Shell model investigation of RaE beta-decay

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Nuclear-Physics40 (1963) 338--346; ~ ) North-Holland Publishing Co., Amsterda~n 4.C

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

SHELL M O D E L INVESTIGATION

OF RaE BETA-DECAY

RICHARD M. SPECTOR t

Centre d'l~tudes Nucldaires de Saclay, Service de Physique Math~matique, Gif-sur-Yvette, S & O, France Received 23 July 1962 Abstract: Starting from Yamada's assumption of accidental cancellation o f matrix dements, the shape correction factor and the electron polarization for RaE beta-decay are investigated. The phenomenologically acceptable range of the two relevant parameters is determined to be -- 1.6 ifr/Sg × r ~< --0.8 and 2.2 ~< 1/~u/i~r ~< 3.1. The first of these ratios, designated A, is investigated theoretically by calculating the necessary shell-model matrix elements using both zero-order and then first-order wave functions for RaE. The results obtained for A with first-order wave functions, which arise from the breaking up of the rigid core of RaE, are seen to resolve the longstanding disagreement between theory and experiment. The same first-order wave functions are also used to calculate the magnetic moment o f RaE and agreement with the experimental value is obtained.

1. Introduction

In an earlier work 1), we indicated briefly that the longstanding disagreement 2- 5) between theory and experiment about the value of the ratio A = iSr/~a x r for RaE beta-decay had been resolved. It is the purpose of this paper to deal explicitly and in greatly expanded detail with the assumptions and procedures involved in the calculation of A as well as to incorporate some minor corrections in the numerical work. The well known t - decay of RaE(Bi21°) to p021°(1- ~ 0 +) is a first forbidden decay which does not have a linear Fermi-Kurie plot 6). In addition to this fact, it is known experimentally 7-11) that the outgoing electron has a polarization P whose magnitude is significantly less than v/c. Both these factors point to the conclusion that this decay does not obey the usual ~ approximation x2) for first forbidden decays. For a long time this proved difficult to understand since in this case the inequality ( a Z / 2 R ) = ~ ~> W 0 is obeyed; this is the usual criterion for validity of the ~ approximation. Here, a is the fine structure constant, Z is the charge of the daughter nucleus, R is the nuclear radius, and Wo is the end point energy of the beta spectrum. The way out of this problem was first suggested by Yamada 13) who assumed that due to an accidental cancellation of nuclear matrix elements, the leading energy independent terms in the Konopinski-Uhlenbeck approximation were small here, and the usually negligible energy dependent terms become important. To see this we note that in the Konopinski-Uhlenbeck approximation, the shape correction factor for the t Present address: Department o f Physics and Astronomy, Rochester University, River Campus Station, Rochester 20, New York. 338

SHELL MODEL OF RaE BETA-DECAY

339

spectrum can be written C O = a+bW+c/W+dW

(1)

2,

where W i s the electron energy in units of met 2 and a, b, c and d a r e complicated (but energy independent) functions of ~, W0 and the relevant first forbidden nuclear matrix elements. Due to the spin and parities of the initial and final states in the RaE decay, only three matrix elements, or two ratios, contribute to formula (1). These we define as

.:,f./f.×.,c-f./,f,.

(.,

Here, Sr means ~.J~*r(J+)z(i)~id~, where ~i and ~kf are the initial and final nuclear wave functions, z + is an isospin operator that changes a neutron to a proton, and the sum overj is over all nucleons. In definition (2), = is the usual Dirac matrix, and the definition of the other integrals is analogous to that of Sr. Normally in formula (1) a dominates all other terms t and C o is very nearly independent of W, but Yamada suggested that this was not so for RaE since accidental cancellation between A and C may cause a to be unusually small. Following his approach we write down the appropriate formula for Co assuming a V - A interaction 14): 2 ..{.0~2Z2

Co oc x 2 + { ( 1 - y ) q x - 4 x ( 1

+ y) (S + ~ )i()-W ~ +_ _

q2 ( I + 2 y 2 ) - ~ q ( 1 - Y

+

p2+e2Z2

2]

"(s+ 2Y i+l)w

+4(1+y)2

V 2+~2Z2 (S+1)2(2S+1) 2 +

(1-2y) 2 (1+S) L I ( p , Z )

(3)

with q = Wo - W,

p2 = W z _ 1,

x = -cy - C-2~ CA \i-~!

,

S'- = 1 -

y =

(x2Z 2,

CvA. CA

Here, C v and CA are the vector and axial-vector coupling constants for beta-decay and we take C v / C A = --1.2. In formula (3), L l ( p , Z ) is a small function which we chose to be 2 L1 = (S + 1)(2S + 1)2 (p2 + e2Z2), so that the part of this term linear in y cancels with the linear y part of the penultimate term in (3). This assures that expression (3) reduces to the simplified form of t It c a n be s h o w n t h a t a is O ($~), b a n d e are O(~1), a n d d is O (t°); hence t h e criterion for t h e validity o f the ~ a p p r o x i m a t i o n .

~40

R . M . SPECTOR

Fujita in the limit c~Z ~ 0, S ~ 1. As it turned out, the results were not sensitive to this assumption since setting L~ = 0 gave very nearly the same answers. We also have need of the polarization expression which is 14)

-

= 1-- ( S + I ) ( 2 S + I ) W C o [ x + ½ q ( 1 - y ) ] .

(4)

With expressions (3) and (4) and the experimental data referred to above, it was possible to investigate, phenomenologically, the values of A and B (the ratio B = C/A = Se/iSr turns out to be more convenient to work with than C) that will fit the experimental results.

2. Phenomenological Results for A and B With the aid o f the IBM 7090 at Saclay, it was possible to revise and extend the calculation reported earlier a). Due to a correction of small numerical errors in the first work, the region of agreement for A and B was slightly modified and the new values are --'1.6 ~< A < --'0.8,

2.2 ~< B/~ <~ 3.1.

(5)

The criterion used to determine the region given in (5) was that formulae (3) and (4) must give theoretical curves that were consistent with the experimental points. Due to the approximations involved in deriving (3) and (4L it is impossible to require any fits to be accurate to more than about 10 ~o. A typical fit to Co and P for a given value of A and B is shown in figs. 1 and 2. There is some variation in the fits over the range (5) but all of them more or less resemble those shown in figs. 1 and 2. In fig. 3 we display the region, marked by the band Z, where acceptable values of A and B must lie. The points shown there mark actual values found by the computer while searching a discrete two-dimensional net. Our values of A and B are not directly comparable to the results o f Fujita 15) since he works in terms of x and y, but the region covered by (5) overlaps for the most part with his. Since Fujita uses a simplified form of (3) and (4), it is not expected that he will get exactly the same values as OURS.

The value of B has been extensively dealt with by many authors 16); the two most reliable predictions are B = 1.06 ~ for ordinary beta-decay by Ahrens and Feenberg 17) and B = 2.46 ~ for a conserved vector current theory of weak interactions by Fujita. These values are drawn as lines y and x, respectively in fig. 3. Clearly, only Fujita's value is in agreement with the phenomenological results. But the value of A which is predicted to lie in the region of - 1.0 has heretofore been a puzzle. The most extensive previous investigation 5) gave a value o f A = + 0.99 in complete disagreement with experiment. In the next two sections we show how this conflict can be resolved.

St-IELL MODEL OF RaE BETA-DECAY

~:1

J ~0

A=-I,20 1~: 2,5~

"~.<

2~

~to=3.z6

zo

A : - 1.20 B: Z.5~

,,WE6NER o : ALIKHANOW o.ULLt~N

w Fig, 1. The solid lines are taken from ref. s) and represent the experimental determination o f the shape correction factor allowing for the uncertainty in our knowledge of W0 which lies between the limits shown. The broken line gives the fit obtained from formula (3) in the text for the indicated values of A and B. Normalization is arbitrarily such that all curves agree at W = 1.72. I.O

0.5

3o

.1

t

1

I

[

i

I

J

I

[

l

1

I

I

]

i

r

f

~

I

~,.

1.5 ~.O 2.5 v¢ 30 Fig. 2. The solid line represents the polarization curve obtained from formula (4) in the text for the indicated values of A and B. 1.0

(x

f E=';.5 - 0.75

1,~E=33 1

-f,O0

I AE=3D I -I 25

I

-1.50

-t. l'z5

A

Fig. 3. Phenomenologically acceptable values of A and B must lie within band Z. Line X shows Fujita's value for B / ~ and line Y shows the value obtained by AF. The energies o f A E are in MeV and are explained in the text.

342

R. M. SPECTOR

3. Evaluation of A to Zero-Order

According to a shell model assignment, RaE has one neutron and one proton outside a closed shell of 82 protons and a closed shell of 126 neutrons. Similarly, Po 2~° has two protons outside these shells. The possible states for these extra two nucleons are many, but the three lowest states of appropriate spin, based on single particle energies in neighbouring nuclei, are given in table 1. TABLE l

The three lowest possible states for the extra two nucleons RaE

Relative energy (keV)

Po21°

Relative energy (keY)

[(h~)p(2g])n]x [(h~)p(i~.)n]1 [(2f~)p(2~)n]l

0 790 900

[(h~)p(h~)p]o [(2f~)p(2f~)p]o [(i~)p(i~)p]o

0 1800 3200

These assignments are from Newby and Konopinski s) (referred to hereafter as NK). Using these three configurations only for each nucleus, N K diagonalized the 3 x 3 energy matrix formed with the aid o f a delta-function Serber exchange potential. In a more simplified notation than that o f table 1, their result was ~k(RaE) = 0.936(h~i÷) + 0.134(h~2g~) + 0.327(2f~2g~), @(Po 21°) = 0.943(h~tq)+0.101(2f~Eg~)-0.317(i~ i~).

(6)

Using (6), it is easy to show that A = +0.99 and that/z = - 0 . 7 6 n.m. for the magnetic moment o f the 1 - state o f RaE. The experimental value is) is p = 10.0442+_0.0002[ in strong disagreement with the value obtained from (6). The wave function for R a E i n expression (6) looks rather suspect since the dominant state, ( ~ i~), is n o t the lowest zero-order state. This was noticed by N K who went on to compute the diagonal matrix elements, only, of the same 3 x 3 matrix but using a finite range Gau~sian potential rather than a delta-function. Their results indicated a reversal o f the dominant states back to the logically expected one upon moving to finite range. We have completed the calculation by including off-diagonal terms to find @(RaE) = 0.963(h~Eg~)+0.E64(h~Lt)-0.049(Ef~2gl),

(7)

which certainly seems more reasonable than (6). The parameters used here were the same as those used by N K for the sake o f comparison. We let the residual nucleonnucleon potential be V ( r ) = Voe-pr2

with

fl-~ = 2 . 5 6 f m -2

SHELL MODFL OF RaE BETA-DECAY

34:3

and Vo = - 60 MeV for triplet interactions and Vo = - 46 MeV for singlet interactions. The single particle wave functions used were the usual appropriately normalized jj shell model wave functions with harmonic oscillator spatial parts. The parameter b in ~9(r) oc e -r2/262 was chosen as b = 2 fin to allow comparison with the results of NK. As a test of the effect of the exchange mixture on the wave function for RaE, four different types were tried: Serber, Rosenfeld, Soper 19) and Peaslee 20). It was interesting that ~(RaE) was to a great extent invariant under the choice of exchange mixture. In our further work, we chose to work with the Serber result, expression (7), but this has no effect on the answers since all four types would produce very nearly t h e same results for A and #. To compute the necessary finite range matrix elements, use was made o f the summation formulae o f N K where the matrix elements are expressed as angular sums over Slater integrals

F, =

qo° r21drl Rl(rl)R'l(ri ,;o °r~dr2R2(r2)R'2(r 2

duPk(u)V(rl2), -1

where the R are the normalized radial parts of the wave functions and the Pk(U) are Legendre polynomials of order k. If V(r12 ) is a Gaussian in form, Ford and Konopinski 21) have shown that it is possible to express the Fk as finite sums over algebraic quantities only. With the aid of the Mercury Computer at Oxford University, it was possible to evaluate the F k in this manner as well as the angular sums of NK. Using expression (7) we obtain A = + 0.27 in disagreement with experiment but # = +0.04 n.m. in excellent agreement with the experimental value. 4. First-Order Contributions to A

Working in much the same way and with much the same philosophy as other authors who have evaluated corrections to the Schmidt values for nuclear magnetic moments 22, z3), we have investigated the contributions to A arising from the breaking up of the dosed shells in RaE and Po zl°. This was done by allowing small amounts of other configurations to be admixed into (7) and the corresponding wave function for Po 21°. This is far more work than in the usual magnetic moment case since many more excited configurations can contribute in first order to the matrix element of ir and a x r than to #~ which has no radial dependence. To make the calculation at all possible it was necessary to use a delta-function potential in computing the coefficients o f admixture into (7). Though this is certainly crude, it does seem to give reasonable answers to the magnetic moment problem and will thus very likely do so here as well. For the strength of our interaction we again chose NK's value of Vtrip = - 6 0 MeV, Vsing = - 4 6 MeV and used their normalization factor so as to preserve the

~4~

R, M. SPECTOR

volume energy of the delta-function potential when compared to a finite range gaussian potential. The problem of determining which excited configurations to admix was a complicated one which was solved only by making some rather crude assumptions. The spin-orbit energies within a given shell were neglected, thus making all levels in one shell degenerate. We also took the energy between shells to be A E = hoe = 6 MeV, a reasonable value for this region 23). Next, it was necessary to restrict excitations to those of one and two particles only. With one exception only, to be described below, these excitations carried energy denominators in first order perturbation theory o f 2hco --- 12 MeV. We considered contributions to the two leading states in (7) and also to the dominant state in Po 21°. The complete set of excitations is shown in table 2. We give here a clarifying description of a typical excitation. Consider the case in Po 21° of an ~ proton being excited to the 4s~ state while at the same time a 3p½ (core) neutron is excited into the 2 4 state. We write h~ ~ 4s~,

3p½ ---, 2g~.

The matrix element o f ir and ~rx r is between a 3P½neutron in RaE and the 4s½ proton in Po 2~°. Most of the excitations shown in table 2 are complicated constructions involving use of 2 and 3 noded wave functions which makes calculation very tedious indeed, though the labour is considerably reduced through the use of a computer. The final numerical values for the coefficients o f admixture fell in the region of 0.01 to 0.1 in amplitude though some fell outside this range. In RaE there is one excitation that is particularly important. This is the spin-orbit excitation o f an h÷(core) proton into the/a; state. The A E f o r this excitation has been estimated 2,, 2s) as anywhere from 2 to 6 MeV; thus, a value of around A E = 4 MeV TABLE 2

The excited configurations whose contributions to A were evaluated by means of first order perturbation theory. RaE

h~(core) ~ h~ (for either 2 ~ or i~- neutron) (core)

h~ ~ -4s½ 3d t 3d~ 2g~ po2~o

2g~ iy

- ~2g~ 3P½, 3p~ 3p b 3p~ 2f~ 3p~, 2f~, 2q

o r i~.

2fv 2fv h~ 2~, hi hi h~

A~

lg](core) -+ 2g~ or i@ lg](core) -+ 2 ~ or i y Only those configurations were chosen which gave non-vanishing matrix elements of i r and (7 × r on the basis of spin and parity considerations.

SHELL MODEL OF RaE BETA-DECAY

3~5

is not unreasonable. Because of this small energy denominator as well as the good overlap between ~ and h ÷ , the contribution o f this excitation to A and # is large. We give here the final value of A, including all first order admixtures, listed in table 2. A =-1.37

for

A = -1.10

for

A=--0.85

for

A E = 3.0MeV, AE=3.5MeV, AE=4.5MeV.

These values are shown by the arrows at the bottom of fig. 3. First order contributions to # were such that the final value of # becomes (with AE = 4 MeV) # = +0.25 n.m. which is in reasonable agreement with the experimental value, especially considering that there are also spin-orbit terms o f about - 0 . 2 5 n.m. contributing as well 26). 5. Discussion and Conclusion Our object has been to show that the experimentally determined value o f A for RaE is not at variance with the shellmodel prediction. Though at first sight there does seem to be an inconsistency, we feel that the further work described in sects. 3 and 4 removes this conflict. The work done in this paper is admittedly not a precise numerical estimate for A but rather an indication of the correct region for A. There are too many factors unknown in nuclear calculations to hope for anything more than we have achieved. The strength and shape of the nucleon-nucleon potential, the precise form o f the nuclear wave functions, the difficulty in choosing which first order configurations to treat and the question of energy denominators are all circumstances which tend to blur our answer. But still the result has shown that with reasonable assumptions about these points, the value of A can be placed in a range consistent with experiment. And our final value for the magnetic moment indicates that there must be a fair amount o f accuracy in our approach. There is no longer any reason to believe that unknown or obscure effects play a significant part in determining the value of A as has been suggested in the past 15). The author would like to thank Dr. R. J. Blin-Stoyle for many illuminating discussions about this work. He would also like to thank Dr. C. Bloch for making possible a three month stay at Saelay, where some of this work was carried out, as well as for his hospitality during that visit.

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346

R. M. SPECTER

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