ββ decay of 76Ge with renormalized effective interaction derived from Paris, Bonn and Reid potentials

Volume 242, number 1

PHYSICS LETTERS B

31 May 1990

[31~DECAY O F 76Ge W I T H R E N O R M A L I Z E D E F F E C T I V E I N T E R A C T I O N D E R I V E D FROM PARIS, BONN A N D R E I D P O T E N T I A L S A, STAUDT, T.T.S. K U O 1,2 and H.V. K L A P D O R - K L E I N G R O T H A U S Max-Planck-lnstitut J~r Kernphysik, D-6900 Heidelberg, FRG Received 5 March 1990

Two-neutrino and neutrinoless double beta decays of 76 Ge are calculated using renormalized effective interactions derived from the Paris, Bonn-A Reid potentials. Core polarization and folded diagrams are considered in our effective interaction. The dependence of the two-neutrino nuclear matrix elements on the strength of the particle-particleinteraction in the QRPA for the various potentials are compared. For the neutrinoless decay our calculated values for T°yz (mv)2 are typically 3 × 1023yreV2. The present extensive study shows that the neutrino mass should be reliably extractable from [~[~-decayexperiments within a factor &three.

N u c l e a r double beta decay is expected to proceed via two decay modes, the two-neutrino m o d e (2v l~13) and the neutrinoless m o d e (0v [~fl). The latter is o f particular interest, b o t h experimentally a n d theoretically, as it is this decay process which is sensitive to the existence o f massive M a j o r a n a neutrinos a n d sets the most stringent limits on the neutrino mass a n d the coupling constants o f the right-handed components o f the charged weak interaction [1,2]. This l e p t o n - n u m b e r non-conserving process is f o r b i d d e n in the s t a n d a r d S U ( 2 ) L × U ( 1 ) gauge model o f the electroweak interaction, in which the neutrinos are massless and there exist no right-handed currents. Massive neutrinos naturally a p p e a r in grand unified theories ( G U T s ) [3]. The nature ( D i r a c or Majorana particle) and the mass o f the neutrino could yield i m p o r t a n t b o u n d a r y conditions for these models. Reliable deductions o f the v-mass from m e a s u r e d 0v J313decay rates are, however, sensitively d e p e n d e n t on predictions o f nuclear matrix elements. Therefore it is o f u p m o s t i m p o r t a n c e to calculate these m a t r i x elements as accurately as possible. Since the nuclear wave functions and the underlying theory fo r treating Work supported in part by US DOE Grant DE-FG0288ER40388. 2 Permanent address: Physics Department, State University of New York at Stony Brook, Stony Brook, NY 11794-3800, USA.

the 0v 131~a n d the 2v ~13 decay are the same, one comm o n l y first calculates the matrix elements for the latter in order to test the nuclear structure theory. This is because there exist experimental results for the 2v 1313processes. Recently considerable progress has been m a d e concerning the understanding o f the suppression m e c h a n i s m of the 2v transition m a t r i x elements [4-7]. 76Ge is one o f the most promising [313emitters for the study o f 0v [313decay. A m o n g the direct (non-geochemical) experiments it gives at present the smallest limit o f the neutrino mass, ( m y ) <2.6 eV [8] corresponding to the half-life limit o f T1°~2>4.7 × 1023yr [9]. O u r interest in this nucleus is further m o t i v a t e d by the ongoing H e i d e l b e r g - M o s c o w 1313 collaboration [10]. This project employs enriched 76Ge as both source a n d detectors, instead o f the c o m m o n l y used natural Ge, a n d thus could substantially i m p r o v e the deduced life-time limits, exploring the half-life o f the 0v I~[~decay up to the 1025 y r range. Calculation o f nuclear [~[3-decay matrix elements is nevertheless a rather sensitive m a t t e r and one is naturally concerned about the c h o i c e o f the effective interaction (Vc~f) to be used in such calculations. In other words one would want to know what is the variation o f these m a t r i x elements due to the use o f various effective interactions. In the present work we shall consider the three realistic n u c l e o n - n u c l e o n

0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )

17

Volume 242, number l

PHYSICS LETTERS B

(NN) interactions - Paris [ 11 ], B0nn-A [ 12,13] a n d Reid [ 14 ]. These three interactions all reproduce the two-nucleon data at low energy ( £ 3 0 0 MeV) remarkably well, the quality of their fits to the data being practically indistinguishable. But there is one important intrinsic difference among them. The strength of the NN tensor force is among t h e most unsettled issues in NN interactions. The recent Bonn-A potential, which is derived from relativistic meson theory and it is not parametrized in terms of local functions as is usually adopted by other NN potentials, has a Considerably weaker tensor force than the Paris and Reid potentials, The predicted deuteron D-state probability PD, a commonly used measure for the tensor force strength, is 6.5%, 5.8% and 4.4% for the Reid, Paris and BonmA potentials respectively. Tensor force plays a highly important role in nuclear structure calculations, as it is well known. Thus it should be of interest as well as useful to investigate the connection between the tensor force strength and the predicted nuclear [3~-decay matrix elements. To proceed we need first to derive from these NN potentials the respective Veermatrix elements for l~l]decay calculations. This is, however, a long and rather difficult task. Here we would want to put up a "best possible" effort in carrying out this derivation. The Pauli exclusion operator in the G matrix will be treated in an essentially exact way in the present work. In earlier works this has been treated only approximately. In addition we shall also consider the renormalization contributions to Verefrom Core polarization and folded di~tgrams. Nuclear structure calculations are almost always carried out in a restricted model space commonly denoted by P. The free NN potentials such as those mentioned above are, however, designed to be used inthe full space P + Q, Q being the complement of P. Hence a first step is to convert the full-space secular equation H T = E T t o a model-space secular equation HerrPT=EPg-'. Here the interaction part of H is V, the free NN interaction, while that of Herr is Vcrr,the model space effective interaction. Starting from a given Vand P, there exist a number of formally exact theories for carrying out the calculation of Veer. One such theory is the folded-diagram method formulated by Kuo, Lee and Ratcliff [ 15,16 ]i This method is relatively convenient and systematic for actual cal18

31 May 1990

culations and is adopted in the present work. A first step here is the calculation of the reaction matrix G defined as G((.o) ..~ Vq- VQ2 p

1

-:

¢o-Q2pT(2p)Q2p

Q2pG(o))

' (t)

where T(2p ) denotes the two-particle kinetic energy. The model space we employed for our ~[3-decay calculations of 76Ge consists of nine proton orbits and nine neutron orbits, each ranging from 0f7/2 to 2sl/2, outside a closed 4°Ca core. We must avoid double counting, in the sense that the correlations included in the intermediate states of the G matrix must not be repeated by those generated by Solving the nuclear secular equation within the above model space. The exclusion operator Q2p in the above equation is mainly for this purpose and we must treat this exclusion operator; which ~projects onto all the two-particle states outside the model space, accurately: This is, however, a rather difficult requirement to enact, a major difficulty being that the G-matrix equation is usually solved in the two-nucleon relative and centerof-mass frame but on this representation the operator Q2p is not diagonal and is very complicated. This problem is further complicated because our:present calculation involves a rather large model space. There exists, however, an essentially exact method in correctly treating the above exclusion operator [ 17,18 ], This method involves the inversion of rather large/'-space matrices and in the past it has been only feasible to apply it to light nuclei such as those in t h e sd shell [ 18]. We have now extended this method to the 76Ge region. With this method the solution of the above model: space G-matrix equation is given exactly as

GT(O)=G:F(O) + aG(o),

(2)

where GTFis the free G matrix defined as GTF(CO)= V+ V o - T(--2p) 1 GTF(O).

(3)

AG(o)) is a correction term defined entirely within the P-space, given as

PHYSICSLETTERSB

Volume 242, number 1 A G (0.)) ~- - - G T F ( O ) ) 1 P2p

e

x

1

P2p [ ( 1/e) + ( 1/e) GTF(tO) ( 1/e) ]P2p

XP2p I GTF((.0) , e

(4)

where e =-o)- T(2p). Pap is the complement of Q2v in the two,particle subspace. Thus we see that the G matrix is now expressed as the sum of two terms; the first term is the free G matrix while the second is calculated using only some simple model-space matrix operations involving the free G matrix. The free G matrix no longer contains the troublesome Q-space projection operator and may be calculated vm momentum-space matrix-inversion methods [ 19]. Indeed the above formalism provides a convenient and essentially exact method tbr calculating the modelspace G matrix. The model-space used in the present work is specified with (nl, n2,//3) given by (6, 15, 28) [18]. (In this notation the nine orbits mentioned above are denoted as orbits 7-15 ). This is a rather sizeable model space and requires a considerable amount of computing effort in carrying out the respective G-matrix calculation. (For comparison, the calculation performed by Krenciglowa et al. [ 18 ] for the sd shell nuclei involved a much smaller P-space of size (nb t/2, /73 ) = ( 3, 10, 21 ). ) We have performed G-matrix calculations as outlined above for the Paris, Bonn-A and Reid potentials for several values for the energy variables co. Based on the above folded-diagram theory, our effective interaction is then expressed as a folded diagram series, grouped according to the number of folds. Namely V¢~=Fo + F , +F2 + F 3 + . . .

(5)

where F, denotes a ( n + 1 ) Q-box term connected with n sets of folded lines. For example, the threetime folded term has the form F3=-oj'ofofo,

(6)

where f stands for a generalized folding operation. In the present calculation we have included in the 0-box all the irreducible diagrams first and second

31 May 1990

order in G, namely the same set of seven diagrams included in the sd shell calculation of Shurpin et al. [ 20 ]. We shall present the details of our calculation in a future publication; for the present let us just mention the various second order diagrams are calculated with respect to a 4°Ca core in consistence with our chosen model space. In fact we have used three schemes to evaluate the effective interactions: ( 1 ) G: Here the effective interaction Veffis taken as the bare G matrix only. (2) G+G2(G (2)): Here we do not include any folded diagrams in Veffand take it as composed of just the three two-body G-matrix diagrams G, G3plh - the core polarization diagram - and G2h. This is in analogy to the choice made sometime ago by Kuo and Brown [ 211. (3) Lesu: This is the "full" calculation where the folded diagram series is summed up to all orders using the Lee-Suzuki iteration method [ 22]. In the Qbox we have included the seven diagrams as mentioned above. The nuclear matrix elements for the [313decays are then calculated within the framework of the protonneutron quasiparticle random phase approximation (pnQRPA) [23 ]. The G matrix is dependent on the energy variable ~o. When using the effective interaction prescribed by ( 1 ) and (2) there is the ambiguity: about what value of co one should use for G. (A value of - 10 MeV is used in the present work. ) This ambiguity is removed in the Lesu case as here the derived V~ffis energy independent [ 15,16,22 ]. The nuclear matrix element of 2v decay is [2 ] M~'~-(7) = ~ (0~-tlt-aII 1~ )
E . + Qf~f~/2+ me - E ~

"

(7) where we have replaced the lepton energy in the intermediate state by Q~/2+ me. The various quantities in the above equation are readily calculated from the pnQRPA energies and amplitudes as described in detail in ref. [ 2 ]. We note that the interaction terms in the pnQRPA secular matrix are written as pp gppVp~,p,n, for the particle-particle interaction and z.~V ~vJ. ph pn,ptrlt for the particle-hole interaction. The strength parameters gpp and gph are introduced in order to study the sensitivity of the calculated t313matrix elements to these two types of interactions. 19

Volume 242, number 1

PHYSICS LETTERS B

Turning to the 0v 131~mode, we write its inverse halflife as

[TO,/z]_,=Cmm((my) ~z+ C~. (t/) 2+ Czz ( 2 ) 2 k me /

...l_Cmrt

( m,,___~)( q) + Cm~.~ lq/l e

tTt e

+c.~ (n) <)~>.

(8)

For a definition of the effective values of the neutrino mass (m.,), the coupling strengths of the righthanded currents ( t / ) , ( 2 ) and the coefficients C~ we refer to ref. [ 8 ]. The coefficients Cxy consist of products of electron phase-space integrals Gk and the nuclear matrix elements M ,

M~= E (0~-Ilt_mt_~C°~IlO+) ,

(9)

where (9mn are the relevant two-body transition operators. Neglecting contributions of right-handed currents, only the matrix elements MGT ov and M °~ 2

(~=¢rl.a2Hm(r),

(gV2=Hm(r)(~)

(10a,b)

contribute, where Hm(r) is the "neutrino potential" [ 8 ] which represents the exchange of a virtual neutrino between two nucleons. The inter-nucleon shortrange correlations and the nucleon finite size effects

31 May 1990

are taken into account in the standard way [ 8,24 ]. Because of the two-body character of the operators (9~2 the nuclear matrix elements can be reduced to a sum of products of two-particle transition densities Z and two-particle matrix elements of the above operators. From the solutions of the pnQRPA equation the densities Z are readily evaluated [ 8 ]. The single-particle energies used in our calculation are obtained from a Coulomb corrected WoodsSaxon potential [25 ], where the depth of the central potential is modified by adding the/-dependent term - 0 . 0 5 l ( l + 1 )MeV. In the subsequent BCS calculation the self-energy term/z is set to zero because this shift in the single-particle energies is presumably already taken into account by the use of the appropriate N- and Z-dependent Woods-Saxon potential. The strengths of the pairing interaction of the BCS calculation are adjusted to experimental even-odd mass differences [ 26 ] for proton and neutron systems separately, by multiplying the respective pp and nn pairing forces in the gap equation with the renotanatization factors gpair and g~air. These and other details concerning the BCS and pnQRPA calculations are the same as described earlier [2,7,8 ]. In table t we present some calculated proton and neutron pairing gaps with g~,~ir= 1 for some shells near the Fermi surfaCe for different effective interactions. It is clearly seen that the Bonn force gives generally about 15% more pairing than the Paris and Reid po-

Table 1 Proton and neutron pairing gaps calculated from various effective interactions and the strength parameters needed to reproduce the experimental odd-even mass differences.

Zip

An

gppair

gnpair

f5/2

P3/2

Pl/2

f9/2

f5/2

P3/2

Pl/7

g9/2

Bonn-bare Bonn-G ~2) Bonn-Lesu

1.483 3,074 1.820

1.643 Z766 1,754

1.724 Z816 1.777

1,366 2.223 1.343

1,753 2,889 1.422

1.428 2.629 1.578

1,454 2.517 1.348

1.177 2,030 1.036

0.956 0.675 0.909

1.157 0,873 1.211

Paris-bare Paris-G ~2) Paris-Lesu Paris--Yukawa

1.247 2,502 1A66 1.219

1,448 2,298 1,432 1.414

1.530 2.361 1,473 1.516

1,189 1.823 1.079 1.195

1.539 2,358 1.075 1.622

1.189 2.106 1.254 1.145

1,222 2,005 1.033 1.190

0,957 1.597 0.771 0,891

1.050 0.761 1.032 1.062

1.283 0.991 1,380 1,319

Reid-bare Reid-G ~z) Reid-Lesu Reid-Yukawa

1.252 2.427 1.425 1.209

1.443 2.242 1.398 1,404

1,525 2.315 1.463 1.503

1.193 1.774 1,092 1.181

1.547 2.303 1.136 1.605

1,197 2,038 L20t 1.131

1.227 1.939 1.201 1.177

0.954 1.544 0.828 0.872

1,048 0.773 1.047 1.066

1.285 1.009 1.350 1.331

20

Volume 242, number 1

PHYsIcs LETTERSB

tentials. For example the neutron g9/2 pairing gaps given by the bare G matrix derived from these three interactions are respectively 1.177, 0.957 and 0.954 MeV. This observation is also seen from the tabulated op~rl ~P~ values, those for the Bonn cases being clearly smaller. For comparison, the results obtained with the G matrix of the Paris potential simulated by s u m of Yukawa potentials [27] are listed in addition. This interaction was applied in the work of refs. [7,8 ]. The same procedure may also be used to derive an effective N N interaction from the Reid potential. We note that the pairing gaps given by the simulated G matrices are quite similar to those by our bare G matrices, where the exclusion operator has been treated in a more accurate way. In general, the nn-pairing force seems to be too small, as indicated by the tabulated g~ ~ values, in all cases except for the second order G matrices (G (2)) of Paris and B o n n potentials. Comparing with the bare-G results, G (2) largety enhances the pairing and this is mainly due to the contribution from the corepolarization diagrams G3plh- We feel that G (2) may have over-enhanced the pairing force. A main effect of the folded diagrams included in the Lesu cases appears to be the reducing ofthis enhancement, rendering the resulting pairing gaps close to the bare-G values. One may also note that the results of the Reid and Paris NN-potentials in table 1 are remarkably similar to each other. Our results for the 2v matrix elements are presented in figs. 1 and 2. Fig. 1 illustrates the dependence of MGT 2v on gpp for different effective interac, tions derived from the Paris potential. We find a similar behaviour as in earlier QRPA studies, namely the matrix elements stay more or less constant for small gp, values but decrease towards zero rapidly near gpp = 1; A further increase of gpp will cause the QRPA to collapse - the occurrence of complex QRPA eigenvalues. It is of interest that for the Lesu V~ffthe collapse of the QRPA equation occurs at significantly larger values of gpp compared to the other forces. I f we set ~,p,n °'pair-2v is substantially more re- - -| then MOT duced for the bare G matrix and the Lesu I%ffwhereas it is enhanced for G (2). For a more detailed discussion we refer to a future publication. Compared to the results of Muto et al. [ 7 ] we obtain somewhat less suppressed matrix elements and the slope is much

31 May 1990 I

f I Veff (Lee-Suzuki)

1

1.0

,-.., O.5 'T

\~seeond

~E

76Ge

~zo . o

ParlYiawa)' orderOG \~ I bare

Paris p o t e n t i a l

-0.5

0.0

0.2

1 0.6

0.4

)

) 0.8

t

I

1.0

12

gpp

Fig. 1.2v matrix elements calculated with different choices for the effectiveinteraction. 1.2

1.0

"7

>~

>p-

I

I

I

I .............

]

/

~'~.......... .........

Par~ potential , d -'~"-'~ , ~ Reidpotential

O.8

8eon-A.e,e.

0.6

I

,

", \\ /

i

i \1

Le -Sozok, v,,,

II

0.4

0.2 0.0

~.

I 0.2

)

I 0.4

)

1.. 0.6 gpp

)

I. 0.8

)

I 1.0

) . 1.2

Fig. 2. 2v matrix elements calculated from renormalized effective interactions derivedfrom Paris, Bonn-Aand Reid potentials.

steeper, especially for the bare G matrix. Fig. 2 compares the 2v matrix elements using the Lesu Vee derived from different N N potentials. The results for the Paris and Reid potentials are quite similar, while the Bonn-A potential yields a considerably smaller matrix element. The 0v transition matrix elements are known to be less sensitive to details of nuclear structure [ 8,24,28 ]. Table 2 contains the calculated values for MF and MoT for different values of the strength of the particle21

Volume 242, number I

PHYSICS LETTERS B

31 May t 990

Table 2 0v transition matrix elements mad half-lives for different effective interactions and particle-particle interaction strengths (blanks indicate the collapse of the QRPA equation). gpp= 0.875 Mm-

ME

gpp= 0.90 T~°)'2(m~)2[yreV 2]

MGT

gpp= 1.00

Mr

Tl°)'2(mv>2[yreV2l

-1.33

5.15× 1023

Bonn-bare Bonn-G ~2) Bonn-Lesu

7.77

- 1.36 4.89X 102~

7.57

10.28

- 1.62 2.88×t0 z3

10.19

Paris-bare Paris-G C2) Paris-Lesu Paris-Yukawa

8.25 6.79 10.91 3.00

-1.49 -1.13 -1.83 -1.17

4.31×102a 6.51×1023 2.5t×1023 2,34× 1024

t0.82 2.65

-1.80 2.567<1023 - 1,14 2.84× 1024

Reid-bare Reid-G (2) Reid-Lesu Reid-Yukawa

8.26 7.61 10.33 3.36

-1.4t -1.13 - 1.72 -1.12

4.36× 1023 5.34× 1023 2.81× 1023 2.03×1024

8.t3 4.38 10.24 3.06

-1,39 -1.08 - 1.69 -1.09

8.11

particle interaction, gpp is expected to be close to unity provided that an appropriate force is chosen. Therefore we compare the present calculation at gpp = 1 with that of ref. [ 8 ] a t gpp = 01875 which was determined from a QRPA calculation of single 15+ decays. We also present the results for an effective interaction simulated by a sum of Yukawa potentials derived from the Reid potential for which we expect a gpp value slightly larger than 0.875 (gpp ~ 0.9 [ 29 ] ). Table 2 indicates that the Lesu Verfyields values of T°'/z ( r n v ) 2 which are smaller by about one order of magnitude compared to the QRPA results of refs. [8,24] which would result in about three times smaller limits of (m,,) at a given half-life. It may be noted that the corresponding value given, by Grotz and Klapdor [4] sometime ago based on a PBCS approach and diagonalisation of a spin-isospin and quadrupole-quadrupole force is 2.6 × 1023 yr eV 2. In contrast to refs. [8,29], we obtain in all cases relatively small values of the tensor matrix element MT, which was also found by Tomoda et al. [ 30 ] in a Hartree,Fock-Bogoliubov calculation with angularmomentum and nucleon number projections. In conclusion we would like to mention the following: The Bonn-A potential yields significantly larger pairing gaps than the Paris and Reid potentials. Although the pairing gaps given by the bare G m a t r i x and those calculated from the Lesu Veffare comparable, the latter interaction apparently has an impor22

Mot

ME

T°)'E(mv)2[yreval

- 1.58 2.94× 1023

9.70

--1,41

3.31×1023

-1.46

4.60

-1.33

1.167<1024

10.44 0.67

--1.66 -1.01

2.79×1023 1.46×102s

7.32

--1.26

5.53×t023

9.85 -1.56 1.53 -0.97

3.13×1023 6.58X1024

4.46×1023

4.51×1023 1.37X!02~" 2.87× 1023 2.37)<1024

rant property in hindering the QRPA collapse. Our calculated 2v matrix elements seemed to be tess suppressed compared with earlier calculations. For the 0v matrix elements our results show rather little variation with respect to the choice of the effective interaction. The main result of the present extensive study is that the neutrino mass can be reliably deduced from 0v ~ - d e c a y experiments within a factor of 2-3. The G-matrix programs were developed together with Dan Strottman and Brian Stout. Their help is gratefully acknowledged. The authors would like to thank also Dr. K. Muto for useful discussions.

References [ 1] M. Doi, T. Kotani and E. Takasugi, Prog. Theor. Phys. Suppl. 83 (1985) 1. [21 K. Muto and H.V. Klapdor, in: Neutrinos. Graduate Texts in Contemporary Physics, ed. H.V. Klapdor (Springer, Berlin, 1988) p. 183. [ 3 ] P. Langacker, in: Neutrinos. Graduate Texts in Contemporary Physics, ed. H.V. Klapdor (Springer, Berlin, t988) p. 71. [4 ] K. Grotz and H.V. Klapdor, Nucl. Phys. A 460 ( 1986 ) 395. [5] P. Vogel and M.R. Zirnbauer, Phys. Rev. Lett. 57 (1986) 3148. [ 6 ] O. Civitarese, A. Faessler and T. Tomoda, Phys. Lett. B 194

(1987) ll.

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[ 7 ] K. Muto and H.V. Klapdor, Phys. Lett. B 201 ( 1988 ) 420; K. Muto, E. Bender and H.V. Klapdor, Z. Phys. A 334 (1989) 177. [8] K. Muto, E. Bender and H.V. Klapdor, Z. Phys. A 334 (1989) 187. [9]D.O. CaldweU, R.M. Eiseberg, D.M. G r u m m and M.S. Witherell, Phys. Rev. Lett. 59 ( 1987) 419. [ 10 ] H.V. Klapdor, A. Piepke, G. Heusser, E. Buchner, A. Miiller, U. Schmidt-Rohr, H. Strecker, S.T. Belyaev, A. Balish, A. Gurov, A. Demehin, I. Kondratenko and V.I. Lebedev, Proc. Intern. Symp. on Weak and electromagnetic interactions in nuclei, (Montreal, May 1989). [ 11 ] M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. C6t6, P. Pir6s and R. de Tourreil, Phys. Rev. C 21 (1980) 861. [ 12] R. Machleidt, K. Holinde and C. Elster, Phys. Rep. 149 (1987) 1. [ 13 ] R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189. [ 14 ] R.V. Reid, Ann. Phys. (NY) 50 (1968) 411. [ 15] T.T.S. Kuo, S.Y. Lee and K.F. Ratcliff, Nucl. Phys. A 176 (1971) 65. [ 16] T.T.S. Kuo, Lecture Notes in Physics, Vol. 144 (Springer, Berlin, 1981 ) p. 248. [ 17 ] S.F. Tsai and T.T.S. Kuo, Phys. Lett. B 39 (1972) 427.

31 May 1990

[ 18 ] E.M. Krenciglowa, C.L. Kung, T.T.S. Kuo and E. Osnes, Ann. Phys. (NY) 101 (1976) 154. [19] G.E. Brown, A.D. Jackson and T.T.S. Kuo, Nucl. Phys. A 133 (1969) 481. [20] J. Shurpin, T.T.S. Kuo and D. Strottman, Nucl. Phys. A 408 (1983) 310. [ 21 ] T.T.S. Kuo and G.E. Brown, Nucl. Phys. A 85 ( 1966 ) 40. [ 22] S.Y. Lee and K. Suzuki, Phys. Lett. B 91 (1980) 173; Prog. Theo. Phys. 64 (1980) 2091. [23] J.A. Halbleib and R.A. Sorensen, Nucl. Phys. A 98 (1967) 542; K. Muto, E. Bender and H.V. Klapdor, Z. Phys. A 333 (1989) 125. [ 24 ] T. Tomoda and A. Faessler, Phys. Lett. B 199 ( 1987 ) 475. [25] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. 1, (Benjamin, New York, 1969). [ 26] A.H. Wapstra and G. Audi, Nucl. Phys. A 432 ( 1985 ) 1. [27] N. Anantararnan, H. Toki and G.F. Bertsch, Nucl. Phys. A 398 (1983) 269. [28 ] J. Engel, P. Vogel, Xiangdong JI and S. Pittel, Phys. Lett. B 225 (1989) 5. [ 29 ] A. Staudt, K. Muto and H.V. Klapdor, to be published. [30] T. Tomoda, A. Faessler, K.W. Schmid and F. Griimmer, Nucl. Phys. A 452 (1986) 591.

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