Nuclear matrix elements in the first forbidden beta decay of 198Au

4.C

I

Nuclear Physics A159 (1970) 143-- 152; (~) 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

BETA

DECAY

OF

tgSAu

H. A. SMITH t and P. C. SIMMS Department of Physics, Purdue Unioersity, Lafayette, Indiana 47907 tt Received 30 July 1970 Abstract: Nuclear matrix elements for the 2-(0.959 MeV ~ - ) 2 + beta transition in 198Au were extracted and comparison was made with CVC theory predictions for the vector matrix element

ratio Acvc= Set/Sir/p. It was found that the ratio Acvc was consistent with the CVC prediction based on the assumption that the Coulomb Hamiltonian is diagonal. However, contributions from off-diagonal matrix elements of the Coulomb Hamiltonian could not be ruled out. Fairly good limits were placed on the nuclear matrix elements in spite of the fact that the Coulombenhanced combinations of matrix elements dominate the transition.

1. I n t r o d u c t i o n

This paper is part of a series devoted to the analysis of first forbidden beta transitions 1-4). The objective of this series is to use the available experimental data on a selected group of isotopes to extract the beta-decay nuclear matrix elements and to obtain the experimental limits on the vector matrix element ratio (Sat~Sir~p). This ratio is interesting because it can be compared to theoretical predictions which are based on the conserved vector current (CVC) theory for beta decay. Furthermore the experimental value of the vector matrix element ratio can provide information on the impurities in isobaric analogue states 2). One would expect that because of the high Z in 198Au the l-approximation s, 6) would hold. That is, Coulomb-enhanced terms would dominate the formulas for the observables. (For 19aAu ~ _- 16.4 > Wo = 2.87, in natural units.) I f the ~-approximation were valid for this nucleus, a complete matrix element extraction would be impossible; and only the matrix element combinations V and Y could be determined. Fortunately there is a lot of accurate experimental data available, so 198Au offers the best possibility of determining the matrix elements for a transition in which V and Y are dominant. The spectrum shape correction factor 7), the beta-gamma directional correlation coefficient 8), and the energy dependence of the beta-gamma circular polarization correlation a) have been accurately measured. Recently a very accurate measurement has been made of the angular distribution of E2 g a m m a rays following the beta decay of oriented nuclei 9). The angular dependence of the beta-gamma circular polarization correlation 1o), the longitudinal 11), and transverse 12) polarizat Present address: Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 87544. tt Work supported by the USAEC under contract no. AT(II-1) 1746. 143

144

H . A . SMITH AND P. C. SIMMS

tion of the beta particle have also been measured; but the matrix elements are not sensitive to these observables. The analysis presented here is more accurate than previous studies of 19SAu for several reasons. In addition to new data from nuclear orientation experiments, finite nuclear size effects, screening of atomic electrons, and higher order matrix elements are included in the analysis. 2.

Theory

2.1. M A T R I X E L E M E N T P A R A M E T E R S A N D CVC P R E D I C T I O N S

The matrix element parameters used in the analysis are those proposed by Kotani 1a) and modified by Simms x). The first forbidden parameters are defined in table 1. The TABLE 1 The first-forbidden matrix element parameters D = ½ctZ+]Wop

Spin change zero

D'v = ~ f

Y5

1 f,;,

W = -- - C A

DV = D'v+Dw Spin change one

D'y = 1 C v f m

x = 1-c fi'Vop

r/

U = -C A

DY = D'y-D(x+u) Spin change two z =

1

--C

r/

A

.~ p

definitions of the higher order parameters (which are included in the analysis but not explicitly extracted) may be found in ref. 1) or ref. 2).

J.gSAu .B-DECAY

145

The calculations of Fujita 14) and Eichler 15), using CVC theory, yield the following result (in natural units) for the vector matrix element ratio [see ref. 2)]:

A ° c ------

f,

= (Wo~2.5)p_+2.4-½¢Z,

(1)

f ir[p for fl~: decay: IVo is the beta transition energy, p and Z are the radius and charge of the daughter nucleus. The Fujita-Eichler result utilizes the Ahrens-Feenberg approximation 1e), which assumes the off-diagonal matrix elements of the nuclear Coulomb Hamiltonian are negligible compared to the diagonal elements. The more general approach of Damgaard and Winther xv) does not rely upon the diagonality of the Coulomb Hamiltonian. Instead, a realistic form for the nuclear Coulomb potential energy is proposed, leading to the following prediction for the vector matrix element ratio:

I,

Acvc = - -

- (W0~2.5)p_+½~tZ(3-2),

(2)

f ir/p where the parameter 2 is defined as:

~.--JP

\P/ .

(3)

f'; P

Acvc

For 2 = 0.6 the two expressions for give the same numerical result. A complete discussion of this parameterization of the vector matrix element ratio is given in ref. 2). 2.2. USEFUL FORMULAS FOR SOME OBSERVABLES The formulas for the general beta-gamma angular correlation functions are presented in detail in ref. 2). The formulas for the E2 gamma-ray angular distribution following the beta decay of oriented nuclei and the electron longitudinal polarization are presented here for completeness. Consider a first forbidden beta decay from an oriented nuclear state with spin and parity/~- to an intermediate state with spin and parity 2 +, followed by an E2 gamma transition to a state with spin and parity 0 +. If the beta particle is not observed and the angular distribution of the gamma rays relative to the orientation direction is measured, the result is: Vr'(0) = 1 - x / A ( 2 I i + l)f2V2P(cos O ) - x / } ( 2 I i ÷ 1) f4U4P4(cos 0).

(4)

146

H. ^ . SMITH A N D

p. c . StMMS

The quantitiesf, are the statistical tensors of Fano i s, t 9), which describe the orientation of the parent nucleus. The parameters U. describe the change in orientation of the initial system caused by the intermediate beta transition. They contain the nuclear matrix elements of the beta transition and the spins of the initial and intermediate states [see ref. 20)]: P.(cos 0) are the nth order Legendi'e polynomials, and 0 is the angle between the axis of the gamma detector and the axis of symmetry dictated by TABLE 2

Beta-gammare-orientationparameters for two important spin sequences 2- ~ 2 + ~ 0 + U2 = 1~o12-1-½1~112-31~212 U+

lO~ol2-~l~tl 2

~

2

3-_..,2+__,0 + U 2 = ~/-~5 (21~, 12+ ½l~2l2)

v,

=

/b I~:I ~ -

F(Z, W)pWq2btfl)dW

flj = 1

For hto~ ujj , see ref.2).

the orientation of the parent nucleus. The U. are used to extract the nuclear matrix elements of the beta transition. The general expression for U. is given below for the spin sequence ff(fl)2+(~)0+:

:=o, 1,2

2

Ic¢:1L

(5)

The quantities [c~l2 are the probabilities that the lepton field will carry away J units of angular momentum. The Icc:l z are normalized such that luol2+l~xl2+lcql2 = 1.

(6)

The symbol in curly brackets is a Wigner 6-j symbol. Explicit expressions for the U2 and the U+ for spin sequences 3-2-0 and 2-2-0 are given in table 2. The U. are normalized so that Uo = 1.0. If, instead of the statistical tensors of Fano, one uses the orientation coefficients, f . , of Tolhoek et al. 19,21,22), the angular distribution in eq. (4) takes the form

W(O) = 1 - ~ f z U2P2(cos 0 ) - ~ f 4 U+P+(cos 0),

(4a)

19SAu #-DECAY

147

SET A SET B

o

I--Z

tr-

8 o.oz 2.8

t_) t~

o 2.6 z 2.4 _o

~

S

E

T

SET

B i O.OI A ,~

2.2 la.l

a:_

u

w 2.0 a. ~ 1.0

O

×

I 2I.O I' 3.0 BETAPARTICLEENERGY (NATU . NITW S)e

Fig. 1. Experimental shape correction factor from ref. ~), with theoretical values for matrix element sets A and B.

o.o LO

I 2.0

I ~ 5.0

BETAPARTC I LEENERGY (NATU .NITS)

Fig. 2. Experimental r 7 directional correlation coefficients from re£ s), with theoretical values for matrix element sets A and B.

v

Z

_o

d -O.35 •

u Z

I~ S E T A -SETB

o_ N

-0.30 r~

-0.25 I

LO

2.0

I

w~0 3.0

BETA PARTICLE ENERGY (NAT. UNITS)

Fig. 3. Experimental energy-dependent r7 circular polarization correlation from ref. a), with theoretical values for matrix element sets A and B.

148

H.A. SMITH A1~ P'. C. SIMMS

where the U.'are given by t U. = (½It)" F(5+'~)! ( 2 I i - n ) l ~ L(a-n)t ( 2 I , + n + l ) t ~ 2 I i + l ) U--..

(5a)

Regardless of the spin sequence of the intermediate beta transition, Uo = Uo. Furthermore, it is clear from eq. (5a) that if I i = 2, U. is equal to On. Thus, for the 2-2-0 spin sequence, one need not ifistinguish between U. and On. However, for the 3-2-0 spin sequence, the Un and Un' are related by V2(3-2-0 ) = 1.08 U2(3-2-0), U4(3-2-0 ) = 0.808 U4(3-2-0). The longitudinal polarizafi0n of the emitted beta particle, with the ( - p / W ) energy dependence removed, is given below for first forbidden beta transitions. i(ll~,,.~(Ad(1)~,~(l)j_ ~f(0) ~,~(0)"I (~,~(0)~2 (~t4(1)~2~ P' =l+21"'"J""tt'"tt . . . . ll'"ll/--k'rt'll/ --\H'll/ / .

-p/W

(7)

c(w)

The matrix element combinations Mtx) k©k~, and m~v are defined in ref, 2). The quantity C(I4") is the spectrum shape.correction factor defined in terms of the matrix element parameters in ref. 2). 3. Analysis and results

The 2-(0.959 MeV fl-)2 + (0.412 MeV 7)0 + t7 cascade in 19SAu was analyzed with the following experimental data being imposed: The spectrum shape correction factor 7) (fig. 1); the beta-gamma directional correlation s) (fig. 2); the beta-gamma TABLE 3 Nuclear orientation data s u m m a r y for agsAu ( 2 - ~ 2 + ~ 0 +)

02(U2) experiment

set A set B

0.795-t-0.007

0.796 0.798

O,(U,) 0.347=t=0.030

0.321 0.327

circular polarization correlation s) (fig. 3); the longitudinal polarization of the electron 11); and nuclear orientation data 9) (table 3). The analysis of 19amu showed that the observables were relatively insensitive to the magnitude of the B~j matrix element, except that dominance of the transition by B~j was not allowed. Furthermore, all of the matrix elements were reduced by about an order of magnitude relative to their maximum possible physical size. * The re-orientationparameters, which are called U. here and in refs. x9.2~.22), are called B. in ref. 20).

a~SAU:#-DEC~y.

, ..

,I~49

The experimental limits on the vector matrix element ratio are consistent with the assumption that the o ff-diag0naI contt~utions fr0m t h e C0ulgmb Hamflt0nian are negligible (see fig. 4). For 2. = 0.6 the experimental limits and the theoretical prediction for the vector matrix element ratio are barely in agreement: A~¢c(2 = 0.6) = 0.707, A~c(Z = 0.6) -- 0.600_+0.125.

•

(Sa) (8b)

For 2 = 1.4 the center value of the experimental limits agrees with the theoretical prediction for A: ALgvc(2 = 1.4) = 0.475, (9a) A~-~c(2 = 1.4) = 0.475_+0.125.

(9b)

0.8 ~_~(lheorelicoi ) 0.7

o.6

'" ~

o.3

e~p

~ ~ ~

0.2 0.1

0.0

I

1.0

I

X

2.0

Fig. 4. Variation o f the experimental limits a n d the theoretical prediction for Acvc as a function o f the p a r a m e t e r ~..

In order to maintain agreement between theory and experiment for A, the parameter 2 must be restricted to the values 0.5 _< 2 _ 2.2. (10) Table 4 shows the extracted limits on the normalized matrix elements. The matrix element parameters are not presented here because the limits of error on the actual matrix elements are somewhat better than on the matrix element parameters. The tighter restrictions on the matrix elements occur because the variations in the scale factor, ~/, are highly correlated with the variations in the matrix element parameters. Table 5 shows typical matrix element parameter sets for two cases: (A) The Bij matrix element approximately equal to the other matrix elements, and (B) The B u matrix element suppressed. Plotted in figs. 1, 2 and 3 and displayed in table 3 are the theoretical values, calculated with sets A and B for the experimental observables used

150

H. A. SMITH AND P. C. SIMMS

TABLE 4 Normalized matrix elements for tgSAu

Matrix element

f

Value

-I-iBlj

0.00524-0.0021

P

0.0270-t-0.0070

I,

0.0419-I-0.0101

0.0795 4-0.0315

f uxr

-- 0.0477 -4-0.0239

~Y

0.0302+0.0018

~V

--0.1)433 4-0.0005

7

r

--0.0379-t-0.0361

TABLE 5 Typical matrix element parameter sets for *gaAu Parameter

Value set A B u not suppressed

Zo

1.0000 1.6025 2.6710 1.984 1.073 -- 1.542 -- 1.41 0.032 0.8

D'yo x u Y Y We ~? t

set B B u suppressed 1.0000 3.9846 12.404 4.931 3.46 --4.93 -- 10.0 0.009 1.90

in t h e analysis. T h e s e p l o t s s h o w t h a t it w o u l d be e x t r e m e l y difficult t o d i s t i n g u i s h e x p e r i m e n t a l l y b e t w e e n sets A a n d B. 4.

Discussion

I n all o f the m a t r i x e l e m e n t sets e x t r a c t e d , the c o m b i n a t i o n s V a n d Y d o m i n a t e d t h e transition. The wealth of experimental data, however, made a more complete matrix e l e m e n t e x t r a c t i o n possible.

19SAul-DECAY

151

A qualitative interpretation of the matrix element results can be obtained using the single-particle shell model. The reduction in the size of the matrix elements is not surprising since the transforming neutron in 19SAu is in a different major shell from the proton which is formed in X9SHg. The Nilsson diagram for neutrons would suggest that the ll9th neutron in 19a--79/o~u119 resides in the 2f~ subshell. However, the ground state spin and parity of 199 s0Hg119(½ ) indicates that the ll9th neutron is in the 3p½ subshell. Therefore, the real wave function for the transforming neutron probably contains both 2ft and 3p~ components. The ground state spin and parity of 197Au and 199Au (both ½+) imlrlY that there is a proton hole in the 2d~ subshell. This possibility is consistent with what one would expect from the Nilsson level diagrams, However, if the proton which is formed in the beta transition is in the 2d~ subshell, the AJ = 0 matrix elements would not be permitted by the available single-particle transitions (2f~--} 2d~ and 3p½ ~ 2d~). Since these matrix elements make a significant contribution to the beta decay of 1soatu,9S--it is unlikely that the newly formed proton is in the 2d~ subshell. In the Nilsson diagram the next state above 2d~ is 3s½. All of the experimental results are consistent with a 3s½ proton being formed in the beta transition. This proton would couple with the 2d~}proton hole to form the 2 + first excited state of 198., 80t'Ig118 . TheAd = O, I matrix elements could occur in the 3p~ ~ 3s~ transition, and the 2f~ --* 3s~ transition would permit the AJ = 2 matrix element to be present. The limits on the parameter 2 in eq. (10) are consistent with the assumption that the nuclear Coulomb Hamiltonian is diagonal (2 = 0.6). However, since 2 can be as large as 2.2, significant contributions to the vector matrix element ratio from the offdiagonal matrix elements of the Coulomb Hamiltonian cannot be ruled out. Ref. 2) contains an extended discussion of the implications that these off-diagonal Coulomb matrix elements have for isobaric analogue states. The values obtained for the matrix elements in the present analysis are quite different from the results of Manthuruthil and Poirier 24) (hereafter referred to as MP) who have reported that the matrix element SBiffp is much larger than the other matrix elements. In two previous papers, Steffen 8,23) came to the opposite conclusion when he found that the Bii contribution was very small compared to the Coulomb enhanced matrix element combinations (V and Y). It is also difficult to understand why the matrix elements presented in MP give the wrong sign for the ratio, A. Even though the size of A cannot be predicted accurately, there is no reason to doubt the theoretical prediction for its sign. The results of the present work are in good agreement with the results of Steffen and the theoretical prediction for A. Since a very accurate nuclear orientation measurement and additional data on the shape correction factor were available, our analysis sets much better limits on the matrix elements than Steffen's analysis does. The different nuclear orientation data used by MP does not explain why their results disagree with ours. When we use the same data that was used by MP, the disagreement is still present.

152

H . A . SMITH AND P . C. SIMMS

There are two points in the paper by MP which could explain the disagreements. First, the paper states that a negative number was used for the nuclear charge Z. The formulas of Morita 26) which were used by MP require a + Z for negaton decay. Second, in fig. 5 of MP the py circular polarization Py(O#~-- 180 °) is shown as a positive quantity. However, the original reference for the circular polarization shows that A: is a positive number. Since P~(180 °) = ALP1(180°), P~ must be negative not positive. All of the experimental data indicate that the 0.959 MeV transition of 198Au is dominated by the Coulomb enhanced matrix element combinations (V and Y). Our results agree with this observation. The results of MP are indirect disagreement, and there are several aspects of their analysis which are unclear and may be incorrect. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ll) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)

P. C. Simms, Phys. Rev. 138 (1965) B784 H. A. Smith and P. C. Simms, Phys. Rev. Cl (1970) 1809 J. Bosken, D. Ohlms and P. C. Simms, to be published H. A. Smith and P. C. Simms, to be published T. Kotani and M. Ross, Phys. Rev. Lett. 1 (1958) 140 T. Kotani, Phys. Rev. 114 (1959) 795 M. D. Parsignault, Compt. Ren. 2.59 (1964) 1515 R. M. Steffen, Phys. Rev. 118 (1960) 763 W. P. Pratt, Nucl. Phys., to be published J. P. Deutsch and P. Lipnik, Nucl. Phys. 24 (1961) 138 (3. Schwarz e t al., Z. Phys. 217 (1968) 465 P. C. Simms, Phys. Rev. 119 (1960) 768 T. Kotani and M. Ross, Pros. Theor. Phys. (Kyoto) 20 (1958) 643 J. I. Fujita, Phys. Rev. 126 (1962) 202 J. Eichler, Z. Phys. 171 (1962) 463 T. Ahrens and E. Feenberg, Phys. Rev. 86 (1952) 64 J. Damgaard and A. Winther, Phys. Lett. 23 (1966) 345 U. Fano, Nat. Bur. of Standards, Washington, D.C., Report No. 1214 (1951) H. A. Tolhoek and J. A. Cox, Physica 19 (1953) 101,673 F. M. Pipkin, Phys. Rev. 129 (1963) 2626 H. A. Tolhoek et al., Physica 20 (1954) 1310 H. A. Tolhoek and J. A. Cox, Physica 18 (1952) 357 R. M. Steffen, Phys. Rev. 123 (1961) 1787 J. C. Manthuruthil and C. P. Poirier, Nucl. Phys. A l l 8 (1968) 657 M. Morita and R. S. Morita, Phys. Rev. 109 (1958) 2048