Is the detailed shape of the nuclear charge distribution relevant to the ft values of superallowed Fermi transitions?

Nuclear

Physics

Al79

(1972) 297-304;

Q North-Holland

Publishing

Co., Amsterdam

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

IS THE DETAILED SHAPE OF THE NUCLEAR CHARGE DISTRIBUTION RELEVANT TO THE ft VALUES OF SUPERALLOWED FERMI TRANSITIONS? H. BEHRENS Institut fiir Experimentelle

Kernphysik

der Universitiit (TH)

und des Kernjorschungszentrums

Karlsruhe

and W. BUHRING II. Physikalisches

Institut der Universitiit Heidelberg

Received 1 October 1971 Abstract: The uniform nuclear charge distribution remains a satisfactory model for calculating electron radial wave functions and nuclear matrix elements even in the case of the superallowed Fermi transitions of light nuclei up to 54Co, where high accuracy is needed to determine the vector coupling constant of nuclear b-decay. The modifications due to the deviation of the realistic charge distribution from this simple model are found to be smaller than previously thought and therefore are negligible, provided that the nuclear radius, which is the most important shape parameter, has its correct value (i.e. the r.m.s. radii of the realistic and the model distribution are the same). Moreover, the dependence on the nuclear radius is fairly small for the light nuclei under consideration.

1. Introduction Superallowed Fermi transitions play an important role for determining the vector coupling constant of nuclear b-decay, theirft values being inversely proportional to the square of the Fermi matrix element MF and of the coupling constant g,. Apart from much effort to improve the accuracy of the experimental values I- 5, of the transition energies and half-lives, also many theoretical papers 6-14) were published in the last few years, which dealt with the calculation of the ft values and the several types of corrections to them. The present paper is devoted to the question of whether or not there is a relevant change in the f t values, if instead of the uniform charge distribution of the nucleus a more realistic one like a Gaussian distribution, for instance, is used. This investigation is based on our recent version 15)of the theory of nuclear /?-decay, which does not contain any restrictions as to the type of nuclear charge distribution and therefore is particularly suitable for this purpose. 297

298

H. BEHRENS

AND W. BUHRING

2. Connection hetweenft value and coupling constant on the basis of earlier work For the superallowed Fermi transitions we can write

/cco[1+~,wl=

2n31n2 &(1+&c)

M:’

The integrated Fermi function is denoted by f rather than f since our Fermi function has not been evaluated at the nuclear radius R but at the origin 16).The half-life is t as usual and a=& is the tine structure constant. The quantity C(W) is the spectrum shape factor averaged over the energies W of the fi-electrons as indicated by the bar. The Fermi matrix element M, is nearly equal to l/z for the superallowed Fermi transitions, so that @=2(1-6,) (2) where 6, is due to the isospin impurities of the initial and final nuclear state of the transition. The attempts to calculate 6, have been reviewed by Blin-Stoyle “). There are more recent papers by Damgaard 6, and Jaus ‘). It follows from all these papers that 6, < 0.005. According to Sirlin *), c and 6,(W) are the model-dependent and the modelindependent radiative correction, respectively ‘. The quantity S,(W) is that part of the radiative correction which depends neither on the structure of the strong nor the weak interaction. We have

where g (W W,, m) is an analytically well-known *) universal function of its arguments: energy w maximum energy W, and rest mass M of the p-electron. It describes the distortion of the P-spectrum by the radiative correction. For the ft value the appropriate average with respect to W is needed, as denoted by the bar. The numerical values “) of 6,(W), which mainly depend on W,, range from 1.5 y0 for “C to 0.8 % for 54Co. The part c of the radiative correction, on the other hand, is model dependent in that the structure of the strong interaction enters as well as the possible existence of an intermediate boson of the weak interaction. Since c does not influence the /Ispectrum it is expected to be equal for all the superallowed Fermi transitions. It may be thought of as a factor which in some sense merely modifies the value of the vector coupling constant. It has not been possible until now to obtain a unique reliable value of c. Some recent work which is of interest in this context has been done by Dicus and Norton 13*14). ’ See also Brene et al. “) and Blin-Stoyle and Freeman”). Eq. (I) implies the assumptton that the radiative correction is multiplicative rather than additive. An investigation of the Za’ terms would decide between these two possibilittes9~“).

NUCLEAR

CHARGE

DISTRIBUTION

299

3. Dependence on the shape of the nuclear charge distributiou The shape of the nuclear charge distribution p(r) enters eq. (1) in two ways, namely via both the integrated Fermi function f and the spectrum shape factor C(W). Besides the usual uniform distribution with radius R we consider the modified Gaussian distribution I*) (4) where

The most important shape parameter is the equivalent uniform radius R, i.e. the radius of a uniform distribution with the same r.m.s. radius. This quantity R is called the nuclear radius. It will be assumed that the nuclear radius is a well-known quantity for each nucleus. Consequently only such charge distributions are considered for comparison, which all have the same nuclear radius. Therefore the modified Gaussian distribution (4) effectively contains only one free parameter A, while the other a then is related to the nuclear radius R and fixed by the relation a=R

2(2+3/I) 5(2+5/I)

(5)

Another type of charge distribution appropriate to the heavier nuclei is the Fermi distribution. It is much closer to the uniform distribution than the modified Gaussian is, and as has been shown earlier in detail ‘*), the same is true for the values of the corresponding Fermi and related functions. So we can here confine our investigation to the modified Gaussian distribution, knowing that the true modifications due to the realistic charge distribution as compared to the uniform one are even smaller. We have calculatedft values +(without screening) for the uniform and the modified Gaussian distribution with several different values of the parameter A. They are shown in table 1. For the light nuclei up to 34C1 the differences are seen to be completely negligible. For “Co there is a 0.2% difference in the most unfavourable case of A =O. This is an overestimate since the Fermi distribution is more suitable for the Co nucleus. Sb the true difference in the ft value due to the deviation of the charge distribution from the uniform one can safely be expected to be smaller than 0.1% and therefore is negligible. It remains to investigate the influence of the shape of the charge distribution on the spectrum shape factor C(W), which contains the so-called second-forbidden terms. Using the notation of our recent paper Is) the form factor coefilcients or nuclear matrix elements ‘F&, (1, m, n, a) and “F,y,( 1, m, n, CT)occur in C(W). The magnitude ’ The values for the uniform distribution have bee.n improved here as compared to our earlier paper16) in so far as more recent experimental results I-‘) for the transition energies and half-lives of “‘C, ‘*O and lernAI have been used. The value used for the nuclear radius IS in the present paper again R = 1.2 Ai fm where A is the mass number.

300

H. BEHRENS AND W. BUHRING TABLE1

Values offt

(without screening) of superallowed Fermi transitions for several different nuclear charge distributions We-l=Ee

r (msec)

[keVJ

St uniform distribution

w

888.1 rfri.8

=0

1809.1 k1.5 3210.6 + 1.0 4459.7 +4.0 5409.0 f2.3 6032.1 k2.2 6609.0 f2.6 7227.lk3.8

26mAl %zf ?Sc 46V “Mn %o

(1275 $)

x lo3

71,057 &36 6346 rfr 5 1565 + 7 683.0+ 1.5 425.9 + 0.8 285.7+ 0.6 193.7* 1.0

modified Gaussian distribution A=0

A=1

A=2

2960

2960

2960

2960

3037 3035 3096 3077 3087 3077 3082

3037 3036 3098 3080 3091 3082 3088

3037 3036 3097 3079 3090 3081 3086

3037 3035 3097 3079 3089 3080 3085

of the relativistic matrix elements “F,:, = “Fe~r(l, M, n, 0) has been investigated in detail by Damgaard 6, and found to be negligible for our purpose t. Neglecting the contributions from a11 the relativistic matrix elements and other small terms one finds by means of the appendix tt of ref. t5) in a similar way as in ref. r6) C(W)=l-

~‘G ‘G&3

~(~R)2+f(qff)2+$(4R)(WR) c “F,&,(L

-$(uZ)(@)

1,1, 1)

“G&J “E&U,

-+z)(WR)

29% 1)

(6)

“F 000 ’ “Fog&

-f(lxz)2

29% 2)

“Go

*

Here 4 = W. - W is the neutrino energy in natural units, as usual, and Z the nuclear charge number of the daughter nucleus. For the positon decays we are interested in, formally a negative value of Z has to be inserted into eq. (6). The correspondence of eq. (6) to our earlier work is explained in the appendix. ’ Using eq. (29) of our recent paper I’), calculating the non-relativistic matrix elements of the superallowed Fermi transitions by the method of Blin-Stoyle”9), and taking account of the fact6) that here We+ 2.5 is equal to the Coulomb displacement energy .4&, we find

VFoLu,m n, a) <0.08 lorZ/. vGL

I ++ Eq. (A.10) with K=O, k,=k,=

I

1. The small contributions

from eq. (A.11) have been neglected here.

301

NUCLEAR CHARGE DISTRIBUTION TABLE2

Values of the nuclear matrix elements relative to the Fermi matrix element for several different nuclear charge distributions Uniform distribution

Modified Gaussian distribution A=0

‘F 000 ’ ‘F O

“Foho(l> LA “Fo80

1)

vFo:o(l> 2,291) “Fo:o “Fo:o(L

vF”

Z&2)

000

A=2

0.6

0.6

0.6

%=0.77143

0.75693

0.76127

0.76288

+$=0.81429

0.84351

0.83629

0.83194

ggj= 1.10952

1.21868

1.18825

1.17229

+=0.6

000

A=1

According to Blin-Stoyle I’), the nuclear matrix elements in eq. (6) can be calculated without reference to the nuclear structure in the case of the superallowed Fermi transitions under consideration. The result is Z(l, m, n, o; r)p(r) r2 dr, where p(r) is the nucleon distribution normalized t to unity and the functions Z(l, m, n, a; r), which depend on the shape of the charge distribution, can be found in our earlier paper “). Because of Z(l, m, n, 0; I) E 1 this equation contains as a special case the well-known result for the Fermi matrix element “FO~e(l, m, n,O)=“F,&=j/3.

(8)

The values of the nuclear matrix elements relative to the Fermi matrix element are shown in table 2 in the case of the uniform distribution and the modified Gaussian distribution for several values of the parameter A. These matrix elements do not depend on the nuclear radius and therefore apply to all the nuclei under consideration. The dependence on the nuclear radius of the spectrum shape factor C(W) is explicitly exhibited by eq. (6). Numerical values of the spectrum shape factor averaged over the j?-spectrum for each nucleus are presented in table 3. The shape dependence is seen to be extremely small, amounting to 0.1% for 54Co with the most unfavourable distribution, which again gives an overestimate. The overall shape dependence of m

’ We have I p(r)r’ dr=l.

Identical distributions

for neutrons and protons have been assumed in

0

eq. (7). More generally p(r) might bereplaced by f [h'pN(r)-zpP(r)], where pP(r) and pN(r) are the distributions of the protons and neutrons, respectively, and fi and 2 are the neutron and proton numbers of the neutron-rich nucleus of the T= 1 isospin multiplet. In extracting this expression from ref. i9) the different conventions for the isospin operators have to be noted.

H. BEHRENS AND W. BUHRING

302

ft C(W) is shown in table 4 and is even smaller than the shape dependence of ft alone, since C(W) and ft are influenced in opposite direction if the shape is modified. In spite of the high accuracy of the experimental data, the shape dependence is still so small for the nuclei of relatively low Z under consideration, that it remains satisfactory to use the simple uniform distribution. TABLE3 Values of C(W)- 1 for superallowed Fermi transitions using several different nuclear charge distributions Uniform distribution ‘OC I40

26mA1 “,%I 42sc ‘?f “Mn s‘%zo

Values offt

-0.000272 - 0.00@474 -0.001323 -0.002257 - 0.003508 -0.004174 -0.004911 -0.005688

Modified Gaussian distribution A=0

A=1

A=2

-0.000313 -0.000552 -0.001551 - 0.002656 -0.004130 -0.004921 -0.005795 -0.006719

- 0.000301 - 0.000530 -0.001486 - 0.002541 - 0.003950 -0.004704 -0.005539 - 0.006420

- 0.000295 -0.000518 -0.001453 -0.002483 -0.003860 -0.004597 -0.005412 -0.006272

TABLE4 C(W) (without screening) for superallowed Fermi transitions using several different nuclear charge distributions Modified Gaussian distribution

Uniform distribution

2959 3035 3031 3089 3066 3074 3062 3064

A=0

A=1

A=2

2959 3035 3031 3090 3067 3076 3064 3067

2959 3035 3031 3089 3067 3075 3064 3066

2959 3035 3031 3089 3067 3075 3063 3066

TABLE5 Values offt C(W) (without screening) for different values of the nuclear radius R = r, A* using the uniform distribution

r. (fm) 'OC I40 26mAI 34C1 42sc 46V “Mn 54co

1.2

1.3

1.4

2959 3035 3031 3089 3066 3074 3062 3064

2959 3035 3031 3087 3064 3071 3059 3061

2959 3035 3030 3086 3062 3069 3056 3058

1.5 2959 3034 3029 3085 3059 3066 3053 3055

NUCLEAR CHARGE DISTRIBUTION

303

The question of the dependence on the nuclear radius has partly been treated in our earlier publication 16) and also by Damgaard (j). Again the ft value and C(W) are influenced in opposite directions by changes of R. For completeness we give in table 5 the variation with the nuclear radius R of 3t C(W) using the uniform distribution. While for 140 the R-dependence is negligible within wide limits, its importance increases for the heavier nuclei. Although the R-dependence is still fairly small for 54Co, it is desirable here to use a nearly correct value. 4. Concluding remarks We have shown that the uniform charge distribution of the nucleus remains a satisfactory model for the superallowed Fermi transitions of relatively small 2 under consideration, since the deviations obtained by using a more realistic distribution are very small according to table 4 and irrelevant in view of the present experimental accuracy. This conclusion supports a statement by Damgaard 6, but is not compatible with results of Halpern and Chern 20) and of Dicus and Norton i3*14) who found larger effects. Their calculations, however, are based on first-order perturbation theory which might give an overestimate of the shape dependence. This seems to be related to the fact that nuclear b-decay has been found is* “) to be much less sensitive to the shape of the nuclear charge distribution than is suggested by the rather large shapedependence of the value of the electrostatic potential at the origin. Appendix CORRESPONDENCE

TO OUR EARLIER WORK

In the case of the uniform distribution nearly corresponds to the first two terms

the eq. (6) for the spectrum shape factor

1+_Mw “~0;0/%0 of eq. (A.6) of our earlier paper 16). The correspondence is not exact, however. Due to the fact that different types of expansions for the electron radial wave functions have been used, some terms are included here which were omitted in the earlier work since they are contained in H4. It should be noted that the earlier treatment 16) of the superallowed Fermi transitions remains satisfactory if combined with the method of Blin-Stoyle lg) for calculating the nuclear matrix elements. The reason is that here the nuclear matrix elements can be related to the shape of the nucleon distribution without reference to the nuclear structure. Then, if the uniform distribution is used for calculating the nuclear matrix elements (and the radius of the neutron distribution is not larger than the radius of the proton distribution), there is strictly no contribution to the radial integrals from the region outside of the nuclear radius. Therefore all the difficulties which had led us to develop our modified version of the theory of nuclear p-decay 15)do not appear here.

304

H. BEHRENS AND W. BUHRING

References 1) J. M. Freeman, J.G. Jenkin, D. C. Robinson, G. Murray and W. E. Burcham, Phys. Lett. 27B (1968) 156 2) J. M. Freeman, J. G. Jenkin and G. Murray, Nucl. Phys. A 124 (1969) 393 3) J. M. Freeman, J. G. Jenkin, G. Murray, D. C. Robinson and W. E. Burcham, Nucl. Phys. A 132 (1969) 593 4) P. de Wit and C. van der Leun, Phys. Lett. 3OB (1969) 639 5) G.J.Clark, J.M. Freeman, D.C. Robinson, J.S. Ryder, W.E. Burchham and G.T.A.Squier, Phys. Lett. 3SB (1971) 503 6) J. Damgaard, Nucl. Phys. A130 (1969) 233 7) W. Jaus, Nucl. Phys. Al62 (1971) 97 8) A. Sirlin, Phys. Rev. 164 (1967) 1767 9) L. J. Logue and B. Chem, Phys. Rev. 175 (1968) 1367 10) W. Jaus and G. Rasche, Nucl. Phys. Al43 (1970) 202 11) N. Brene, M. Roos and A. Sirlin, Nucl. Phys. B6 (1968) 255 12) R. J. Blin-Stoyle and J. M. Freeman, Nucl. Phys. AlSO (1970) 369 13) D.A. Dicus and R. E. Norton, Phys. Rev. D l(l970) 1360 14) D.A. Dicus, preprint 15) H. Behrens and W. Biihring, Nucl. Phys. Al62 (1971) 111 16) H. Behrens and W. Biihring, Nucl. Phys. A106 (1968) 433 17) R. J. Blin-Stoyle, Isospin in nuclear p-decay, Isospin in nuclear physics, ed. D. H. Wilkinson (North Holland, Amsterdam, 1969) 18) H. Behrens and W. Btihring, Nucl. Phys. A150 (1970) 481 19) R. J. Blin-Stoyle, Phys. Lett. 29B (1969) 12 20) T.A. Halpern and B. Chern, Phys. Rev. 175 (1968) 1314