+l)-~ ¢ l';mz,ml)C(Ia,l,,l';O,O)l L 4rc(2/'+1) A C(Iz'll'

x f da,[Y~m'*m~(P)]*[Yff~+m']. d

(93 )

CONTINUOUS GAMMA RADIATION

133

Using eq. (89) we get

fd~,[~. (r)Y;~(~)] m4 A

m3

[g;~m2( p ) ~(~ml ;1 (r)]

[(2/, + 1)(213 + 1)7 ~ [(21z + 1)(2/1 + 1)] } L -J L

= X

*

m,,

C(I2, I,, l; m2, ma)C(la, la, I; O, 0)C(/2, 1,, l; O, O)am4+m3,

m2+ml

"

(94)

Appendix 2 EVALUATION OF R A D I A L INTEGRALS

Erdelyi ~7) developed the following equation, relating integrals of products of Whittaker functions M.,b(x) to hypergeometric series developed by Lauricella 18):

fo :e-s'rn-1 I-~ Mk,, mj-~(Ij r)dr = ~-[ (lj)ms(s + ½L)-M-"F(M+n) J

J

x r a ( M + n ; m i - k j ; 2mj; I/(S+½L)),

(95)

where

L = ~, lj,

M = Z m j,

J

J

(96)

Fa(a; b i; ej; ~j)

= Z (a)ml+m2+ma(bl)ma(b2)m2(b3)ma (~1)ml (~2)m2 (~3)m3 ~ (e~)ml(C2),.~(~3)m3 ml! m2! m3!

(97)

with (a). - r(a + n) _ a(a + 1 ) . . . (a + n - a). r(a)

(98)

Utilizing the relationships jL(k' r) =

1

(re)~ "

1

F(L+ 1) 22L+* (i)L(2ik'r)

Mk,re(x) =

X~ %

Mo, L+,(2ik'r),

~ F(½ + m - - k, 1 + 2 m ; x),

(99)

(100)

we may express our radial integrals in the form of eq. (95). The hypergeometric series developed by Lauricella obeys the following transformations:

R. E. STRUZYNSKI AND F. POLLOCK

134

f o r d = 3,

FA(a; bj; cj; ~j) =(l_~l)-.Fa(a;cl_bl,b2,b3;c.i;

h

=(l_~:)-.FA(a;bl,c2_b:,b3;cj;

h, 1-~2

=(l_~3)-aFA(a;bl,b2,c3_ba;cj;

7~,

1- 3

~2 ,

~3 )

~2 ~3 ) fiz-l' 1--~2

~2 ,

~3

1- 3

)

"

(101)

,

(lO2)

The Lauricella series is related to a Horn series by the equation 19)

H3, 2(albl , b2, b3lcldl , dell1, ~z, ~3) = r ( 1 - a ) ( F ( c ) F ( i ~ - b 3 ) $;b3FA a+b3lbj[di' 1+b3-c15~, -

+ r(b3)r(l-a-c) ~;cfA a+clbi' c[dl, d2, 1+c-b31~, where the Horn series is defined by

H3,2(a; bj; c; di; ~j) = z(a)'~+~'~-'~3(b~)m'(b2)m~(b3)"~(c)m~ ~7~ ~_~_~~ mj (dl)ml(d2)m2 ml! m2! ma[' w i t h i = 1,2and j = 1, 2, 3. The Horn series was evaluated on the Univac 1106 computer.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

J. K. Knipp and G. E. Uhlenbeck, Physica 3 (1936) 425 F. Bloch, Phys. Rev. 50 (1936) 272 C. S. Wu, Phys. Rev. 49 (1941) 481 T. B. Novey, Phys. Rev. 89 (1953) 672 P. Bolgiano, L Madansky and F. Rasetti, Phys. Rev. 89 (1953) 679 N. Starfelt and N. L. Svantesson, Phys. Rev. 97 (1955) 708 H. Langevin-Joliot, Ann. der Phys. 2 (1957) 16 R. R. Lewis and G. W. Ford, Phys. Rev. 107 (1957) 756 W. Gold and L. Spruch, Phys. Rev. 113 (1959) 1060 R. V. Mau, Ann. de Phys. 6 (1961) 1493 G. Felsner, Z. Phys. 174 (1963) 43 J. D. Jackson, S. B. Treiman and H. W. Wyl4, Z. Phys. 150 (1958) 640 B. Persson, Nuclear Physics 55 (1964) 49 W. R. Johnson and R. T. Deck, J. Math. Phys. 3 (1962) 319 A. Sommerfel4 and A. W. lVIaue, Ann. der Phys. 22 (1935) 629 M. E. Rose, Elementary theory of angular momentum (Wiley, New York, 1961) p. 57 A. Erdelyi, Nieuw Arch. voor Wisk. Amsterdam 20 (1939) 1 G. Lauricella, Bediconti Cireoto Math. Palerno 7 (1893) 11 A. Erdelyi, Proc. Royal Soc. Edinburgh AG-2 (1948) 378

(103)