Coulomb corrections in allowed beta transitions

Nuclear Physics 4 (1957) 206---212; ( ~ Not

North-Holland Publishing Co., Amsterdam

to be reproduced by photoprint or microfilm without written permission from the publisher

C O U L O M B C O R R E C T I O N S IN A L L O W E D BETA TRANSITIONS t J. D. J A C K S O N , t t S. B. T R E I M A N

a n d H. W . W Y L D

JR.

Palmer Physical Laboratory, Princeton University Princeton, New Jersey Received 8 April 1957 A b s t r a c t : Complete expressions, w i t h all C o u l o m b effects t a k e n into a c c o u n t , are g i v e n for t h e d i s t r i b u t i o n f u n c t i o n s for allowed b e t a decay, i n c l u d i n g recoil, n u c l e a r o r i e n t a t i o n , a n d electron p o l a r i z a t i o n effects. T h e e x p r e s s i o n s a r e general in t h a t no a s s u m p t i o n s are m a d e as to i n v a r i a n c e w i t h r e s p e c t to space inversion, c h a r g e c o n j u g a t i o n , or t i m e reversal. Simple a r g u m e n t s b a s e d o n t h e b e h a v i o u r of t h e v a r i o u s m o m e n t a a n d a n g u l a r m o m e n t a u n d e r s p a c e i n v e r s i o n a n d t i m e reversal are u s e d to e l u c i d a t e w h i c h c o m b i n a t i o n s of v e c t o r s will a p p e a r w h e n t h e different s y m m e t r y laws are violated, a n d t h e m o d i f i c a t i o n s of t h e s e a r g u m e n t s n e c e s s a r y b e c a u s e of t h e C o u l o m b i n t e r a c t i o n in t h e final s t a t e are p o i n t e d out. V a r i o u s t e s t s of t i m e reversal i n v a r i a n c e , p r e v i o u s l y s u g g e s t e d , are r e - e v a l u a t e d in view of t h e r e s u l t s for t h e C o u l o m b corrections.

1. I n t r o d u c t i o n In a recent report b y the present authors 1) general distribution functions, including recoil, nuclear orientation, and electron polarization effects, were given for allowed beta transitions. These calculations were quite general in that no assumptions as to invariance with respect to space inversion, charge conjugation, or time reversal invariance were made. However, Coulomb corrections were ignored in this work. We wish here to give complete results, including all Coulomb corrections, for allowed transitions ttt. 2. Rate f o r m u l a e

The rate formulae are similar in form to those given previously x). For completeness we repeat them here using the same notation as in ref. 1).. For the distribution in electron and neutrino directions and electron energy for an allowed transition from oriented nuclei we find t T h i s w o r k w a s s u p p o r t e d in p a r t b y t h e Office of N a v a l R e s e a r c h a n d t h e U.S. A t o m i c Energy Commission. t t Visiting G u g g e n h e i m Fellow, o n leave of a b s e n c e f r o m McGill U n i v e r s i t y , Montreal, Canada. t i t T h e c o m p l e t e l y general d i s t r i b u t i o n for a n e x p e r i m e n t i n c l u d i n g recoil, n u c l e a r orient a t i o n , a n d electron p o l a r i z a t i o n s i m u l t a n e o u s l y c o n t a i n s t e r m s in a d d i t i o n to t h o s e p r e s e n t in e q u a t i o n s (1), (2), a n d (3). See t h e following p a p e r b y Ebel a n d F e l d m a n . * T h e indices e a n d v refer t o electron a n d n e u t r i n o , respectively. T h e s y m b o l s E, p, d e n o t e energy, m o m e n t u m a n d a n g u l a r c o o r d i n a t e s of t h e leptons; m is t h e electron m a s s . T h e initial nucleus, of c h a r g e Z T 1, h a s a n a n g u l a r m o m e n t u m J ; ] is a u n i t v e c t o r in t h e direct i o n of J . 2O6

COULOMB

CORRECTIONS

oJ ( ( J ) lEe, Qe, D~)dEe d~ge d~2, -Pe'P,

m

+7" F(--FZ, Ee) is

IN

ALLOWED

F(±Z, E,)

TRANSITIONS

207

pe Ee (E ° - Ee) 2 dEe dDe dO,

(p,.j)(p,.J)lFJ(J+l)-3((d.j)2)]

Fpe'p,

I De

BETA

p.

PeXpqt

A-~e +B-E-~, +D EeE, JI"

Here the usual Fermi function for negatons and positons respectively 3). The distribution in electron and neutrino direction and electron polarization for allowed beta decay from non-oriented nuclei is

eo(alEe , ~e , ,-Q~)dEedQed~,

--

F(:kZ, E~) (2~)~

Pe Ee (E ° - Ee) 2dEe dQe d~Q,

x ½~ {1-Fa Pe "P, + b Ee E, E~

I p~

P"

(~)

Pe (Pe" P.~

Pe X P']/

Ee-Fm \E-----~,]

EeE, ]l

The distribution in electron energy and angle and electron polarization for allowed beta decay from oriented nuclei is eo((J), alEe, Q e ) d E e d Q e -

.F(-FZ, Ee)peEe(EOEe)2dEed2,.2e

x ~ l + b N + T e e. A - T + G o +*"

7 - + e Ee+m ~ J " ~

(3)

+R-7- x ~

The coefficients a, b. . . . including all Coulomb corrections are listed in the appendix. In order to obtain results for the two component neutrino t t theory a, 4) one must set C s = --Cs, C v = --Cv . . . . in these formulae. As can be seen from eq. (2) or eq. (3) the longitudinal polarization of electrons emitted in ordinary beta decay is G Ve

P = - -

c m

(4)

l+b Z-= This effect has been found experimentally b y Frauenfelder et a/. 5) for negatons in the decay of Co S° and b y Page and Heinberg 6) for positons in the decay of Na 22.

208

J . D. JACKSONj S. B. T R E I M A N AND H. W. V~I*rLD J R .

3. D i s c u s s i o n

The expressions given above and in the appendix are perfectly general in that no assumptions concerning invariance with respect to space inversion, charge conjugation, or time reversal have been made. Invariance with respect to space inversion implies that the coupling constants C' vanish (or alternatively the.C vanish). Invariance with respect to charge conjugation implies t h a t the constants C are real and the constants C' pure imaginary, up to an overall phase. Invariance under time reversal would imply that all coupling constants C, C' are real, again up to an overall phase. A more physical statement of what is involved in these criteria can be given: The terms which violate space inversion invariance in eqs. (1), (2), and (3) are those which change sign under inversion (momentum vectors p change sign under inversion, spin vectors a, J do not). If Coulomb corrections are ignored, the terms which violate time reversal invariance are those which change sign under time reversal (all vectors, p, a, J change sign under time reversal). If p represents a linear momentum (either Pe or p,) and S represents an angular momentum (either J or a), the combinations of vectors which appear in the angular distributions are of the following types: Pl"P~, Sl" S2, S- p, S. (Pl × P~), Sl" (82 × P), as well as the bilinear forms (Sx • Pl)(Px " P~) and ($1 • P)($2 • P). In the absence of the Coulomb field, it is clear t h a t the first and second types violate neither space inversion (P) nor time reversal invariance (T). By the SchwingerLtiders-Pauli theorem 7), they therefore do not violate charge conjugation invariance (C). Consequently such terms can appear in the beta spectrum even if all symmetry properties are valid, as is well known from "old" beta decay theory. This can be explicitly verified by inspection of the Z = 0 part of their coefficients, a and N, in the appendix. The third type (S • p) violates space inversion invariance, but not time reversal invariance, and so will appear in the beta spectrum only if P and C are violated. Again this can be explicitly seen from the form of the coefficients A, B, G, and H. By similar arguments, it can be shown that terms of the form S • (Px x pz) will appear only if T and C are violated, while Sx" ($2 × P) occurs only if P and T are not conserved. The coefficients D, L and R, respectively, contain these requirements. The presence of the Coulomb field alters these arguments. The effect of interaction in the final state is to cause a breakdown of the arguments about time reversal based on the properties of the various vectors. The space inversion arguments with vectors are, of course, still valid. Inspection of the coefficients in the appendix shows that if a given combination of vectors behaved in a certain way under time reversal and charge con-

COULOMB CORRECTIONS I N A L L O W E D B E T A T R A N S I T I O N S

209

jugation, the Coulomb correction linear in ~Z will behave in the opposite way. Thus, for example, for the J • Pe term, with coefficient A, the Z = 0 part is present i~ P and C are not conserved, while the Coulomb correction is present only if P and T are violated. Similarly, for J - (Pe×P,), the Z = 0 part tests T and C, while the Coulomb correction is present even if P, C, and T are separately conserved. Before the discovery of parity violations in beta decay s) it was generally thought that the vector and axial vector coupling constants were small compared to the scalar and tensor coupling constants 9-13). However, the most precise experiment of this sort, that ot Sherr and Miller 13), shows only that the Gamow-Teller part of b is zero, and according to the appendix (eq. (A.5)) this proves only that Re (CTCA*~-C'TCtA *) = O. If [CA!, [C'Ai should turn out to be comparable to JCTJ, tC'TJ, while at the same time b = O, there would seem to be a clear indication of violation of time reversalinvariance, barring of course a fortuitous cancellation of terms in Re (CTCA*'~-CtTCtA*). Whether or not ]CA], ]C'A], ]Cvt, ]C'v] are really small compared to tCsl, ]C'sl, ICTI, [C'Tt can in principle be tested by further refinement of the standard electron neutrino angular correlation experiments 10-12). On the other hand if ]CAI, IC'AJ, ]CvI, [C'v] are negligibly small compared to ICsl, IC'sl, ICTJ, IC'TI the Coulomb corrections to A and G (eqs. (A.7) and (A.10)) will be small. This m a y make it difficult to detect possible violations of time reversal invariance by observation of the momentum dependence of A and G 14, is). Consider now the terms involving D, L, and R. It has been suggested 1) that a search for any one of these three terms in beta decay experiments would provide a test for the violation of time reversal invariance. It must be noted, however, that the Coulomb corrections to D, L, and R do not violate time reversal invariance. Thus if the test is to be definitive (in the absence of measurement of the energy dependence of the coefficient) the Coulomb corrections must be small. If the vector and axial vector coupling constants are small relative to the scalar and tensor coupling constalits, this is indeed the case for D, but not for L and R. Thus the recoil experiment on oriented nuclei as described by eq. (1) and discussed in ref. 1) is very likely a much more sensitive test for violation of time reversal invariance than the experiments described by eqs. (2) and (3).

Appendix In the following formulae the upper signs refer to negaton decay and the lower signs to positon decay. ]MFr2 is the conventional Fermi nuclear

210

J . D . JACKSON, S. B. T R E I M A N AN D H. W. W Y L D J R .

matrix element with selection rule AJ = 0, no nuclear parity change; and [MGT[2 is the Gamow-TeUer matrix element with selection rule A J = 0, 4-1, no nuclear parity change, J = 0--> J ' = 0 forbidden. J and J ' are the angular momenta of the original and final nuclei, ~],j is the Kronecker delta symbol and

1 1 ~j,j :

J--->J'=]-I

j+l

J--->]' = J

J

J--->J'=

f+l

1 (2J-l)

Aj, j =

j+l J(2J-1) ( ] + 1 ) ( 2 J + 3)

(A.1)

1+1

J--->J'

=J-1

J--->J' = J

(A.2)

]-->]'= j+l.

Z is the atomic number of the final nucleus, ~ is the fine structure constant, and y = (1--~2Z2) ½.

= IMFI=(ICsI=+ ICvl=+ IC'sl=+ IC'vl =) -4-IMGTI=([CTI=+ ICAt 2-4- [C'wl2+ [C',l =)

(A.3)

~Zm } a~ = IMFI2 [-ICs['~+]Cv]2-]C's]2-t-ICvl~]23 Pe 2 Im (CsCv*+C'sC'v*) M 2 ~Zm -~-~-{ [[CTI~-[CA[2"-[-[C'T]2-1C'AI2j "Jr- ~be 2Im (CTCA*+C'TC'A*)) (A.4)

b~ = :j=2yRe[IMg2(CsCv*+C'sC'v*)+IMGTJ2(CTCA*+C'TC'A*)]

(A.5)

c~ = IM~I~A m EIG~I~-- IC, I~+ IC'~I~-- IC',I~4- ~Zm z lm (C~ C** + C'T G'** )I P. (A.6)

A~= IMGT[*Ij,]I4-2 Re(CTC'T*--CAC'**)+~;7 2 Im (CTC'**-i-C'TCA*)1 +O],]MFMGTVT-~ I2 Re (Cs C'T*+C's CT*--CvC'**--C'vC **) o~Zm 2 Im ± p--~-

(CsC'A*+C'sCA*--CvC'T*--C'vCT*)1

(A.7)

cOULOMB

C O R R E C T I O N S IN A L L O W E D

211

B E T A TRA!~SITIONS

--bI,IMFMGTv ] _ [(CsC'T*+C'sCT*+CvC'A*+C'vC~,*) J+l ~:~m W, (c~C,A.+C,sCA.+CvC,r.+C,vCT*)] } __

j

(h.8)

i . , ic,

o~Zm ~ 2Re(CsCA *--Cv C~*+ C's C'A*--C'vC'T*)]

(A.9)

/,e

G~ = [MFI~ I +2 Re +IMG~:I~[5:~, Re

(CsC's*--CvC'v*)+ ~,Zm p----~2 Im (CsC'v*+C'sCv*)1

(CTC'T*--CAC'A*)+~Zm l~e- 2 Im (CTC,A*+C'.rC~*)I(A.10)

H~=2Re I lMFl'[--(CsC'v*+C'sCv*)~ ?m (CsC,s,+CvC,v,)l

+L~ [c~c,..+c'~c,,*~:~ (c~c'~*+c,,c',:)] }

(A.11)

(E~--~Re {iM,,l'l:Fc~c'¢:~CvC'¢+c~c',,*+C'~C,,*)(A.12)

K} = 2 \ ~ 1

+ I~

(!C~:C'~*±C,,C'A*--C~C',,*--C'~CA*))

L * = IMF[~ [2Im(CsCv*+C'sC'v*)~

~Zm -~ (}Cdz--iCvi*+ iC,sl~_ iC,vl~)l

[MGT[2I--2Im(CTCff+C'TC'A*)+~Zm (ICT[2+-W-

[CAlS.~_[C,Ti2_

IC,AI2)]

(A.aS)

N~= 2 Re ( }M~T}2~],:Il rm (Ic..rl"+ Ic.,,l"+ Ic'..r{"+ Ic',,,.I~) + (CTC*+C'TC',,.*)]

4:_~,,,MFMGT~IIf_~ [(CsC,,+CvCT,4_C,sC'h*-f-C'vC'T(A.14) *) + 7m ~ (c~ cT* + Cv c** + c'~ c'T* + c',, c',,*)] }

212

J.

D.

JACKSON,

S. B.

TREIMAN

AND

I-I. W .

WYLD

JR.

{[MGT[2~j,j[½(]CT]2~-[CA]2--{-[C'TI2--~[C'A[2)x~( C T C A * - ~ - C ' T C'A*)I

Q~= 2 ( ~ ) R e

I

--Sj,]MFMGTV J [(CsCA*+CvCT*+C'sC'A*+C'vC'T*)

(A.15)

(CscT* + Cv cA* + c's c'T* + c 'v c '** )] } pe i

+~,,IMFMGTV[--~[2 Im (CsC'A*+C'sCA*--CvC'T*--C'vCT*)(A.16) ¢¢Zm T --

Pe

2 Re (Cs C'w*+ C's Cw*--Cv C'A*-- C'v CA*)I-

The product MFMGT which appears in some of the formulae above is defined by M

(~g(J', M'), ~(J, M) ) ( W(J, M), a 3 ~(J', M') ) = ~MM'61I"x/J (j + 1~ MF MGT

(A.17) where bY(J, M) and ~ ( ] ' , M') are the initial and final nuclear wave functions. If the nuclear Hamiltonian is invariant with respect to time reversal, MFMGT is real. Furthermore IMFMGTI = ]Mp][MGTI. The sign of M F MGT must, however, be determined by explicit calculation of the left hand side of eq. (A.17).

References 1) J. D. Jackson, S. B. Treiman and H. W. Wyld Jr., Phys. Rev. 10b (1957) 517 2) See, e.g., J. M. B l a t t and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, New York, 1952) p. 682 3) T. D. Lee and C. N. Yang, Phys. Rev. 105 (1957) 1671 4) L. Landau, Nuclear Physics 3 (1957) 127 5) H. Frauenfelder, R. Bobone, E. von Goeler, N. Levine, H. R. Lewis, R. N. Peacock, A. Rossi and G. DePasquali, Phys. Rev. 10b (1957) 386 6) L. A. Page and M. Heinberg, Phys. Rev. (to be published) 7) G. Liiders, Mat. Fys. Medd. Dan. Vid. Selsk. 28 (1954) no. 5; W. Pauli, Niels Bohr and the development of physics (Pergamon Press, London, 1955) (See also ref. 14)). 8) C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Phys. Rev. 105 (1957) 1413 9) E. J. Konopinski, Beta- and G a m m a - R a y Spectroscopy, ed. K. Siegbahn (North Holland, Amsterdam, 1955) p. 311 10) J. M. Robson, Phys. Rev. 100 (1955) 933 11) B. M. Rustad and S. L. Ruby, Phys. Rev. 97 (1955) 991 12) W. P. Alford and D. R. Hamilton, Phys. Rev. 105 (1957) 673 13) R. Sherr and ~ . H. Miller, Phys. Rev. 93 (1954) 1076 14) T. D. Lee, R. Oehme and C. N. Yang, Phys. Rev. 106 (1957) 340 15) R. 13. Curtis and R. R. Lewis, Phys. Rev. (to be published}