GIANT RESONANCES AT HIGH SPINS AND FINITE TEMPERATURE AND PHASE TRANSITIONS IN NUCLEI

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GIANT RESONANCES AT HIGH SPINS AND FINITE TEMPERATURE AND PHASE

TRANSITIONS IN NUCLEI

P. Ring

To cite this version:

P. Ring. GIANT RESONANCES AT HIGH SPINS AND FINITE TEMPERATURE AND PHASE TRANSITIONS IN NUCLEI. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-247-C6-253.

�10.1051/jphyscol:1984629�. �jpa-00224231�

G I A N T RESONANCES A T H I G H S P I N S AND F I N I T E TEMPERATURE A N D PHASE T R A N S I T I O N S I N N U C L E I

P. Ring

Physik-Department der Technischen Universitüt München, 0-8046 Garching, F. R. G.

Résumé - On examine l ' i n f l u e n c e d e s t r a n s i t i o n s de phase dans d e s noyaux f i n i s s u r l e s é n e r g i e s d e s r é s o n a n c e s g é a n t e s p a r l a t h é o r i e de l a r é p o n s e l i n é a i r e dépendant de l a t e m p é r a t u r e dans un système t o u r n a n t .

A b s t r a c t - The i n f l u e n c e o f phase t r a n s i t i o n s i n f i n i t e n u c l e i on t h e peak e n e r g i e s o f g i a n t resonanceç i s ç t u d i e d by tempe- r a t u r e dependent l i n e a r r e s p o n s e t h e o r y i n a r o t a t i n g frame.

R e c e n t l y one h a s found e x p e r i m e n t a l e v i d e n c e f o r g i a n t r e s o n a n c e s based on h i g h l y e x c i t e d r o t a t i n g compound s t a t e s /1,2/, which a r e b u i l t a f t e r heavy i o n f u s i o n r e a c t i o n s . The s t u d y o f t h e s e g i a n t r e s o n a n c e s i n h o t r o t a t i n g systems i s i n t e r e s t i n g by s e v e r a l r e a s o n s : Temperature and a n g u l a r v e l o c i t y p r o v i d e two new d e g r e e s of freedom, which g i v e u s , i n g e n e r a l , a d d i t i o n a l i n s i g h t i n t o n u c l e a r i n t e r a c t i o n s and n u c l e a r s t r u c t u r e . An example, which s h a l l b e s t u d i e d i n t h i s c o n t r i b u t i o n , a r e phase t r a n s i t i o n s i n f i n i t e n u c l e i , which e v e n t u a l l y o c c u r a t h i g h e x c i t a t i o n e n e r g i e s o r a t l a r g e a n g u l a r v e l o c i t i e s and which can c a u s e s h i f t s i n t h e f i n e s t r u c t u r e o f t h e g i a n t d i p o l e r e s o n a n c e .

I n o r d e r t o s t u d y t h e d i p o l e a b s o r p t i o n c r o s s s e c t i o n o f a heavy nuc- l e u s w i t h a g i v e n s p i n and a g i v e n e x c i t a t i o n e n e r g y , we u s e t h e c r a n - k i n g approximation a t f i n i t e t e m p e r a t u r e s . I t a l l o w s u s t o d e s c r i b e t h e s y s t e m by r o t a t i n g t e m p e r a t u r e dependent g e n e r a l i z e d Hartree-Fock- Bogoliubov (HFB) d e n s i t i e s and u . For t h e h o t r o t a t i n g n u c l e u s t h e y

a r e q u a s i s t a t i c and can b e found from the s o l u t i o n o f t e m p e r a t u r e de- pendent HFB e q u a t i o n s . The a b s o r p t i o n of d i p o l e photons c a u s e s i n such a system a time dependence. I n t h e l i m i t o f s m a l l a m p l i t u d e s we f i n d harmonic v i b r a t i o n s , whose a m p l i t u d e s show a resonance b e h a v i o u r a s a f u n c t i o n o f t h e incoming photon energy. I t c a n be c a l c u l a t e d i n l i n e a r r e s p o n s e t h e o r y f o r r o t a t i n g s u p e r f l u i d systems a t f i n i t e t e m p e r a t u r e s /3/. One t h u s f i n d s e x p r e s s i o n s f o r t h e a b s o r p t i o n c r o s s s e c t i o n i n t h e i n t r i n s i c system. They have t o b e t r a n s f o r m e d t o t h e l a b o r a t o r y frame. T h i s t r a n s f o r m a t i o n c a u s e s a n a d d i t i o n a l s p l i t t i n g /4/.

A l t o g e t h e r t h e microscopie i n v e s t i g a t i o n of g i a n t r e s o n a n c e s a t h i g h s p i n s and f i n i t e t e m p e r a t u r e s i n v o l v e s t h e r e f o r e t h r e e s t e p s

i ) A s o l u t i o n o f t h e c r a n k e d t e m p e r a t u r e dependent HFB e q u a t i o n s f o r a r e a s o n a b l e n u c l e a r f o r c e . I n t h e f o l l o w i n g t h i s s t e p i s approximated by a t e m p e r a t u r e dependent S t r u t i n s k i - M e t h o d b a s e d on a r o t a t i n g Saxo- Woods p o t e n t i a l /5/ whose d e f o r m a t i o n p a r a m e t e r s a r e v a r i e d . P a i r i n g

c o r r e l a t i o n s a r e t a k e n i n t o a c c o u n t by s o l v i n q t h e gap e q u a t i o n i n t h e r o t a t i n g frame ( d e t a i l s a r e g i v e n i n r e f . 3 ) . We t h u s o b t a i n a n g u l a r momentum and t e m p e r a t u r e dependent mean f i e l d s . We o b s e r v e phase t r a n - s i t i o n s a s a f u n c t i o n of a n g u l a r momentum and o f t e m p e r a t u r e .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984629

C6-248 JOURNAL DE PHYSIQUE

I n F i g . 1 . ~ $ ~ 3 h o w a s an example t h e p a i r i n g f i e l d s f o r p r o t o n s and n e u t r o n s i n E r : A t z e r o t e m p e r a t u r e t h e n e u t r o n gap v a n i s h e s f o r s p i n s l a r g e r t h a n 30 k and t h e p r o t o n gap v a n i s h e s f o r s p i n s l a r g e r t h a n 50 k . For h i g h e r t e m p e r a t u r e s t h i s p a i r i n g c o l l a p s e i s s h i f t e d

Fig. 1 - Gap p a r a m e t e r s f o r n e u t r o n s ( a ) and p r o t o n s ( b ) f o r t h e nuc- l e us I 6 % r a s a f u n c t i o n of a n g u l a r momentum a t v a r i o u s t e m p e r a t u r e s . The d e f o r m a t i o n p a r a m e t e r s a r e k e p t f i x e d .

t o lower a n g u l a r momenta. A t T - 0 . 5 MeV p a i r i n g v a n i s h e s a l r e a d y a t s p i n z e r o .

For h i g h e r t e m p e r a t u r e s we a l s o o b s e r v e changes i n t h e s h a p e s . We show i n F i g . 2 a s an example t h e n u c l e u s 5 8 ~ r , which undergoes a phase t r a n s i t i o n £rom p r o l a t e t o o b l a t e s h a p e s a l r e a d y a t z e r o t e m p e r a t u r e . A t h i g h e r t e m p e r a t u r e s (T - 2.5 MeV) t h e d e f o r m a t i o n v a n i s h e s a l r e a d y

f o r s p i n z e r o . We t h e n a r e l e f t w i t h a c l a s s i c a l l y r o t a t i n g h o t drop- l e t .

ii) I n t h e second s t e p we s t u d y o s c i l l a t i o n s w i t h s m a l l a m p l i t u d e s around t h e t e m p e r a t u r e and a n g u l a r momentum d e p e n d e n t e q u i l i b r i u m s h a p e s de t e r m i n e d i n t h e f i r s t s t e p . The t e m p e r a t u r e d e p e n d e n t l i n e a r r e s p o n s e e q u a t i o n s a r e s o l v e d f o r s u p e r f l u i d s y s t e m s a s a f u n c t i o n of t h e e n e r g y of t h e a b s o r b e d d i p o l e photon. For r e a l i s t i c f o r c e s t h e s e e q u a t i o n s i n v o l v e a n e x t r e m e l y l a r g e c o n f i g u r a t i o n s p a c e , b e c a u s e n e a r l y a l 1 symmetries a r e b r o k e n . We t h e r e f o r e r e s t r i c t o u r s e l v e s t o a s e p a r a b l e d i p o l e - d i p o l e i n t e r a c t i o n , which h a s been w i d e l y used i n t h e l i t e r a t u r e /6-12/. The s o l u t i o n o f t h e l i n e a r r e s p o n s e e q u a t i o n c o r r e s p o n d s i n t h i s c a s e t o t h e i n v e r s i o n of a complex I g l m a t r i x f o r p o s i t i v e s i g n a t u r e and a complex 2 x 2 m a t r i x f o r n e g a t i v e s i g n a t u r e . A s i t i s well-known £rom t h e d i p o l e r e s o n a n c e a t t h e ground s t a t e , n u c l e a r d e f o r m a t i o n s i n d u c e a f i n e s t r u c t u r e of t h i s r e s o n a n c e . For a x i a l l y symmetric p r o l a t e s h a p e s one h a s a mode, which c o r r e s p o n d s t o a v i b r a t i o n o f p r o t o n s a g a i n s t n e u t r o n s p a r a l l e l t o t h e symmetry a x i s and two d e g e n e r a t e modes p e r p e n d i c u l a r t o t h e symmetry a x i s . For t r i a x i a l s h a p e s t h i s degeneracy i s removed.

I n F i g . 3 we show t h e a n g u l a r momentum and t e m p e r a t u r e dependence o f

t h e s e t h r e e peak e n e r g i e s f o r f i x e d d e f o r m a t i o n . The p a i r i n g c o l l a p s e

can be c l e a r l y r e c o g n i z e d i n a s h i f t of t h e K=O mode. To u n d e r s t a n d

t h i s s h i f t b e t t e r we show i n F i g . 4 t h e change of t h e r e s o n a n c e p e a k s

i n t h e harmonic o s c i l l a t o r mode1 a s a f u n c t i o n o f t h e p a r a m e t e r A f o r

1 5 ' ~ r

-60 -60

T=O

F i g . 2 - E n e r g y s u r f a c e s f o r 1 5 8 ~ r i n t h e s h a p e p a r a m e t e r s /3 ^and .

They c o r r e s p o n d t o c o n s t a n t e n t r o p y S and c o n s t a n t a n g u l a r momentum 1.

The c o n t o u r l i n e s d e s c r i b e an e n e r g y d i f f e r e n c e o f 2 MeV and a b s o l u t e v a l u e s f o r t h e e n e r g i e s a r e g i v e n i n MeV. The e n t r o p y v a l u e s S = 5 8 . 5 and S = 9 7 . 5 c o r r e s p o n d on t h e a v e r a g e t o t h e t e m p e r a t u r e s T = 1 .5 and T = 2 . 5 MeV.

A selfc.

16 A frozen A f rozen

-, Z: 14

F i g . 3 - The t e m p e r a t u r e dependence ( f o r I = O ) and t h e a n g u l a r momentum

dependence ( f o r T=O) of t h e p e a k e n e r g i e s i n 1 6 4 ~ r . The d e f o r m a t i o n s

/ 3 = 0 . 3 , g=O a r e k e p t f i x e d .

C6-250 JOURNAL DE PHYSIQUE

a s p h e r i c a l c a s e @=O) and a w e l l deformed c a s e ( P = 0 . 3 ) . We f i n d s h i f t s by s e v e r a l MeV between A = O and A = 2 MeV.

1 6 4 ~ r y = O , w=O - _1.3 19 -

12 - - - - - T z 1 . 5

- _0.8

I

I

O 0 . 5 1 1.5 2

A , = A , = A (MeV)

F i g . 4 - The dependence o f t h e g i a n t d i p o l e r e s o n a n c e peak e n e r g i e s on t h e gap p a r a m e t e r f o r v a n i s h i n g a n g u l a r v e l o c i t y .

I n F i g . 5 we show t h e i n f l u e n c e of t h e s h a p e changes i n 1 5 8 E r on t h e f i n e s t r u c t u r e of t h e GDR. To a v e r y good a p p r o x i m a t i o n t h e s p l i t t i n g i s d e t e r m i n e d b y t h e s h a p e and f o r a q i v e n d e f o r m a t i o n i t c a n b e c a l - c u l a t e d i n t h e o s c i l l a t o r model /6-9/.

Woodi Saxon selfc.

Oicillator mode1

w $ 0

10 " w - O

1 (h)

F i g . 5 - The p o s i t i o n o f t h e p e a k s Emax of t h e GDR i n 1 5 8 ~ r a s a func- t i o n of t h e s p i n . F u l l l i n e s c o r r e s p o n d t o t h e r e a l i s t i c s e l f c o n s i s - t e n t c a l c u l a t i o n . Dashed and d a s h e d l d o t t e d l i n e s a r e o b t a i n e d £rom t h e harmonic o s c i l l a t o r model.

i i i ) The t h i r d s t e p i s a t r a n s f o r m a t i o n t o t h e l a b o r a t o r y system. I t i n d u c e s an a d d i t i o n a l s p l i t t i n g of t h e p e a k s , which y i e l d s p r a c t i c a l l y o n l y a b r o a d e n i n g / 1 2 / . A s l o n g a s one c a n n o t r e s o l v e t h e a n g u l a r d i s - t r i b u t i o n s , i t w i l l t h e r e f o r e b e v e r y h a r d t o o b s e r v e t h e s e e f f e c t s e x p e r i m e n t a l l y .

The p h a s e t r a n s i t i o n s from s u p e r f l u i d t o normal f l u i d s y s t e m s , from

f i n i t e system and t h e r e f o r e a l 1 t h e s e phase t r a n s i t i o n s a r e smeared o u t more o r l e s s , i f a d d i t i o n a l c o r r e l a t i o n s a r e t a k e n i n t o a c c o u n t . E v e n t u a l l y , t h e c r i t i c a l a n g u l a r v e l o c i t i e s o r t e m p e r a t u r e s a r e s h i f - t e d . I t i s t h e r e f o r e n o t c l e a r , i f a l 1 t h e s e t r a n s i t i o n s can b e found i n r e a l i s t i c n u c l e i .

I n Fig. 6 we show f o r example, t h a t t h e p a i r i n g c o l l a p s e i n 1 6 4 ~ r a t T=O d i s c u s s e d i n F i g . 1 i s c o m p l e t e l y smeared o u t , i f an improved v a r i a t i o n a l p r i n c i p l e w i t h number p r o j e c t e d cranked HFB s t a t e s i s used /13/. Even a t t h e h i g h e s t p o s s i b l e a n g u l a r momenta (1 1 7 0 ) we s t i l l have a gap of 3 0 0 keV.

- ^{I I I l 1 I I I}

% E - 1 6 4 ~ , - - _- - ^{A prot (SCC) -}

A neut. (SCC)

- - - A prot (PNP) -

-A neut. (PNP)

-

- -._- _{- -} ^-

- 1 -

1

I 1

4 20 36 52 1 68

Fig. 6 - P a i r i n g p a r a m e t e r s A a s a f u n c t i o n o f t h e a n g u l a r momentum

£rom a s e l f c o n s i s t e n t c a l c u l a t i o n w i t h (PNP) and w i t h o u t ( S C C ) number p r o j e c t i o n .

S i m i l a r t h i n g s c o u l d happen f o r t h e p a i r i n g c o l l a p s e w i t h i n c r e a s i n g t e m p e r a t u r e . Fig. 7 shows a model c a l c u l a t i o n f o r 4 p a r t i c l e s i n a

Fig. 7 - P a i r i n g p a r a m e t e r fL i n a model of 4 p a r t i c l e s i n a j13/2 s h e ï i

i n t e r a c t i n g v i a a p a i r i n g p l u s q u a d r u p o l e i n t e r a c t i o n . E x a c t r e s u l t s

a r e compared w i t h a HFB a p p r o x i m a t i o n , which shows an e a r l y p a i r i n g

c o l l a p s e .

C6-252 J O U R N A L DE PHYSIQUE

j 1 3 / 2 s h e l l . The e x a c t r e s u l t i s compared w i t h a HFB a p p r o x i m a t i o n /141 Again t h e m e a n f i e l d t h e o r y shows much t o o e a r l y p a i r i n g c o l l a p s e . I n t h e same s p i r i t one s h o u l d c a r r y o u t an a n g u l a r momentum p r o j e c - t i o n f o r n u c l e i i n t h e r e g i o n o f s h a p e c h a n g e s . An example a r e s o - c a l l g d &' -sof t n u c l e i i n t h e O s - r e g i o n , where a n g u l a r momentum p r o j e c - t i o n i n d u c e s t r i a x i a l minima /15/.

We t h u s h a v e a nurnber o f c a s e s i n t h e r e g i o n o f p h a s e t r a n s i t i o n s , where s i m p l e minded m e a n f i e l d t h e o r y seems t o b r e a k down i n f i n i t e n u c l e i . I t i s t h e r e f o r e o f g r e a t i n t e r e s t t o s t u d y t h o s e c a s e s e x p e r i - m e n t a l l y . The f i n e s t r u c t u r e o f g i a n t r e s o n a n c e s i s c e r t a i n l y one

t o o l , which can e v e n t u a l l y b e u s e d f o r t h i s p u r p o s e i n t h e f u t u r e .

The n u m e r i c a l r e s u l t s p r e s e n t e d i n t h i s p a p e r a r e b a s e d on c a l c u l a - t i o n s i n c o l l a b o r a t i o n w i t h 3.L. E g i d o , M. F a b e r , S. I w a s a k i , H . J . Mang a n d L.M. Robledo.

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