DEEP HOLE AND HIGH-LYING PARTICLE STRENGTH FUNCTIONS

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DEEP HOLE AND HIGH-LYING PARTICLE STRENGTH FUNCTIONS

V. Soloviev

To cite this version:

V. Soloviev. DEEP HOLE AND HIGH-LYING PARTICLE STRENGTH FUNCTIONS. Journal de

Physique Colloques, 1984, 45 (C4), pp.C4-69-C4-83. �10.1051/jphyscol:1984407�. �jpa-00224072�

JOURNAL DE PHYSIQUE

Colloque C4, supplément Tome 45, m a r s 1984 page C4-69

DEEP HOLE AND HIGH-LYING PARTICLE STRENGTH FUNCTIONS

V.G. S o l o v i e v

Joint Institute for Nuclear Research, Dubna, U.S.S.R.

Résumé - Les principes fondamentaux du modèle quasiparticule-pho- non du noyau sont présentés. Les résultats des calculs de la frag- mentation des états à une et deux quasiparticules sont comparés avec les données expérimentales pour des noyaux sphêriques. Le changement de la fragmentation de la sous-couche 1 h 11/2 dans la transition du noyau sphérique au noyau déformé est étudié. La distribution de la force des résonances de charge-échange dans des noyaux sphêriques et déformés est calculée.

Abstract - The basic assumptions of the quasiparticle-phonon nuc- lear model are expounded. The results of calculation of the frag- mentation of one- and two-quasiparticle states are compared with the experimental data for spherical nuclei. The change of the fra- gmentation of the 1hii/² subshell as one moves from a spherical to a deformed nucleus is studied. The strength distribution of the charge-exchange resonances in spherical and deformed nuclei is calculated.

The strength distribution (fragmentation) of single-particle and colle- ctive states in atomic nuclei is intensively studied experimentally and theoretically. A great progress is associated with the study of the fragmentation of deep hole and proton particle states in spherical nuclei and of collective states of the giant resonance-type in light, medium and heavy nuclei.

With increasing excitation energy the level density of atomic nuclei grows and their structure becomes more complicated. Thus, a transition proceeds from simple low-lying to very complicated states at interme- diate and high excitation energies. It is almost impossible to measure and describe the characteristics of each of many thousand levels, the wave functions of which contain a large number of components. Therefore, the problem is to investigate the laws of complication of the struc- ture with increasing excitation energy. The study of the fragmentation of one- and two-quasiparticle states is a first step in solving this problem. An increasing interest in the study of the fragmentation of deep hole neutron and proton states stems from that the spreading wi- dths Tt are considerably larger than the escape widths l~t . An expe- rimental discovery of many multlpole, spin-multipole and charge-exchan- ge resonances raised a problem about the strength distribution of the- se collective states. This problem turned out to be more important be- cause in some cases the strength of the collective mode was apparently less than half of the predicted value.

An effective method for describing the fragmentation of one-quasipar- ticle and one-phonon states has been developed within the quasipartic- le-phonon nuclear model /1-3/. In this report we expound the basic as- sumptions of the model, compare it with other models and demonstrate

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984407

C4-70 J O U R N A L DE PHYSIQUE

t h e a d v a n t a g e s of t h e q u a s i p a r t i c l e - p h o n o n n u c l e a r model. The fragmen- t a t i o n o f one- and t w o - q u a s i p a r t i c l e s t a t e s i n s p h e r i c a l n u c l e i a r e d e s c r i b e d i n t h e framework of t h e model. The r e s u l t s a r e compared w i t h t h e e x p e r i m e n t a l d a t a . S p e c i f i c p r o p e r t i e s o f t h e f r a g m e n t a t i o n o f charge-exchange c o l l e c t i v e s t a t e s i n s p h e r i c a l and d e f o r ~ ~ e d n u c l e i a r e d i s c u s s e d .

1

-

BASIC ASSUMPTIONS OF THE QUASIPARTICLE-PHONON NUCLEAR MODEL

The f r a g m e n t a t i o n o f o n e - q u a s i p a r t i c l e , one-phonon and q u a s i p a r t i c l e

@ phonon s t a t e s i s c a l c u l a t e d w i t h i n t h e q u a s i p a r t i c l e - p h o n o n n u c l e a r model. The c h a r a c t e r i s t i c s o f n u c l e a r s t a t e s a t low, i n t e r m e d i a t e and h i g h e x c i t a t i o n e n e r g i e s , determined by t h i s f r a g m e n t a t i o n , a r e c a l c u - l a t e d . The model i s based on t h e f o l l o w i n g assumptions:

i ) The RPA one-phonon s t a t e s a r e used a s a b a s i s . A l l t h e model pa- r a m e t e r s a r e f i x e d w h i l e c o n s t r u c t i n g t h e phonon b a s i s .

i i ) The q u a s i p a r t i c l e - p h o n o n i n t e r a c t i o n a f f e c t s t h e f r a g m e n t a t i o n of q u a s i p a r t i c l e and c o l l e c t i v e motion and t h u s t h e c o m p l i c a t i o n o f t h e s t r u c t u r e of n u c l e a r s t a t e s w i t h i n c r e a s i n g e x c i t a t i o n e n e r g y .

iii) The e x c i t e d s t a t e wave f u n c t i o n s a r e g i v e n a s a s e r i e s i n t h e number of phonon o p e r a t o r s ( i n odd-A n u c l e i t h e y a r e added by t h e qua- s i p a r t i c l e o p e r a t o r ) . I n t h e above c a l c u l a t i o n s t h e expansion i s l i m i - t e d by two phonons.

i v ) The s t r e n g t h f u n c t i o n method i s uSed t o c a l c u l a t e t h e reduced t r a n s i t i o n p r o b a b i l i t i e s , t r a n s i t i o n d e n s i t i e s and o t h e r n u c l e a r cha- r a c t e r i s t i c s w i t h o u t s o l v i n g t h e r e l e v a n t s e c u l a r e q u a t i o n s .

The Hamiltonian of t h e q u a s i p a r t i c l e - p h o n o n n u c l e a r model i n c l u d e s t h e a v e r a g e f i e l d a s t h e Saxon-Woods p o t e n t i a l and t h e p a i r i n g , m u l t i p o l e - - m u l t i p o l e , s p i n - m u l t i p o l e

-

s p i n - m u l t i p o l e i s o s c a l a r and i s o v e c t o r i n c l u d i n g charge-exchange i n t e r a c t i o n s . A g e n e r a l d e s c r i p t i o n of t h e model Hamiltonian f o r t h e deformed n u c l e i i s g i v e n i n r e f s . / l , 2 / and f o r t h e s p h e r i c a l n u c l e i i n r e f s . / 3 , 4 / .

I n t r a n s f o r m i n g t h e model Hamiltonian by t h e c a n o n i c a l Bogolubov t r a n - s f o r m a t i o n , one moves from t h e nucleon o p e r a t o r s t o t h e q u a s i p a r t i c l e o p e r a t o r s o(tm _d and

dg, .

The p a i r s o f o p e r a t o r s

$,.,, dimz

^{and dd;rn2 $,m,}

a r e e x p r e s s e d t h r o u g h t h e phonon o p e r a t o r s , whereas t h e q u a s i p a r t i c l e o p e r a t o r s remain i n t h e formo(.+ d.,,,

.

Such a n i n c l u s i o n o f t h e phonon

dm d

o p e r a t o r s overcomes t h e d i f f i c u l t i e s w i t h t h e d o u b l e c o u n t i n g o f some diagrams, which t a k e p l a c e i n t h e n u c l e a r f i e l d t h e o r y / 5 / . The phonon c r e a t i o n o p e r a t o r i s

~i +

..

Q,;~ = ' t zQBX / Y . . A ^d~ r i d ~ ; ~ p ~ - ( - ~ l P y / / i ~ ~ ~ ; A ; u ) } ,

where

The s e c u l a r e q u a t i o n d e t e r m i n i n g t h e e n e r g i e s

CdAi

of t h e one-phonon s t a t e s i s

f c- i d i ) =

( G + ~ ) ( X ' ( ~ ) + X ' ( ~ ) ) - ~ ~ X , X ' ( ~ ) X ' ( ~ )

-

/ = o ^{+ ( 2 )}

A A L 1.

Then by 2,

, a??

^{and X o}

, 8,

w e d e n o t e t h e i s o s c a l a r and i s o v e c t o r c o n s t a n t s of t h e m u l t i p o l e and s p i n - m u l t i p o l e f o r c e s ,

f o r t h e m u l t i p o l e f o r c e s , ( 3 )

a

, . (-) 2

T

(f

l d , d z j % , j , ) & j , j , for the spin-multipole forces(3') single-particle reduced matrix elements of the multipole and spin-multipole operators, 7 s /I,P implies the summation over neutron ( n ) and proton ( p ) single-particle states: 2/,.$$)=

d d

@ ^f U.,Q.;

of<)=

?,./.ai?8.8./

,

^where

d i ' d d dd d d d d

3

^and

^0.

are the Bogolubov tran- d

sformation coefficients, & . ^and

&.

., are the one- and two-quasiparticle

energies;

d 44

To describe charge-exchange resonances we introduce charge-exchange multipole and spin-multipole phonons in the form /6/

The secular equation for finding the charge-exchanqe one-phonon ener- gies

R A i

^is

ai(+)

i t - 2

z M ( f l A i ) % ( / - x M )(j-xAi'-')-(xm ^{) = o ,}

^{( 6 )}

where

Taking into account the secular equations for one-phonon states, the model Hamiltonian can be transformed into

where

QiPi

Q V i / .

JOURNAL DE PHYSIQUE

Formulae f o r

Hcsv

and/fcspp d i f f e r from ( 1 4 ) and ( 1 5 ) a s i n s t e a d o f t h e r e d u c e d m a t r i x e l e m e n t s o f t h e m u l t i p o l e o p e r a t o r s t h e y c o n t a i n t h o s e o f t h e s p i n - m u l t i p o l e o p e r a t o r s .

W e p r e s e n t a g e n e r a l scheme o f c a l c u l a t i o n s w i t h i n t h e q u a s i p a r t i c l e - -phonon n u c l e a r model. The wave f u n c t i o n of a n odd-A s p h e r i c a l n u c l e u s i s

where

41,

i s t h e ground s t a t e wave f u n c t i o n o f a d o u b l y e v e n n u c l e u s . Than, we c a l c u l a t e a n a v e r a g e v a l u e o f f f M ( 9 ) o v e r t h e s t a t e ( 1 6 ) and u s e t h e v a r i a t i o n a l p r i n c i p l e . A s a r e s u l t , we f i n d t h e s e c u l a r equa- t i o n f o r t h e e n e r g i e s

3,

o f t h e s t a t e s V

where

r l ~ j l i ) = ( ~ ~ * * o c J$ /HM,,,,e /[*oc;,~&~?r~

and t h e e q u a t i o n f o r t h e

D .

f u n c t i o n s , which i s g i v e n , f o r i n s t a n c e , i n r e f s . / 7 , 5 / . d

The e x c i t e d s t a t e wave f u n c t i o n o f a d o u b l y e v e n s p h e r i c a l n u c l e u s i s w r i t t e n a s

i . e . we u s e o n l y t h e two-phonon t e r m s . Then, w e c a l c u l a t e a n a v e r a g e v a l u e o f

HM

o v e r ( 1 8 ) , and u s i n g t h e v a r i a t i o n a l p r i n c i p l e t h e s y s t e m o f e q u a t i o n s 1s d e r i v e d (see r e f s . / l , 9 , 1 0 / ) . I t h a s b e e n shown i n r e f . / l o / t h a t a l a r q e s e t o f d i a g r a m s i s summed i n t h e q u a s i p a r t i c l e - p h o n o n n u c l e a r model. T h i s set of d i a g r a m s i s c o n s i d e r a b l y l a r g e r t h a n t h e s e t of d i a g r a m s summed i n r e f s . / l l - 1 3 / . Our model h a s no d i f f i c u l t i e s d u e t o t h e c a n c e l l a t i o n between p a r t i c l e and h o l e a m p l i t u d e s , which become more i m p o r t a n t i n heavy n u c l e i i n t h e models d e v e l o p e d i n r e f s . / l l - 1 3 / . By u s i n g t h e s t r e n g t h f u n c t i o n method o n e c a l c u l a t e s i m m e d i a t e l y a p r o - p e r * u n c t i o n i n a c e r t a i n e n e r g y i n t e r v a l i n s t e a d o f s o l v i n g t h e c o r - r e s p o n d i n g s y s t e m s o f e q u a t i o n s and f i n d i n g t h e e n e r g i e s a n d wave f u n c - t i o n s f o r e a c h s t a t e .

2

-

FRAGMENTATION O F ONE-QUASIPARTICLE STATES

Many p a p e r s /14-18/ h a v e been d e v o t e d t o t h e e x p e r i m e n t a l and t h e o r e - t i c a l s t u d y o f t h e f r a g m e n t a t i o n o f o n e - q u a s i p a r t i c l e s t a t e s . E a r l i e r , t h e f r a g m e n t a t i o n o f o n e - q u a s i p a r t i c l e s t a t e s h a s b e e n c a l c u l a t e d i n s t u d y i n g t h e l o w - l y i n g s t a t e s (see, f o r i n s t a n c e , r e f . / l / ) . The f r a g - m e n t a t i o n o f o n e - q u a s i p a r t i c l e s t a t e s f a r from t h e Fermi e n e r g y h a s b e e n c a l c u l a t e d i n r e f . / l 9 / i n deformed n u c l e i w i t h i n t h e q u a s i p a r - t i c l e - p h o n o n n u c l e a r model. The f i r s t m i c r o s c o p i c c a l c u l a t i o n s o f t h e f r a g m e n t a t i o n o f p a r t i c l e s t a t e s i n s p h e r i c a l n u c l e i h a v e b e e n aimed a t t h e s t u d y o f S

-

and p -wave n e u t r o n s t r e n g t h f u n c t i o n s . T h e s e c a l - c u l a t i o n s w e r e performed w i t h t h e e x c i t e d s t a t e wave f u n c t i o n i n t h e form of ( 1 6 ) . The f r a g m e n t a t i o n o f d e e p h o l e s t a t e s h a s been c a l c u l a - t e d i n r e f . / 2 1 / , where t h e c o u p l i n g w i t h t h e f i r s t q u a d r u p o l e and o c - t u p o l e phonons h a s b e e n t a k e n i n t o a c c o u n t e x p l i c i t l y w h e r e a s w i t h o t h e r c o n f i g u r a t i o n s by i n t r o d u c i n g a n e f f e c t i v e w i d t h , and i n r e f . / 1 1 / w i t h i n t h e n u c l e a r f i e l d t h e o r y . The f r a g m e n t a t i o n o f d e e p h o l e s t a t e s h a s b e e n c a l c u l a t e d i n r e f . / 2 2 / w i t h t h e wave f u n c t i o n ( 1 6 ) , i n which

F = O ,

and a l a r g e number o f phonons. The f r a g m e n t a t i o n o f one- q u a s i p a r t i c l e s t a t e s i n many s p h e r i c a l n q c l e i h a s b e e n c a l c u l a t e d i n r e f s . / 2 3 - 2 6 / w i t h t h e wave f u n c t i o n ( 1 6 ) t a k i n g i n t o a c c o u n t t h e t e r m s

-dtq+Q' w i t h a l a r g e one-phonon b a s i s .

The f r a g m e n t a t i o n o f l o w - l y i n g h o l e s t a t e s h a s f i r s t b e e n measured ex- p e r i m e n t a l l y / 2 7 / i n 1960. The o n e - n u c l e o n t r a n s f e r r e a c t i o n s t u r n e d o u t t o b e e f f e c t i v e f o r o b t a i n i n g t h e s p e c t r o s c o p i c f a c t o r s which cha- r a c t e r i s e t h e s t r e n g t h d i s t r i b u t i o n of h i g h - l y i n g p a r t i c l e a n d d e e p h o - l e s t a t e s i n s p h e r i c a l n u c l e i ( s e e r e f s . / 1 4 , 2 3 - 3 9 / ) . The p o l a - r i z e d p a r t i c l e beams t u r n e d o u t t o b e v e r y u s e f u l f o r t h e assignment t o t h e s p i n s o f o n e - q u a s i p a r t i c l e f r a g m e n t e d s t a t e s / 4 0 , 4 1 / .

Now we g i v e t h e f o r m u l a e f o r c a l c u l a t i n g t h e f r a g m e n t a t i o n o f h o l e and p a r t i c l e , n e u t r o n and p r o t o n s t a t e s i n odd-A s p h e r i c a l n u c l e i . The s t r e n g t h f u n c t i o n d e s c r i b i n g t h e f r a g m e n t a t i o n o f t h e s u b s h e l l

,

h a s t h e form

where

CrL'

e n t e r s i n t o ( 1 6 )

, g(2)

i s d e f i n e d by ( 1 7 )

,

^and L1 i s t h e a v e r a g i n g p a r a m e t e r . The s p e c t r o s c o p i c f a c t o r , summed o v e r t h e e n e r g y i n t e r v a l A E

,

f o r t h e e x c i t a t i o n o f t h e h o l e s t a t e

j

i n t h e one- n u c l e o n t r a n s f e r r e a c t i o n i s

c2s = / 2 j + / ) q 2 1 $ r l ) d f .

^{( 2 0 )}

I n c a l c u l a t i n g t h e s p e c t r o s c o p i c f a c t o r f o r a p a r t i c l e s t a t e i s s u b s t i t u t e d by

U? .

The f r a g m e n t a t i o n o f o n e - q u a s i p a r t i c l e s t a t e s i s d e t e r m i n e d by t h e s p e c t r o s c o p i c f a c t o r s

CZS ,

c e n t r o i d e n e r g i e s and w i d t h s

w h e r e

3 ,

i s t h e ground s t a t e e n e r g y o f a n odd-A n u c l e u s . These va- l u e s ar&. &ompared w i t h t h e c o r r e s p o n d i n g e x p e r i m e n t a l d a t a . The c a l c u - l a t i o n s w i t h i n t h e q ~ a s i ~ a r ' t i c l e - p h o n o n n u c l e a r model, which a r e p r e - s e n t e d below, h a v e b e e n p e r f o r m e d w i t h t h e wave f u n c t i o n ( 1 6 ) . The

(2-5) .?02 t e r m s o f t h e t y p e di

a +

and t h e ( 1'5) 104 t e r m s o f t h e t y p e d f

a i d

w e r e i n c l u d e d i n t o t h e wave f u n c t i o n ( 1 6 ) . I n c o m p a r i s o n w i t h r e f . / l l / and o t h e r p a p e r s we t a k e i n t o a c c o u n t t h e . t e r m S - d i 4 ' Q ' ,

C4-74 JOURNAL DE PHYSIQUE

p l a y i n q a n i m p o r t a n t r o l e i n some c a s e s i n s p h e r i c a l n u c l e i .

P r e v i o u s l y , t h e d i s t r i b u t i o n o f t h e s i n g l e - p a r t i c l e s t r e n g t h f o r t h e s t a t e s f a r from t h e Fermi e n e r g y h a s b e e n t h o u g h t o f t o b e Of t h e B r e i t - Y i g n e r form w i t h t h e c e n t e r c o r r e s p o n d i n g t o t h e q u a s i p a r t i c l e e n e r g y . The f i r s t c a l c u l a t i o n s / l 9 , 2 0 / o f t h e f r a g m e n t a t i o n o f one-qua- s i p a r t i c l e s t a t e s h a v e shown t h a t t h e d i s t r i b u t i o n o f t h e i r s t r e n g t h h a s a complex f o r m , which d i f f e r s f r o m t h e B r e i t - W i g n e r o n e . T y p i c a l e x a m p l e s o f t h e f r a g m e n t a t i o n o f n e u t r o n h o l e s t a t e s i n ''5.5'n

,

c a l c u - l a t e d i n r e f . / 2 3 / , and o f p r o t o n p a r t i c l e s t a t e s i n ' 4 5 E ~

,

c a l c u l a t e d i n r e f . / 2 5 / , are g i v e n i n f i g . 1. More t h a n 902 o f s t r e n q t h o f t h e s e s t a t e s i s e x h a u s t e d up t o a n e n e r g y o f 14 MeV. I t i s s e e n from f i g . 1 t h a t t h e s t r e n g t h d i s t r i b u t i o n h a s s e v e r a l p e a k s . They a r e d u e t o l a r - ge m a t r i x e l e m e n t s c o r r e s p o n d i n g t o t h e p o l e s o f t h e s e c u l a r e q u a t i o n . C o n s i d e r t h e energy: d e p e n d e n c e o f t h e w i d t h

rf .

^{I t} b a s b e e n assumed i n some p a p e r s ( s e e , f o r i n s t a n c e r e f . / I 8 / ) t h a t

r' -

^E:

^.

The a v a i l a - b l e e x p e r i m e n t a l and t h e o r e t i c a l d a t a o n t h e f r a g m e n t a t ~ o n o f one-qua- s i p a r t i c l e s t a t e s d i d n o t r e v e a l a n y r e g u l a r i t i e s o f - t h e d e p e n d e n c e o f rt o n Ex

.

^{I t i s} o b v i o u s t h a t w i t h i n c r e a s i n g Ex t h e r e g i o n o f l o c a t i o n o f t h e most p a r t o f t h e s u b s h e l l s t r e n g t h i n c r e a s e s . The w i d t h

r t ,

d e t e r m i n e d b y f o r m u l a ( 2 2 ) , s p e c i f i e s t h e s t r e n g t h d i s t r i + b u t i o n i f i t i s of t h e B r e i t - W i g n e r f o r m o r h a s o n e maximum. I f t h e s t r e n q t h distribution h a s s e v e r a l maxima, a s i n f i 1 , t h e

r i

- v a l u e s t u r n o u t t o be l e s s i n f o r m a t i v e . F o r i n s t a n c e , i n 3 4 Z r more t h a n 9 0 % of t h e /,$?9/2 s u b s h e l l s t r e n g t h i s c o n c e n t r a t e d i n t h e ground s t a t e . Ta- k i n g i n t o a c c o u n t t h e r e s t p a r t o f t h e 1 ~ 9 1 ~ s t r e n g t h l y i n g above a n d u s i n g f o r m u l a (2_2), one g e t s t h e v a l u e o f f f c l o s e t o t h a t o f f t i n

1 1 9 ~ n , i n which Ex =6.5 YeV a n d t h e s t r e n g t h i s d i s t r i b u t e d o v e r many l e v e l s i n t h e i n t e r v a l 4-5 MeV.

The c a l c u l a t i o n s w i t h i n t h e q u a s i p a r t i c l e - p h o n o n n u c l e a r model d e s c - r i b e w e l l t h e f r a g m e n t a t i o n o f o n e - q u a s i p a r t i c l e s t a t e s i n s p h e r i c a l n u c l e i . The c a l c u l a t i o n s h a v e no f r e e p a r a m e t e r s . S i n c e t h e r e i s no f i t o f t h e e x p e r i m e n t a l d a t a , o n e c a n n o t e x p e c t a c o m p l e t e a g r e e m e n t

F i g . 1. F r a g m e n t a t i o n o f n e u t r o n h o l e s t a t e s i n I l 5 s n and o f p r o t o n p a r t i c l e s t a t e s i n 1 4 5 ~ u .

6

Ex

MeV

F i g . 2. E x p e r i m e n t a l / 4 1 / ( s h a d e d ) and t h e o r e t i c a l / 2 3 / h i s t o g r a m s o f t h e s t r e n g t h d i s t r i b u t i o n o f n e u t r o n s u b s h e l l s i n 1 1 1 ~ n

o f t h e o r y w i t h e x p e r i m e n t . A t y p i c a l a g r e e m e n t o f t h e c a l - - c u l a t i o n s w i t h e x p e r i m e n t i s shown i n f i g . 2. Almost s u c h a n a g r e e m e n t o f t h e c a l c u l a - t i o n s / 2 4 / w i t h t h e experimen- t a l d a t a / 4 2 / i s o b s e r v e d f o r t h e f r a g m e n t a t i o n o f t h e p r o - t o n s u b s h e l l f ~ i n ~ 2 0 7 ~ 1 . / ~ The c o m p a r i s o n o f t h e s p e c t - r o s c o p i c f a c t o r s C*S

,

^sum-

med o v e r AE, i s shown i n t a b l e 1 . The e x p e r i m e n t a l and c a l c u l a t e d v a l u e s o f C Z S f o r 5 7 ~ 0 a r e o b t a i n e d by Pugach, V a a n e r , Vdovin e t a l . F o r 3 9 ~ r t h e e x p e r i m e n t a l d a t a a r e t a k e n from r e f . / 3 3 / and t h e c a l c u l a t i o n s from r e f . / 2 6 / ; f o r I l 5 s n from r e f s . / 4 0 / and / 2 3 / . r e s n e c t i v e l a . . . . 1t i s s e e n from t h e t a b l e

-

t h a t t h e c a l c u l a t e d v a l u e s o f C'S f o r

/d5/2

i n 5 7 ~ 0 and f o r 2p i n I 1 5 s n a g r e e f a i r 1 w e l l w i t h t h e e x p e r i m e n t a l d a t a . The l f 7 1 2 ~ t r e n g t h d i s t r i - b u t i o n i n 96Zr i s a n example o f a d i s c r e p a n c y between t h e o r y and expe- r i m e n t . The d e s c r i p t i o n c a n b e somewhat improved by f i t t i n g t h e p a r a - m e t e r s o f t h e Saxon-Woods p o t e n t i a l .

The a v a i l a b l e e x p e r i m e n t a l i n f o r m a t i o n o n t h e f r a q m e n t a t i o n of one- q u a s i p a r t i c l e s t a t e s i n s p h e r i c a l n u c l e i i s s c a r c e . The c a l c u l a t i o n s a r e r e s t r i c t e d t o s e v e r a l s t a t e s i n a few n u c l e i . T h e r e f o r e , o n e can- n o t make c o n c l u s i o n s a b o u t t h e r e g u l a r i t i e s o f t h e f r a g m e n t a t i o n . We c a n o n l y make t h e f o l l o w i n g r e m a r k s a b o u t t h e f r a g m e n t a t i o n o f one- q u a s i p a r t i c l e s t a t e s i n : s p h e r i c a l n u c l e i .

i ) About ( 7 0 - 9 0 ) % o f t h e s i n g l e - p a r t i c l e s t r e n g t h i s c o n c e n t r a t e d i n e a c h ground o r l o w - l y i n g s t a t e o f a n odd-A n u c l e u s . The f r a g m e n t a t i o n d e p e n d s on t h e o n e - q u a s i p a r t i c l e e n e r g y . W i t h i n c r e a s i n g e n e r g y o f t h e o n e - q u a s i p a r t i c l e s t a t e t h e f r a g m e n t a t i o n grows, and t h e s i n g l e - p a r - t i c l e s t r e n g t h i s d i s t r i b u t e d o v e r s e v e r a l l e v e l s .

i i ) The form o f t h e o n e - q u a s i p a r t i c l e s t r e n g t h d i s t r i b u t i o n d i f f e r s from t h e B r e i t - W i g n e r o n e , and t h e d i s t r i b u t i o n h a s s e v e r a l p e a k s . iii) The f r a g m e n t a t i o n d e p e n d s on quantum numbers o f t h e s i n g l e - p a r - - t i c l e s t a t e and on t h e p r o p e r t i e s o f c o l l e c t i v e v i b r a t i o n a l s t a t e s 05 t h e r e l e v a n t d o u b l y e v e n n u c l e u s . Under o t h e r e q u a l c o n d i t i o n s , t h e s u b s h e l l s w i t h s m a l l d a r e f r a g m e n t e d s t r o n g e r t h a n t h e s u b s h e l l w i t h l a r g e 4

.

T h i s c a n b e s e e n from f i g s . 1 and 2 . The f r a g m e n t a t i o n o f o n e - q u a s i p a r t i c l e s t a t e s i r l t h e n u c l e i w i t h c l o s e d s h e l l s i s w e a k e r .

I t i s i n t e r e s t i n g t o s t u d y how t h e f r a g m e n t a t i o n of t h e s u b s h e l l s t r e - n g t h c h a n g e s a s o n e p a s s e s f r o m s p h e r i c a l t o deformed n u c l e i . Due t o t h e d e f o r m a t i o n , e a c h s u b s h e l l i s s p l i t t e d , and t h u s t h e o b t a i n e d f r a g - m e n t a t i o n i s l a r g e . A f u r t h e r f r a g m e n t a t i o n i s c a u s e d by t h e q u a s i p a r - t i c l e - p h o n o n i n t e r a c t i o n . Now,we c o n s i d e r t h e f r a g m e n t a t i o n o f t h e n e u t r o n lhll/2subshell i n t h e s p h e r i c a l 4 5 ~ m and deformed ~m n u c l e u s . The c a l c u l a t i o n s h a v e b e e n p e r f o r m e d by Vdovin, Malov, Nguen Din Vin and S o l o v i e v w i t h t h e p a r a m e t e r s o f t h e Saxon-Woods p o t e n t i a l g i v e n i n r e f . / 2 / . i n I 5 l s m w i t h t h e phonons b e l o n g i n g t o 1 5 2 ~ m a t t h e d e f o r - m a t i o n pi! =O. 3,

p,,

= 0 . 0 4 , a n d i n 1 4 5 ~ m w i t h t h e phonons b e l o n g i n g t o

1 4 6 ~ m , p , = ~ y =

⁰

.

Due t o t h e d e f o r m a t i o n t h e /hl* s t r e n g t h i s d i s t r i - b u t e d a s f o l l o w s : 90% i s c o n c e n t r a t e d i n t h e s i n g l e - p a r t i c l e s t a t e s

n/i

15053, 9/i C5143, 7/2-1523], 5 6 f 5321, 3/2- [ 5 4 1 1 , V i L5501; 6% i n t h e p a r - t i c l e s t a t e s n e a r t h e Fermi e n e r g y , and 4% i n t h e p a r t i c l e s t a t e s i n

C4-76 J O U R N A L DE PHYSIQUE T a b l e 1 . sums o f s p e c t r o s c o p i c f a c t o r s

i n t h e e n e r g y i n t e r v a l s AE A E , MeV

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5 7 ~ 0 4.11-5.35 5.35-7.06 7 -06-9.16

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F i g . 3 . D i s t r i b u t i o n o f t h e s u b s h e l l fhri/2 s t r e n g t h i n t h e s p h e r i c a l I45sm and deformed I51sm n u c l e i

t h e r a n g e of t h e n e u t r o n b i n d i n g e n e r g y . Due t o t h e d e f o r m a t i o n 90%

o f t h e lh,,,? s t r e n g t h i s f r a g m e n t e d i n t h e e n e r g y i n t e r v a l 7 MeV. The fragmentation o f ih,f/2 ( s e e f o r m u l a ( 1 9 ) ) , c a l c u l a t e d w i t h t h e wave f u n - c t i o n ( 1 6 ) w i t h F'O and A ⁼ 0 . 5 MeV t a k i n g i n t o a c c o u n h e q u a s i p a r - t i c l e - p h o n o n i n t e r a c t i o n , i s e x h i b i t e d i n f i g . 3. F o r r5rSm t h e e n e r - g i e s o f h o l e s t a t e s a r e p l o t t e d i n t h e r i g h t s c a l e and o f particle s t a - tes i n t h e l e f t s c a l e . The s i n g l e - p a r t i c l e s t a t e s 15051 a n d 9/t-i5141

( p e a k s a r e a t a n e n e r g y o f 1 and 2.5 MeV) a r e n o t p r a c t i c a l l y f r a g - mented. The main p a r t o f o t h e r h o l e s t a t e s t r e n g t h i s d i s t r i b u t e d i n t h e i n t e r v a l 1-3 '?eV. Note, t h a t a v e r y s m a l l p o r t i o n o f t h e fhr,/2stre- n t h i s i n t h e s t a t e s 5h'15237 and '12-

f

5211 o b s e r v e d e x p e r i m e n t a l l y i n

1 2 1 . 1 5 3 ~ m . I t i s s e e n from f i g . 3 t h a t t h e main p a r t o f t h e i,,/, s t r e - n g t h i n 1 4 5 ~ m i s f r a g m e n t e d i n t h e e n e r g y i n t e r v a l 2-3 MeV and t h e r e s t p a r t ( 3 8 % ) a t h i g h e r e n e r g i e s . These c a l c u l a t i o n s a r e aimed a t d e m o n s t r a t i n g t h e g e n e r a l r e g u l a r i t i e s of t h e f r a g m e n t a t i o n o f sub- s h e l l s i n s p h e r i c a l and deformed n u c l e i r a t h e r t h a n a t e l u c i d a t i n g t h e e x p e r i m e n t a l d a t a / 3 8 , 3 9 / on t h e /hflI2 f r a g m e n t a t i o n i n t h e Sm i s o t o p e s . Based on t h e i n v e s t i g a t i o n o n e may c o n c l u d e t h a t : f i r s t , t h e s i n g l e - p a r t i c l e s t r e n g t h i n a s p h e r i c a l n u c l e u s i s f r a g m e n t e d s t r o n g e r t h a n i n t h e deformed n u c l e u s ( d u e t o a l a r g e c o l l e c t i v i t y o f v i b r a t i o n a l s t a t e s ) , and s e c o n d , t h e s t r e n g t h o f e a c h s u b s h e l l i n a deformed nuc- l e u s i s f r a g m e n t e d s t r o n g e r t h a n i n a s p h e r i c a l n u c l e u s ( d u e t o a s t a b l e d e f o r m a t i o n ) .

The e x p e r i m e n t a l and t h e o r e t i c a l i n v e s t i q a t i o n s o f t h e f r a q m e n t a t i o n o f o n e - q u a s i p a r t i c l e s t a t e s show t h a t t h e s t r e n g t h i s c o n c e n t r a t e d i n a c e r t a i n e n e r g y i n t e r v a l and t h e s t r e n g t h d i s t r i b u t i o n c o n t a i n s p e a k s up t o Ian e n e r g y of 15-20 MeV. T h i s f a c t i s n o n t r i v i a l , s i n c e p r e v i o u s - l y it was assumed t h a t above t h e n e u t r o n b i n d i n g e n e r g y t h e s t a t i s t i c a l r e g u l a r i t i e s p r e d o m i n a t e , w h e r e a s n o n s t a t i s t i c a l e f f e c t s a r e v e r y s m a l l . A f u r t h e r s t u d y of t h e f r a g m e n t a t i o n of o n e - q u a s i p a r t i c l e s t a t e s r e - q u i r e s a n e x t e n s i o n o f t h e r e g i o n o f e x p e r i m e n t a l i n v e s t i g a t i o n b y i n - c r e a s i n g t h e number o f n u c l e i s t u d i e d and b y g o i n g t o w a r d s h i g h e r ex- c i t a t i o n e n e r g i e s . I t would b e i n t e r e s t i n g t o s t u d y t h e f r a g m e n t a t i o n o f t h e d e e p e s t h o l e s t a t e s v i a h i g h r e s o l u t i o n ( e

, e P

) and (

P,

2 p ) e x p e r i m e n t s .

3

-

FRAGMENTATION OF TWO-QUASIPARTICLE STATES

The e x p e r i m e n t a l i n f o r m a t i o n o n t h e f r a g m e n t a t i o n o f t w o - q u a s i p a r t i c l e s t a t e s i n s p h e r i c a l n u c l e i i s g a i n e d from t h e s p e c t r o s c o p i c f a c t o r s o f t h e o n e - n u c l e o n t r a n s f e r r e a c t i o n s and f r o m t h e two-nucleon t r a n s f e r r e a c t i o n s / 1 4 , 4 3 , 4 4 / . I n r e c e n t y e a r s t h e e x p e r i m e n t a l d a t a from t h e

( p

, t

) and (o( ,6He ) r e a c t i o n s h a v e b e e n e n r i c h e d /45-47/.

Now we g i v e t h e f o r m u l a e f o r d e s c r i b i n g t h e f r a g m e n t a t i o n o f two-qua- s i p a r t i c l e s t a t e s w i t h i n t h e q u a s i p a r t i c l e - p h o n o n n u c l e a r model, which h a v e been, o b t a i n e d i n r e f . / 4 8 / . I n t h e RPA t h e t w o - q u a s i p a r t i c l e com- p o n e n t

d,i2

i n t h e s t a t e

ff;piYo

w i t h s p i n J = A i s d e t e r m i n e d by t h e phonon a m p l i t u d e

%/~i~d,l~.

I n t h e c a l c u l a t i o n s o f t h e f r a g m e n t a t i o n o f one-phonon s t a t e s w i t h t h e wave f u n c t i o n ( 1 8 ) and model H a m i l t o n i a n

( 9 ) , we g e t

a n e x p l i c i t form o f t h e s t r e n g t h f u n c t i o n i s g i v e n i n r e f . / 4 8 / . The wave f u n c t i o n o f a n odd-A t a r g e t - n u c l e u s i s t a k e n i n t h e f o r m

- -

and t h e wave f u n c t i o n o f t h e f i n a l s t a t e w i t h s p i n

J f

i s t a k e n i n t h e form ( 1 8 ) . The s t r e n g t h f u n c t i o n f o r t h e s p e c t r o s c o p i c f a c t o r o f t h e n u c l e o n p i c k u p r e a c t i o n o n t h e s u b s h e l l i s

Summing o v e r

Jf ,

which form t h e s t a t e s a n d

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we g e t

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!25)

ddo J,

The r e s u l t s o f c a l c u l a t i o n o f t h e f r a g m e n t a t i o n o f t w o - q u a s i p a r t i c l e s t a t e s h a v e b e e n compared w i t h t h e e x p e r i m e n t a l d a t a f o r some s p h e r i - c a l n u c l e i i n r e f . / 4 8 / . I n t h i s p a p e r we s h a l l d i s c u s s t h e new r e s u l t s . The f r a g m e n t a t i o n o f t h e " v a l e n c e p a r t i c l e - h o l e " s t a t e s i s s t u d i e d v i a t h e p i c k u p r e a c t i o n . We s h a l l c o n s i d e r t h e s p e c t r o s c o p i c f a c t o r s o f t h e " v a l e n c e p a r t i c l e - h o l e " s t a t e s i n 2 0 6 ~ b , t h e e x p e r i m e n t a l d a t a f o r which h a v e b e e n o b t a i n e d i n r e f . / 4 3 / from t h e 207pb ( ' H e , & ) 2 0 6 ~ b r e - a c t i o n . The e x p e r i m e n t a l and c a l c u l a t e d s p e c t r o s c o p i c f a c t o r s summed i n t h e e n e r g y i n t e r v a l s AE a r e g i v e n i n t a b l e 2 . I t i s s e e n from t h e t a b l e t h a t t h e c a l c u l a t i o n s d e s c r i b e f a i r l y w e l l t h e f r a g m e n t a t i o n o f t w o - q u a s i p a r t i c l e s t a t e s , t h o u g h p r o v i d e a somewhat l a r g e r c o n c e n t r a - t i o n of s t r e n g t h a t low e n e r g i e s . A c c o r d i n g t o r e f . / 4 3 / 35% o f t h e

{ 3 p f I 2 , { h f f i

\

s t r e n g t h i s c o n c e n t r a t e d i n t h e i n t e r v a l ( 7 . 4 - 1 0 . 6 ) MeV, t h e c a l c u l a t i o n s g i v e a v a l u e two t i m e s l a r g e r / 4 9 / . I t i s o f i n t e r e s t t o f i n d o u t where t h e r e s t p a r t o f t h i s s t r e n g t h i s . The c a l c u l a t e d

C4-78 J O U R N A L DE PHYSIQUE T a b l e 2. D i s t r i b u t i o n o f t h e two-

q u a s i p a r t i c l e s t r e n g t h i n 206pb

Ex

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o f t h i s s t a t e i s 8 . 6 MeV, t h a t i s v e r y c l o s e t o t h e e x p e r i - m e n t a l v a l u e 3 . 9 MeV.

I n t h e two-nucleon t r a n s f e r r e a c - t i o n s o n e c a n o b s e r v e r e s o n a n c e - l i k e s t r u c t u r e s , w h i c h c a n b e com- p a r e d w i t h t h e c a l c u l a t e d f r a g - m e n t a t i o n o f t w o - q u a s i p a r t i c l e s t a t e s . I n r e f . / 4 5 / a s t r u c t u r e a t t h e e n e r g y 8 . 0 MeV w i t h t h e w i d t h f f 2.20

+

0 . 1 5 MeV h a s b e e n o b s e r v e d i n t h e 1 16sn ( d

,

6 ~ p ) 1 l 4 ~ n r e a c t i o n . The a n a l y s i s o f t h e an- g u l a r d i s t r i b u t i o n i n d i c a t e s a n e x c i t a t i o n o f t h e s t a t e s w i t h

5" = 6' w i t h a p o s s i b l e a d m i x t u r e o f s t a t e s w i t h J = = 8+. The firag- m e n t a t i o n o f t h e two-quasi a r t i c l e s t r e n g t h w i t h ='J 6+ and 8' i n I l 4 s n , c a l c u l a t e d by Voronov/50/, i s shown i n f i g . 4. I t f o l l o w s from t h e c o m p a r i s o n o f f i g . 4 w i t h t h e e x p e r i m e n t a l d a t a t h a t t h e o b s e r v e d a t 8 P?eV s t r u c t u r e i n 1 1 4 ~ n may b e d u e t o t h e f r a g - m e n t a t i o n o f t h e t w o - q u a s i p a r t i c - l e s t a t e s 6 1 j 9 / 2 r

j

^{w i t h}

J R = 8' a n 6+ and {lfg/Z, jd5/2L) w i t h f"= 6+. I n t h e ( p

,

f ) r e - a c t i o n c r o s s s e c t i o n s mea-sured i n r e f . / 4 6 / many n u c l e i c o n t a i n w i d e s t r u c t u r e s a t ( 7 - 9 ) MeV. A c c o r d i n q t o t h e c a l c u l a t i o n s made b y Voro- nov, t h e s e s t r u c t u r e s a r e c a u s e d by t h e f r a g m e q t a t i o n o f two-quasi- p a r t i c l e s t a t e s . Thus, a w i d e s t r u c t u r e w i t h Ex = 8 . 3 YeV i s o b s e r v e d i n 9 4 ~ 0 . The r e s u l t s o f c a l c u l a t i o n o f f o u r t w o - q u a s i p a r t i c l e s t a t e s o f t h e t y p e o f " v a l e n c e p a r t i c l e - h o l e " and " h o l e - h o l e " a r e p r e s e n t e d i n f i g . 5. I t i s s e e n from t h i s f i g u r e t h a t t h e s t r u c t u r e a r o u n d 8 . 3 MeV may b e d u e t o t h e e x c i t a t i o n o f t h e c o n f i g u r a t i o n s { 2 d a , !fa ) a n d

{

^it912, .jtp12

.

The n a t u r e of t h i s s t r u c t u r e and t h e s t r e n g t h d l s t r l b u t l o n of t h e {.?ds/z, ! f ~ / ~ s t a t e c a n b e e x p l a i n e d by m e a s u r i n g t h e 9 5 ~ o ( p d ) 5 4 ~ 0 r e a c t i o n c r o s s s e c t i o n .

9

, ^MeV

F i g . 4 - , S t r e n g t h f u n c t i o n s o f two- u a s l p a r t l c l e n e u t r o n s t a t e s i n l 7 4 ~ n

F i g . 5 - S t r e n g t h f u n c t i o n s o f two- u a s i p a r t i c l e n e u t r o n s t a t e s i n 92p10

I n r e c e n t y e a r s t h e f i r s t a t t e m p t s have b e e n made t o s t u d y t h e f r a g - m e n t a t i o n o f t w o - q u a s i p a r t i c l e s t a t e s i n s p h e r i c a l n u c l e i . Many i n v e s - t i g a t i o n s o f t h e one- and two-nucleon t r a n s f e r r e a c t i o n s s h o u l d b e ma- d e f o r a c o n s i s t e n t s t u d y of t h e f r a g m e n t a t i o n o f t w o - q u a s i p a r t i c l e s t a t e s i n s p h e r i c a l and deformed n u c l e i .

4 - PWGPTENTATION OF CHARGE-EXCHANGE COLLECTIVE STATES

The f r a g m e n t a t i o n o f s i n g l e - p a r t i c l e and c o l l e c t i v e vibrational mo- t i o n s i s m a i n l y due t o t h e q u a s i p a r t i c l e - p h o n o n i n t e r a c t i o n and i s de- s c r i b e d i n t h e framework o f t h e same model / 2 , 7 - 1 1 , 1 8 / . We s h a l l de- m o n s t r a t e how t h e f r a g m e n t a t i o n o f c h a r g e - e x c h a n g e c o l l e c t i v e s t a t e s , which i s i n t e n s i v e l y s t u d i e d i n r e c e n t y e a r s / 5 1 , 5 2 / , i s d e s c r i b e d w i - t h i n t h e q u a s i p a r t i c l e - p h o n o n n u c l e a r model.

I n s t u d y l n g t h e g i a n t m u l t i p o l e r e s o n a n c e s i n l d e f o r m e d n u c l e i , it h a s b e e n shown / 2 / t h a t t h e R P A - c a l c u l a t i o n s g i v e c o r r e c t v a l u e s o f t h e i r w i d t h s . They a r e s l i g h t l y i n f l u e n c e d by t h e 2p - 2 h c o n f i g u r a t i o n s / 5 3 / . The r e a s o n i s t h a t t h e f r a g m e n t a t i o n o f t h e s i n g l e - p a r t i c l e s p h e r i c a l s t a t e s d u e t o t h e d e f o r m a t i o n i s s t r o n g e r t h a n t h a t d u e t o t h e q u a s i - p a r t i c l e - p h o n o n i n t e r a c t i o n . T h e r e f o r e , we c a l c u l a t e i n t h e RPA t h e

( p , f 7 ) and ( 1 7

,p

) s t r e n g t h f u n c t i o n s . The s e c u l a r e q u a t i o n s and s t r e n g t h f u n c t i o n s of t h e ( p

,

n ) and ( ~7

,

^,D ) r e a c t i o n s f o r t h e e x c i - t a t i o n o f c h a r g e - e x c h a n g e s t a t e s o n d o u b l y e v e n deformed n u c l e i a r e g i v e n i n r e f . / 5 4 / . The s t r e n g t h f u n c t i o n s f o r t h e e x c i t e d I f s t a t e s i n c l u d i n g t h e Gamow-Teller (GT) r e s o n a n c e on 2 3 9 ~ c a l c u l a t e d w i t h

,pO

⁼ 17/A MeV, d = 0 . 5 MeV and o t h e r c o n s t a n t s o f t h e model from r e f . / 2 / a r e shown i n f i g . 6 . I n t h e deformed n u c l e i t h e GT r e s o n a n c e c o n s i s t s o f t h e components w i t h I R K = I f 0 and 1 f 1 , which a r e s p l i t t e d by 0 . 6 MeV i n 1 6 6 ~ r and b y 0 . 3 MeV i n 2 3 8 ~ . I n t h e G T s t r e n g t h d i s t r i - b u t i o n o n e c a n s e p a r a t e t h e low-energy p a r t , t h e r e g i o n o f maximum a r o u n d Ex = 20 MeV and t h e h i g h - e n e r g y p a r t w i t h c o n t r i b u t i o n s 2 7 % , 56% and 1 6 % , respectively, a b o u t 1 % o f s t r e n g t h i s a t 50 MeV. Almost t h e same d i s t r i b u t i o n o f s t r e n g t h i s o b s e r v e d i n a l l c a l c u l a t e d n u c l e i . The c e n t r o i d e n e r g y o f t h e low-energy p a r t i n 2 3 % ~ i s 14 MeV, i t i s below t h e maximum o f t h e GT r e s o n a n c e b y 6 MeV. The r a t i o B(p,n)/B(n,p) i s e q u a l t o 20 f o r t h e r a r e - e a r t h n u c l e i a n d t o 57 f o r 2 3 8 ~ . The model i n d e p e n d e n t sum r u l e i s e x h a u s t e d b y ( 9 8 - 9 9 ) % .

The s t r e n g t h f u n c t i o n s f o r t h e s p i n - d i p o l e c h a r g e - e x c h a n g e r e s o n a n c e s w i t h A" ^s 0 - , I - , 2- on deformed n u c l e i h a v e b e e n c a l c u l a t e d by S o l o - v i e v and Sushkov. The s t r e n g t h f u n c t i o n o f t h e ( p , f 7 ) r e a c t i o n o n

1 6 6 ~ r , c a l c u l a t e d a c c o r d i n g t o r e f . / 5 1 / w i t h def

'=

^{0 . 7 5 47ir/<}

aef=O = 1 7 / A MeV- a n d d = 0 . 5 MeV, i s shown i n f i g . 7. The c e n t - r o i d e n e r g y IS E X = 24 MeV,

Ex

f o r d i f f e r e n t components i s : 28 MeV f o r 0 - , 25 MeV f o r 1- and 21 MeV f o r 2 - . The s p i n - d i p o l e s t r e n g t h i s d i s t r i b u t e d o v e r t h e e n e r g y i n t e r v a l s a s f o l l o w s : 6% up t o 11 MeV, 20%

w i t h i n 11-20 MeV, 30% w i t h i n 20-25 MeV, 41% w i t h i n 25-33 MeV and 3%

a b o v e 33 MeV. A c c o r d i n g t o t h e c a l c u l a t i o n s 70% o f t h e s p i n - d i p o l e s t r e n g t h i s c o n c e n t r a t e d i n t h e i n t e r v a l o f 15 MeV. The e n e r g y r e g i o n of l o c a t i o n o f t h e s p i n - d i p o l e c h a r g e - e x c h a n g e s t r e n g t h i n deformed n u c l e i i s q u i t e l a r g e . I t e x c e e d s c o n s i d e r a b l y t h e r e g i o n o f c o n c e n t - r a t i o n o f t h e GT s t r e n g t h .

A s i s known / 9 , 1 8 / , t h e w i d t h s o f t h e g i a n t r e s o n a n c e s i n s p h e r i c a l n u c l e i a r e c a u s e d b y t h e f r a g m e n t a t i o n o f one-phonon s t a t e s d u e t o t h e c o u p l i n g w i t h t h e Z p - 2 h c o n f i g u r a t i o n s . While s t u d y i n 9 t h e c h a r g e - exchange r e s o n a n c e s i n s p h e r i c a l n u c l e i , o n e s h o u l d c a l c u l a t e t h e f r a - g m e n t a t i o n o f

n p

phonons. The method o f i n c l u d i n g n p phonons i n t o t h e q u a s i p a r t i c l e - p h o n o n n u c l e a r model h a s b e e n d e v e l o p e d i n r e f . / 5 5 / . The e q u a t i o n s f o r d e s c r i b i n g t h e f r a g m e n t a t i o n o f np phonons i n sphe- r i c a l n u c l e i were o b t a i n e d i n a g e n e r a l form. The c a l c u l a t i o n s were p e r f o r m e d w i t h t h e wave f u n c t i o n i n t h e form

C4-80 JOURNAL DE PHYSIQUE

5 10 15 20 2 5 E,MeV

F i g . 6

-

S t r e n g t h f u n c t i o n s o f t h e ( n

,

p ) and ( p

,

n ) r e a c t i o n s w i t h t h e e x c i t e d GT r e s o n a n c e o n 23%

60

LO

20

10 20 30

E

MeV

F i g . 7

-

S t r e n g t h f u n c t i o n o f t h e I p

,

ⁿ ) r e a c t i o n w i t h t h e e x c i t e d s p i n - d i ? o l e s t a t e s o n 1 6 6 ~ r

I n r e f . / 5 5 / a t r a n s i t i o n h a s b e e n made t o t h e sysbem o f app- r o x i m a t e e q u a t i o n s , which i s now u s e d f o r t h e d e s c r i p t i o n o f t h e ( P

,

f? ) and ( f?

,

p ) s t r e n - g t h f u n c t i o n s f o r e x c i t a t i o n o f t h e c h a r g e - e x c h a n g e s t a t e s . A s i s known, l e s s t h a n 50% o f t h e GT r e s o n a n c e s t r e n g t h i s f o u n d e x p e r i m e n t a l l y i n s p h e r i c a l n u c l e i . I t i s assumed i n some p a p e r s ( s e e r e f s . / 5 1 , 5 2 , 5 6 , 5 7 / )

t h a t t h e missbng o f t o t a l GT s t r e n g t h i s c a u s e d b y a n a d m l x t t u r e o f e x c i t a t i o n s o f t h e A

-

i s o b a r

-

n u c l e o n h o l e t o t h e p r o t o n - p a r t i c l e

-

n e u t r o n - h o l e GT s t a t e s . However, o n e d i d n o t s u c c e e d i n e x p l a i n i n g s u c h a l a r g e m i s s i n g o f t h e GT s t r e - n g t h . I n r e f . / 5 7 / t h e i n f l u e n c e o f t h e 2 , ~ -2h c o n f i g u r a t i o n s o n t h e GT r e s o n a n c e i n 2 0 8 ~ b h a s b e e n c a l c u l a t e d by t a k i n g i n t o a c c o u n t t h e f i r s t v i b r a t i o n a l 3- and 2+ s t a t e s . However, a v e r y s m a l l number o f t h e

2'-2h

c o n f i q u r a t i o n s was u s e d i n t h e - s e c a l c u l a t i o n s . T h e r e f o r e , it i s r e a s o n a b l e t o c a l c u l a t e t h e f r a g m e n t a t i o n o f np phonons w i t h a l a r g e number o f two- phonon s t a t e s i n t h e wave f u n c - t i o n ( 1 6 ) and t h u s t o f i n d o u t t o what e x t e n t t h e f r a g m e n t a - t i o n i s r e s p o n s i b l e f o r t h e m i s s i n g o f GT s t r e n g t h . Such c a l c u l a t i o n s h a v e b e e n p e r f o r - med by Kuzmin and S o l o v i e v . The r e s u l t s o f c a l c u l a t i o n s of

t h e ( ,D

, n

) r e a c t i o n o n 1 2 4 ~ e w i t h e x c i t a t i o n o f t h e GT re-

s o n a n c e a r e shown i n f i g . 8 . A r a t h e r l a r g e phonon b a s i s w i t h t h e wave f u n c t i o n ( 2 6 ) h a s b e e n u s e d i n t h e c a l c u l a t i o n s . I n t h e R P A c a l c u l a t i o n s t h e GT s t r e n g t h i s d i s t r i b u t e d a s

~ O ~ ~ O W S : 27% up t o 11 MeV, 58% w i t h i n 13-16 M e V , 9 8 w i t h - i n 16-20 MeV, and 6 % above 20 N e V . I f t h e q u a s i p a r t i c l e - phbnon i n t e r a c t i o n i s t a k e n i n t o a c c o u n t t h e maximum a l m o s t d o e s n o t s h i f t . The GT s t r e n g t h i s d i s t r i b u t e d a s f o l l o w s : 30% up t o 11 MeV, 30% w i t h i n 13-16 MeV, 20% w i t h i n 16-20 MeV and 20% above 20 MeV. Thus, t h e GT s t r e n g t h d e c r e a s e s from 60%

t o 30% a r o u n d maximum. Up t o a n e n e r g y o f 20 MeV 80% o f t h e GT s t r e n g t h i s e x h a u s t e d .

These c a l c u l a t i o n s a r e c o n t i - nued, a l a r g e s p a c e o f one-pho- non s t a t e s i s t a k e n i n t o a c c o -

I I 2 ' ~ e u n t , and t h e GT a n d s p i n - d i p o -

B ( ~ , n l , MeV'' l e r e s o n a n c e i s c a l c u l a t e d f o r s e v e r a l n u c l e i i n c l u d i n g t h e

1

n u c l e i w i t h u n f i l l e d s h e l l s .

1

^1'

A f u r t h e r i n v e s t i g a t i o n o f t h e f r a g m e n t a t i o n o f charge-exchan-

I

g e r e s o n a n c e s i n s p h e r T c a l and

d e k r m e d n u c l e i i s o f g r e a t i n t e r e s t .

I I . F i g . 8

-

S t r e n g t h f u n c t i o n o f

5 10 ^{t h e (} p

,

n ) r e a c t i o n w i t

92Zhe

l 5 E,

MeV e x c i t e d GT r e s o n a n c e o n Te.

CONCLUSION

The s t u d y o f t h e f r a g m e n t a t i o n o f one-, t w o - q u a s i p a r t i c l e and one- phonon s t a t e s i s a n i m p o r t a n t s t a g e i n i n v e s t i g a t i n g t h e c o m p l i c a t i o n o f t h e n u c l e a r s t a t e s t r u c t u r e w i t h i n c r e a s i n g e x c i t a t i o n e n e r g y . The f r a g m e n t a t i o n o f i s o b a r - a n a l o g s t a t e s i s o f g r e a t i n t e r e s t , t h e e x p e r i m e n t a l i n f o r m a t i o n o n it i s v e r y s c a r c e / l 4 / . One c a n e a s i l y c a l - c u l a t e t h e f r a g m e n t a t i o n o f i s o b a r - a n a l o g s t a t e s u n d e r t h e assump- t i o n t h a t i s o s p i n i s a good quantum number. The f r a g m e n t a t i o n o f t h e q u a s i p a r t i c l e B phonon s t a t e s i s c a l c u l a t e d w i t h i n t h e q u a s i p a r - t i c l e - p h o n o n n u c l e a r model. The f r a g m e n t a t i o n o f t h e s e s t a t e s i n f l u - e n c e s , f o r i n s t a n c e , t h e p a r t i a l s t r e n g t h f u n c t i o n s f o r

r

- t r a n s i - t i o n s t o t h e l o w - l y i n g s t a t e s i n s p h e r i c a l n u c l e i s t u d i e d i n r e f 8 / . The d i s t r i b u t i o n o f t h e 1ggI2 @ 2; s t r e n g t h w i t h J"= 912' i n i14Sn, c a l c u l a t e d i n r e f . / l 6 / , i s shown i n f i g . 9 . I t i s s e e n from t h e f i - g u r e t h a t a l a r g e r p a r t o f t h i s s t r e n g t h i s l o c a l i z e d i n t h e e n e r g y i n t e r v a l 5 MeV. The s t u d y o f t h e f r a g m e n t a t i o n o f t h e q u a s i p a r t i c l e p l u s phonon s t a t e s i s o f g r e a t i n t e r e s t . How a r e m a n y - q u a s i p a r t i c l e a n d many-phonon s t a t e s f r a g m e n t e d a t i n t e r m e d i a t e and h i g h e x c i t a t i o n e n e r g i e s ? T h i s q u e s t i o n h a s been r a i s e d s e v e r a l y e a r s a g o / 5 8 , 5 9 / , b u t s t i l l t h e r e i s n o ' e x p e r i m e n t a l i n f o r m a t i o n o n t h e f r a g m e n t a t i o n o f m a n y - q u a s i p a r t i c l e s t a t e s .

I n c o n c l u s i o n it i s w o r t h m e n t i o n i n g t h a t o n e s h o u l d n o t e x p e c t a c o m p l e t e a g r e e m e n t o f t h e o r y w i t h e x p e r i m e n t w h i l e d e s c r i b i n g n u c l e a r s t a t e s a t i n t e r m e d i a t e and h i g h e x c i t a t i o n e n e r g i e s . The f i t b y chan- g i n g t h e p a r a m e t e r s i s n o t v e r y v a l u a b l e . The n u c l e a r many-body prob- l e m i s s t a t e d s o t h a t o n e s h o u l d t a k e i n t o a c c o u n t t h e d i f f e r e n c e o f t h e p r o p e r t i e s o f t h e s y s t e m o f 238 n u c l e o n s from t h e s y s t e m o f 240

0.2 '.-

2 o r 235 o r 1 9 0 n u c l e o n s . T h i s

\D~;,~~/2;9)) , M e V - ' p r o b l e m i s h i g h l y d i f f i c u l t .

T h e r e a r e l a r g e d i f f i c u l t i e s i n s o l v i n g t h e t h r e e - and f o u r - n u c l e o n problem. V a r i o u s n u c l e a r models p r o v i d e a f a i r l y good

:/h

d e s c r i p t i o n t i o n o f o n e - q u a s i p a r t i c l e , o f t h e f r a g m e n t a - two- q u a s i p a r t i c l e and one-phonon

5 10 s t a t e s , and t h e s e models a r e t o

be f u r t h e r improved.

P i g . 9

-

S t r e n g t h d i s t r i b u t i o n o f t h e [lgt e 2: ]9,2+ s t a t e i n 1 1 9 ~ n ,

JOURNAL DE PHYSIQUE

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