MICROSCOPIC AND SEMI-CLASSICAL TREATMENTS OF OCTUPOLE DEFORMATION IN THE LIGHT ACTINIDES

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MICROSCOPIC AND SEMI-CLASSICAL

TREATMENTS OF OCTUPOLE DEFORMATION IN THE LIGHT ACTINIDES

R. Chasman

To cite this version:

R. Chasman. MICROSCOPIC AND SEMI-CLASSICAL TREATMENTS OF OCTUPOLE DEFOR- MATION IN THE LIGHT ACTINIDES. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-167- C6-179. �10.1051/jphyscol:1984620�. �jpa-00224221�

JOURNAL DE PHYSIQUE

Colloque C6, suppl6ment a u n06, Tome 45, juin 1984 page C6-167

MICROSCOPIC AND SEMI-CLASSICAL TREATMENTS OF OCTUPOLE DEFORMATION I N THE LIGHT ACTINIDES+

R.R. Chasman

Chemistry Division, Argonne NationaZ Laboratory, Argonne, IZZinois 60439, U.S.A.

Rdsum6

-

Une comparaison avec les rdsultats expgrimentaux est donnge.

A b s t r a c t Microscopic and semi - c l a s s i c a l d e s c r i p t i o n s o f octupole deformation are compared. New semi-classical r e s u l t s , obtained w i t h t h e use o f a Woods- Saxon p o t e n t i a l are presented. Comparisons w i t h experiment are made.

I. I n t r o d u c t i o n

The l o w - l y i n g K r = O - r o t a t i o n a l bands i n t h e even-even Ra and Th isotopes have been known 111 f o r many years and c l e a r l y s i g n a l t h e presence o f s t r o n g octupole c o r r e l a t i o n e f f e c t s i n t h i s mass region. However, e a r l y c a l c u l a t i o n s /2/ using t h e s h e l l c o r r e c t i o n method o f S t r u t i n s k y 131 d i d n o t f.ind any n u c l i d e s i n t h i s mass region w i t h a r e f l e c t i o n asymmetric ground s t a t e shape.

Another approach t o t h e study o f t h i s region comes from t h e c a l c u l a t i o n s o f Vogel and Neergaard /4/. They found i t necessary t o go beyond t h e RPA, t a k i n g i n t o account particle-phonon i n t e r a c t i o n diagrams, i n order t o get any reasonable s o r t o f d e s c r i p t i o n o f t h e even nuclides i n t h i s mass region. This approach i s not a b l e t o handle deformation, but i t does extend t h e range o f i n t e r a c t i o n s t r e n g t h s t h a t can be handled, r e l a t i v e t o t h e RPA. T h e i r r e s u l t s suggest t h a t t h e even- even nuclides o f the A-224 mass r e g i o n can be understood as being very anharmon- i c a l l y v i b r a t i o n a l , i n an octupole sense.

My o r i g i n a l i n t e r e s t 151 i n octupole c o r r e l a t i o n e f f e c t s came from t r y i n g t o understand t h e l o w - l y i n g 0+ e x c i t e d s t a t e o f 234U. I n 234U, one o b t a i n s an e x c i t a t i o n energy o f t h e f i r s t O+ e x c i t e d s t a t e o f -1350 keV w i t h a conventional p a i r i n g f o r c e model. Experimentally, t h i s s t a t e i s found a t 810 keV. One gets roughly t h e same resuLt w i t h o t h e r choices o f p a i r i n g f o r c e m a t r i x elements, when t h e s i n g l e p a r t i c l e l e v e l s are adjusted t o f i t t h e observed e x c i t a t i o n energies i n neighboring odd nuclides. Using a c a l c u l a t i o n a l approach t h a t places p a i r i n g i n t e r a c t i o n s and octupole i n t e r a c t i o n s on an equal f o o t i n g , I found t h a t t h e c a l c u l a t e d e x c i t a t i o n energy o f t h e 0+ e x c i t e d s t a t e i n 234U i s lowered t o -900 keV, i n reasonably good agreement w i t h t h e experimental value o f 810 kev. Q u j t e r e c e n t l y , Piepenbring /6/ has shown t h a t t h e i n c l u s i o n o f octupole c o r r e l a t i o n e f f e c t s leads t o a f a i r l y good desc, i p t i o n o f t h e 0+ e x c i t e d s t a t e s i n t h e mass r e g i o n 2 2 4 4 ~ 2 2 8 , w i t h t h e exception o f Zz8Ra. H i s c a l c u l a t i o n s were c a r r i e d out making use o f an multi-phonon basis set, w i t h octupole phonons generated from a TDA c a l c u l a t i o n .

I n 1980, we extended /7/ our approach t o i c l u d e t h e c a l c u l a t i o n o f s t a t e s i n odd

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984620

C6-168 JOURNAL DE PHYSIQUE

mass nuclides. We also extended t h e c a l c u l a t i o n t o i n c l u d e t h e quadrupole- quadrupole p a r t i c l e - h o l e i n t e r a c t i o n as w e l l as t h e p a i r i n g f o r c e and t h e octupole-octupole i n t e r a c t i o n . I n an odd mass system, t h e s i g n a t u r e o f octupole deformation i s a p a r i t y doublet. This doublet c o n s i s t s o f a p a i r o f s t a t e s having t h e same spins but opposite p a r i t i e s , almost degenerate i n energy, w i t h a l a r g e E3 m a t r i x element connecting t h e two l e v e l s . I n our study, we found several instances i n which t h e c a l c u l a t i o n p r e d i c t s p a r i t y doublets. The most n o t a b l e case was a 5/2+ ground s t a t e doublet i n 229Pa, w i t h a s p l i t t i n g p r e d i c t e d t o be l e s s than 1 keV. The l o w - l y i n g s t a t e s o f 229Pa were not known a t t h a t time. The c a l c u l a t i o n a l s o gave a 3/22 doublet, w i t h a 23 keV s p l i t t i n g , as t h e ground s t a t e o f 227A~, i n good good agreement w i t h t h e known l e v e l s . An ex- perimental study o f t h e s t r u c t u r e o f 229Pa was made by my colleagues, Ahmad e t a1

.,

I n 1981, M o l l e r and Nix /9/ noted t h a t t h e ground s t a t e b i n d i n g energies o f t h e nuclides near A-224 are increased by more than 1 MeV, when t h e octupole degree o f freedom i s introduced i n t o t h e parameterization o f nuclear shapes. These r e s u l t s were obtained u s i n g t h e s h e l l c o r r e c t i o n method /3/. The s i n g l e p a r t i c l e energy l e v e l spectrum used i n t h e i r c a l c u l a t i o n s was generated from a f o l d e d Yukawa p o t e n t i a l /lo/. The r e s u l t i s i n marked c o n t r a s t w i t h t h e f i n d i n g s o f Ref. 2), i n which a modified o s c i l l a t o r p o t e n t i a l was used t o generate t h e s i n g l e p a r t i c l e spectrum. The f i n d i n g o f Mol l e r and Nix has been expanded i n t h e work o f Leander e t al. /11/, who have made a most valuable survey o f t h e l i g h t a c t i n i d e region.

Using t h e f o l d e d Yukawa p o t e n t i a l , w i t h t h e S t r u t i n s k y method, they have found many nuclides i n t h i s region w i t h a r e f l e c t i o n asymmetric ground s t a t e shape.

I n my c o n t r i b u t i o n t o t h i s ldorkshop, I s h a l l (1): discuss t h e microscopic treatment o f r e s i d u a l i n t e r a c t i o n s t h a t I have used t o study octupole i n t e r - a c t i o n s (2): present some new s h e l l c o r r e c t i o n c a l c u l a t i o n s o f t h e p r o p e r t i e s o f n u c l i d e s t h i s r e g i o n using a Woods-Saxon p o t e n t i a l t o generate t h e s i n g l e par- t i c l e energy l e v e l s ( 3 ) : make some connection between t h e c a l c u l a t i o n s and t h e experimental s t u d i e s o f t h i s mass region. Because t h e r e are no n u c l i d e s w i t h r e f l e c t i o n asymmetric ground s t a t e shapes obtained w i t h t h e m o d i f i e d o s c i l l a t o r p o t e n t i a l , and many such n u c l i d e s c a l c u l a t e d w i t h t h e f o l d e d Yukawa p o t e n t i a l , i t i s worthwhile t o i n v e s t i g a t e t h i s phenomenon w i t h another p o t e n t i a l .

11. Microscopic C a l c u l a t i o n o f Octupole C o r r e l a t i o n E f f e c t s

The Hamiltonian t h a t we use i n our study o f octupole deformation / 7 / i s

3 0 +

-V3

1 .

1

1 - 1

'

where sk a r e t h e s i n g l e energies obtained from a deformed r e f l e c t i o n symmetnc Woods-Saxon p o t e n t i a l . The p a i r i n g f o r c e m a t r i x elements, G. _are not constant; they come from a density dependent d e l t a i n t e r a c t i o n . ~ h i i ' $ L t o f m a t r i x elements /12/ explains many features o f t h e a c t i n i d e s a t low and h i g h spin. The quadrupole m a t r i x elements t h a t we use a r e p r o p o r t i o n a l t o t h e m a t r i x elements o f r2Y(2,0) and t h e octupole m a t r i x elements are p r o p o r t i o n a l t o t h e m a t r i x elements o f r3Y(3,0). The quadrupole s t r e n g t h was adjusted t o give t h e observed onset o f quadrupole deformation i n t h e l i g h t a c t i n i d e s . The octupole s t r e n g t h was adjusted t o o b t a i n t h e known energies o f t h e l o w - l y i n g 1- s t a t e s i n t h e Th isotopes.

To s o l v e t h i s H a m i l t o n i a n , we e x p l o i t /5,7/ t h e f a c t t h a t t h e i n t e r a c t i o n i s c y l i n d r i c a l l y symmetric and separable. We denote t h e deformed o r b i t a l s w i t h a g i v e n v a l u e of T and n as an n group. O r b i t a l s w i t h b o t h p o s i t i v e and n e g a t i v e p a r i t y a r e i n c l u a e d i n t h e group. Many c o n f i g u r a t i o n s can be c o n s t r u c t e d by p u t - t i n g p a r t i c l e s i n t h e o r b i t a l s o f a g i v e n n group, and t h e number o f c o n f i g u r a - t i o n s r i s e s r a p i d l y w i t h t h e number o f doubly degenerate o r b i t a l s i n t h e n group.

F o r purposes of i l l u s t r a t i o n , we c o n s i d e r an n group w i t h o n l y two doubly degen- e r a t e N i l s s o n o r b i t a l s ; e. g. 5/2+ and 5/2-. There a r e s i x c o n f i g u r a t i o n s t h a t a r e r e l e v a n t f o r t h e d e s c r i p t i o n o f 0+ and 0- s t a t e s .

C o n f i g u r a t i o n Occupied O r b i t a l s P i c t u r e P a r t i c l e # Q 11

I f we a r e i n t e r e s t e d i n t h e odd mass n u c l i d e w i t h an D v a l u e o f 5/2, t h e r e are f o u r c o n f i g u r a t i o n s t o consider.

O r b i t a l s 1 and 2 i n t h i s i l l u s t r a t i o n have p o s i t i v e p a r i t y ; 3 and 4 have n e g a t i v e p a r i t y . O r b i t a l 1 and 3 have p o s i t i v e 52; 2 and 4 n e g a t i v e 52. I n an even-even nucleus, t h e t o t a l w a v e f u n c t i o n i s o f t h e f o r m

C6-170 JOURNAL DE PHYSIQUE

P a r i t y and p a r t i c l e number a r e n o t conserved w i t h i n any one o f t h e s2 groups;

however, we p r o j e c t s t a t e s o f good p r o t o n number, n e u t r o n number, and p a r i t y t o c o n s t r u c t t h e w a v e f u n c t i o n y(Ng,Np, n). T h i s i s denoted by t h e p r o j e c t i o n o p e r a t o r P(N ,N ,r). The amplitudes C(a,TZ) a r e o b t a i n e d by m i n i m i z i n g t h e energy o f t(e P f u l l y p r o j e c t e d wavefunction. We s o l v e t h e s e t o f coupled e q u a t i o n s

I f we want t o s o l v e f o r t h e 5/2 s t a t e s o f an odd p r o t o n nucleus, t h e r e l e v a n t w a v e f u n c t i o n i s o f t h e f o r m

I n p r a c t i c e , we can handle up t o f i v e doubly degenerate l e v e l s i n a (0, T,) group. I n t h e even p a r t i c l e number group, t h i s amounts t o 252 c o n f i g u r a t i o n s w i t h 142 independent amplitudes. There a r e 210 independent amplitudes i n an odd p a r t i c l e number group. By v a r y i n g t h e octupole, quadrupole o r p a i r i n g s t r e n g t h s , we generate many d i f f e r e n t wavefunctions w i t h d i f f e r i n g c o l l e c t i v e p r o p e r t i e s . Our f i n a l s o l u t i o n i s a 1 in e a r combination o f such wavefunctions, t a k i n g t h e i r n o n - o r t h o g o n a l i t y i n t o account. The s t r u c t u r e o f t h e s e s o l u t i o n s i s s u f f i c i e n t l y r i c h t o d e s c r i b e s t a t e s t h a t a r e s p h e r i c a l , v i b r a t i o n a l o r deformed i n t h e s e t h r e e degrees o f freedom.

The c a l c u l a t i o n s /7/ t h a t we have made f o r odd mass a c t i n i d e s show t h e onset o f o c t u p o l e d e f o r m a t i o n f o r s e v e r a l values o f s2 i n b o t h odd p r o t o n and odd n e u t r o n n u c l i d e s . The r e s u l t s o f t h e c a l c u l a t i o n s a r e shown i n f i g u r e s ( 1 ) and (2). I n Fig. (3), we show t h e ground s t a t e d o u b l e t spacing measured by Ahmad e t a l . /8/

f o r 229Pa.

111. S h e l l C o r r e c t i o n C a l c u l a t i o n of Octupole Deformation

The S t r u t i n s k y method /3/ i s w e l l known and I s h a l l n o t d i s c u s s i t here i n any d e t a i l . The t o t a l energy o f a nucleus i s c a l c u l a t e d as a f u n c t i o n o f t h e n u c l e a r shape. The energy i s t a k e n t o be t h e L i q u i d Drop energy w i t h m i c r o s c o p i c s h e l l and p a i r i n g c o r r e c t i o n s ; i. e.

where v i denotes a s e t o f d e f o r m a t i o n parameters. EShell i s t h e d i f f e r e n c e i n energy between t h e a c t u a l s e t o f occupied l e v e l s and a s u i t a b l y smoothed

Figure 1. Microscopic c a l c u l a t i o n of p a r i t y doublets i n odd N nuclides. The numbers beside t h e arrows a r e proportional t o ~ ~ ~ ( 3 , 0 ) > ~ . E x c i t a t i o n energy i n keV.

Figure 2 . Microscopic c a l c u l a t i o n o f p a r i t y doublets i n odd Z n u c l i d e s .

JOURNAL DE PHYSIQUE

L o g f t ^IEC(O/0)

- -

2 2 9 ~ 58min

'

JEC-8O5

Q E C = 1.32 MeV

Figure 3. Low l y i n g l e v e l scheme o f 229Pa.

d i s t r i b u t i o n o f l e v e l s . We have use an 8 t h order Hermite polynomial, w i t h a w i d t h o f 10 MeV f o r our smoothing function. The q u a n t i t y Epa., i s t h e d i f f e r e n c e i n p a i r i n g c o r r e l a t i o n energy between t h e t r u e set o f l e v e l s and t h e s u i t a b l y smoothed set. We n o t e t h a t t h e smoothed s e t o f l e v e l s i s e s s e n t i a l l y independent o f t h e shape f o r t h e purpose o f a p a i r i n g c a l c u l a t i o n . For our p a i r i n g strength, we use G=21/A f o r neutrons and 30/A f o r protons. The p a i r i n g c a l c u l a t i o n s a r e c a r r i e d out using 15 p a i r s o f p a r t i c l e s and 30 doubly degenerate l e v e l s . We go s l i g h t l y beyond t h e BCS approximation t o c a l c u l a t e t h e p a i r c o r r e l a t i o n energy, u s i n g t h e method o f c o r r e l a t e d q u a s i - p a r t i c l e s 1131. The advantage o f t h i s method i s t h a t i t gives good c o r r e l a t i o n energies when t h e l e v e l spacings a r e large. The BCS method gives t o o l i t t l e c o r r e l a t i o n energy i n such cases; i. e.

when t h e s h e l l e f f e c t s are l a r g e s t . The s i n g l e p a r t i c l e energy l e v e l spacings are generated from a Woods-Saxon p o t e n t i a l f o r t h e n u c l i d e 224Ra. For t h e p r o t o n o r b i t a l s , we took r0=1.27 fm ,a=0.65 fm, and a s p i n o r b i t r a d i u s o f .85r0. To c a l c u l a t e t h e neutron s i n g l e p a r t i c l e energies, we choose rg=1.33 fm and a=0.72 fm, w i t h t h e s p i n - o r b i t r a d i u s parameter equal t o ro. Thls set o f parameters gives a good r e p r e s e n t a t i o n o f t h e l e v e l spacings i n t h e m i d - a c t i n i d e region.

Our p a r a m e t e r i z a t i o n o f t h e deformation d i f f e r s s l i g h t l y from t h a t used i n t h e m o d i f i e d osci 1 la t o r c a l c u l a t i o n s . We i n t r o d u c e deformation v i a t h e t r a n s - formation

r2 -+ r 2[exp(%) s i n P o + exp

(- 2)

k>zk

1

I n our c a l c u l a t i o n s , we have used a g r i d w i t h t h e set o f p o i n t s

We have a l s o used the smoothing r e l a t i o n s /11/

I n f a c t these choices o f v5 and V6 reduce t h e e f f e c t i v e magnitudes o f v3 and v4, r a t h e r than smoothing t h e nuclear surface.

I n Figures 4-8, I present some comparisons o f t h e energy gains associated w i t h octupole deformation, as w e l l as t h e magnitude o f t h e octupole deformation obtained w i t h Woods-Saxon l e v e l s and t h e f o l d e d Yukawa p o t e n t i a l /11/. I n t h i s s e r i e s o f f i g u r e s , t h e v e r t i c a l a x i s i s t h e p r o t o n number and t h e h o r i z o n t a l a x i s i s t h e neutron number o f t h e n u c l i d e being studied. The numbers i n Fig. ( 4 ) are t h e d i f f e r e n c e i n b i n d i n g energy obtained f o r a r e f l e c t i o n asymmetric shape r e l a t i v e t o t h e b i n d i n g energy obtained f o r symmetric shapes, given by /11/ a f o l d e d Yukawa p o t e n t i a l . A p o s i t i v e number means t h a t t h e r e f l e c t i o n asymmetric shape i s more t i g h t l y bound. There i s a l a r g e region where t h i s energy gain i s g r e a t e r than 0.5 MeV, and many cases i n which i t i s s u b s t a n t i a l l y l a r g e r than 1 MeV. I n Fig. (5), we show t h e magnitude of E a t these minima. The c a l c u l a t e d octupole deformation e f f e c t s are l a r g e s t f o r 132 and 134 neutrons. To put these energy gains i n t o perspective, i t i s worth n o t i n g t h a t t h e energy gained i n going from a s p h e r i c a l shape t o t h e r e f l e c t i o n symmetric shapes o f t h e m i d - a c t i n i d e r e g i o n i s 10-15 MeV, compared w i t h t h e -1 MeV s h i f t s associated w i t h octupole deformation. This i s a comparatively small e f f e c t and d i f f i c u l t t o c a l c u l a t e . I n Figs. (6) and (7), we present new r e s u l t s t h a t we have obtained w i t h a Woods- Saxon p o t e n t i a l . The energy gains are displayed i n Fig. (6) and t h e associated deformations v3 i n Fig. (7). The most s t r i k i n g f e a t u r e o f a comparison w i t h t h e f o l d e d Yukawa r e s u l t s i s t h a t t h e region o f n u c l i d e s c a l c u l a t e d t o be octupole deformed i s much smaller. The increase i n b i n d i n g energy due t o octupole de- formation i s a l so considerably s m a l l e r i n most instances. However, t h e r e g i o n t h a t shows maximum octupole c o r r e l a t i o n e f f e c t s i s much t h e same i n both cases i.e. a t 132 and 134 neutrons. I have a l s o i n c l u d e d r e s u l t s f o r Z=94 and Z=96 and f i n d energy minima f o r shapes w i t h octupole deformation i n isotopes o f Pu and Cm. The maximum energy gain i n these elements occur a t 130 and 132 neutrons. I n F i g (8), we d i s p l a y t h e energy gains associated w i t h octupole deformation i n t h e odd mass n u c l i d e s o f t h i s region, as obtained w i t h t h e Woods-Saxon p o t e n t i a l . Both t h e Woods-Saxon and f o l d e d Yukawa p o t e n t i a l s give r e f 1 e c t i o n asymmetric minima near A-224, b u t t h e e f f e c t s are considerably more pronounced i n t h e c a l c u l a t i o n s using a f o l d e d Yukawa p o t e n t i a l . The d i f f e r e n c e s are s u r p r i s i n g i n view o f t h e s i m i l a r i t y o f t h e shapes o f t h e two p o t e n t i a l s .

I V . Comparison w i t h Experiment

The p o s i t i o n o f t h e l o w - l y i n g 1- s t a t e serves as a f i r s t i n d i c a t o r o f octupole deformation e f f e c t s i n even-even nuclides. I n Fig. (9), we show t h e known e x c i t a t i o n energies o f 1- s t a t e s i n t h i s region. There are a few n u c l i d e s i n which the 1- s t a t e i s below 300 keV, and none i n which t h e e x c i t a t i o n energy i s l e s s than 200 keV. I n t h e case o f s t r o n g octupole deformation, we expect t o see a ground s t a t e r o t a t i o n a l band sequence o f 0+, 1-, and 2+. So f a r as I know, t h e r e are no n u c l i d e s i n which t h e 1- s t a t e i s below t h e 2+ state. There are several cases i n which a 1- s t a t e has been found s l i g h t l y above o r s l i g h t l y below t h e 4+ l e v e l . These data argue a g a i n s t permanent octupole deformation i n t h e even-even nuclides. On t h e o t h e r hand, t h e r e i s no evidence /14/ f o r a two

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F i g u r e 4. Energy g a i n s a s s o c i a t e d w i t h r e f l e c t i o n asymmetric shapes as c a l c u l a t e d /11/ w i t h a f o l d e d Yukawa p o t e n t i a l . The v e r t i c a l a x i s g i v e s t h e p r o t o n number ; t h e h o r i z o n t a l a x i s , t h e n e u t r o n number. Energy i n MeV.

F i g u r e 5. Magnitude o f o c t u p o l e d e f o r m a t i o n o b t a i n e d /11/ w i t h f o l d e d Yukawa p o t e n t i a l .

F i g u r e 6. Energy g a i n a s s o c i a t e d w i t h r e f l e c t i o n asymmetric shapes c a l c u l a t e d w i t h a Woods-Saxon p o t e n t i a l . Energy i n MeV.

F i g u r e 7. Magnitude of o c t u p o l e d e f o r m a t i o n o b t a i n e d w i t h Woods-Saxon p o t e n t i a l .

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F i g u r e 8. Energy g a i n associated w i t h r e f l e c t i o n asymmetric shapes i n odd Z n u c l i d e s . Energy i n MeV.

phonon O+ octupole s t a t e a t t w i c e t h e energy o f t h e 1- s t a t e i n t h e cases when t h e 1- energy i s l e s s than 300 keV. This means t h a t t h e octupole c o r r e l a t i o n e f f e c t s are much stronger than v i b r a t i o n a l ; i. e. we are i n a t r a n s i t i o n a l r e g i o n f o r these even-even nuclides. These data suggest t h a t octupole c o r r e l a t i o n e f f e c t s are stronger i n nuclides w i t h 136 and 138 neutrons than we estimate w i t h she1 1 c o r r e c t i o n methods.

The next data we consider are t h e s i n g l e p a r t i c l e s t a t e s o f t h e odd mass nuclides. The observed p a r i t y doublets and t h e ground s t a t e spins are c o n s i s t e n t w i t h t h e l e v e l orderings t h a t one c a l c u l a t e s w i t h t h e microscopic approach /7/

and w i t h t h e r e s u l t s one o b t a i n s /11,15/ a f t e r i n t r o d u c i n g octupole deformation i n t o t h e s i n g l e p a r t i c l e p o t e n t i a l . The l e v e l orderings t h a t I o b t a i n w i t h a Woods-Saxon p o t e n t i a l are q u i t e s i m i l a r t o those obtained w i t h t h e f o l d e d Yukawa p o t e n t i a l . Studies /16/ o f t h e s t r u c t u r e o f 225Ac show t h a t t h e c o r i o l i s m a t r i x elements are reduced i n t h i s n u c l i d e by roughly an order o f magnitude, r e l a t i v e t o t h e values observed i n t h e m i d - a c t i n i d e region. This r e s u l t i s c o n s i s t e n t w i t h c a l c u l a t i o n s /15/ o f s i n g l e p a r t i c l e p r o p e r t i e s t h a t i n c l u d e octupole de- formation. I n t h e l i m i t o f octupole deformation, t h e value o f g should be t h e same f o r p o s i t i v e and negative p a r i t y members of a p a r i t y doubret. This con- d i t i o n i s r a t h e r w e l l f u l f i l l e d 1171 f o r t h e 3/22 ground s t a t e doublet of 227A~. On the o t h e r hand, t h e absolute value o f t h e decoupling parameter should be t h e same f o r p o s i t i v e and negative p a r i t y members o f a 1/22 band i n t h e l i m i t o f o c t u p o l e deformation. Such i s not t h e case f o r t h e e x c i t e d s t a t e 1/2+ and 1/2- bands o f 227A~. This d i f f e r e n c e i n t h e 3/22 and 1/2 bands was a n t i c i p a t e d i n t h e microscopic /7/ c a l c u l a t i o n s .

I n a study 1181 o f 225Ra, Sheline e t al. noted t h a t t h e decoupling parameters o f t h e 1/2+ and 1/2- bands were much c l o s e r i n magnitude than c a l c u l a t i o n s using a r e f l e c t i o n symmetric p o t e n t i a l would suggest. However, t h e absolute values o f t h e two decoupling parameters s t i l l d i f f e r from each o t h e r s u b s t a n t i a l l y . I n t h i s same nuclide, t h e 1/2+ ground s t a t e i s populated 1191 s t r o n g l y i n a one

nucleon t r a n s f e r reaction. The angular momentum decomposition o f t h e 1/2+

i n t r i n s i c s t a t e s c a l c u l a t e d i n r e f l e c t i o n symmetric p o t e n t i a l s shows no appreci

-

a b l e J=O component; i. e. t h i s s t a t e should n o t be populated. Recently, I have made an angular momentum decomposition o f t h e s i n g l e p a r t i c l e s t a t e s i n a p o t e n t i a l w i t h octupole deformation. The i n t r o d u c t i o n o f octupole deformation

Figure 9. E x c i t a t i o n energy o f 1 - s t a t e s i n even-even n u c l i d e s . Energy i n keV.

( v 3 = .OY) gives a wavefunction w i t h a J=O component l a r g e enough t o e x p l a i n t h e observed p o p u l a t i o n o f t h e 1/2+ state.

I n t h e l a s t month Rose and Jones r e p o r t /20/ spontaneous 14C decay o f 2 2 3 ~ a . This suggests a p i c t u r e o f octupole deformation i n t h e ground s t a t e o f 223Ra. We envisage t h i s n u c l i d e as c o n s i s t i n g o f a Pb l i k e p a r t i c l e and a C l i k e p a r t i c l e a l l s u i t a b l y smoothed, as sketched i n Fig. 10.

F i g u r e 10. Sketch o f 2 2 3 ~ a as Pb and C cores.

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On t h e whole, t h e data i n odd mass n u c l i d e s i s c o n s i s t e n t w i t h a p i c t u r e o f octupole deformation f o r some values o f a. I n t h e microscopic c a l c u l a t i o n s /7/, we f i n d t h a t t h e presence o f an odd p a r t i c l e i n an a p p r o p r i a t e o r b i t a l can p o l a r - i z e the nuclear core strongly. We found t h a t t h e octupole c o r r e l a t i o n e f f e c t s a r e stronger i n these s t a t e s than they are i n nearby even-even nuclides. I n t h e s h e l l c o r r e c t i o n c a l c u l a t i o n s these d i f f e r e n c e s are not so apparent. Both t h e f o l d e d Yukawa and Woods-Saxon based semi-classical d e s c r i p t i o n o f t h i s mass regions i n d i c a t e t h a t t h e l i g h t U isotopes might be an i n t e r e s t i n g r e g i o n i n which t o search f o r octupole deformation. The Woods-Saxon c a l c u l a t i o n s i n d i c a t e p o s s i b l e octupole deformation i n l i g h t Pu and Cm isotopes, as w e l l .

F i n a l l y , I should l i k e t o note /21/ t h a t science has been a n t i c i p a t e d by a r t . A semi-classical d e s c r i p t i o n o f octupole deformation was given by t h e French a r t i s t Daumier e x a c t l y 150 years ago. H i s d e s c r i p t i o n o f t h i s phenomenon can be seen i n Fig. (11).

I thank I. Ahmad, B. W i l k i n s and G. Leander f o r h e l p f u l discussions on various aspects o f t h i s work.

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*Work performed under t h e auspices o f t h e O f f i c e o f High Energy and Nuclear Physics, D i v i s i o n of Nuclear Physics, United States Department o f Energy under c o n t r a c t number W-31-109-ENG-38.

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