HIGH ANGULAR MOMENTA AND YRAST STATES

HAL Id: jpa-00216583

https://hal.archives-ouvertes.fr/jpa-00216583

Submitted on 1 Jan 1976

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

HIGH ANGULAR MOMENTA AND YRAST STATES

Z. Szymanski

To cite this version:

JOURNAL DE PHYSIQUE Colloque C5, supplément au n° 11, Tome 37, Novembre 1976, page C5-39

HIGH ANGULAR MOMENTA AND YEAST STATES t

Z. Szymanski

Institute for Nuclear Research ul. Hoza 69, 00-681 Warsaw, Poland

Résumé: Les changements dans la structure nucléaire qui peuvent exister dans l'état de rotation très rapide sont discutés.

Abstract: Changes in nuclear structure that may occur in the state of a very fast rota-tion are discussed.

Recent experimental progress in the heavy ion physics opens many possibilities for the produc-tion of atomic nuclei with very large angular mo-menta. The high spin nuclear states are very inte-resting since they may reveal new features of nu-clear structure emerging in the presence of a fast rotation.

In the nucleus rotating with large angular mo-mentum, considerable changes are expected to occur in nuclear shape, coupling scheme, moments, struc-ture of the spectrum etc. [l-6J. As an illustration of the changes induced by a fast rotation let us discuss the neutron deficient nucleide '°uYb. This is an example of the nucleus that may be considered as prolate deformed and superfluid in its ground state. Fig.1 illustrates the calculated energy sur-faces |_4J for several values of spin I plotted in the (e,y) deformation plane (e-Nilsson elongation parameter, y-non axiality parameter). Fig.2 repre-sents the trajectory in the (e,y) plane of the mi-nimum equilibrium shape as a function of the total angular momentum I. As,we can see from figs.l and 2, the 1 6 0Yb nucleus undergoes a change in shape from

prolate to triaxial at I * 40. In this region (i.e.for 0 < I £ 40) it is also expected that the rotational alignment of many nucleonic orbits will lead to the destruction of the superfluid correla-tions between the nucleus, so that the superfluid state is replaced by the normal state at some value of I.

At I <v 50 a second shape transition from triaxial to the oblate shape is expected. Finally, at I ^ 70 a subsequent transition is predicted that leads to the very elongated (superdeformed) shape (via tri-axial), terminating eventually by nuclear fission.

i.o .i=55$\\(7 \ l*M y£5sv\\({ \

0 ftt 02 03 Ot 05 M 0 41 U 03 «* OS 0*

i f l i y. Jill'

0 t l 02 03 04 05 M 0 01 01 U W 03 M "•Y6 Z ^ v > * " °Y* / I N/* 1.70 - A - ^ / X < \ ''K v C S~~-^ \ o oi u as OA as 06 o at 02 a3 a« 05 « e c

Fig.1 Potential energy surface for various spin values in the nucleus '°^Yb plotted in the (e,y) plane (ref.[4]).

The occurrence of this transition has been predic-ted by Cohen, Plasil and Swiatecki on the basis of the classical treatment of the rotating liquid drop model [l]. This transition should, therefore, occur ass a common feature of nuclei independent of the

Fig.2 T r a j e c t o r y i n t h e ( ~ , y ) p l a n e of t h e minimum e q u i l i b r i u m shape f o r t h e nucleus 1 6 0 ~ b . ( r e f . 141 )

.

F i g . 3 Schematic p l o t of shapes p r e d i c t e d a l o n g t h e y r a s t l i n e o c c u r r i n g i n v a r i o u s t y p e s of n u c l e i . The r e g i o n s a r e : ( I ) p r o l a t e - s u p e r f l u i d , ( 2 ) t r i a x i a l (3) o b l a t e , ( 4 ; t r i a x i a l t e n d i n g t o p r o l a t e (super- deformed), ( 1 ' ) o b l a t e s u p e r f l u i d . o r i g i n a l c o n f i g u r a t i o n of t h e nucleus i n i t s ground s t a t e , a t I = 0. On t h e o t h e r hand, t h e e x i s t e n c e and l o c a t i o n of t h e o t h e r r e g i o n s depend v e r y es- s e n t i a l l y on t h e o r i g i n a l s h e l l s t r u c t u r e i n t h e nucleus i n q u e s t i o n . Some of t h e p o s s i b i l i t i e s a r e shown s c h e m a t i c a l l y i n f i g . 3 . I n 6 t h i s review t a l k we s h a l l a t t e m p t a s h o r t d i s - c u s s i o n o £ , v a r i o u s phenomena o c c u r r i n g , o r p r e d i c - t e d i n t h e d i f f e r e n t r e g i o n s of a n g u l a r momenta. ,, I.

I TIE LOW SPIN REGION OF PROLATE NUCLEAR DEFORMA-

TXON

This r e g i o n i s t h e o n l y one where t h e d i s t i n c t

g e m l i n e s d e - e x c i t i n g - t h e quanta1 s t a t e s i n t h e nucletrs have been observed i n experiments. Conse- q u e n t l y , t h e s t r u c t u r e of t h e whole n u c l e a r spec-

trum i s f e l a t i v e l y most f a m i l i a r .

AS i s well known, mariy r o t a t i o n a l n u c l e i exhi- b i t a sudden i n c r e a s e i n t h e dement of i a e r t i a i n t h e r e g i o n of 10

6

I

6

20 accompanied by a l o c a l d e c r e a s e i n t h e v a l u e of t h e a n g u l a r v e l o c i t y LO

of t h e c o l l e c t i v e r o t a t i o n [7] ( s e e a l s o t h e re- view a r t i c l e s [8-121). The phenomenon which i s known a s t h e back-bending e f P e c t can be explained by t h e c r o s s i n g of t h e two r o t a t i o n a l bands charac-

t e r i z e d by two d i f f e r e n t moments of i n e r t i a . I n most of t h e egperiments o n l y the y r a s t i . e . lower l y i n g p a r t 6 of each band have been observed. I t i s a n i n t e r e s t i n g t a s k f o r e x p e r i m e n t a l i s t s t o s e a r c h f o r t h e remaining p a r t s of t h e two bands. The pos- s i b i l i t y of e x c i t i n g t h e whole spectrum depends es- s e n t i a l l y on t h e amount of mixing between t h e two bands. I n p a r t i c u l a r , t h e m u l t i s t e p Coulomb e x c i ta- t i o n could b e used f o r t h i s purpose.

An i n t e l t e s t i n g example of t h e c r o s s i n g bands i s provided by t h e simultaneous o b s e r v a t i o n of t h e c r o s s i n g of b o t h , t h e ground s t a t e band and t h e be- t a v i b r a t i o n a l band by t h e t h i r d band. Then t h e two back-bending e f f e c t s a r e sometimes observed [13-151

.

The p h y s i c a l p i c t u r e of t h e back-bending e f f e c t depends on t h e n a t u r e of t h e upper band. Various assumptions concerning t h e n a t u r e of t h i s band have been i n t r o d u c e d w i t h i n r e c e n t few y e a r s

.

The idea- l i z e d assumption that. i t i s a n unpaired (nonsuper- f l u i d ) band l e a d s d i r e c t l y t o t h e p i c t u r e of a pha- s e t r a n s i t i o n from t h e s u p e r f l u i d t o normal system i n t h e s p i r i t of t h e Mottelson-Valatin e f f e c t [16]

.

Various s i m p l i f i e d models have been i n t r o d u c e d i n o r d e r t o v i s u a l i z e t h e s t r u c t u r e of t h e n u c l e a r wa-

ve f u n c t i o n and c o u p l i n g scheme a t t h e p o i n t of t h e phase t r a n s i t i o n [17,18].

Another assumption r e l a t e s t h e n a t u r e of t h e up- per band w i t h t h e cwo q u a s i - p a r t i c l e s t a t e s composed o u t of t h e n u c l e o n i c o r b i t s t h a t a r e most s e n s i t i v e t o t h e C o r i o l i s i n t e r a c t i o n i n t h e r o t a t i n g nucleus [19] ( s e e a l s o the review a r t i c l e s [20-231). As a r e s u l t of t h i s i n t e r a c t i o n t h e o r b i t s a r e a l i g n e d w i t h t h e n u c l e a r r o t a t i o n a x i s ( r o t a t i o n a l a l i g n - ment model). The low Q o r b i t s belonging t o h I J l 2 ,

HIGH ANGULAR MOMENTA AND YRAST STATES C5-4 1

p l e s of such c o n f i g u r a t i o n s .

A number of o t h e r p o s s i b i l i t i e s f o r t h e n a t u r e of t h e second band have been a l s o d i s c u s s e d . L e t us mention t h e a n a l y s i s i n terms of b a n d - h y b r i d i s a t i o n

[24] l e a d i n g t o t h e K = 1 assignment of t h e upper band

.

Another approach 1251 d i s c u s s e s t h e p o s s i b i - l i t y of t h e p a i r i n g v i b r a t i o n a l band.

The model of r o t a t i o n a l alignment of Stephens and Simon h a s been a l s o e x t e n s i v e l y a p p l i e d i n c a s e of t h e odd-A n u c l e i , where t h e alignment of t h e s i n g l e p a r t i c l e nucleon o r b i t l e a d s t o t h e v e r y s u c c e s s f u l p i c t u r e f o r t h e s t r u c t u r e of t h e spec- t r a . The r o t a t i o n a l alignment h y p o t h e s i s of a n odd p a r t i c l e l e a d s , t h e r e f o r e , t o t h e unique explana- t i o n of t h e e f f e c t s b o t h i n odd-A and even n u c l e i . The alignment mechanism of t h e back-bending seems a l s o t o be f a v o u r e d by t h e s t r a i g h t k i n e m a t i c s i n a simple model t r e a t i n g t h e r o t o r Hamiltonian and t h e p a i r i n g f o r c e e x a c t l y b u t i g n o r i n g t h e s i n g l e p a r t i c l e s p l i t t i n g [26].

some o f t h e HFB approaches a s a p p l i e d t o t h e inves- t i g a t i o n of t h e back-bending t r a n s i t i o n have been checked a g a i n s t simple s o l v a b l e models L38-40J. The r e s u l t s s e m t o prove t h a t t h e HFB t h e o r y a s w e l l a s some of t h e r e f i n e d methods u s i n g t h e a n g u l a r momentum p r o j e c t i o n a r e q u i t e a d e q u a t e f o r t h e des- c r i p t i o n of n u c l e a r r o t a t i o n , perhaps e x c e p t t h e c l o s e s t v i c i n i t y of t h e t r a n s i t i o n p o i n t . Some of t h e r e s u l t s f o r such c o n f r o n t a t i o n a r e i l l u s t r a t e d i n Fig.4.

The r e s u l t s of HFB seem t o work i n f a v o r of t h e r o t a t i o n a l alignment model a s c o n t r a s t e d t o t h e hy- p o t h e s i s of a p u r e phase t r a n s i t i o n of t h e p a i r i n g

type. A s t h e n u c l e a r a n g u l a r momentum i n c r e a s e s the- r e i s f i r s t a continuous and smooth decrease i n pai- r i n g c o r r e l a t i o n s accompanied by t h e smooth i n c r e a - s e of t h e n u c l e a r moment of i n e r t i a

.

A t c e r t a i n v a l u e of a n g u l a r momentum t h e f i r s t two n u c l e o n i c o r b i t a l s decouple from t h e p a i r e d system and a l i g n t h e i r a n g u l a r momenta w i t h t h a t of t h e r o t o r . This s h a r p t r a n s i t i o n l e a d s t o t h e back-bending e f f e c t , and o b v i o u s l y , t o t h e sudden d e c r e a s e of p a i r i n g . N e v e r t h e l e s s , i t f o l l o w s from t h e c a l c u l a t i o n t h a t t h e p a i r i n g c o r r e l a t i o n s do n o t go immediately t o z e r o a f t e r t h i s t r a n s i t i o n [33]. S i n c e t h e remai- o =

0.0197

n i n g p a i r i n g c o r r e l a t i o n s a r e v e r y weak a t t h i s

100 G

.

= 0.0750 p o i n t , one c a n perhaps suppose t h a t a f u r t h e r d i s -

=

0.1000

t i n c t p a i r i n g phase t r a n s i t i o n w i l l n o t show up a t a l l above t h e back-bending i n any conspicuous way.

1

I n o r d e r t o f o l l o w more c l o s e l y t h e mechanism

51

80

_{of t h e sudden t r a n s i t i o n a t t h e back-bending p o i n t}

t h e behaviour of t h e c r a n k i n g model a t t h e two bands c r o s s i n g should be examined more c a r e f u l l y

60

E X O C ~ 135-361

.

A SCC 2

0 SCC I

+

RPA 2. THE INTERMEDIATE TRIAXIAL REGION

o VAPN

1

Fig.4 The back-bending p l o t of moment of i n e r t i a

3

v e r s u s wZ(ref. [39] )

.

The e x a c t t r e a t m e n t of t h e t r a n s i t i o n ( b l a c k d o t s ) i s compared w i t h two ver- s i o n s of t h e HFB t h e o r y (SCC2 and SCCl + RPA) and t h e p r o j e c t i o n method (VAPN).

N e v e r t h e l e s s , t h e development of t h e model of ro- t a t i o n a l alignment h a s t o be completed by a more r i - gorous t r e a t m e n t of t h e p a i r i n g f o r c e . For t h i s r e a - son, t h e Hartree-Fock-Bogolibov t h e o r y a s a p p l i e d t o t h e r o t a t i n g system h a s been d i s c u s s e d e x t e n s i v e - l y w i t h i n r e c e n t few y e a r s 127-44. The v a l i d i t y of

Above t h e back-bending r e g i o n , when some of t h e n u c l e o n i c o r b i t s have been a l r e a d y a l i g n e d w i t h ro- t a t i o n of t h e whole n u c l e u s and t h e p a i r i n g c o r r e -

l a t i - o n s seem n o t t o have any e s s e n t i a l i n f l u e n c e on t h e n u c l e a r motion, t h e r e i s a good chance t h a t t h e n u c l e u s may become t r i a x i a l . For example, i n t h e n u c l e i d e I6OYb, t h i s r e g i o n i s p r e d i c t e d t o oc- c u r f o r a n g u t a r momenta roughly between f = 40 and I = 50, a s c a n b e s e e n from Figs.1 >and 2. The nu- c l e a r c o u p l i n g scheme c o r r e s p o n d i n g t o t h e f a s t ro- t a t i o n of a t r i a x i a l r o t o r h a s been d i s c u s s e d by A.Bohr and B.R. M o t t e l s o n [ 4 1 ] . One may assume t h a t

Z. SZYMANSKI

Fig.5 Theoretically calculated bands associated with the

wobbling motion in the triaxial 6 stem

in

the region I=40

to 6 0 for the nucleus 1 1 8 ~ e (ref.f4]).

Fig.6 Illustration of the level crossings. Relation between angular velocity w energy

E

and spin I in case of 'kg. (ref. 1 4 1 ) .

H I G H ANGULAR MOMENTA AND YRAST STATES C5-43

3cd a s a s m a l l p e r t u i h g t i o n . A s a c h a r a c t e r i s t i c consequence of t h i s c o u p l i n g scheme t h e e x c i t e d l e - v e l s form c h a n n e l s o r bands p a r a l l e l t o t h e y r a s t l i n e . The E2 t r a n s i t i o n s w i t h i n each channel a r e s t r o n g l y enhanced, w h i l e t h e i n t e r c h a n n e l t r a n s i - t i o n s a r e s t r o n g l y forbidden. The d i s t a n c e between t h e channel i s determined by t h e wobbling frequency

r h e r e 3 ,

,

J2

and

J3

denote t h e p r i n c i p a l moments of i n e r t i a i n t h e t r i a : r i a l n u c l e u s

(3,

>32

)33)

1.

F i g . 5 i l l u s t r a t e s t n c o r e r g i n g p i c t u r e of t h e chan- n e l s i n t h e ' 1 8 ~ e n u c l e u s . The moments of i n e r t i a

3,

,

j2

and

z3

have been c a l c u l a t e d 141 : f o r t h e sha- pe parameters (E,Y) c o r r e s p o n d i n g t o t h e c a l c u l a - t e d y r a s t t r a j e c t o r i e s .

I n t h e s i n g l e

article

t r e a t m e n t [4] of t h e nu- c l e o n i c motion i n t h i s r e g i o n many l e v e l s c r o s s . T h i s l e a d s t o t h e d i s c o n t i n u i t i e s i n t h e energy E and a n g u l a r momentum I c u r v e s p l o t t e d v e r s u s angu- l a r v e l o c i t y of r o t a t i o n w. Consequently, t h e cur- v e s of er,ergy v e r s u s a n g u l a r momentum e x h i b i t gaps. These gaps have been a r t i f i c i a l l y averaged o u t i n t h e numerical t r e a t m e n t s developed s o Ear [3-61. F i g . 6 i l l u s t r a t e s [4] t h e t y p i c a l s i t u a t i o n of t h e l e v e l c r o s s i n g i n t h e n u c l e u s 2 4 ~ g .

One h a s t o emphasize t h a t t h e c o u p l i n g scheme d e s c r i b e d above i s v a l i d only i n c a s e of t h e l a r g e a n g u l a r momentum. For lower a n g u l a r momenta t h e f u l l v e c t o r s t r u c t u r e of t h e t r i a x i a l r o t o r h a s t o b e c o n s i d e r e d . T h i s l e a d s f o ' d i f f e r e n t c o u p l i n g scheme [4 1-43]

.

3 THE OBLATE REGION

When t h e a n g u l a r momentum i n c r e a s e s , more and more n u c l e o n i c o r b i t s a l i g n w i t h t h e a n g u l a r momen- tum of t h e n u c l e u s , and f i n a l l y t h e n u c l e a r shape becomes o b l a t e w i t h r e s p e c t t o t h e r o t a t i o n a x i s . I n c a s e of 160yb n u c l e i d e t h i s p o i n t c o r r e s p o n d s t o 3.

a n g u l a r momentum I = 50, a s s e e n from Fig.2. I n t h i s r e g i o n , the a n g u l a r momentum of t h e n u c l e u s i s of t h e p u r e s i n g l e p a r t i c l e o r i g i n . Owing t o t h e a d d i t i o n a l ( a x i a l ) symmetry, a n i n t e r e s t i n g p a t t e r n of t h e y r a s t l i n e may p r e d i c t e d . A t c e r t a i n " o p t i -

mal" p o i n t s t h e energy E may e x h i b i t l o c a l minima a s f u n c t i o n of a n g u l a r momentum I. The minima cal- l e d a l s o t h e " y r a s t t r a p s t ' may l e a d t o t h e e x i s t e n - ce of t h e isomers c h a r a c t e r i s t i c f o r t h e h i g h s p i n s t a t e s i n n u c l e i . Fig.7 i l l u s t r a t e s t h e s t r u c t u r e of t h e y r a s t l i n e f o r 1 5 0 ~ d a s an example c a l c u l a - t e d [ 4 ] w i t h t h e N i l s s o n p o t e n t i a l . The energy i n between two optimal s t a t e s is c a l c u l a t e d a s a par- t i c l e - h o l e type.

I 1 I

K O ~ d

-

r

lplh transitin DIIEWtEZl

r

-

Energy levels (parity)

a* "Optimal" contiguratiw (straight F.SJ

2 . SZYMANSKI

E

(w.)

,

,

tw fxr'rmg -err pnr rrn9 l ' n c / u d e d

!

An interesting situation occurs in some strongly neutron deficient nuclei that are spherical or obla- te even in the low spin region. The prolate or tri- axial regions such as described above for the l6Oyb nucleide, do not exist in these nuclei. In the very low spin part of the yrast line the superfluid cou- pling scheme characteristic for the pairing force may exist. When I increases, more and more nuclear

orbits become aligned and, consequently, more pairs coupled by the short range interaction are broken. The situation can be analysed by applying the BCS

theory with simple blocking of those states that contribute to the angular momentum of the nucleus. Calculation of the yrast states along this line both

"optimal" and the particle-hole type are in progress with the inclusion of the pairing force as described above L44-j. A very preliminary result for the nu- cleus 1 5 2 ~ y is shown in Fig.8. It is interesting to note that an isomeric high spin state in this nu- clpus has been experimentally observed at the ener- Fig.8 Calculated single-particle yrast states with

the wood-

saxon

potential in case of the 1 5 2 ~ ~ gy the MeV and the spin I lying bet- cleus (very prelimonary results [44] ). ween 14 and 17 [45]

.

Bibliography

?~ased on collaboration with : G.. Andersson, R.Bengtsson, S.E. Larsson, P.Mzller, S.G.Nilsson, I.Ragnarsson, S.Iberg, J.Dudek, B.Nerlo-Pomorska, K.Pomorski and J.Cerkaski. (Lund-Warsaw collabora- tion)

.

[I]

S.Cohen, F.Plasil and N-J. Swiatecki, Ann. of Phys. N.Y.

82,

557 (1974).

[21 A.Bohr and B.R. Mottelson, Proceedings of the Nobel Symposium on Superheavy Elements,Ronneby, June 1974, Eds. S.G. Nilsson and N.R. Nilsson, Stockholm 1974 see also Physica Scripta

&

(1974).

[4] G.Andersson, R.Bengtsson, S.E.Larsson, $.Leande P.Mgller, S.G.Nilsson, I.Ragnarsson, S.Aberg, J.DUdek, B.Nerlo-Pomorska, K.Pomorski and Z. Szymanski to be published.

[53 K.Neergaard and V.V.Pashkevich, Phys. Letters 59B, 218 (1975).

-

[6] K.Neergaard, V.V.Pashkevich and S.Frauendorf, Nucl. Phys.

171 A.Johnson, H.Ryde and J.Sztarkier, Phys.,Letters 34B, 605 (1971);A.Johnson, H.Ry8e and S.A.Hjorth

-

Nucl. Phys.

m,

753 (1972).

483 R.A.Sorensen, Rev. Mod. Phys.

5,

3 5 3 (1973). [9] A.Johnson and Z.Szymanski, Phys. Reports E,183

(1 973)

[~o]J.

~rumlinde, Nukleonika

2

[I l >.Faessler, invited lecture, XIV Int. Winter Meeting on Nucl. Phys. in Bormio, Italy (1976).

[12]

c

.Mayer-~gricke, invited talk, Sympsium on Nuclear Structure, Balatonfured,Hungary, Sep-

tember 1975.

[13] D.Ward, R.L.Graham, J.S. Geiger and H.R.Adrews Phys. Lett.

E ,

39 (1973).

T.L.Khoo, F.M.Berntha1, J.S.Boyno and R.A. Warner, Phys. Rev. Lett.

2,

1146 (1973).

[I&]

Y.E1 Masri, P.Monsen, J.Steyaert and J.Vervier

Proc. Int. Conf. Nuclear Physics, ~Enich, 1973 North Holland,American Elsevier vol.1, 185. R.M.Lieder, H.Beucher, W.E.Davidson, A.Neskakis C.~aver-~zricke. Y. El Masri. P.Monsen.

J.Steyaert and J.Vervier, Phys.Lett.

s,

161 (1974).

H.R.Andrews, D.Ward, R.L.Graham and J.S.Gbiger Nucl. Phys.

N,

141 (1974).

H.Beucher, W.F.Davidson, R.M.Lieder, A.Neskakis and C.Mayer-~Zricke, Proc. Int. Conf. Reactions between Complex Nuclei, Nashville, 1974 North Holland, American Elsevier vo1.l 189. [15] Y.E1 Masri, R. Janssens, C.Miche1, P.Monsen,

J.Steyaert and J.Vervier, Jeitschrift fur Phy- sik, to be published.

[16] B.R.Mottelson and J.G.Valatin, Phys. Lett.

₂

511 (1960).

1171 R.A. Sorensen, Proc. Orsay Coloqium on Interme- diate Nuclei, July 1971.

R.A.Sorensen, Phys. Lett.

m,

376 (1972).

-

-

1181 J.Krumlinde and Z. Szwanski, Phys. Lett.= 157 (1971).

J.Krumlinde,and Z.Szymanski, Ann. of Phys.N.Y. 79, 201 (1973

-

HIGH ANGULAR MOMENTA AND YRAST STATES C5-45

[20] F.S.Stephens, invited talk, Proc. Int. Conf. Nuclear Physics, dsnich, 1972 North Holland, Pn~erican Elsevier ~01.2, 367.

[21] F.S.Stephens, Proc. of the 5-th Summer School on Nucl. Phys. 1972, Rudziska, Poland. [22] F.S.Stephens,

Rev.

Mod. Phys.

5,

43 (1975). [23] R,M,~iamond, invited lecture, Proc. of the 8-th

S m e r School on Nucl. Phys. Masurian Lakes, 1975, Nukleonika

1,

29 (1 976).

1241 R.A.Broglia, A.Molinari, G.Pollarolo and T.Regge, Phys. Lett. 50B, 295 (1974) and PhysL Lett,

z,

113 (1975).

[25] A,Faessler, Proc. of the 7-th Summer School on Nuclear Physics, Masurian Lakes 1974,

Nukleonika

20,

57 (1975).

[26] J.Krumlinde and Z.Szyrnanski, Nucl.phys. A 221, 93 (1974).

[27] H.R.Dalafi, B.Banargee, H.J.Mang and P.Ring Phys.Lett.

s,

27 (1973).

1281 K.Kumar Physica Scripta 6 270 (1972). [29] B.Banargee, H.J.Mang and P.Ring, Nucl. Phys.

A215, 366 (1973).

-

[30] P.C.Bhargava, Nucl. Phys.

x;

258 (1973).

f31] P.C,Bhargava and D. J.Thouless, Nucl. Phys. A215, 515 (1973).

-

[32] A.Faessler, K.R.Sandhya ?evi, F.Grummer,K.W. Schmid and R.R.Hilton, Nucl.Phys.

B,

106 (1976).

[33] A.Faessler et al. Proc. Conf. on Highly Exc. States in Nuclei, Sept. 1975, ' ~clich, See also

ref.1271and 1291.

1341 A.L.Goodman, Nucl. Phys.

8230,

466 (1974), and preprint, 1976.

[35] I.Hamamoto, preprint, May 1976 and private com- munication June 1976.

[36] R.Sorensen, preprint 1976.

[57] E.R.Marshalek, Nucl. Phys. to be published. [38] S.Bose, J.Krumlinde and E.R.M&rshalek, Phys.

Lett.

a,

136 (1974).

[39] S.Y.Chu, E.R.Marshalek, P.Ring, J.Krumlinde and J.O.Rasmussen, Phys.Rev.

G,

1017 (1975). [40] J.Krumlinde and E.R.Marshalek, preprint 1976. [41] A.Bohr and B.R.Mottelson, Nuclear Structure

vo1.2, Benjamin, New York 1975.

[42] J.Meyer-ter-Vehn, Nucl .Phys.

m,

1 1 1 and 141 (1975).

1431 J. Gizon, A.Gizon and Meyer-ter-Vehn Nucl. Phys. to be published (1976) and contribution to this Conference.

[44] Further results of the Lund-Warsaw collabora- tion.