Quasi-particle description of 2+ and 3− states of doubly even spherical nuclei

1.D.2

I

Nuclear Physics A l l 4 (1968) 449--462; (~) North-HollandPublishing

Co., Amsterdam

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

QUASI-PARTICLE DESCRIPTION OF 2 + A N D 3 - STATES OF D O U B L Y EVEN SPHERICAL NUCLEI R. J. L O M B A R D t++

Department of Physics and Institute for pure and applied Sciences, University of California, San Diego, La Jolla, California Received 8 December 1968 Abstract: The description of the first 2 + and 3- levels of spherical nuclei 90 ~< A _< 150 and 186 < A < 206 is performed in the framework o f the BCS theory. The residual interaction is taken as the superposition of the pairing force and of the multipole force. The short-range p r o t o n - n e u t r o n correlations are neglected. In addition to the valence nucleons, the core particles are taken into account; it allows the computation of the transition probabilities without any effective charge. Theoretical predictions concerning the E2 ÷ and E3- energies, the B(E2) and B(E3) reduced transition probabilities and the gyromagnetic factors g2 + are in good agreement with experiment.

1. Introduction

The microscopic description of the vibrational states of doubly even spherical nuclei in terms of quasi-particle excitations has become very usual in nuclear spectroscopy a). The greatest part of the work has been devoted to single closed shell nuclei. However numerous experimental data have been reported concerning low-energy spectra of spherical nuclei away from closed shell. Therefore it is tempting to try to reproduce some general features over a wide class of nuclei. The purpose of this paper is to show to what extent this program may be achieved in a simple way, that is in using crude assumptions with regards to the correlations between particles and the residual interaction. The present study is concerned with the lowest 2 + and 3 - states. The description of these levels is obtained in the framework of the BCS approximation. The residual interaction is chosen as the conventional pairing plus quadrupole (or octupole) forces superposition. The proton-neutron short range correlations are neglected. The long range part of the force is assumed to act equivalently between all pairs of nucleons in the nucleus. In this way, the calculation is not restricted to the valence nucleons but includes adjacent major shells. Although the schematic model cannot be trusted in specific cases, it yields general trends which will probably remain unchanged by use of more realistic forces. It also provides a simple way of studying the importance of the core effects in terms of particle-hole type excitations. + This research was supported by the United States Atomic Energy Commission. ++ Permanent address: I.P.N. Division de Physique Th6orique, laboratoire associ6 au C N R S , 91-ORSAY F R A N C E . 449

450

R . J . LOMBARD

In sect. 2 the results corresponding to the energy spectra will be presented. The transition probabilities and magnetic moments are discussed in sects. 3 and 4, respectively.

2. Energyspectra As usual, the spherical shell-model Hamiltonian constitutes the starting point. The short-range correlations are taken into account by means of the Bogolyubov-Valatin quasi-particle transformation

a+m = Ujtl;mJl-(-)J-mVj~lj_m

(l)

with 1

connecting the particle (a +, a) and quasi-particle 0/% ~) operators. The U and V coefficients are obtained by solving the gap equations for neutrons and protons separately. The ground state of doubly even nuclei is then assumed to consist in the product of the two BCS states. Neglecting the proton-neutron short-range correlations should not affect seriously the results since for almost all the nuclei we shall consider that protons and neutrons are filling different major shells. Also it has been shown by Camiz et al. 2) that, as far as only the T = 1 part of the interaction is taken into account, the product of the two BCS states constitutes the lowest energy state. The gap equations need to be solved for the valence nucleons only, since it is a sufficient approximation to assume V = 1 for the core-particles. The pairing constants are adjusted in order to fit the odd-even mass differences. They are found to be of the order of Gp ~ 26/A MeV and G, ~ 23/A MeV. In the next step, the excitation energies are obtained by a diagonalization in the subspace of two quasi-particles. If the creation operator of a pair of quasi-particles coupled to J and M is denoted by

AfM(ab) -

1 E (Jorn.jbmbJdM)~l~ ~I+' x/1 + bob

(2)

where the index a = (n o, Jo), the wave functions of the nuclear states are given by IJM) = E q~Ju(a, b)Af~t(a, b)l Vo)

(3)

a,b

with

I%> = 10)pl0)n, where the sum runs over all possible configurations (a, b) in the chosen subspace. For a multipole force

~

= Zz X (-)",-~r~(~,),-~ Y;"(~),

(4)

/.t

assuming Zpp = Z"n = Z"p = Z and neglecting the exchange terms, the secular prob-

451

QUASI-PARTICLE DESCRIPTION

lem takes the very simple form (approximation I): Fx = 2 2 + 1 - ~ ' I(allraYallb)12

Z~

(UaVb-[-UbVa)2

T,b E, + E b - E~

(5)

1 + 6.b

Further, if ground-state correlations are taken into account (quasi-RPA) the resulting secular equation is very similar to (5) (approximation II):

I(allr~Y~Jlb)[ z S,~ - 22+1Zx - 22",b E.+Eb-E~/(E.+Eb) 0,,

(U. Vb+ Va Ub)2

(6)

1+6.b

o,

2.5

1.0

0 ~ 90

,,

L

"'~

°

"\°'" L 100

, IF0

--i-- lheor i20

A

Fig. 1. The E2+ and E3- energies for 90 ~ A ~ 124 nuclei. For each isotope set, the experimental value used to determine Zz is encircled.

Both approximations have been applied to the 2 + and 3- levels. The parameters of the calculation, particularly the single-particle energies, are taken from a previous work devoted to single closed shell nuclei 3). The results corresponding to F~ are presented in figs. 1-5 for nuclei generally accepted as non-deformed nuclei: 90 _
452

R,J. LOMBARD

However in the case of the 2 + level the ground-state correlations have the tendency to lower its energy too rapidly as one goes away from closed shell nuclei. This point will be made more clear in the next section, for the particular case of the Sm isotopes. J

3 E (MeV) / j-,. .... --Q-.... ~=~._~ o ....

Te

//

P

//

//

P

///

J---~..~

.d//

Xe

Bo

/,..o

. . . . .

-

-

o

-

~

----c-

/ ~

tneor

Fig. 2. The E2+ and E3- energies for Te, Xe and Ba isotopes. For each isotope set, the experimental value used to determine Zz is encircled.

2.0I E(MeV)\,'~\ ,,, Q",x\\x\xx . 1.5

~X\

\~ \\\

',,

1

\

3--

/

m

....

'

t hi'or./

A Fig. 3. The E2+ and Ea- energies for N d and Sm isotopes (A ~ 150). For each isotope set, t h e experimental value used to determine Zz is encircled.

QUASI-PARTICLEDESCRIPTION

453

1.5 E (MeV)

,@--- - - -

1.0

0.5

a,~ ~ -

- - - --o

3-

0~

=-

Pt

Os

2+ •

, exp.

---c--- th~on

186

I~B

~90

1~2

lh

1~6

1~8

A

Fig. 4. The E2+ and E3- energies for Os and Pt isotopes. For each isotope set, the experimental value used to determine g~ is encircled.

E (MeV) 3.0

.~--

o--- - z S z _ _ _ 2 ~

2.0

Pb Hg

I.O

c- .

.

.

.

.

_--c----

•

~

---o---

t

exp

theor.

Fig. 5. The E2+ and E~- energies for Hg and Pb isotopes. For each isotope set, the experimental value used to determine Z.~ is encircled.

3. Transition probabilities The inclusion of the core configurations into the description of the vibrational states allows the computation of electromagnetic transition probabilities without introducing any effective charge. The ground-state transitions are very simply expressed in terms of the secular functions: B(E2; J ~ 0 +) = e2 (F~)2

FI(E~)

=

e z (S~)2

(7)

454

R. J. LOMBARD

for approximation I and II, respectively. The B(E2) are proportional to F~ (or S~) which is the square of the proton contribution to the secular function; F' and S' denote the derivative of F and S with respect to E~, taken at the eigenvalue. The results are plotted in figs. 6-12 and compared to experiment. The B(E2) are expressed in singleparticle units. As far as the B(E2) are concerned, the experimental data are taken from B (E2; 2 + ~ 0 + )

/ Bsp

I00

o //

zo / /

// //

(k

"%

"---'---'---"-"---"'''"---'---'---" -

Zr 0

---

o-

_

-o

-

-

exp,

-o

o- --~, M0

. . . .

theor.--o-Ru

:

.Pd

I I00

90

Cd

Sn

o---o---o-

--o---o---o-

I I10

--o

1 120

A

Fig. 6. B(E2) values in single-particle units. The lower theoretical estimations correspond to approximation I. Higher theoretical values result from approximation II.

B (E2; 2 + "-" O+)/Bsp

100

o----

-o . . . .

-o-

-o-

--o-

-o-

-o

Xe

T e

Bo

,

exp.

---o--I

118

I

120

I

I

I

I

[

I

I

I

122

124

126

128

130

132

134

136

th~ A

Fig. 7. See caption fig. 6.

the compilation by Grodzins and Stelson i 0). The B(E3) values have been measured by McGowan and Stelson s) in case of Pd and Cd; other values concerning Zr and Mo are derived from the work of Kim and Cohen s). Similarly to what is found for closed shell nuclei 3), the experimental values lie in between the predictions of both approximations. It is however obvious that the extension of the configuration space (approximation I) is not sufficient by itself to provide large transition probabilities.

QUASI-PARTICLEDESCRIPTION

455

It is more specially the case for the B(E2). Therefore ground-state correlations are needed in order to get enough coherence. 100

B (E2; 2 + ~

O +) /Bsp

10 -..~ .......

~:=_-===-:8-

Nd

Sm •

exp.

---c---thdo~ i

i 144

142

i 146

i 148

Fig. 8. B(E2;2+~0

/ 150

See caption

A

fig. 6.

+)/Bsp

100

10

"Q'x\\~ o- . . . .

~- . . . .

"~---

-o

Hg

% •

exp.

---o-_-thdon

186

188

1;0

1;2

i

194

__1

Fig. 9.

196

1;8

See caption

200

282

204

A

fig. 6.

The B(E3) are poorly known experimentally. A large part of them has been derived from scattering data using the formula B(Ea)/B~.p.

= ( 9 e 2 / 7 7 0 Z 2 fl 2 ,

(8)

456

R.J.

LOMBARD

where fl, the deformation parameter, is related to the inelastic scattering cross section [ref. 11)]. The B(E3) are more sensitive to shell effects and their study should be very fruitful, particularly in heavy nuclei (A ~ 200). A peculiar feature of approximation II is that the function S~ tends to zero with Ex and therefore the transition probability tends to infinity. At this limit however, B( E 5 ; 3 - ~

0+)/ Bsp

100 . . . . . . . L - -'~ /

I0

----~2q~

_ , n - - - - - - o - - -- - O -

o-

Zr

Mo

Pd

-

-c-

- - c -

Cd

I00

-

-o-

__c-

Sn

I

I

90

-- - o

-

_o

~ exp. . . . . . th~or. I

120

110

A

Fig. 10. B(E3) values in single-particle units. T h e lower theoretical estimations c o r r e s p o n d to a p p r o x i m a t i o n I. H i g h e r theoretical values result f r o m a p p r o x i m a t i o n ]I. 100

B (E3; 3 - - "

o- . . . .

-o-----"°"----'43-

O+ ) / B s p

. . . .

-o- ---

-o-

- - - --o--

__

-o

10

Te

•

exp.

___~___ th&or. l

118

i

120

i

122

i-

124

l

126

I

128

i

130

i

132

A

Fig. 11. See caption fig. 10.

the validity of the approximation is doubtful. This property appears also in the liquid drop model. For harmonic surface vibrations described in terms of collective variables, the Hamiltonian is usually taken as 12) Hcol I = 1 Z

B,I~,~,,,Iz+½ Z C~.l~.~.,.I2-

(9)

QUASI-PARTICLEDESCRIPTION

457

The energy of the " o n e - p h o n o n " state is given by

hoo~ = h VC~

(10)

and the transition probability by 1

B(E2, 2

~ 0 +) = const, x/Bz Cz

-

h2 c o n s t .htnzBz --.

In the liquid drop model, the mass parameter Bz = (4zc3/2)AMR2 is independent of the energy and B(E2) tends to infinity as h~oz approaches zero. l

B (E 3 ; 3-"'~ 0+) / Bsp

100

~'"

C~ . . . . . .

,o

.o /~

.~/

.'-0-..

. . . . .

Nd

-0-

. . . . . .

<9

Sm .

exp.

---o---th~0n 1

142

±---

144

t

.

146

.

.

.

.

1

148

....

L

150

A

Fig. 12. See caption fig. 10.

This particular behaviour will be illustrated in the 2 + state of the even Sm isotopes.

Since 144Sm possesses a neutron closed shell whereas aSZsm and a54Sm are known to be deformed, it provides the opportunity of studying the effect throughout a transition region. As far as approximation II is concerned, the single-particle energies are taken from the x44Sm calculation and kept fixed for all the other isotopes. The coupling constant Z2 is adjusted for each nucleus in order to fit the experimental energy E2 +. The renormalization of Z2 is needed already for 146Sm. In other words adding two neutrons to 144Sm increases the secular function Sz in such a way that it does not cross the straight line Z2 = const, between the origin and the first pole. Note that under the same conditions approximation I yields a pretty good fit to the E2 + energy for all the Sm isotopes with a fixed Zz [ref. 13)].

458

R , J . LOMBARD

In the liquid drop model, the role of X2 is played by surface tension x2: C2 = 4 R 2 x 2 - - ~ 1 3 (ZEe2/go).

c,--,~.~

(11)

Xa <2-

i44

146

IZ~8

1501521

154 A

Fig. 13. Variation of Z2 and ,% against A. The values are adjusted in order to fit the experimental E2+ energies.

1000

B (E2}/Bsp 500

q .\, ,,\~

----0-.-- RPA •

---c---

exp,

phonon

",~, "',,%, 100

50

10

0.1

0.2

0.4

0.6 0.8 1.0

E2+ [MeV]

210

Fig. 14. B(E2) values plotted against the Ea+ energy for the Sm isotopes. The experimental data are compared to the prediction of the RPA and the hydrodynamical model (one-phonon state).

This parameter has to be adjusted in each case to reproduce the 2 + energies, much in the same way as XzThe variation of X2 and x2 against A is shown in fig. 13. The corresponding B(E2) are plotted against E2 + and compared to experiment in fig. 14. Both models are re-

QUASI-PARTICLEDESCRIPTION

459

p r o d u c i n g t h e e x p e r i m e n t a l s i t u a t i o n r a t h e r w e l l a n d a p p e a r in this a s p e c t to be a l m o s t equivalent.

4. Gyromagnetic factors It w a s k n o w n l o n g a g o t h a t t h e m a g n e t i c m o m e n t o f n u c l e a r states is v e r y sensitive t o t h e d e t a i l o f t h e w a v e f u n c t i o n s . T h e r e f o r e it c o n s t i t u t e s a p o w e r f u l test o f m o d e l s . F o r a s t a t e o f a n g u l a r m o m e n t u m J , t h e g y r o m a g n e t i c f a c t o r in a p p r o x i m a t i o n I is TABLE 1 A Zr

90 92 94 96

A

Mo

Cd

Nd

gz+

A

1.05 0.07 0.14 0.14

Ru

g2+ Pd

g2+

96 98 100 102 104

0.49 0.45 0.45 0.48 0.51

0.55 0.44 0.304-0.03

A

gz+

exp

102 104

0.48 0.50

92 94

1.27 0.36

96

0.33

106

0.53

98 100

0.34 0.36

108 110

0.57 0.58

A

g2+

A

g~+

106 108 110 112 114 116

0.41 0.44 0.47 0.48 0.47 0.45

112 114 116 118 120 122 124

0.16 0.11 0.076 0.058 0.015 --0.036 --0.076

A

g2+

142 144 146 148

1.11 0.11 0.13 0.16

exp Sn 0.304-0.12 0.44±0.06

exp Sm 0.3 -4-20 % 0.3 t 2 0 % 0.24=]=20%

exp

A

g~÷

144 146 148 150

1.13 0.22 0.23 0.24

0.45 4-0.06 0.354-4-0.03

given by

gs --

1

~ ~P~(Ja , jn)[(ga+gb)J(J + 1 ) + ( g . - g b ) { j ~ ( j a + 2J(J + 1) Jo~b

1 ) - - j b ( j b + 1)}],

(12)

w h e r e q~s(J~,J~) r e p r e s e n t s t h e w e i g h t o f t h e c o m p o n e n t (Ja,Jb) to t h e state v e c t o r [ J M } . T h e g~ a r e t h e s i n g l e - p a r t i c l e g y r o m a g n e t i c f a c t o r s . I n t h e p r e s e n t w o r k t h e y

460

R.J. LOMBARD

are t a k e n from the Schmidt values. There are a n u m b e r of possible corrections arising from the velocify dependence of nuclear forces, the relativistic energies of the n u c l e o n s at the F e r m i surface a n d m e s o n exchange currents. Nevertheless they are expected to be small a n d our crude m o d e l does n o t w a r r a n t for such refinements. Note t h a t gs depends o n the U a n d V coefficients of the B o g o l y u b o v - V a l a t i n can o n i c a l t r a n s f o r m a t i o n only t h r o u g h the weights ~o. This is due to the d i a g o n a l character a n d the time-reversal p r o p e r t y of the magnetic m o m e n t operator. A n o t h e r conTABLE 2 Te

g~+

118 120 122 124 126 128 130 132

0.50 0.41 0.32 0.28 0.26 0.26 0.31 0.40

a) Ref. so).

exp

0.438 ±0.092 a) 0.431 ±0.036 0.585±0.036 0.872±0.144 0.932 zk0.146

0.314-0.03 b) 0.224-0.05

b) Ref. 2~). TABLE 3

Os

Pt

A

g2+

exp.

A

g2+

186 188

0.28 0.30

o.274±0.019 0.2804-0.021

194 196

0.41 0.44

190

0.32

0.280±0.03

198

0.47

192

0.34

0.30 ±0.04

200 202 204

0.50 0.54 0.61

A

g~+

A

g2+

188 190 192 194 196 198

0.32 0.33 0.36 0.38 0.40 0.43

206 204 202 200

0.32 0.23 0.16 0.03

Hg

exp Pb 0.24 0.32 0.28 0.30

21_0.04 4-0.05 ±0.04 ~.0.04

exp.

o.38±O.ll o.55--O.ll 0.73 ±0.27 0.77±0.23 0.24±0.25

sequence is that the validity of eq. (12) is n o t restricted to the case of the multipole force b u t is still the same for m o r e realistic interactions. The g~ d e p e n d only o n l a n d j q u a n t u m n u m b e r s . C o n s e q u e n t l y the magnetic m o m e n t is mostly d e t e r m i n e d by the mixing of configurations a m o n g the single-particle states which are closed to the F e r m i surface. I n contrast to t r a n s i t i o n probabilities the core c o n t r i b u t i o n s have little influence.

QUASI-PARTICLE DESCRIPTION

461

The results are summarized in tables 1-4 and compared with experiments 14-25). Calculation concerning single closed shell nuclei are also reported. They show large deviations from the value Z / A predicted by the hydrodynamical model, whereas the other nuclei have g2 * which are generally closed to this value. The g2 + of Te isotopes appear of particular interest since the first measurement 2 o) has yielded values "substantially greater" than Z/A, whereas in a previous theoretical estimation, Kisslinger and Sorensen 1) have obtained rather low g2+. In the present calculation we let the 2d~ and lg¢ proton single-particle energies vary smoothly with A, according to the position of these levels in the neighboring nuclei. It increases the calculated g2÷ and brings some of them in good agreement with new experimental results 22). The theoretical values could be further increased in choosing the neutron single-particle energies more properly. However such a treatment seems rather unjustified, more especially in view of the oversimplified picture of the nucleus implied here. Our knowledge of 3- state gyromagnetic factors is quite poor. There is only one available experimental datum which is in disagreement with the theoretical estimation. TABLE 4 Isotopes

g3-

exp.

88Sr l°6pd 114Cd 11nSn 14nNd 2°6Pb ~°sPb

1.10 0.48 0.57 0.12 0.57 0.30 0.50

0.080±0.124

5. Discussion

The present work shows that the schematic model is able to reproduce the general trends of the first 2 ÷ and 3 - states over a wide class of nuclei. The core configurations bring an important contribution to the secular functions and are necessary to describe the evaluation of the E2÷ and E a - energies against .4 without varying the singleparticle energies or the coupling constants from one nucleus to the other. The results are in good agreement with experiment, although they cannot be considered as very confident in each specific case. The assumption concerning the ground-state wave function, i.e., the neglect of the proton-neutron short range correlations seems reasonable in first approximation, even in nuclei such as the Te, Ba or Xe isotopes in which both types of particles are filling the same major shells. The study of the transition probabilities indicates the need of ground-state correlations in order to get sufficient B(E2) or B(E3) rates. It is interesting to note that the static quadrupole moment associated with the 2 + state turns out to be rather small

462

R.J. LOMBARD

in the schematic model: -0.1(114Cd), ~ +0.03016Sn), ~ +0.2(19aHg)in e. b.

These results are obtained in approximation I. They contrast with the large observed values (26, 27) (although some of them have large errors and are still compatible with small Q2 +). It reflects the fact that approximation I yields B(E2) which are systematically lower than experiment. On the other hand, approximation II gives Q2 + ~ 0. It suggests that the RPA, which is proved to be so useful in doubly magic nuclei 28), might be unsuitable for other nuclei and that other types of ground state correlations are needed. I would like to express many thanks to Miss M. T. Commault for expert assistance in carrying out the numerical calculations. References 1) L. S. Kisslinger and R. A. Sorensen, Revs. Mod. Phys. 35 (1963) 853; A. M. Lane, Nuclear theory (Benjamin, New York, 1964); S. Yoshida, Nuc]. Phys. 38 (1962) 380; C. J. Veje, Mat. Fys. Medd. Dan. Vid. Selsk. 35, No. 1 (1966) 2) P. Camiz, A. Covello and M. Jean, Nuovo Cim. 42 (1966) 199 3) R. J. Lombard and X. Campi-Benet, Nucl. Phys. 83 (1966) 303 4) Nuclear Data Sheets 5) F. K. McGowan, R. L. Robinson, P. H. Stelson and L. C. Ford, Nucl. Phys. 66 0965) 97 6) H. Morinaga and N. Lask, Nucl. Phys. 67 (1965) 315 7) M. Sakai, T. Yamazaki and H. Ejiri, Nucl. Phys. 74 (1965) 81 8) Y. S. Kim and B. L. Cohen, Phys. Rev. 142 (1966) 788 9) G. Gerschel, M. Pautrat, R. A. Rici, J. Teillac and J. Vanhorenbeeck, Congr6s Intern. de Physique Nucl6aire, Paris (1964) Vol. II, page 501 10) P. H. Stelson and L. Grodzins, Nucl. Data I (1965) 21 11) J. S. Blair, Phys. Rev. 115 0959) 928 12) J. M. Arafijo, in Nuclear Reaction, Vol. II, ed. by P. M. Endt and P. B. Smith (North-Holland Publ. Co., Amsterdam, 1962) 13) X. Campi-Benet and R. J. Lombard, Phys. Lett. 19 (1965) 502 14) L. Keszthelgi, I. Berkes, I. Deszi, B. Molnar and L. Pocs, Phys. Lett. 8 0964) 195 15) L. Keszthelgi, J. Berkes, 1. Dezsi and L. Pocs, Nucl. Phys. 71 (1965) 662 16) K. Auerbach, K. Siepe, J. Wittkemper and H. J. Korner, Phys. Lett. 23 (1966) 367 17) P. Gilad, G. Goldring, R. Herber and R. Kalish, Nucl. Phys. A91 (1967) 85 18) I. Benfzvi, P. Gilad, G. Goldring, R. Herber and R. Kalish, Nucl. Phys. 96 (1967) 138 19) S. K. Bhattacherjee, J. D. Baumann and E. N. Kaufmann, Phys. Rev. Lett. 18 (1967) 222 20) R. R. Borchers, J. D. Bronson, D. E. Murnick and L. Grodzins, Phys. Rev. Lett. 17 (1966) 1094 21) D.E. Murnick, L. Grodzins, R. R. Borchers, J. D. Bronson and B. Herskind, BAPS I2 (1967) 528 22) S. K. Bhattacherjee, J. D. Bowman and E. N. Kaufmann, Phys. Lett. 2413 (1967) 651 23) J. Murray, T. A. McMath and J. A. Cameron, Can. J. Phys. 45 (1967) 1813 24) Proceedings of the Int. Conf. on hyperfine interactions detected by nuclear radiation. Pacific Grove, Calif., USA (August 1967) 25) L. Grodzins and G. C. Pramila, BAPS 12 (1967) 534 26) J. J. Simpson, D. Eccleshall, M. J. L. Yates and N. J. Freeman, Nucl. Phys. 94 (1967) 177 27) A. Winther, Proc. Int. Conf. on hyperfine interactions detected by nuclear radiation. Pacific Grove, Calif., USA (1967) 28) V. Gillet, Proc. Int. School of Physics "Enrico Fermi", XXXVI, Varenua (1965)