Cascade process in muon capture and gamma-neutrino angular correlations

ANNALS

OF PHYSICS

92, 2543 (1975)

Cascade Process in Muon Gamma-Neutrino Angular V. Department

of Nuclear International

Physics, Centre

Capture and Correlations

DEVANATHAN

University of Madras, and for Theoretical Physics,

Madras

600 025, India

Trieste,

Italy

Madras

600 025, India

AND

P. R. Department

of Nuclear

Physics, Received

SUBRAMANIAN University

of Madras,

December

20, 1974

A nuclear cascade process resulting from muon capture and subsequent deexcitation by gamma emission is considered. Possible observable quantities such as the final nuclear polarization and the gamma-neutrino angular correlation coefbcients are investigated. It is shown that the nuclear transition with the spin sequence 0 + JI -+ 0 is very favorable for study since the angular correlation coefficients in this case are not affected by the uncertainties in the gamma decay matrix elements. For a more restricted transition with JI = 1, there exist some simple relations connecting the asymmetry coefficient of the recoil nucleus emitting the gamma ray and the gamma-neutrino angular correlation coefficients. These relations are shown to be independent of both the nuclear structure and the muon capture coupling constants. Numerical results are presented for the nuclear transition 160(0+, p.s.) “5 l”N(l-, 392 KeV) z r6N(O-, 120 KeV).

I.

INTRODUCTION

Recently the polarization of the final nuclear state 12B(1+) in the reaction pL- + W(O+) - v, + 12B(l+) has been investigated both theoretically [l] and experimentally [2]. This reaction is comparatively simple for study since the contribution resulting in the nuclear transition to the ground state of 12B(1’) is dominant [3, 4] when compared to the other transitions to the low-lying bound states of 12B, stable against nucleon emission. In the case of other reactions, not so favourable, wherein the transition probability to the low-lying excited states, stable against nucleon emission, is comparable to the ground state transition rate, the process 25 Copyright All rights

0 1975 by Academic Press, Inc. of reproduction in any form reserved.

26

DEVANATHAN

AND

SUBRAMANIAN

must be viewed as a two-step cascade transition, first to an excited state by muon capture and then its subsequent deexcitation to the ground state by gamma emission. The purpose of this article is to consider such a cascade process and investigate the possible observable quantities, viz., the final nuclear polarization and the gamma-neutrino angular correlation coefficients. For obtaining the polarization of the final nuclear state at the end of this twostep cascade, an integration over the neutrino and the gamma-ray directions have to be made. This imposes the condition that the final nucleus, as in the case of a single step muon capture process [l], can have only a vector polarization for the spin sequence 0 -+ JI ---f Jf with Jf b 1. An expression for the final nuclear polarization has been derived but it will be very difficult to have this polarization retained for experimental measurements [2]. The other observables in the two-step cascade process in muon capture are the gamma-neutrino angular correlations. In a recent experiment Miller et al. [5] have investigated the Y-V angular correlations by observing the energy distribution of the doppler-broadened y-ray using a method suggested by Grenacs et al. [6]. The doppler shift of the y-ray energy due to nuclear recoil from neutrino emission is given by E = E,(l - u,, cos &J, (1) where E is the doppler-shifted y-ray energy, E, , the y-ray energy in the rest frame of the emitting nucleus, u0 , the nuclear recoil velocity and 19~”is the angle between the gamma and neutrino momenta. Theoretical investigations of these correlations have been made earlier by Oziewicz and Pikulski [7], Popov [8], and Bukhvostov and Popov [9, lo]. In this article it is shown that if the angular integrations over the neutrino and the gamma-ray directions in the cascade process are not performed, the y-ray yield can be expressed in terms of certain observable correlations involving the three vectors: the gamma ray momentum k, the neutrino momentum Y, and the muon polarization P. It is advantageous to study the cascade with the spin sequence 0 -+ J, -+ 0 since in this case the correlation coefficients are independent of the gamma decay mechanism. For the particular case of J, = 1, two interesting relations independent of nuclear structure and muon capture coupling constants are obtained between the asymmetry coefficient of the recoil nucleus emitting the gamma ray and the y-v angular correlations. These relations are true even if the second-class currents exist. Hence measurements of the correlation coefficients give us a handle to find the nuclear recoil asymmetry and present an alternative to other efforts in the field [2, 111. In Section II, a general method of obtaining the nuclear transition probability and spin orientation in a cascade involving a two-step process is given. Section III deals with the interaction Hamiltonians responsible for muon capture and the

CASCADE PROCESS IN MUON CAPTURE

27

subsequent emission of a gamma ray. In Section IV, an expression for the spin orientation of the final nuclear state in a cascade process in muon capture is derived. When an integration over the recoil direction and the photon direction is made, the final nucleus is found to have only a vector polarization. The evamation of the reduced nuclear matrix elements is given in Section V. The theory of y-v angular correlations is given in Section VI. In Section VII we obtain some simple relations connecting the asymmetry parameter cy of the recoil nucleus resulting from muon capture and the y-v angular correlation coefficients. Numerical results are presented in Section VIII for the reaction 160(0+, g.s.) 2

16N(1-, 392 KeV) -L

16N(O-, 120 KeV).

The p- capture rate for this process is independent of g,/g, but dependent on gA/gv . Hence measurement of the y-v angular correlation coefficients will enable one to calculate g,/g, independent of g,/g, . The correlation coefficients are sensitive to g,.,/gv and nuclear structure. Both the single particle j-j coupling shell model and the Gillet and Vinh Mau [12] wave functions are used. Momentumdependent terms in the muon capture interaction have to be included since their inclusion drastically changes the values of the correlation coefficients. II. NUCLEAR

TRANSITION PROBABILITY AND SPIN ORIENTATION IN A CASCADE PROCESS

Consider a cascade process resulting in a nuclear transition from an initial spin state Ji to one of the possible intermediate states JI by muon capture and then to the final state with spin J, by gamme deexcitation. Assuming the initial nucleus to be unpolarized, the density matrix pr for the final state of the nucleus is defined such that its element is given by

where Ha denotes the muon capture interaction Hamiltonian and Hb the interaction Hamiltonian for the gamma emission. In the study of y-v angular correlations wherein the intermediate nuclear state is known, J, = J,‘. Although we are considering a specific cascade process, the method outlined below is very general and is applicable to any nuclear cascade involving a two-step process and is also easily amenable to extension to a many-step process. The interaction energy is an invariant and hence the interaction Hamiltonians Ha and H,, must have the structure

28

DEVANATHAN

AND

SUBRAMANIAN

Ha =

c (AyA}* U?(n), n,A,mA

(3)

Hb =

c {By}* %Y*w$

(4)

P’;“(n).

In the above expansions, II denotes the nucleon index and Ura, VP refer to spherical tensor operators in the nucleon coordinates responsible for nuclear transition. The spherical tensor components ADA and B YY refer to the radiated field and, in our particular case, they are spherical harmonics with arguments denoting, respectively, the neutrino and the gamma directions. The tensor moments (2”;~) of the spin orientation of the final nuclear state is given by [l] CT?)

= Tr(T,“Kp,YTr

pf ,

(5)

where

x (JIM,,

1c Us’ n

1J,M,)*

{A?}*

AT’{By”V}* By?‘.

(6)

The symbol CS denotes a summation over the 15 variables y, y’, h, h’, .7, , Ji, m ’ mh, m,‘, MI, M,‘, A4, , M,’ and Mi . To obtain Eq. (6), the expansions (3; a2 (4) for the interaction Hamiltonians Ha and Hb are used. Applying the Wigner-Eckart theorem1 to separate the reduced matrix elements, and after several regroupings of the Clebsch-Gordan coefficients, we perform the summation over magnetic quantum numbers to obtain the following result: T@‘,n”Kp,) = &

z

; [(A,* x A,,), x @,,* x B,,J rl;

x (- 1)JrJi (- l)v+n-2 +& U(JIXr,,X’; x U(JFJ4;

X

U(AyKy’; 1 JiA) U(JfyJI,A; Jig) JrK)
X
6pr>

It C u,(n) IIJO

X * *, n

12

(7)

1 For angular momentum coefficientsand reduced matrix elements, we follow the notation and definition of M. E. Rose, “Elementary Theory of Angular Momentum,” Wiley, New York, 1957.

29

CASCADE PROCESS IN MUON CAPTURE

where & stands for a summation over the nine variables JI , J,‘, y, y’, X, h’, fl, L?, and r. The quantity [K] is defined by [K] = (2K + l)liz,

(8)

and the coefficient U(abcd; ef) is related to the Racach coefficient W(abcd; qf). U(abcd; ef) = [e][,f] W(abcd; ef).

(9)

Equation (7) is very general and can be applied to obtain the nuclear transition probability and the spin orientation in any cascade involving a two-step process. By substituting K = 0 and mK = 0, one can obtain Tr pr which is a measure of the cross-section for this cascade process. Putting y = y’ = 0, we obtain the result for the single-step process W”,“Kpf)

=

* x A,,),mK(-l)J’-Ji

& z

&

&

U(J,XJ,h’; J,K)

(A,

x (Jf II 1 u,(n) II JXJf

n

II1 U,,(n) IIJi>*. n

(10)

This is identical with Eq. (9) of [l] when the tensor quantities ADA are included in the definition of the interaction Hamiltonian.

III.

THE INTERACTION

HAMILTONIANS

The interaction Hamiltonian for the muon capture process involves both lepton and nucleon operators. Summing and averaging over lepton spins, we obtain certain bilinear combinations of nuclear matrix elements involving the following single-nucleon transition operators [ 1, 13-161: t, = exp( --iO * r) y(r) 0, ,

(11)

with 0, (I = 1, 2, 3, 4) denoting 01 = 1,

0, = Q,

OS = P,

0, = CI . p.

o is the Pauli operator for the nucleon, and p(= -iV) is the momentum for the proton. I represents the muon wave function in 1s orbit.

(12)

operator

2 Throughout this paper, the system of units in which fi = c = muon mass = 1 is followed unless otherwise explicitly mentioned. Also the notations used in [l, 16, 20, 211 are generally followed.

30

DEVANATHAN

AND

SUBRAMANIAN

Using the Rayleigh expansion for exp(-i9 the expression (3):

t3=

c

t, =

-

{Y,(8) x V}?,

ial{Yz(9}*

C - iut(lA,rn

* r), we can rewrite t, in the form of

l)“-’ #

{ Yzm(C)}* {(Y,(fi) x V)n x o}?,

(16)

with aI = 47r(i)-lj,(vr)

y(r).

(17)

The vector spherical harmonic Yz is defined by Yyy = c C(ZlX; mqm,) ytmg*. ?n

(18)

P is a unit vector in the spherical basis. In Eqs. (13)-( 18) the nucleon index is suppressed but it is understood that a sum over all the nucleons is to be made to obtain the nuclear operators. Using the one-body operators 13-(16), the appropriate bilinear combination of nuclear muon capture matrix elements that occur in Eq. (6) can be evaluated using standard techniques. Following Eisenberg and Greiner [17], the interaction Hamiltonian for the gamma emission process can be written as

M4

= rl24)y

(&/ y l)! !

I

(2y + l)(Y + 1) 1’2 I . b

E is the index which denotes either electric (e) or magnetic (m) transitions, and n is the nucleon index. p represents the circular polarization of the gamma ray and 77 is given by rl=

1, I /*,

E = e. E = m.

CASCADE

PROCESS

IN

MUON

31

CAPTURE

The rotation matrix Dy,,,,(I& 8,0) rotates the coordinate system from a frame of reference in which the gamma direction has polar and azimuthal angles 8 and 4, respectively, to a frame of reference in which the z-axis is along the direction of the gamma ray. The nucleon operators corresponding to the electric and the magnetic transitions are !2y(r,

e) = e,rYYTy(i),

where e, is the effective charge of the nucleon, L is the orbital angular momentum operator, X is the nucleon magnetic moment (.X = 2.79 for proton and .%? = - 1.91 for neutron) and M is the nucleon mass. In Eqs. (21) and (22) the nucleon index is suppressed. The single-nucleon matrix elements involving the transition operators Q>(r, e) and sZymy(r, m) can be evaluated as outlined in [18] and the results are given below.

(23)

.c (rYel\

e, (Ji(Ji + 1)Y2 (- ljzd-~d-zf+~,

:r: U(Jil Jf y -

x ~JilM~MWl

y+1

1; J,y) U(i&Jf y - 1; J& !Y ; 1 ;

;I

+ ($

I7

-

1 (24)

where (r7) is the radial integral 0-9 = J and { } denotes the 9jsymbol. 595/92/I-3

G,z,@>

rzun&)

r2

dr,

(25)

32

DEVANATHAN AND SUBRAMANIAN

IV. POLARIZATION OF THE FINAL NUCLEAR STATE Substitnting the interaction Hamiltonian (19) for gamma emission into Eq. (7) and integrating over the photon directions, we obtain

x U(J,yJ,,A;

JpY) U(J#‘Jfy;

JI,K)

x * x (JI II c u,(4 II J,>k IIc u&4 II Ji>* kLA,mK 11

.

71

(26)

In Eq. (26) the summation is over the seven variables J, , Ji, y, A, A’, (1 and SZ’. The integration over the neutrino direction is also to be performed. This imposes the condition that K = 1, thereby restricting the final nuclear orientation to a vector polarization PN defined by

Piv = [JAJ, + l)/311’2
(27)

when Ji = 0 and Jf > 1. The final result is similar in form to that obtained in [ 11,

The quantities &’ and a involve the muon capture coupling constants and the nuclear matrix elements of this two-step process. They are not given explicitly here because of the lengthy expressions involved but can be obtained in a straightforward way from Eq. (26). V. NUCLEAR MATRIX ELEMENTS To evaluate the nuclear matrix elements we have to assume some model. Let us consider the initial nuclear state to be the ground state of a closed-shell nucleus I O), the intermediate and the final states to be one-particle one-hole states ( JiMi)

= ( 0).

I JIMI> =

c

(29)

Wp.hJ~ ; mp, -mH , M~)(-ll)‘~-“~ a~.mpajH,mH I Oh

(30)

WphJf ; m,

(31)

mP*mH

I J&f& = c

%.%l

, -mh

, Mf)(-ll)ih-“h

a&,pj,,,,,

I Q

CASCADE PROCESS IN MUON CAPTURE

33

where a, a+ are, respectively, the annihilation and creation operators. Writing the transition operators in the occupation number representation, (32) (33) c,d

the nuclear matrix elements can be expressed in terms of single-particle elements. (JIM, i C U?(n) j 0) = LLd (j tJ11 11

VI. GAMMA-NEUTRINO

p

11UAi'lj HP

ANGULAR

A,J,6m,,,M, ’

matrix

(34)

CORRELATIONS

In Section IV we were interested in determining the final nuclear polarization at the end of a cascade resulting from muon capture and subsequent deexcitation by gamma emission. Therein the neutrino and the y-ray directions were not observed and hence angular integrations were performed over these directions. In this section we show that if the angular integrations are not performed the y-ray yield can be expressed in terms of certain observable correlations involving the three vectors: the gamma-ray momentum k, the neutrino momentum Y and the muon polarization P. The y-ray yield Y in a given direction with the neutrino making an angle Olcv in the nuclear cascade process with the spin sequence Ji “5 J, L Jf can be written as Y = (27~)-~k2v2 j / Q j2 cl+,, ,

(36)

I HI, I JAG’)*

(37)

where

x (JfMf

*.

34

DEVANATHAN

AND

SUBRAMANIAN

The integration in Eq. (37) is over the azimuthal angle 4” of the neutrino emission which is not observable. Substituting the appropriate interaction Hamiltonians, the matrix elements can be evaluated as outlined in Section V. Following the notations used in [ 1, 161 and including the momentum-dependent terms in the muon capture interaction Hamiltonian, we find that in addition to all the terms given in expression (19) of [ 161 an extra term GA g (1 + P . O){iM, . (4 x MS*) - i(C x Ma) * MS*) also contributes for the present process. We obtain

~IQl%=;~[~

G,2an+ GA2bA+ (Gp2- ~GPGA)CA

A

- 2Gy g dA + 2(Gp - GA) $ eA + 2GA $$,I

1 PA(cos 8,“) d&

where a, = c R(y, ~‘)(i)~‘-” [J][J’] C(JJ’A; 00) U(JIJJIJ’; J& 8

I(J,, J, 0, J, 0, J, Ji)

x I(JI, I’, 0, I’, 0, I’, Js) 6A&,, , bA = - c

R(y, y’)(#‘-Z f#-

(39)

[X][h’] C(JJ’A; 00) U(AJXJ’; l-$

s x U(J&A’;

Jill) Z(JI, J, 0, J, 1, A, Ji) I(J,, I’, 0, J’, 1, A’, Ji),

c,, = 1 R(y, y’)(i)l’-’

8

X

(40)

[J][J’] C(Jlh; 00) C(J’lx,; 00) C(Ah’A; 00)

U(JIXrlh’; JZA) I(JI 3 Jy0, J, 1, A, Ji) I(J, 3I’, 0, I’, 1, A’, Ji),

dA = c R(y, y’)(i)z’+l-z [fl[J’] C(Jh’A; 00) C(J’lh’; 8

(41)

00) U(J,JJ$; J&t)

x ICJ,, J, 0, J, 0, J, Ji) ~(JI , I’, 1, h’, 0, x’, Ji) a,, ,

(42)

CASCADE

PROCESS

en = C R(Y, r’)(l). z,+1--1(-l)“‘-1’ x U(JI/\JIZ’; J&

IN

35

CAPTURE

[Q[xl] C(llh; 00) C(Xl,A; 00)

Z(JI , Z, 0, I, 1, h, Ji> Z(JI , Z’, 1, A’, 1, I’, Ji),

z’+l-z (- l)‘fl-” x C(91A;

MUON

00) U(J,AJ,h’;

[I][I’][h][h’][S]

(43)

C(ZI’Y; 00)

JiA) Z(JI, I, 0, 1, 1, A, Ji) Z(JI, I’, 1, A’, 0, A’, Ji)

(44)

gLA = 43r c R(y, y’)(i)z’-z

-##

s

C(IlX; 00) C(Al’L; 00) U(J,AJ,h’;

J+4)

x U(X2’Al; Lx’) Z(J,, I, 0, I, 1, A, Ji) Z(J,, I’, 0, Z’, 1, A’, Ji), ALA = 4~ c R(y, y’)(i)l’+l-z

(-I)“‘-1’

(- l)L+l+” f#-

s x U(J,XJ,Z’; J&

U(l’I/11;

C(II’L;

(45)

00)

LA) Z(JI, l, 0, I, 1, h, Ji) Z(J, , I’, 1, h’, 1, I’, Ji).

(46) In the above equations C, denotes summation over the six variables y, y’, Z, I’, h and A’. The quantities R(y, y’) and Z(J, , I, q, A, N, J, Ji) are given below: R(y, y’) = C 47r( - 1)” (- lj+’ u

x U(JvJd;

b,b;t $

I

$$

C(YY’4

FL>- P., 0)

Jd) (JT II Q, II JXJf IIQ,, II JI>*,

(47)

The notation [K] is defined by Eq. (8). PZ(x) is the Legendre polynomial of order 8 in X. The muon capture effective coupling constants GY, GA , G, , etc., are as defined in [l, 161. The circular polarization of the y-ray is denoted by p. In Eq. (36), an integration over the angle 96,must be performed. Before doing so, we shall discuss the kinematics of the problem. We have three vectors: the y-ray momentum k, the neutrino momentum Y and the muon polarization P. The angle between k and v and the angle between k and P are observables but the angle between the two planes, the one containing k and v and the other containing k

36

DEVANATHAN

AND

SUBRAMANIAN

and P, is not observable. The angle 4” is exactly the angle between these two planes and an integration has to be performed over this angle. Integrating, we obtain -257 (4% J PA(COS e,,> d$” = 2?TP~(COS e,,>, s0 P

2zz(P *i) P,(cos O,,) dcj$ = 2~ c {C(AlL;

00)}2 (P . k) PL(cos 0,,),

(50)

L

s0

2R(YL(;>

x Y,@)}:

dqbv = C(LlA;

(P . li) PL(cos f&J.

00) y

(51)

Even if the muon is not polarized, an integration over #+ is still necessary since by observing the energy distribution of the doppler-shifted y-ray, only the polar angle of neutrino emission (k&J can be observed. Substituting Eqs. (49)-(51) into Eq. (38), we obtain a simple form J I e 12dh = Ic SC~,P~(COSedj A

-

c aLn(p . Q ~~(~0s ed, LA

(52)

where dA = m Gy2aA + GA2bA + (Gp2 - 2GAGP) cA - 2GygG dA I + ~(GP gLn = 7~ Gv2aA I

GA) g en + ~GA $ft/, GA2b, + Gp2cn - 2G y 6 dA 2G, k$ fA1 [C(AlL;

+2GpeeA-M

+ ; IG,(G,

(53)

00)12

- W g,n - GA !&t hm/ [L][l]

C(LlA;

00).

(54)

The correlation function W(&) is given by

w(ed

= 1 V,P~(COS e,,) + (P . fc) c gLpL(cos L A

where VA and VL are the correlation

=

-

c

A

(55)

coefficients defined by

g<* = =4&4J, %L

e,),

(~LAP3,

A = even, L = odd.

(56)

Parity Clebsch-Gordan coefficients and parity conservation in strong interactions imply that d is even in Eqs. (39)-(44) and L odd in Eqs. (45) and (46). From Eqs. (50) and (51), it also follows that A is even in Eqs. (45) and (46). Hence it is clear from Eq. (52) that the even correlation coefficients arise from the first term and the odd correlation coefficients arise from the second term. The contributions

37

CASCADE PROCESS IN MUON CAPTIJRE

to the y-ray yield from the odd correlation coefficients are suppressed when P * ir is zero either because the muon is unpolarized (P = 0) or because the y-ray is observed at right angles to the muon polarization (P # 0). In this case only the first term in Eq. (52) is nonvanishing and the correlation function involves only even correlation coefficients. If we consider a simple cascade process with the nuclear spin sequence 0 S J, -Zr,0, the expressions (39)-(46) become very much simplified. A factor (--1Y I b, l”l@ II Q+ II JAI 2 is common for all the terms a,, bA , etc., and hence for the correlation coefficients this factor occurs both in the numerator and in the denominator and consequently gets canceled. Hence the correlation coefficients are independent of the y-decay mechanism whereas the y-ray yield depends upon the y-decay interaction Hamiltonian. Therefore it is advantageous to study the spin sequence 0 --f JI --f 0 which does not require a knowledge of the y-decay matrix elements. In the following section we obtain interesting relations between the asymmetry coefficient of the intermediate nuclear state and the y-v angular correlation coefficients, taking into account the complete set of nuclear tensor operators. These relations are algebraically proved to be independent of nuclear structure and coupling constants of the muon capture interaction and they are true even if second class currents exist. VII.

SIMPLE RELATIONS BETWEEN ASYMMETRY ANGULAR CORRELATION COEFFICIENTS

AND

The expressions for a, , bA , etc., given in Eqs. (39)-(46) become much simplified for the special case 0 + JI -+ 0. Let us denote them by a,‘, bA’, etc., the prime being introduced to denote the special case 0 + J, + 0. It can be easily seen that ,aZ,obtained from Eq. (53) for this special case is proportional to the muon capture rate for the nuclear transition 0 -+ JI . Further the asymmetry coefficient ‘y. [3, 14, 19, 20, 211 of the recoil nucleus with spin J, is given by 01= ~&&.

(57)

If we further restrict ourselves to the particular case of J, = 1, we can obtain the following simple relations.

(iii) (iv)

c2’ = -co’,

(v)

d2’ = -a,,‘, e2’ = -e,‘,

(vi)

fi’ = f,‘/2.

(581

38

DEVANATHAN

AND SUBRAMANIAN

It is to be noted that a,’ and d,’ are zero when ninI = +l and c,,’ and e,,’ are zero when 7ri7rI = - 1. We have proved algebraically these relations using the Tables of Clebsch-Gordan and Racach coefficients. In proving&’ = f0’/2, we have made use of the following relation3 given by Devons [22] and Uberall [23]: = L(L + 1) - I(I + 1) - L’(L’ + 1) + Z’(1’ + 1) 2(s(s + 1) k(k + l)}“” (59) The correctness of Eq. (59) has been checked numerically for particular values of the arguments in the 9jsymbol. Using Eqs. (58), we prove algebraically that 3cu+ 4gz + 1 = 0,

77j7r, = il.

(60)

This relation holds good for both possible changes of parity rrjrrl = &l. When ?riTi-~= - 1, ar and hence V, are strongly dependent on gA/gv and nuclear structure but independent of gP/gA . Hence a measurement of V, will enable one to calculate gA/gy if the nuclear structure is known. For the parity change ~T~TQ= + 1, cyand hence %‘S are sensitive to gP/gA and hence measurement of V, presents a nice possibility of the determination of g,,/gA , knowing gA/gv . It is a matter of simple algebra to prove from Eqs. (56) and (57) that 01is a linear combination of the odd correlation coefficients for the case Orri + J;’ + On, with IT~~Q= (-l)J~ In this case the terms g,,, and hLA obtained from Eqs. (45) and (46) for this special case are zero. This yields a relation which is independent of nuclear models and coupling constants of the muon capture interaction. For the particular case with JI = 1, ginI = - 1, we have proved algebraically that 301+ 3g1 - 2%Y3= 0,

Tri27,= -1.

(f-51)

However this relation breaks down for the nuclear transition 0 -+ 1 + 0 with parity change rrirr, = + 1. The relations (60) and (61) are independent of both the nuclear structure and coupling constants of the muon capture interaction. They are true even if second class currents exist. Hence measurements of the correlation coefficients give us a handle to find the asymmetry 01 and present an alternative to other efforts in the field [2, 111. From Tables I-IV, one can easily verify the relations (60) and (61) for the case lsO(O+, g.s.) L

16N(1-, 392 KeV) Y,

lsN(O-, 120 KeV).

* Both [22,23] contain some misprints and they are corrected here.

CASCADE

PROCESS

IN

TABLE Variation Model

of Q, with

MUON

39

CAPTURE

I gA/gr

Ratioalb

I

Model

II

&?A!&

(4 -1.10 -1.15 -1.16 -1.17 -1.18 -1.19 -1.20 -1.21 -1.22 -1.23 -1.24 -1.25 -1.30

(b)

0.5855 0.6209 0.6277 0.6344 0.6410 0.6474 0.6538 0.6601 0.6663 0.6724 0.6784 0.6843 0.7126

0.7757 0.7990 0.8035 0.8078 0.8122 0.8164 0.8206 0.8248 0.8288 0.8328 0.8368 0.8407 0.8593

(a)

(b)

0.6498 0.6836 0.6901 0.6964 0.7026 0.7088 0.7148 0.7207 0.7265 0.7323 0.7379 0.7435 0.7700

0.8339 0.8548 0.8588 0.8627 0.8666 0.8704 0.8741 0.8778 0.8815 0.8850 0.8885 0.8920 0.9085

a Columns (a) and (b) are obtained, respectively, with momentum-dependent terms of the muon capture interaction. b Equation (61) can be verified from Tables I, III and IV. TABLE Variation Model

of %, with

and

without

the

II gA/gr

Ratio’,*

I

Model

II

gA/&’ (4

-1.10 -1.15 -1.16 -1.17 -1.18 -1.19 -1.20 -1.21 -1.22 -1.23 -1.24 -1.25 -1.30

-0.0121 0.0175 0.0231 0.0287 0.0341 0.0395 0.0449 0.0501 0.0552 0.0603 0.0653 0.0703 0.0938

(i Same as the first footnote of Table b Equation (60) can be verified from

(b)

(a)

(b)

0.1464 0.1658 0.1695 0.1732 0.1768 0.1804 0.1839 0.1873 0.1907 0.1940 0.1973 0.2006 0.2161

0.0415 0.0697 0.0751 0.0803 0.0855 0.0906 0.0957 0.1006 0.1055 0.1102 0.1149 0.1196 0.1416

0.1949 0.2123 0.2157 0.2189 0.2222 0.2253 0.2285 0.2315 0.2345 0.2375 0.2404 0.2433 0.2571

I. Tables

II and

IV.

-

40

DEVANATHAN

AND TABLE

Variation

SUBRAMANIAN III

of %s with gA/gv

Model

Ratio”

I

Model

II

&llbv (4 -1.10 -1.15 -1.16 -1.17 -1.18 -1.19 -1.20 -1.21 -1.22 -1.23 -1.24 -1.25 -1.30 a Same

(b)

0.4024 0.3965 0.3954 0.3943 0.3932 0.3921 0.3910 0.3900 0.3890 0.3879 0.3869 0.3859 0.3812 as the footnotes

0.3707 0.3668 0.3661 0.3654 0.3646 0.3639 0.3632 0.3625 0.3619 0.3612 0.3605 0.3599 0.3568

of Table

@>

0.3917 0.3861 0.3850 0.3839 0.3829 0.3819 0.3809 0.3799 0.3789 0.3780 0.3770 0.3761 0.3717

0.3610 0.3575 0.3569 0.3562 0.3556 0.3549 0.3543 0.3537 0.3531 0.3525 0.3519 0.3513 0.3486

I.

TABLE Variation Model

(4

of 01 with

IV gA/gv

Ratioasb

I

Model

II

&TAkV (4 -1.10 -1.15 -1.16 -1.17 -1.18 -1.19 -1.20 -1.21 -1.22 -1.23 -1.24 -1.25 -1.30

-0.3172 -0.3566 -0.3641 -0.3716 -0.3789 -0.3861 -0.3931 -0.4001 -0.4070 -0.4138 -0.4204 -0.4270 -0.4585

(b)

(a>

(b)

-0.5285 -0.5544 -0.5594 -0.5643 -0.5691 -0.5738 -0.5785 -0.5831 -0.5876 -0.5921 -0.5964 -0.6008 -0.6215

-0.3886 -0.4263 -0.4334 -0.4404 -0.4474 -0.4542 -0.4609 -0.4675 -0.4739 -0.4803 -0.4866 -0.4928 -0.5222

-0.5932 -0.6165 -0.6209 -0.6253 -0.6296 -0.6338 -0.6380 -0.6420 -0.6461 -0.6500 -0.6539 -0.6578 -0.6761

a Same as the first footnote of Table I. B Equations (60) and (61) can be verified

from

Tables

I-IV.

CASCADE PROCESS IN MUON CAPTURE

VIII.

41

NUMERICAL RESULTS FOR A CASCADE PROCESSIN MUON CAPTURE BY I60

In the light of a recent experimental proposal by Deutsch [ll], we present the numerical results for the correlation coefficients in the case of a simple nuclear cascade 160(0+, g.s.) L

16N(1-, 392 KeV) Y,

16N(0-, 120 KeV).

Since nirI = (- l)“~, the p- capture rate for this process is independent of gp/gA but dependent on gA/gv . Hence the measurement of the y-v angular correlation coefficients will enable one to calculate g,/g, independent of g,/g, . We have estimated separately the contribution from the momentum-independent and the momentum-dependent terms of the muon capture interaction. The simple j-j coupling shell model (Model I) and the Gillet and Vinh Mau wave functions (Model II) are used in the study. In Tables I, II, III and IV, we present the numerical results4 obtained for %I , %??z,V, and a(, respectively, in Models I and II with and without the momentumdependent terms for various values of g,/g, ratio. It is found that %s is almost independent of nuclear structure and g,/g, . The effects of nuclear structure and momentum-dependent terms of the muon capture interaction are very prominently seen in 9, and CL.The correlation coefficient %?1is fairly large and is sensitive to nuclear structure and momentum-dependent terms of the muon capture interaction. The correlation coefficients V, and ?Z1 decrease with gA/gv whereas 01 and ?Fs increase with g,/g, , in conformity with Eqs. (60) and (61). The inclusion of momentum-dependent terms decreases the values of %I and %?Zbut increases the values of %?sand 01.

IX. CONCLUSIONS Hitherto there has been no direct attempt to measure the asymmetry of the recoil nucleus in muon capture because of the intrinisic experimental difficulties. Since it is possible to measure experimentally [2, 211 the avarage polarization and the longitudinal polarization of the recoil nucleus, we have shown recently [20] that there is an interesting relation 01- 2P, = 1, connecting the asymmetry parameter oi and the longitudinal polarization PL of the recoil nucleus. This gives an indirect way of obtaining cyfrom a measurement of PL . This relation has been proved to be independent of both the nuclear structure and muon capture coupling 4 We use the same set of values for the coupling constants as in 1211 and we choose a value 1.76 Fermi for the oscillator length parameter.

42

DEVANATHAN

AND SUBRAMANIAN

constants using the Fujii-Primakoff effective Hamiltonian [13]. Subsequently, Bernabeu [24] has shown that this relation stems from the rotational invariance and the definite helicity of the neutrino. In this article we have obtained similar relations connecting 01and the Y--V angular correlation coefficients in the special case of a cascade process 0 14; 1 % 0. The correlation coefficients are experimentally observable quantities [5, 6, 111 and hence Eqs. (60) and (61) give us alternative method of obtaining 01. Since these relations are also found to be independent of both the nuclear structure and muon capture coupling constants, it is an open question whether it is possible to relate these results to some fundamental symmetry principles.

ACKNOWLEDGMENTS We are greatly indebted to Professor J. P. Deutsch, Professor R. E. Welsh, Dr. Z. Oziewicz, and Dr. A. Pikulski for having drawn our attention to the problem of gamma-neutrino angular correlations and also for many fruitful communications and discussions. One of us (V.D) expresses his thanks to Professor Abdus Salam, Professor P. Budini, the International Atomic Energy Agency, and UNESCO for kind hospitality at the International Centre for Theoretical Physics, Trieste, where a part of this work was done.

REFERENCES 1. V. DEVANATHAN,

R. PARTHASARATHY,

AND P. R. SUBRAMANIAN,

291. 2. A. Possoz, D. FAVART, L. GRENACS, J. LEHMANN, AND C. SAMOUB, Phys. Lett. 50B (1974), 438. 3. L. WOLFENSTEIN, NUOOO Cimento 13 (1959), 319. 4. G. H. MILLER,

M.

ECKHAIJSE,

F. R. KANE,

P. MACQ,

P. MARTIN,

Ann. Phys. (N.Y.) 73 (1972)’ D. MEDA,

L. PALFFY,

AND R. E. WELSH,

J. JULIEN,

Phys. Lett. 41B

(1972), 50. M. ECKHAUSE, F. R. KANE, P. MARTIN, AND R. E. WELSH, Phys. Rev. Lett. 29 (1972), 1194. 6. L. GRENACS, J. P. DEUTSCH, P. LIPNIK, AND P. C. MACQ, Nucl. Inst. and Methods 58 (1968), 164. 7. Z. OZIEWICZ AND A. PIKULSKI, Acta Ph.ys. Polon 32 (1967), 873. 8. N. P. POPOV, Sov. Phys. JETP 17 (1963), 1130. 9. A. P. BUKH~OSTOV AND N. P. POPOV, Sov. J. Nucl. Phys. 6 (1968), 589. 10. A. P. BUKHYOSTOV AND N. P. POPOV, Nucl. Phys. Al47 (1970), 385. 11. J. P. DEIJTSCH, private communications. 12. V. GILLET AND N. VINH MAU, Nucl. Phys. 54 (1964), 321. See also Erratum, Nucl. Phys. 57 (1964), 698. 13. A. FUJII AND H. PRIMAKOFF, Nuovo Cimento 12 (1959), 327. 14. V. DEVANATHAN AND M. E. ROSE, J. Math. Phys. Sci. (India) 1 (1967), 137. 15. V. DEVANATHAN, in “Lectures in Theoretical Physics” (A. 0. Barut and W. E. Brittin, Eds.), Vol. lOB, p. 625, Gordon and Breach, New York, 1968. 5. G. H. MILLER,

CASCADE

PROCESS

IN

MUON

43

CAPTURE

16. V. DEVANATHAN, R. PARTHASARATHY, AM) G. RAMACHANDRAN, Ann. Phys. (N.Y.)

72

(1972),

428.

17. J. M. EI~ENBERGAND W. GREINER, “Excitation Mechanisms of the Nucleus,” North-Holland, Amsterdam, 1970. 18. A. DE-SHALIT AND I. TALMI, “Nuclear Shell Theory,” Academic Press, New York, 1963. 19. M. MORITA, Phys. Rev. 161 (1967), 1028. 20. P. R. SUBRAMANIAN AND V. DEVANATHAN, Phys. Rev. C2 (1975), to appear. 21. V. DEVANATHAN AND P. R. SUBRAMANIAN, Phys. L&t. 53B (1974), 21. 22. S. DEVONS AND L. J. B. GOLDFARB, in “Handbuch der Physik” (S. Fluugge, Ed.), Vol. XLII, p, 545, Springer, Berlin, 1957. 23. H. OBERALL, “Electron Scattering from Complex Nuclei,” Part B, p. 804, Academic Press, New York, 1971. 24. J. BERNABEU, TH. 1950 CERN preprint, Nov. 20, 1974. We are grateful to Dr. L. Palffy for sending us a copy of the preprint.