Magnetic susceptibility and M1 transitions in208Pb

Nuclear Physics A318 (1979) 162-172; ©North-Holland Pxbllahtrtp Co., Amsterdam Not to be reproduced by photoprlnt or microfilm without writtaa permiaedon from the publisher

MAGNETIC SUSCEPTIBILITY AND Ml TRANSITIONS IN

z° 8 Pb

M . TRAINI, E. LIPPARINI, G . ORLANDIM and S. STRINGARI

Dipartintereto dt Matematica e Fisica, Untoersttd di Trento, Italy Received 10 April 1978 (Revised 16 October 1978)

Abetract : Ml transitions in ~ °a Pb are studied by evaluating energy-weighted and inverse energy-weighted sum-rules . The role ofthe nuclear interaction is widely discussed: It is shown that the nuclear potential increases the energy-weighted sum rule and lowers the inverse energy-weighted sum rule, with respect to the predictidn of the pure shell model. Values of strengths and excitation energies are compared with experimental results and other theoretical calculations .

1. Introduction Much experimental t-4) and theoretical s-1°) work has been recently devoted to the study of M1 transitions in z°BPb. In a pure shell model picture one should expect strength at two levels : 5 .6 and 5.8 MeV . In fact, in ~° aPb, only the h~ht and i~.-i,~ single particle states can be excited by the magnetic dipole operator so that the predicted energies correspond to the spin-orbit splittings of the h- and i-levels, which are experimentally well known. On the other hand, the experimental situation for M1 transitions is very different; in fact, a large concentration of M1 strength has been found around 8 MeV [refs. s-4)] . This means that correlations due to the nuclear interaction are essential to reproduce experimental values ; these effects have been considered and evaluated in various microscopic approaches [for a recent review see ref. e)] . In this work we shall analyze the role of the nuclear interaction in M 1 excitation by using a sum-rule approach. Owing to the analyticity of the results, this method permits separating the côntribution of the nuclear potential from the spin-orbit one, which is connected with the spin-orbit splitting, and to analyze in detail various types of interactions . The M1 transitions are essentially due to the isovector operator In fact the contributions of the isoscalar and orbital parts to the total strength can be neglected io). We shall evaluate two sum rules in the framework ofthe RPA : the energy-weighted sum rule

MAGNETIC SUSCEPTIBILITY

163

and the inverse energy-weighted sum rule 9

~ (En-Eo)-1~<0~_ +Z) ~ Qi T3~ni~ 2. (2) n i Here ~n~ and En are the eigenstates and eigenvalues of the nuclear Hamiltonian Ho, ~c_ _ -4.7 n.m, and Qs and is are the third components of spin and isospin Pauli matrices . The former sum has been widely studied in the past lo " 11) ; the latter, which is directly connected to the static magnetic susceptibility, has been recently considered in the study of the magnetic properties of the Skyrme interaction iz" is). The sum rule B_ 1 (ML)j is particularly interesting because it can be used to study not only the M1 excitation but also magnetic core polarization effects in nuclei like s°'Pb, Z ° 9Bi etc. ls). A knowledge of the transition strengths B_ 1(M1)j and B,(M1)j will permit. us to evaluate a mean excitation energy through the relation B_ 1 (M1)j =

4~

=

B~(ML)j B_ 1 (M1)j

2. . The sascephbility sam rule In this section we evaluate the magnetic susceptibility by performing a restricted HF calculâtion. We suppose that the main effect of the external magnetic field .~E° interacting with the nucleus through the Hamiltonian H;, _ -~3.~E° is to rearrange the spins of nucleons through a simple spin transformation changing the single particle wave functions, by means of the following unïtary transformation: ~9(1JM)

In. eq. (4)

a,rM

-. tfiq(1JJ'M) _

(aün~+v9Q3)~9(~~~

is an appropriate .coefficient to ensure the normalization of fir' : auM

_

L

2M 21+1

v+

1

2M

2 CC21 + 1)

z 1/

vZ'

where v is a parameter as a function of which we shall solve the restricted HF problem by minimizing the energy on the deformed Slater determinant ~>/r') built up with the single particle wave functions ~~(1JJ'M), where q indicates a neutron or proton . Transformation (4) has already been used to study magnetic properties of microscopic interactions [spin stability is) and magnetic moments of the Skyrme interaction ia)] . It operates only ôn nucleons belonging to spin unsaturated shells (ham for protons and i,~ for neutrons in z°8Pb) which gain a spin. +r

8 lg(la + 1) v9 ~9 (1JJ'M~3~q(1JJ'M) = SQ = i ~ 3 219 +1 ' M=_ r

M . TRAIHI et ul.

164

where lQ is the angular momentum of the polarized shell (in Z°SPb In = 6; IP = 5). The energy variation induced by transformation (4),

consists of two terms : the first term arises from the variation of the mean value of the nuclear Hamiltonian Ho ; this contribution is quadratic in v (linear terms vanish in nuclei with time reversal invariant wave functions). The second term, linear in v, is due to the interaction -~3.1E°, which gives the following contribution : One has

8E = ZAnnVn+2`4PPVP+flnpVnVp+AnVn+APVP,

(7)

where 8 In(In + 1) 3 21n +1 -(~

1 8 1p(Ip +1) 3 21P + 1

while Ate, depends on the nuclear Hamiltonian Ho. The values of vn, vP which minimize thé énergy are _ - APAnp -AnApP _ AAnp- ApAn n vn Ann`4pp - Anp `4nn`4PP-f1nP ' ~ ~Vp A knowledge of the values vn, vP, and thén of the deformed state (eq. (4)) enables us to evaluate the magnetic susceptibility : In fact the magnetic moment induced in the nucleus is the mean value of the operator ua on the deformed state ~~'i. One has

=

1 A~APp -1-Ap Ann -2A  APAP Z , Ann`4 PP-f1nP

(10)

and therefore the magnetic susceptibility is given by 1 «~i~3l~~i

X=

.~Éo

_

~2

A,~,A pp -~ApA n -2A APAnp `4nnAPP-A~P

2lnlp(ln+lxlp+l) lô(1n+1)Z A + Ip(IP+1)Z A (21n+1)Z PP (21p +1) Z n°+ (21n+ 1x2 1p+1) AnP . = ~_+1)Z 9 2 `4nnAPP-f1nP

( 11 )

MAGNETIC SUSCEPTIBILITY

165

We reca11 14) that the HF susceptibility ~ is related to the RPA inverse energyweighted sum rule by (12)

X~ _ ~(B-r(M1)T~ern~

so that a method to evaluate ~ gives the evaluation of the RPA inverse energyweighted sum rule. We note that eq. (11) has been obtained by performing a restricted HF calculation [this restriction is due to the fact that eq. (4) does not define the most general Slater determinant] ; since the HF method is a variational approach, our prediction for X consequently underestimates the value of ~F and gives a lower bound to (B-1(M1)T~,,~

n

p

n p n +lxlp +

1~(1 +1)Z A . + lp(1 + 1)Z A 21 1 (l l) pp (21p +1)2 n° + (21n + 1x21p+ 1) Anp 8 1 (21n+1)Z z

B_ 1 (M1)T =

7G

(u + i)

a

Ann`4 FP -`4nP

.

(13)

(The calculation for X is an exact HF calculation if only spin-orbit forces are considered, Possible discrepancies are due to exchangé terms in the nuclear potential and are estimated to be less than 10 ~.) Our problem is now to evaluate explicitly the coefficients Aqq. entering in eq. (11). To do this we can use the two-body Hamiltonian : z Fl o = ~ ~ + ~ at lt '.at + ~ V(rt~xW +MP;~-HPi~+BPi~). (14) t

t

t
When evaluated on the most general Slater determinant its value is given by ~~~Ho~~i = ~ ~

q

{O1 '

Ozpq(rl, rz) - i ~. agi'2 x VZpq(rla, rZd) ~ <~iQidi }., =~2=,dr n a

+ z E ~ v(r12)lpq(rl)pq'(r2IlW -ZMagq.-HlSgq.+ZB)+Sq(rl)' SQ (rzx q9

2MSgq'+ZB)

- pq(rl, rs)pq'(rz, rlJl2wsgq' - M-2H+uVgq') - Pq(rlß+rz~Pq'(rz 6,r1QKQIQIQ~i' <61?Id)(ZWS gQ -Z~}~l~z, where pq(rl~, rzd) _ ~ ~y(r1~lY'ty(r2d)~ t

Pq(rv rz) _ ~ ~q(r1Q~tQ(rzQ)~ t,e t

(15)

16 6

M . TRAINI et al.

The energy variation we are looking for is induced by transformation (4) through the change ofthe density matrix appearing in the expressiôn for the energy of eq. (15). The changes in the density matrix p4(riQ, rzd) are éxplicitly given by 8p4(riv' rztr

[14(Q+Q~p~(rlo', rzß) +i((rz x Vz)a-(ri x Oi)a)P Q(ri 6~ rzd)] ) - v4 219+ 1 +v4

(214+1)z

[IQVa'P 4(ri~, rzd)-14(14+1)p4(rlo, rz6')

-il4(Q'(r l x Vi)s - ~rz+Vz)3)P Q(ri~, rz~) `+ zl(Q(ri x Oi)s-6~(rz x Oz)s)PQ(ri~~ rz~) - z{(ri x Oi)a-(rz x Oz)a)zp4(ri~~ rzd)]~ After inserting expression (16) in the formula for the energy (eq. (15)), we get the following result for the energy variation : SE -- - ~ 3v4a414(IQ+ 1) 4 11 .(I +lxl .+1) 1 r . ~v4v4 + (214 +1x21q +1) ~z J V(r)R"` °(rl)R"i°,(rz~i~z(-zMa44- +iB) 3nz ~ ,l v4v9 (214 + x214,+1) R"~°(ri)R"t°(rz)R "i

°,(ri)R"t°,(rz)

âRzr z +(R' x {rrz C2(R4 + ier4)r)z) dd``° iz (Rzr z- . r)z)z _dz P d z z (R t Pi _ Rz- âr z z z dCdP` rlrz (R r -(R ~ r) ) du du + rir2 d u z duz°'

du

. 1 ~~-2(Rz - i z ~t_ Rzrz - (R r)z dzPi ar du duz°) 14(14+ 1 )P~° + q -~ q rlrz + ) + rirz

+3141q(14 +

+. ~z

lxlq + 1)P,°P,°, ~ (ZWS4q -?Flkir l drz

~ { J v4 (21V+r1)z 49 4

R"i°(ri)R"i°(rz)R"~°-(ri)R"t°,(rz) rTz

dP dP `°' x x(214+214-1xRzrz -(R~ r)z) -s du du

1 2

MAGNETIC SUSCEPTIBILITY

167

z - _8 R -ârz (Rzrz-(R . r)z) dPi °- dzP~° du du z 3 rl rz +? 3

(Rzrz

r)z)z dPi d 3 Pi° °' ~ drldrz+ (4 ~ ~)~ (iWag9 -iH) du du J

aR~ rlrz

4 1

vv

,

V(r)

R"i (ri)R"i (rz)R"~ (ri)R"i (rz)izz

dP dP -M-ZH+BS 99.}dr ldr z , x (Rzrz -(R ~ r)z) ~° i°' du du (iWa9q

(17)

where R;,j(r) is the single particle radial wave function, R = ~ r l +r z), r = rl-rz and P~(u) is the lth Legendre polynomial with u = r l ~ r z lr,r z . Eq. (17~ by comparison with eq. (7), permits the evaluation of the coefficients A"", APP and A" P . We note that for polarized shells with high values of 1 9 the angular geometry greatly depresses all the terms of eq. (17) but the first two terms ; explicit calculations performed with expression (17) have shown that one gets an accurate estimate of B_ 1 (M1) (in z SPb) with the following simplified expressions for A 99 , : °

1 lz(1 +1)z (' z z V(r)R"i°(rl)R",°(rz~rldrz, A99 __ _ a 9 19(19 + 1)+(B-M) _ ~z (21 9 + 1)z J C1

B

A"P

(21n+lX2llP+1) ~z

(18)

J .V(r)R"in(rl)R"1p(rzklrldrz .

The previous procedure can be repeated using a different nuclear Hamiltonian, e.g. the effective density-dependent Skyrme interaction

where . vlz = t°( I+x°Piz~(ri-rz)+it~~a(ri-rzxVi-Oz)z+(Oi'Oi)za(ri-rz)~ +tz(Vi -Vz)b(ri - rzxOi -Oz)+ 6ta( 1 +x3Plz~(ri - rz)P~z(ri +rz)) +4iW°( 1 Q

+Q

Z (Vi -VZ) x S(r l - rzXVl - Vz)~ ) ~

Of course one must consider the appropriate expression for the energy density when time reversal simmetry requirements are not imposed. In fact transformation (4) can change some time odd quantities like spin density and momentum density. The general expression of the energy has been already derived in ref.'s) and the variations induced by transformation (4) calculated in ref. 13). For technical

M . 1TtAINI et al.

168

details we refer to the previous works. One finds for the coefficients A99 . : A~

(21qq + - 8W° +2t 1

1)z

~ {4t°(1-xo) l J

fRiQdr + 3 Wo

Ri,rzdr+3ts(1-xs) z

~ J dr ~°

i9(l9 +11)

3{514+519 +2)

~R1 gdr+ ~~~R~Q~ rz dr~

z

1°(1° +1)lp(1P+1)

where p9

J

RiQP~

+ p9)Rigrdr

,J

1 = R~inR~iDr R~ 4t°x° z dr+3t3x3 A°a (21° +1X21p +1) ~{

- 3 Wo

J

R,z,~R~Pdr-~ti + tz)

J

aR~pparzdr

R~ioR°iP~ t

is the density of the q-nucleons and p is the total nuclear density. 3. The energy-weighted sum rule

In this section we recall some results concerning the energy-weighted sum rule. According to a theorem due to Thouless 1 fi), one obtains the energy-weighted sum rule in the random phase approximation by evaluating the double commutator of the nuclear Hamiltonian with the excitation operator on the HF ground state ; explicitly one has B1(M1)Î = ~

CY'~Lt43 ~ ~ HO~ P 3~~~Y'i

_ - ~ ~t_ +i) z ~~~'~ aili ~ i

Qi~~i

9 ~ (Qt - Qi) z (ii _TÎ)z V(r i1XBP°I -HP,i)~~ i~ (2~) +z)~_ 327c (P i
MAGNETIC SUSCEPTIBILITY

169

27i~-+2)z[anjnlln+l)+Clpjp(jp+1)] Bl(M1)Î 8n3 ~

+ z)zH

Rzr _ R r)z ' zrzr du~ du~° V(r)R,~a(rl)R~ip(rz)R~tp(ri)R~ta(rz~i~z~ J iz

(21)

Similarly to the case of the susceptibility sum rule [eqs. (13) and (18)], the energyweighted sum rule permits a very simple analysis of the role ofthe two-body nuclear interaction Ho (entering only through the Heisenberg exchange potential) . For the Skyrme interaction one obtains 1 z)

41C (P

+ z)zWo J Cln(in + 1)R~n

d (P + Pn) +

ip(ip + 1)RnIP

dr (P + Pp)J

rdr .

(22)

In expression (22) only the spin-orbit term contributes to B1(Ml)j so that in the framework of the Skyrme interaction this result corresponds to the Kurath sum rule : 4. Numerical results and conclusions The experimental situation for the M1 resonances presents sôme uncertainties as to the magnitude and distribution of the observed strength. One of the biggest difficulties consists in the definitive assignment of spin and parity to the observed states. With the aim of helping the reader to correct or add to any future experimental information, we have listed in table 1 recent results obtained at Princeton z ), Argonne a) and Oak Ridge 4). These data refer to different and contiguous energy regions. covering the interval between 7.06 and 9.5 MeV. All the measured levels in these regions are good candidates for a J~ = 1 + assignment. We have not reported resonance fluorescence experiments because the assignment JR = 1 + is very uncertain l '). From the table one can see the relative contribution to Bl(M1)j and B_ 1(M1)j from the two strong states (7.06 and 7.98 MeV) and the group of levels below and above 7.98 MeV. In table 2 we list the numerical values for the total strengths Bl(M1)j and B_ 1(M1)j as well as the mean energy E = Bl(M1)j/B_ 1 (M1)j. In the first column we report the predictions of the pure jj coupling model (spin-orbit contribution only). In the remaining columns we compare our theoretical results for different kinds of interactions with other results 5 " e. 9) and with the total experimental strengths. The RPA calculations of refs. S" e) have been performed in a one-particle one-hole

170

M . TRAINI et al. TABLE 1

Observed M 1 transition strengths and energies E (MeV)

Bo(M1) (eh/2mc)z

7.06

16 .8

7 .41 7 .46 7 .50 7 .51 7 .53 7 .54 7 .55 7 .58 7 .62 7 .65 7 .66 7 .70

0.53 1 .26 0.96 2.19 0.4 0 .07 1 .28 0.09 0 .11 0 .09 0 .41 0 .68

7 .98

9 .0

8 .22

1 .2

8.37 8.6 8.69 9.01 9.142 9.30

0 .93 1 .35 1 .23 1 .56 1 .39 ~ 0 .84

B,(M1) (MeV ~ (ei~/2mc)z)

z)

3 .93 9 .4 7 .2 16 .45 3 .01 0 .53 9.66 0.68 0.84 0.69 3.14 5.24

0 .07 0 .17 0 .13 0 .29 0 .05 0.01 0.17 0 .01 0 .01 0 .01 0.05 0 .09

a)

72

1 .1

3,4)

9 .86

0 .15

3 .4)

7 .78 11 .61 10 .69 14 .06 12 .71 7 .81

0.11 0.16 0.14 0.17 0.15 0.09

3)

2

Numerical values for the total strengths B Ml Pure shell model

COP

CAL

288

336

8 .7 5 .7

5 .5 7 .8

Ref.

2 .4

119

TABLE

B,(M1)j (MeV ~ (eh/2mc)z) B_ 1 (M1)j E(MeV - '(eh/2mc) z)

B_,(M1) (MeV - ' ~ (eh/2mc)z)

, B_

M1)j and for the mean energy

Slil°°

SKa

Ref.')

Ref. a)

Ref.

288

~ 327

341

367

134

248

326

6 .0 6 .9

6 .2 7 .2

6 .3 7 .4

6.6 7.5

2 .1 5 .0.

3 .6 8 .3

5 .5 7 .7

9)

Exp

approximation. Vergados used Kuo's matrix elements and no effective operators. Ring and Speth used a density-dependent residual interaction as proposed by the Migdal group and effective operators. The differences in the previous calculations are connected with the spin dependent part of the particle-hole interactions used. In ref. 9) the contribution of two-particle two-hole correlations has been investigated .

17 1

MAGNETIC SUSCEPTIBILITY

The results of the present work have been obtained by using different nuclear interactions : the COP and the CAL forces which have been introduced by Gillet and Sanderson ia) to perform RPA calculations in doubly closed shell nuclei and two density-dependent Skyrme forces, here denoted as SIII°° [ref. 19)] and SKa [ref. 2°)]. These last two interactions, widely used in HF calculations, produce stable spin-saturated ground states i2) and give satisfactory magnetic properties .in the lead region ia). TnsLe 3

Parameters of the nuclear interactions used in the present work

COP CAL

Vo

p

W

M

H

B

-40 -40

1 .68 1 .68

0.4 0.175

0.4 0.575

0.4 0

-0 .2 0.25

to . SIII°° SKa

-1128.75 -1602.78

Wo

0.45 -0.02

395 570.88

-95 -67.70

14000 8000

120 125

1 -0.286

1 i

In table 3 we list the parameters for the different interactions. Integrals entering in the expressiôns for B_ 1(Ml)j and Bt(M1)j have been calculated using harmonic oscillator wave functions R,~(r) and a Fermi two-parameters density p. The values fog ao and ap have been chosen to reproduce the experimental spin-orbit splittings. About the use of COP and CAL forces, the following remarks are in order here . For these forces one cannot invoke the RPA sum-rule theorems of Marshalek and Da Providencia (eq. (12)~ and Thouless [see discussion above eq. (20)] because these interactions cannot be used in Hartree-Fork calculations due to the way in which they were built. By using them we estimate, with an uncorrelated ground state, the contribution of the residual interaction .to Bl(M1)j and B_,(M1)j . Analogous calculations for M1 transitions in lead have been performed in ref. lo). The RPA correlations would be taken into account by considering, for example, the Brink-Boeker at) interaction which is to be used in Hartree-Fock calculations and for which RPA theorems are fulfilled. Unfortunately, to .simplify the notation, the authors of ref. 21) have taken B = H = 0. Thus, physically, it makes no sense to give predictions of the BrinkBoeker force for quantities that are sensitive to B and H, as is the case in the present work . Conversely, it is with the type of approach proposed in this work that one can further determine the parameters of the Brink-Boeker force to fit experimental data, without loss of agreement with previously fitted quantities . Some indications can be obtained from an analysis of table 2. Since the Kurath sum rule (spin-orbit only, col. 1) does not saturate completely the experimental energy-weighted sum rule Bl(M1)j, one can conclude from eq. (20) that H .must be

172

M. TRAINI et al.

positive in our notation for the potential (eq. (14)). This means that the Heisenberg. exchange force must be attractive in the isospin singlet state and repulsive in the triplet one. In the case of the inverse energy-weighted sum rule [B_1(M1)T], the pure shell model prediction overestimates the experimental values ; so that the nuclear interaction must reduce this sum -rule. This fact puts constraints on the values of the Bartlett and Majorana exchange forces [see eqs. (13) and (18)]. The RPA sum-rule theorems can be invoked for the two density~ependent Skyrme interactions SIII°° and SKa. .The only approximation we made with these fôrces was to replace the exact Hartree-Fock ground state by some approximation to it. (Use, for example, of a Fermi two-parameter density instead ofthe HF density.) Therefore one must consider our results for Bl(M1)T and B_ 1(M1)T with these forces as RPA calculations . By referring to table 2 one can see that all the forces used in the present work (except CAL which has H = 0) increase the value of Bt (M1)j and lower that of B_ t (MI)T with respect to the jj coupling model predictions. This effect ofthe nuclear potential is reflected in the value of the mean energy E = B1 (M1)T/B_ 1(M1)T which increases with respect to the pure shell model predictions. Finally we note that the Kurath sum rulé is not strongly affected by the nuclear potential so that it must be considered a good estimate of the linear energy-weighted sum rule. References

1) S. Hanna, Giant multipole resonances, Proc . Int. Sçhool on electro- and photnuclear reactions, Erice 1976, ed . S. Costa and C. Shaerf (Springer-Verlag, Berlin, 1977) p. 295 2) S. J. Freedman, C. A. Gagliardi, G. T. Garvey, M. A. Oothoudt and B. Svetitsky, Phys . Rév. Lett . 37 (1976) 1606 3) R. J. Holt and H. E. Jackson, Phys . Rev. Lett . 36 (1976) 244 ; R. M. Laszewski, R. J. Holt and H. E. Jackson, Phys . Rev. Left . 3g (1977) 813 4) D. J. Horen, J. A. Harvey and N. W. Hill, Phys. Rev. Lett . 38 (1977) 1344 ; S. Raman, M. Mizumoto and R. L. Macklin, Phys. Rev. Lett . 39 (1977) 598 5) J. D. Vergados, Phys. Lett . 36B (1971) 12 6) P. Ring and J. Speth, Phys . Lett. 44B (1973) 477 7) A. Bohr and B. Mottelson, Nuclear structure, vol. 2 (Addison-Wesley, Reading, Mass., 1975) 8) .J . Speth, E. Werner and W. Wild, Phys . Reports 33C (1977) 129 9) J. S. Dehesa, J. Speth and A. Faessler ; Phys . Rev . Lett . 38 (1977) 208 10) E. Lipparini, S. Stringari, M. Trairai and R. Leonardi, Nuovo Cim. 31A (1976) 207 11) D. Kurath, Phys. Rev. 130 (1963) 525 12) S. Stringari, R. Leonardi and D. M. Brink, Nucl. Phys . A269 (1967) 87 13) E. Lipparini, S. Stringari and M. Trairai, Nucl . Phys. A291 (1977) 157 ; A293 (1977) 29 14) E. R. Marshalek and J. da Providencia, Phys . Rev. C7 (1973) 2281 15) Y. M. Engel, D. M. Brink, K. Gceke, S. J. Krieger and D. Vautherin, Nucl . Phys . A249 (1975) 215 16) D. J. Thouless, Nucl . Phys. 21 (1960) 225 17) R. M. Del Vecchio, S. J. Freedman, G. T. Garvey and M. A. Oothoudt, Phys. Rev. C13 (1976) 2089 18) V. Gillet, Nucl . Phys . 51 (1964) 410 ; V. Gillet and E . A. Sanderson, Nucl . Phys . 54 (1964) 472; A91 (1967) 292 19) M. Beiner, H. Flocard, Nguyen Van Gisi and P. Quentin, Nucl . Phys. A238 (1975) 29 20) H. S. KShler, Nucl. Phys. A25g (1976) 301 21) D. M. Brink and E. Bceker, Nucl . Phys. A91 (1967) 1