Nuclear matrix elements in the first-forbidden β-decay of 125Sb

Nuclear Physics A202 (1973) 602-608;

@

North-Holland

Publishing

Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

NUCLEAR MATRIX ELEMENTS IN THE FIRST-FORBIDDEN /I-DECAY OF 12sSb J. S. SCHWEITZERt Tundem Accelerator

Laboratory,

and P. C. SIMMS

Purdue University,

Received 26 September (Revised

11 November

Lafayette,

Indiana 47907,

USA tt

1972 1972)

Abstract: Nuclear matrix elements in the %+(444 keV jz?-)%-(177 keV r)+p-7 cascade in r*‘Sb have been extracted from the available experimental data. A significant cancellation in the Coulomb-enhanced matrix element combination permits good limits of error to be. placed on the individual matrix elements, even though there is not a large amount of experimental data available.

1. Introduction First-forbidden p-decays have been studied with the objective of determining the nuclear matrix elements of the transition from the experimental observables. An attractive feature of such studies is the possibility of determining as many as six primary nuclear matrix elements between the same two nuclear states. If nuclear matrix elements can be obtained for nuclei differing by only a few nucleons, insight can be obtained into the systematics of a region. The antimony isotopes provide an excellent example of a region where the experimental data are sufficient for determining the nuclear matrix elements in the firstforbidden decays of a few nuclei. We have previously determined ‘) the nuclear matrix elements for /?- decays of lz2Sb and rz4Sb. Because of the large log-0 value (logft = 9.3) and the significant energy dependence of the P-7 directional correlation coefficient 2q3), it was expected that the nuclear matrix elements for the $+(4&l keV /.I-)%- transition in 12’Sb could also be extracted from the experimental data. Data from nuclear orientation measurements 4, ‘) were used for the first time to restrict the possible matrix elements in this transition. The analysis was based on the formulas presented by Biihring “) and modified by Simms ‘). These formulas include higher-order matrix elements which arise from the radial expansion of the electron radial wave functions, screening corrections *), and the effects of finite nuclear size. The matrix element parameters used in the analysis are the normalized parameters presented by Simms ‘), which are based on the paramf Present address: California Institute of Technology, Pasadena, California 91109. tt Work supported by the National Science Foundation. 602

lzssb FIRST-FORBIDDEN

B-DECAY

603

eters originally proposed by Kotani ‘). A description of the analysis program and the method for determining the limits of error for the individual matrix elements may be found in refs. ‘, lo), In addition to requiring agreement with the experimental data, possible sets of matrix elements can be required to agree with the Damgaard-Winther formulation ‘i) for the ratio of two vector-type matrix elements: A

‘“’

z -

sa-- W,p+(3-l)$aZF2Sp,

f irjp

for fir decay (in natural units). Here Wo, 2 and p are the endpoint energy, charge of the daughter nucleus, and the nuclear radius, respectively. The parameter I is defined as: ‘ir

12

E

r

2

J0 --

s-

pp

.

ir

P

The theoretical and experimental evidence for the validity of eq. (1) has been discussed in ref. r2). By including ncvct a restriction is placed on the most important higher-order matrix elements which appear in the analysis r2). A determination of ncvc can yield information on the importance of the off-diagonal matrix elements of the Coulomb Hamiltonian lo),

2. Fomwlas for using nuclear orientation data The formulas for most of the observables in first-forbidden /?-decay have been presented previously 6S7*13). However, the expressions for describing the decay of oriented nuclei have been given for only particular spin sequences. In addition, there have been two different forms for describing the angular distributions of y-rays following the unobserved #?-decay of the initial nuclei, and the relationship between them has often not been clearly established. Therefore, the general formulas are presented here for completeness. If oriented nuclei with angular momentum IO decay by unobserved B-decay to an excited state with angular momentum Ii, the angular ~stribution of de-excitation y-rays between the states lIi> and 11,) is described by

The maximum value of I which will appear in the summation is given by 1mm = min (21,) 23,,, ,2L),

where J,,,,, is the maximum angular momentum carried away by the lepton field, i.e.

J. S. SCHWEITZER AND P. C. SIMMS

604

for first-forbidden b-decays J,,,,, = 2; and L is the angular momentum carried away by the y-ray. In most experimental situations, the polarization of the y-ray is not observed so that information is obtained only from terms with even values of A. The fA(Z,) are statistical tensors 14) which describe the initial orientation: fA(ZO)= (2Z0+ l)+ 2 (- l)‘o+m(z, -mz,m]l0)u,, m

(4)

where the a,,, are the relative populations of nuclei in the magnetic substate m. The D,(j) describe the change in orientation of the nuclei caused by the unobserved j?transition ’ 5), and they are the only terms which depend on the nuclear matrix elements of the p-transition:

w> =(-

1)r0+r1+A[(2Zo+ 1)(2Z, + 1)-J) cJ

(- 1)” 1;

;o 1

1

;) ]c$]?

(5)

The ]cl, 1’ are the normalized probabilities that the lepton field carries away J units of angular momentum. The IaJj2 for first-forbidden B-decay are given by:

IcrJ12 = pJ =

OJ

Bo+P1+PI’

j-;P(Z, W)pWq2b$:‘dW

(For b$O,’see ref. i”).) The other quantities in eq. (3) are the Legendre polynomials PA(O), where 0 is the angle of the observed y-radiation relative to the orientation axis, and the FA(LL’Z2 I, ) which are the F-coefficients for the y-transition (the F-coefficients must be replaced by appropriate combinations of F-coefficients if multipole-mixing is present). There has been some confusion caused by the use of another form for the angular distribution: Wfl) = Y$ol(z,)f,(lo)U,(B)F,(LL’z,

ZlPdQ.

(6)

The orientation coefficients fA (I,) [ref. i “)I are not statistical tensors. However, they can be expressed in terms of theyA( PA

= SA(ZO)/~A(ZO)~

(7a)

where o,(Z,) is given by

4Zo)

=

21 z” (21, + 1)(21+ 1)(2Z, -n)! 1 0 (2Z,+I+l)!

0 [

1* *

Pb)

The o,(Z,) which appear in eq. (6) are obtained from eq. (7b) by substituting Z1for IO. Since eqs. (3) and (6) describe the same angular distribution, they must be equivalent. This yields the requirement that 4Z ,)fA(ZO)Un(B) = UZO) UP),

(8)

~T3b FIRST-F~RB~DDEN

@-DECAY

605

for all values of /1. By using eq. (7a), this condition can be seen to be satisfied if

which can be written explicitly as:

(9) The reasons for possible confusion are now apparent. If Z, and Z1 are the same, then U,(p) and o&3) are identically equal. From the form of eq. (9), it is also clear that when lZo-I1 ) is one the difference between Un(/3) and U,(j.?) is often not large enough to create an obvious disagreement. In particular, for the &decay of oriented “‘Sb nuclei to the 4- state, u, (8) = 0.95G

(B),

U*(B) = 0.98704 (/I). If a calculated value of U, (U,) is compared with an experimental result which was Uz (U2), it will cause inaccuracy in the extracted nuclear matrix elements. Once the form of the angular distribution used in an experiment (eq. (3) or (6)) has been determined, the nuclear deorientation parameters can be correctly calculated by using eq. (5), and eq. (9) where applicable. The calculated and experimental values can then be accurately compared, regardless of which parameters (U,(8)or u, (8)) were used in the experimental analysis. 3. Analysis and resuIts Though the two measurements of the directional correlation coefficient ‘=“> are similar, they were used separately in the attempt to find valid solutions for the p-decay nuclear matrix elements. As expected, there were no appreciable differences in the results, so only the most recent measurement ‘) was used in the remainder of the analysis. In addition to the nuclear orientation data 4* ‘), a restricted-energy-range shapecorrection factor measurement was used I’). The constancy of the shape factor is only reported over a limited energy range, and there is an indication from the present analysis, as well as the previous one ‘), that deviations are expected at low electron energies and near the endpoint energy. A final consideration in the analysis involves the mixed nature of the 177 lceV y-ray. Deduced values for 6(E2/Ml) for this y-transition have been in the range of -0.5 to - 1.7. Preliminary attempts to determine nuclear matrix element solutions were made forvarimrs values of 6 witbin this raw. There were no appreciable changes in the matrix element solutions as a function of the mixing ratio 6. A complete analysis was then performed using ~(E2~Ml) = -0.93.

606

J. S. SCHWEITZER

AND P. C. SIMMS

Initially, no agreement was required between ACXP

=

D’y,/x

and ncvc (given in eq. (1)). In this portion of the analysis 3, was set equal to 0.6. The limits of error obtained for the nuclear matrix elements obtained in this analysis are shown in table 1. An attempt was made to see if limits could be set on the parameter 1. The only restriction which could be placed by the present data was that 1”could not be more positive than 1.5. This is in contrast to the earlier result “) that the solution for this isotope could not yield agreement with ~Icvc if I was set at 0.6. If nexp and &vc are required to agree within +20 % and A is limited to the range from + 1 to - 1, then j ir/p is restricted to the upper half of the uncertainty range shown in table 1. The limits of uncertainty on the other matrix elements are unchanged. TABLE

1

Extracted nuclear matrix elements for lzsSb with 1 = 0.6 Matrix element

s i&j -

Value 0.1624&0.0168

P

QXT s-

I

P ir -

-0.1263~0.0087

0.0520&0.0510 0.0347&0.0092

I: rly

-0.0031 co.0021

We investigated the effect of further experimental data on the present solution. It would be difficult to measure the spectrum shape-correction factor accurately enough to further restrict the limits on the nuclear matrix elements. However, if the uncertainties on the directional correlation coefficient could be reduced by a factor of 2, all of the limits of error on the matrix elements would be significantly reduced. Furthermore, limits on the parameter I could probably then be obtained. While the individual A J = 1 matrix elements shown in table 1 are similar to the previous results 3), the size of the Coulomb enhanced matrix element combination Y is substantially smaller in the present analysis. In addition, the Bi, matrix element is an order of magnitude larger in our solution. These changes in the nuclear matrix elements were required for agreement with the nuclear orientation data, which was not used previously. The reasonably good limits of error which have been placed on the individual matrix elements are due to the extremely large cancellation in Y. Significant cancellation in Y has also been found for the other antimony isotopes whose nuclear matrix elements have been determined ‘).

l=Sb

FIRST-FORBIDDEN

607

#?-DECAY

4. TXaei&on Since “‘Sb has only a g, proton outside a closed shell, it may be possible to obtain a qualitative description of the expected nuclear matrix elements for this transition

by using a simple shell-model picture. The Q- state would then involve an odd neutron in the h, orbital. A calculation of the matrix elements using these single-particle wave functions was performed, using the method described by Delabaye and Lipnik ‘*). The results of the calculation for the non-relativistic matrix elements are shown in table 2, normalized to the B,j matrix element. These results clearly indicate that a single-particle shell-model description of these states is inadequate. A description of the $- level in lz ‘Te must apparently be more complicated. Kisslinger 19) has shown that certain important features of this state can be satisfactorily explained if the $- state is assumed to be formed from three quasiparticles of the h, orbit. Subsequent experimental data on the g-factor 20) and the B(E2) value deduced

A comparison of the experimentally

Matrix element ratio

TABLE 2 determined matrix element ratios with the predictions of a singleparticle calculation Single-particle prediction

-0.83

0.89

Resuft of present work 0.3203cO.316

-0.778&O.t13

from a lifetime measurement 2’) were in reasonably good agreement with the threequasiparticle description. It should be noted, however, that the B(M1) value obtained in ref. 21) indicates additional impurities in this state, thus leading to a further complication in any attempt to calculate the nuclear matrix elements for the /?-decay to this level. The lack of any significant hp strength in the f- level in 12’Te is confirmed by the ’ 24Te(d, p) lz5Te data ‘“). No evidence of the h, orbit was found in this experiment. If the -u- level in ‘25Te is assumed to be a pure single-particle level, then the experimental’results indicate that no more than at most 10 % of the wave function of the SW level is due to the h, orbit. Since the ma~itudes of the matrix elements in table 1 are not reduced by this large a factor, it is necessary to include contributions from the three-quasiparticle portion of the wave function to account for the extracted matrix elements. In addition, this admixture could explain the disagreement with the matrix element ratios of table 2. A detailed calculation of the @decay matrix elements expected when the e- state is described by both one and three quasiparticles would be useful to see if the results for the matrix elements can be accounted for.

608

J. S. SCHWEITZER

AND P. C. SIMMS

References 1) H. A. Smith, J. S. Schweitzer and P. C. Simms, to be published 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)

L. DuBard, L. D. Wyly, Jr., and C. H. Braden, Phys. Rev. 150 (1966) 1013 M. L. Narasimha Raju, V. Seshagiri Rao and D. L. Sastry, Phys. Rev. C2 (1970) 566 H. R. Andrews, T. F. Knott, B. Greenebaum and F. M. Pipkin, Phys. Rev. 169 (1968) 978 K. S. Krane, J. R. Sites and W. A. Steyert, Phys. Rev. C4 (1971) 565 W. Biihring, Nucl. Phys. 40 (1963) 472 P. C. Simms, Phys. Rev. 138 (1965) B784 W. Btihring, Nucl. Phys. 61 (1965) 110 T. Kotani and M. Ross, Prog. Theor. Phys. Jap. 20 (1958) 643 H. A. Smith and P. C. Simms, Phys. Rev. Cl (1970) 1809 J. Damgaard and A. Winther, Phys. Lett. 23 (1966) 345 I. S. Schweitzer and P. C. Simms, Nucl. Phys. Al98 (1972) 481 A. de Beer, H. M. W. Booij, 5. H. Stuivenberg and 5. Blok, Nucl. Phys. A160 (1971) 351 U. Fano, National Bureau of Standards, Washington, DC, report no. 1214 (1951) F. M. Pipkin, Phys. Rev. 129 (1963) 2626 H. A. Tolhoek and J. A. Cox, Physica 18 (1952) 357, 19 (1953) 101; J. A. Cox and H. A. Tolhoek, Physica 19 (1953) 673 G. M. Palmer, A. P. Arya and C. L. Walls, Bull. Am. Phys. Sot. 14 (1969) 569; A. P. Arya, private communication M. Delabaye and P. Lipnik, Nucl. Phys. 86 (1966) 668 L. S. Kisslinger, Nucl. Phys. 78 (1966) 341 D. W. Cruse, K. Johansson and E. Karlsson, Nucl. Phys. Al54 (1970) 369 A. Marelius, J. Lindskog, 2. Awwad, K. G. Valivaara, S. E. HPgglund and J. Pihl. Nucl. Phys. Al48 (1970) 433 A. Graue, J. R. Lien, S. Reyrvik, 0. J. Aaroy and W. H. Moore, Nucl. Phys. Al36 (1969) 513