Nuclear matrix elements of the first forbidden beta transitions in the decay of Ba139, Ce141, La140 and Pr142

I i 4"C

I

Nuclear Physics 65 (1965) 21--32; (~) North-Holland Publishing Co., Amsterdam Not to

be reproduced

by photoprint or microfilm without written permission from the publisher

NUCLEAR MATRIX ELEMENTS OF THE FIRST FORBIDDEN BETA TRANSITIONS I N T H E D E C A Y O F Ba t39, Ce 14z, La 14° and e r 142 P. LIPNIK t Centre de Physique Nucl~aire, Louvain, Belgique a n d J. W. SUNIER Department of Physics, University of California, Los Angeles, California tt

Received 10 August 1964

Abstract: The ratio of the coordinate type matrix elements involved in the beta decay of Ba13', Cex-x, Lax*° and Pr x4* have been computed with the help of the jj coupling of the shell model. The theoretical functions for the observable quantities are compared with the available experimental data and exhibit a qualitative agreement. The ratio Ao = --fiTs/~fa • r is computed to be A0 ~ 1 for the Ce1.I [-(fl)Prm½ + transition and 1.3 =
1. Introduction The study o f the forbidden beta transition yields valuable information a b o u t the wave functions o f the initial and final nuclei. However, it is rather tedious to extract this information f r o m the experimental data because: (i) the n u m b e r o f nuclear parameters involved in the observable quantities is large, (ii) destructive interferences can occur between the different terms governed by those parameters, (iii) the accuracy o f the experiments is relatively poor. O n the other hand, the theoretical c o m p u t a t i o n o f nuclear matrix elements encounters serious difficulties: (i) the estimation o f relativistic matrix elements is doubtful, (ii) the calculation is very sensitive to configuration mixing ,(iii) it has to be m a d e in a single-particle representation. The theory o f the conserved vector current (CVC) sets restrictive conditions for the vectorial part o f the beta interaction. It allows the reduction o f the n u m b e r o f independent nuclear parameters 1) t h r o u g h a relation between the matrix elements Si~ and j'r. This relation turns out to be independent o f the nuclear potential. N o similar situation exists for the axial vector part o f the interaction, and it has been shown 2) that the ratio o f the matrix elements j'i~5 and Sa • r undergoes considerable fluctuations. These fluctuations m a y very well be responsible for the b r e a k d o w n o f the ~ approximation 2, 3), in particular for A J = 0 transitions. Furthermore, a similar t Chercheur agr66 h I'I.I.S.N., Belgique. tt Work supported in part by the U.S. Office of Naval Research. 21

22

P. LIPNIK AND J. W . SUNIER

criterion can be developed for the IAJ[ = 1 transitions 2), in terms of the ratio

A1 = $ia x r/~r. The purpose of this paper is to check these considerations by a comparative study of the beta transitions involved in the decays of Ba 139, Ce 141, La ~4° and Pr 142. These nuclei are characterized by N = 83 and the assumption of an initial neutron state (f~) should be quite reliable. We shall compute the ratios of the nuclear matrix elements involved in the jj coupling model, using the general calculus of Rose and Osborn 4). The commonest observable quantities will then be computed and compared with the available experimental data. 2. Nuclear Matrix Elements 2.1. M O M E N T U M TYPE MATRIX ELEMENTS

Two types of operators connecting the initial and final nuclear state are to be considered. In the notation of Rose and Osborn 4) they are e/~(r) = r'YT(0,

(1)

T~(r, a) = ~ COL2; - m ', m + m')~c'~+m'(r)@"lm'(a). rn'

In these equations, the degree L of the solid spherical harmonic ~L gives the order of forbiddenness of the beta transition, and C is a Clebsch-Gordan coetficient. The other notations used in the literature differ either by 5, 6) a factor in in q/L(r) or by 7) a different coupling of ~L(r) and °2/1(a), or also by normalization constants. The reduced matrix elements of these operators have been evaluated in the jj coupling model 4) and are given in terms of radial integrals. We write the single particle wave function of the final state as If> = 11½j> = ~(r) E C(1½j; #-~, z)Y~-'(0Z~.

(2)

In this equation ~ ( r ) is the real radial wave function and Z~ the spin eigenfunction. The wave function of the inital state has the same structure and is labelled by ~ ' , l', #',j'. With this description one obtains for the reduced matrix elements, according to eqs. (12) and (16) of ref. 4) <1½Jll@/zlll,½j,> =

(_)v+j'-j

~!2/+1)(21'+ 1)(2j'+ 1)] ~r k

2~z

A

o

½

= (_)t'+j'-j I9(2/+ 1)(2/'+~ 1)(2j' + 1)']/&

x[(2).+l)(2L+l)]÷(:

L I'0) : j ' 0

½

,~L.

(4)

NUCLEAR MATRIX ELEMENTS

23

rr+ Z~(r)~'(r)dr.

(s)

Here o~-r. is the radial integral

=f

The connection with the usual cartesian tl) representation is given in table 1 in terms of the reduced matrix elements for the first-forbidden beta transitions (L = 1, 2 = 0, 1, 2). In the same table, the nuclear parameters w, x, u, z defined by Kotani s), are also given. TABLE 1 Nuclear matrix elements in cartesian and spherical representation for first-forbidden beta transitions

l ~11 r+ = - . / - ~4n \2j'(2J+



fix = --Cvfr

= --X/-$~ ( 2 J + 1 ~ ~r \2J' + 1/


nu =

cAfie×r

= ~zrX/ff. ( 2 J + l ~ \2J' + 1/



~/z =

( CAtB j J

~w=

CAf o ' r

=

(2J+17 \2J--~!

(fllCnY2dii)

2.2. RELATIVISTIC MATRIX ELEMENTS t

The relativistic matrix elements can be given in terms of those of the coordinate type which have the same transformation properties 1,9, 1o). For the vectorial part of the interaction, Fujit a 1) has shown that the following relations do not depend on the nuclear potential/according to the CVC theory: /

fi =

-Acvc~fr=-

(2.4+ W ~ - W f ) ~ f r , .

(6)

In these formulas 2~ = ~Z/R is the Coulomb energy at the nuclear surface expressed in natural units (h = m~ = c = 1); 0~is the fine structure constant; Z and R are the charge and radius of the daughter nucleus; WrWf is the total energy transfer in the decay. A similar relation holds for the axial vector part of the interaction:

f i?5 = -(Acvc+nucl. potential term) ~ f o. r = -Ao~ f a" r.

(7)

* We adopt the notation of Konopinski and Uhlenbeck, which differs from that of Kotani by the sign of et.

24

P. LIPNIK AND ~, W . SUNIER

The theoretical estimate of Ao depends on the choice of the nuclear potential. Pursey 9) finds Ao g 2 whereas Ahrens and Feenberg xo) give A o ,~ 1. In a previous paper, we proposed 2) large fluctuations of A o by considering the f t values of the first-forbidden A J = 0 beta transitions. 2.3. RATIOS OF THE NUCLEAR MATRIX ELEMENTS If we express the relativistic matrix elements in the way defined under 2.2. we get rid of the different radial integrals and remain with only one type for a given order of forbiddenness. This is only true, however, if the wave functions of the nuclear states do not contain configuration mixing. TAm.E 2

Ratio of nuclear matrix elements for first-forbidden beta transitions for odd-mass nuclei (.t7coupling shell model) (a)_W= j ' a ' r u

CAX

(b)

"~ia x r

Ir

Cv u - Sia

x

z_ r

j = l-½

j = l+½

A j = Al = +_1

j'=

l'-½

l'+½

Al = l ' - l

1

1

Al

21'

[

j'=

Aj =0

= 1

- [1- 4-~,2]½~ -1

F2(/'+/-2)l+r

I

j A-t

Aj =O

IBu Sia x r

(c) u

Al = - 1

i

[2~]

~"

(a) (b)

F2(/n_- i).l ½

L

3r 2

J

(c)

A j = Al = +_1

i

(b)

[~2(/'+/+4)I+ I

(c)

In table 2, we give the ratio of the nuclear matrix elements o f the first-forbidden beta transitions. The primed characters label the initial state.

NUCLEAR MATRIX ELEMENTS

25

The matrix element ]'r undergoes an important reduction for the Aj = 0 transitions. The other coordinate type matrix elements have almost the same order ofmaguitude. Table 2 is valid for odd-mass nuclei and pure single-particle configurations. For the two-nucleon configurations of doubly-odd nuclei, an appropriate decoupling 4'12) of the angular momenta can always reduce the computation to the simple formula of table 2. 2.4. THE ~ APPROXIMATION 3) In the i-approximation the leading terms in the transition probability are linear combinations of the zero-order and first-order matrix elements:

V=CA[f'vs+¢f,'"J

= -C.¢(Ao-1)f. ,,

Y= +CvfiO,+,[Cvfr-cAfio×r]

(8)

= -Cv,(Acvc-l-l.2A~)fr.

If the approximation is valid, which is reflected by the statistical shape of the betaspectrum, V and Y are of the order of ~ and therefore roughly ten times larger than the coordinate type matrix elements I$~r" rl ~ ISrl ~ Ij'~× rl ~ SBel. We see from eq. (8) that the breakdown of the ~ approximation is equivalent to Ao~l

and/or

This is the case for the Ao value predicted by Ahrens and Feenberg lo) and for the transitions IAjl = 1, l ' - I = AI = 1. If such "cancellation effects" do not occur in V and Y, and if the second order matrix element J'B~j is not enhanced by a "selection rule effect", a measurement of the ~ and circular polarized y correlation will give the ratio V/Y. This ratio can then be used to extract the value of Ao by a modeldependent calculation.

3. Computation of the Observable Quantities Using an extension of the Konopinski-Uhlenbeck approximation, Kotani s) has computed the general expressions of the observable quantities related to the firstforbidden beta decay. They are given as functions of the electron energy IV. In this paper, we shall be concerned with the computation of a) the shape factor correction C I ( W ) = l + a W + b / W + c W 2, b) the longitudinal polarization PL(W), C) the fl-y directional correlation N(W, O) = 1 +e(W)P2(cos 0), d) the fl and circular polarized y correlation No(W, O) = I +og(W, O)(P/W)cos O. For a given choice of the matrix elements ratios w/u, x/u, z/u defined in table 2, and of Acv c and Ao given in eqs. (6) and (7) of subsect. 2.2, the calculation of the numerous coefficients involved in the above expressions is obvious. We refer for that purpose to the eqs. (A3)-(A25) of ref. s).

26

P. LIPNIK AND J. W. SUNIER

4. Beta Decay of Ba 139, Ce 141, La 14° and Pr "2 For each of the considered nuclei, we have a single neutron outside the closed shell N = 82. The ground state spin of Ba 139 and Ce ~4~ strongly support the neutron configuration (f~) for the initial states. For the final states, the proton configurations d~ and g~ are involved. In the daughter nuclei La ~39 and Pr 141, the pure M1 gamma transitions can be interpreted as single particle ones. It is interesting to point out the inversion of the spins of the ground state and of the first excited state between L a 139 and Ce 141. This is due to the filling of the sub-shell (g~)p. From this point of view La ~4° should be a transition nucleus of interest.

139 5(;BOa3

~

141

SaCe a3 - ~ 7/~1i)

}~li) 2.17 29% 7.2

0.435 70% 6.9

2.34 71% 6.9

~--0,580

30% 7.7

' 4 ,,,, LQ 139

,,) P r 141

Fig. 1. D e c a y s c h e m e o f B a laa a n d C e m a c c o r d i p g to ref. zz). 4.1. T H E

B a xsa A N D

Ce 141 N U C L E I

The decay schemes of the Ba 139 and Ce 141 nuclei are given in fig. 1. We shall consider the single-particle states li) = ](f~).),

If) --I(d~)p).

[ff)

=

[(g])p).

TABLE 3 R a t i o s o f the c o o r d i n a t e t y p e m a t r i x e l e m e n t s f o r t h e f i r s t - f o r b i d d e n b e t a t r a n s l•h•o n s o f B a la9 a n d C e m

Transition W

~0" " r

u

~ia x r

_x = u

u

Cv~r CA~ia × r

Sia × r

(f÷). ~ (g~)p

(f~). ~ (d~)p

0.99

0

0.104

0.833

-0.79

1.9

27

NUCLEAR. MATR=P~ E L E M E N T S

The ratios of the nuclear matrix elements corresponding to these single-particle states are summarized in table 3. We see immediately that the transitions -~- ~ ~r+ should exhibit a considerable deviation from the ~ approximation. 4.1.1. The-~- --* -~+ transitions. These transitions are in relatively good agreement with the ¢ approximation if we take into account that for Ba 139 the basic condition >> Wo is not very well fulfilled: ~ = + 12.9, W o = 5.58. The spectrum shape faca (m¢2) "t

(

Bo 139

~/--~ >

0.2

J 1.05

0.1

1.0 I 0.95

0

'

;

'

2

P

0. .

.

. 1

A.

Ao

O (mc2) "1

(PL/-~")

Ce 141

0.2 ~..

I.O3

0.1

i

0

i

I

|

i

2

A.

i

0

i

i

i

I

2

Ao

Fig. 2. Coefficient a o f the spectrum shape factor C I ( W ) ~ 1 q - a W and mean longitudinal polarization, as functions o f the nuclear matrix elements ratio Ao. (a) transition Ba139½-(~)Laa88~+. (b) Transition Cem½-(fl)Prl¢l~ +.

tors are o f the type CI(W),,~ l+aW. The longitudinal polarization of the beta particles is roughly PL(W) = --P/W. The anisotropy of the fl-? directional correlation in the decay of Pr i.1 is small. The coefficient og(W, 0) of the fl and circularly polarized 7 correlation depends very slightly on the electron energy W and on the emission angle 0pr In figs. 2a and 2b, we give the coefficient a of CI(W) and the mean polarization (PL(W)(--P/W)-1) as functions of the nuclear parameter Ao, for Ba 139 and Ce 141 respectively. In fig. 3, we present the coefficient of the/~-? directional correlation Ce 14177+ (?)~ + Pr 141, computed at the m a x i m u m beta energy Wo, in ~- (fl)~function of Ao. In the same figure, we show the dependence o f o~(Wo, zr) as a func-

28

P. Xal~mKA N D

j. w.

strNmR

tion of Ao. The agreement with the experimental data 13) in the decay of Pr 141 is satisfactory. The measured value of og(W, 0) corresponds to Ao~l. (

(W.}

oJ (W.,TT)

002

0.5

0

0

Ce 141

-0.5

002

o

'

'

;

~

,

A°

2

.A°

Fig. 3. Coefficients of the I%7 directional and the ,8- and circularly polarized 7 correlations in the ½-(/3)½+ decay of Ce x4x, as functions of the nuclear matrix elements ratio A0. The experimental data a r e taken from ref. xa). C'(W)

Bo 139

C'(W)

¢.(.~)-,

1.0 r

Ce

~.(-~)-,

~.(.-})-,

141

1.0

"

0,5

0.5

~

p~.(.~}-, c'(w)

i

I

I

I

2

3

4

5

i

W(mc 2 )

1.0

r

15

t

I

2.0

i

W(mc 2 )

Fig. 4. Spectrum shape factor and longitudinal electron polarization as funct.ons of the electron energy IV. (a) Transition BaZS~-(~)LaZ89~+ (b) Transition CeZ4Z½-(fl)Pr141~+

This value is compatible with the measured shape factor (1.4 standard deviation). 4.1.2. The ½----~+ transitions. In these cases, the matrix element Y undergoes an important cancellation and takes the values Y=-5.3CvSr

and

Y=-2.2CvJ'r

for Ba 139 and Ce TM, respectively. The shape factors show a strong dependence on the beta particle energy IV. The longitudinal polarization deviates considerably from the value (-P/W). The coefficient co(W, 0) of the/~ and circularly polarized ? correlation is nearly independent of 0 but varies strongly with the electron energy.

NUCLEAR

MATRIX

29

ELEMENTS

In figs. 4a and 4b we give the shape factors and the longitudinal polarization as functions of Wfor Ba 139 and Ce 14~ respectively. In fig. 5 we present the energy dependence of the coefficients of the fl-y directional and the fl and circularly polarized y correlation in the cascade Bala9@-(fl){+(y){+La 139. It would be very interesting to measure og(W) for Ba 139 and PL(W) for Ce 141. (W,~) 0.5

((W)

8o 139

0.10

-0.5

-I.0

i

i

i

i

2

3

4

5

W (mcz)

Fig. 5. Coefficients o f the fl--~ directional and the fl and circularly polarized 7 correlations in the ~-(fl)t+ decay o f Ba xs°, as functions of the electron energy IV. Lo 140 22 2"

I~ ,.,1~, ~ 0~ ¢e 140

~{W) 0.1

c'(w)

}ft~

}

2

0.05 0

- 005

~2

-0.1 2

3

4

W (mc 2)

Fig, 6. Coefficient o f the fl-y directional correlation and spectrum shape factor of the 3-(fl)2 + decay of La 14°, as functions o f the electron energy IV. The experimental data are taken from refs xs, 17) and normalized at W = 4 in the case of Cx(W).

We must point out that the Fermi analyses of the beta spectra of Ce 1.1 and B a la9 given in the literature mostly report statistical spectrum shapes for both the ~ - ~ 3 + and -~- --* {+ transitions 19-21, 2a). However, the small energy difference between

30

P. LIPNIK AND J. W. SUNIER

both end points ( ~ 150 keY) and the poor statistical accuracy of the/~-? coincidence spectra make it impossible to draw definite conclusions from these data. 4.2. THE La14° NUCLEUS The partial decay scheme of this nucleus is given in fig. 6. Let us consider the initial and final wave functions as built up of a mixture of d÷ and g~ proton configurations li) = Ctl(f~)n, (d~)p; 3-)+Pl(f~)n, (g~)p; 3 - ) , If) = £l(d~) 2 ; 2+)+/~'[(gc)p2 ; 2 + ) , where ~,/?, ~',/~' are the parameters of the mixture. According to Rose and Osborn (sect. 3 of ref. 4)) we have computed the ratios of the matrix elements for the pure cases ~ = ~' = 0 and/~ =/~' = 0. The results are given in table 4. TABLE 4

Ratio of the matrix elements for the outer beta transition of La1~° Transition

(f~)n --~ (g-l-)p = c~'

x U

_

Cv ~ r CA~ i a x r

Z

Snij

u

Si~ x r

(f~-)n "-4 (d~)p /~ = fl'

0.104

0.833

1.42

0.2

The computed quantities corresponding to the parameters of table 4 are reported in fig. 6, together with the experimental data. The transition (f~)n ~ (d,~)p shows the characteristics of a cancellation effect and is ruled out by the experimental measurements. The transition (f~)n ~ (g~)v deviates from the ~ approximation, too. This can be attributed to the relative enhancement o f the matrix element SBij combined with the reduction of the matrix element J'r. The high value of the end-point energy Wo also implies that the ~ approximation should not be very good in that case. The experimental data 14-17) strongly support a major contribution of the transition (f~)n ~ (g~)p, although the agreement with the theoretical curves is only qualitative. 4.3. THE Pr14~ NUCLEUS The partial decay scheme 22) is given in fig. 7. We take the wave functions in analogy with the case of La 1.° and limit ourselves for the pure transition (fi)n ~ (g÷)p" This is supported by the fact that the experimental data 15) agree satisfactorily with

NUCLr~rt MATRIX ELm~mNTS

31

the ~-approximation. We obtain thus the following ratios:

~r.r

w_ u

_

3.24,

Sia x r

x_

- - C v ~ r _0.104, CASi~ × r

z _

~B~j -- 1.83.

u

~i~r x r Pr 142 2"

O" Nd t4;~

0.05

0,5

0

0

- 005

-

0

1

2

0.5

A.

I

2

/~..

Fig. 7. Coefficients o f t h e fl-7 directional a n d the ff a n d circularly polarized 7 correlations in t h e 2-(fl)2 + decay o f Pr m , as f u n c t i o n s o f t h e nuclear m a t r i x elements ratio-do. T h e curves are c o m p u t e d for t h e m e a n electron energy c o r r e s p o n d i n g to the experimental d a t a o f ref. la).

The observable quantities are computed as a function of Ao and show the following characteristics: a) the shape factor CI(W) is constant except for the value Ao ~ 1, where the total deviation amounts to 20 ~ ; b) the coefficient co(0, W) of the fl and circularly polarized ~ correlation is independent of 0pr and exhibits an energy dependence only for Ao ~ 1: co(n; Ao = 1; W = 1) = - 0 . 3 0 and co(n; A o = 1; W = Wo) = --0.14. In fig. 7 we have plotted the coefficients ~(W = 1.6) and co(n, W = 1.45) as functions of Ao, together with the experimental data 18). F r o m the fl and circularly polarized ), correlation we get 1.3 < Ao < 2. This choice is still compatible with the experimental value of the anisotropy of the fl-~ directional correlation.

32

P.

LIPNIK

AND

J. W .

SUNIER

5. Discussion and Conclusion I n each o f the investigated cases, a qualitative a g r e e m e n t can be ~eached between the e x p e r i m e n t a l d a t a a n d the c o m p u t e d o b s e r v a b l e quantities. T h e limited a m o u n t o f e x p e r i m e n t a l i n f o r m a t i o n m a k e s the theoretical c o m p u t a t i o n with m o r e detailed wave functions i n a d e q u a t e for the present time. It is interesting e n o u g h to see h o w a single p a r t i c l e picture can l e a d one to p i c k o u t the m a j o r n u c l e o n configurations f r o m the observed b e h a v i o u r o f p a r t i c u l a r b e t a transitions: L a 1*°, P r 1*2. O n the other h a n d our c o m p u t a t i o n p o i n t s to the interest o f m e a s u r e m e n t s o f the l o n g i t u d i n a l p o l a r i z a t i o n o f the b e t a rays in the t r a n s i t i o n Cel*l-~ - (fl)Pr 1*1 {+. A careful s t u d y o f the inner b e t a t r a n s i t i o n o f Ba la9 w o u l d be very interesting, too. I n the case o f the A J = 0 b e t a transitions, the c o m p a r i s o n o f the theoretical p r e d i c t i o n s with the exp e r i m e n t a l d a t a c o u l d be very m e a n i n g f u l to d e t e r m i n e the r a t i o A o = - i S y s / ~ J ¢ " r a n d its fluctuations f r o m one nucleus to another. I t is a pleasure to t h a n k Professors J. I. F u j i t a a n d T. K o t a n i a n d D r . D. B o g d a n f o r a s t i m u l a t i n g exchange o f letters. W e also w a n t to a c k n o w l e d g e v a l u a b l e s u p p o r t a n d discussions f r o m the N u c l e a r S p e c t r o s c o p y G r o u p o f the E T H in Z u r i c h , where this w o r k was initiated.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ll) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

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