DAMPING OF HIGHLY EXCITED VIBRATIONS IN HEAVY NUCLEI

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DAMPING OF HIGHLY EXCITED VIBRATIONS IN HEAVY NUCLEI

J. Wambach, B. Schwesinger

To cite this version:

J. Wambach, B. Schwesinger. DAMPING OF HIGHLY EXCITED VIBRATIONS IN HEAVY NU- CLEI. Journal de Physique Colloques, 1984, 45 (C4), pp.C4-281-C4-296. �10.1051/jphyscol:1984421�.

�jpa-00224087�

JOURNAL DE PHYSIQUE

Colloque C4, supplPment au n03, Tome 45, mars 1984 page C4-281

DAMPING OF HIGHLY EXCITED VIBRATIONS I N HEAVY NUCLEI

J

.

.

' ~ n s t i t u t fZir K e m p h y s i k , Kern forschungsan Zage JiiZich, 0-51 70 JuZich, F.R. G.

*Department o f Physics, S t a t e U n i v e r s i t y o f New Yark a t Stony Brook, Stony Brook, NY 11794, U.S.A.

R6sum6 - Nous pr6sentons un modgle d e c r i v a n t l ' a t t 6 n u a t i o n de 1 1 i n t e n s i t 6 des t r a n s i t i o n s 1 p a r t i c u l e-1 t r o u dans des e x c i t a t i o n s plus complexes. Les canaux 2 p a r t i c u l e s - 2 t r o u s sont t r a i t & e x p l i c l t e m e n t t a n d i s que l e s c o n f i g u r a t i o n s d'ordre p l u s grand 1 e sont approximativement. Nous empl oyons pour 1 a f o n c t i o n de r6ponse l i n g a i r e une expression g q u i v a l e n t Z l a d i a g o n a l i s a t i o n de l ' i n t e r - a c t i o n r6sSduelle dans l e s sous-espaces I p - l t e t 2p-2t combin6s. Les ' p l i c a - f&ns pour l e s e x c i t a t i o n s de p a r i t 6 n a t u r e l l e e t non n a t u r e l l e dans 'r!l e t

Pb sont discut6es.

A b s t r a c t

-

I m p l i c a t i o n s f o r n a t u r a l and unnatural p a r i t y e x c i t a t i o n s i n " ~ r and b$'' are discussed.

I - INTRODUCTION

The microscopic understanding o f damping o f nuclear c o l l e c t i v e motion has a t t r a c t e d t h e o r e t i c a l i n t e r e s t i n the l a s t few years. Considerable progress has been made es- p e c i a l l y f o r simple nuclear e x c i t a t i o n s l i k e s i n g l e - p a r t i c l e and v i b r a t i o n a l s t a t e s (see r e f . 1 f o r a recent review).

Generally two damping mechanisms can be i d e n t i f i e d f o r v i b r a t i o n s :

( i ) pure mean f i e l d damping which gives r i s e t o a spreading o f I p - l h s t r e n g t h due t o s h e l l s t r u c t u r e ("fragmentation w i d t h " ) and a broadening above t h e continuum thresh- o l d due t o p a r t i c l e emission ("escape w i d t h " ) .

( i i damping from r e s i d u a l two-body c o l l i s i o n s , which couple the l p - l h doorway s t a t e s t o nuclear compound s t a t e s ("spreading width").

With i n c r e a s i n g mass number mean f i e l d e f f e c t s become l e s s important 121, a t l e a s t f o r 1 ow energy resonances, and the decay 1 argely proceeds v i a more complicated s t a t e s . T h e i r l e v e l d e n s i t y increases r a p i d l y w i t h e x c i t a t i o n energy and microscopic d e s c r i p t i o n s o f damping become d i f f i c u l t because o f t h e enormous number o f s t a t e s t o be taken i n t o account.

We present a model which t r e a t s l p - l h and 2p-2h s t a t e s e x p l i c i t l y and h i g h e r com- pound s t a t e s i n an approximate way. I n t h e r e s t r i c t e d space of l p - l h and 2p-2h e x c i - t a t i o n s a d i a g ~ n a l i z a t i o n i t e r a t e s t h e p-h i r r e d u c i b l e diagrams o f order v2 t o a l l Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984421

C4-282 JOURNAL DE PHYSIQUE

orders. This procedure i s e q u i v a l e n t t o a d i a g o n a l i z a t i o n o f V i n t h e f u l l subspace / 3 / . Since a l l 2p-2h c o n f i g u r a t i o n s i n a model space o f two major s h e l l s above and below t h e Fermi surface are r e t a i n e d t h e model i s e s p e c i a l l y powerful f o r d e s c r i b i n g the high energy d i s p e r s i o n o f strength.

I n s e c t i o n I 1 we g i v e a b r i e f o u t l i n e o f t h e theory and s e c t i o n I 1 1 presents r e s u l t s f o r and 2 0 8 ~ b . F i r s t , n a t u r a l p a r i t y e x c i t a t i o n s are discussed. We a l s o present t h e i s o b a r i c analog s t a t e (IAS) which i s an approximate e i g e n s t a t e o f the nuclear hamiltonian. This c o n s t i t u t e s a s t r i n g e n t t e s t f o r models o f s t r e n g t h dispersion.

Furthermore i s o s p i n i n t e r f e r e n c e e f f e c t s enhanced by 2p-2h c o u p l i n g a r e considered f o r the quadrupole response i n '08Pb. I n the sector of unnatural p a r i t y e x c i t a t i o n s we show r e s u l t s f o r " t h e t w i s t mode", a mode propagatin v i a transverse zero sound.

The h i g h energy d i s p e r s i o n o f Ganow-Teller s t r e n g t h i n "Zr and '08Pb i s a l s o d i s - cussed.

I 1 - OUTLINE OF THE MODEL

The d i s t r i b u t i o n o f nuclear t r a n s i t i o n s t r e n g t h S t o a weak e x t e r n a l p e r t u r b a t i o n

i s given by the l i n e a r response f u n c t i o n

- ^Im - - -

x npnh

1

w i t h

Here H i s the f u l l nuclear hamil tonian, I

>

s t a t e s involved. E m p i r i c a l l y , however, t h e s t r e n g t h f u n c t i o n exhi b i t s gross s t r u c - t u r e s due t o the f a c t t h a t Q i s a one body operator and t h e main c o n t r i b u t i o n t o SQ comes from n=nl=l. Therefore t o a very good approximation we only need t h e i n v e r s e o f Cnn1 i n t h e space o f l p - l h e x c i t a t i o n s . I n terms of a p r o j e c t o r P onto t h i s space

we o b t a i n

The operator 1-P p r o j e c t s onto t h e space complementary t o l p - l h , i.e. contains 2p-2h and higher e x c i t a t i o n s . I n t h e s p e c i f i c c a l c u l a t i o n presented below the l p - l h model space i s spanned by 2 major s h e l l s above and below the Fermi surface ( w i t h d i s c r e - t i z e d continuum s t a t e s ) . The hamiltonian H has been s p l i t i n t o a mean f i e l d p a r t Ho and a r e s i d u a l i n t e r a c t i o n V. V can only couple np-nh s t a t e s w i t h n<2 t o t h e l p - l h doorway, i - e . t h e entrance channel f o r t h e decay i s given by t h e Zp-2h s t a t e s only.

We t r u n c a t e 1-P a t t h i s l e v e l

and t r e a t the more complicated s t a t e s n=n1>2 on t h e average by i n t r o d u c i n g a f i n i t e imaginary p a r t i A i n t h e 2p-2h propagator (A = 3 MeV i s used). Furthermore V i s neg- l e c t e d i n t h e 2p-2h space which i s a reasonable approximation a t h i g h frequencies since the l e v e l d e n s i t y P 2 p 2 h ( ~ i s l a r g e and the d e t a i l e d energy d i s t r i b u t i o n u n i m p o r t a n t . A f i n i t e i A i n t h e 2p-2h propagator a l s o compensates f o r t h e neglected r e - s i d u a l i n t e r a c t i o n i n t h e 2p-2h space which r e d i s t r i b u t e s the s t r e n g t h c o u p l i n g t o the l p - l h states. The i n t e r a c t i o n diagrams which are i t e r a t e d by eq. (2.4) are given i n Fig. 1. To order v2 we have selfenergy i n s e r t i o n s on the p a r t i c l e o r h o l e l i n e (which are diagonal i n t h e l p - l h i n d i c e s ) and ph-linked diagrams i n which e i t h e r a ph-pair (bubble diagram) o r a pp- o r hh-pair ( l a d d e r diagram) i s exchanged between ph-states. Since t h e s e t o f v2 diagrams i s complete one can show t h a t SQ(w) i s p o s i - t i v e d e f i n i t e f o r a l l frequencies w>O.

Fig. 1

-

We have chosen a phenomenological nuclear hamiltonian H o = - m A n2 + U ( r )

where U(_r) i s a Woods-Saxon w e l l . The nucleon e f f e c t i v e mass m* i s an a d j u s t a b l e parameter. V i s approximated by an antisymmetrized zero range i n t e r a c t i o n

w i t h 1 in e a r d e n s i t y dependence

The parameters o f V and m*/m used a r e 1 i s t e d i n Table 1.

Table 1 - Strength of t h e r e s i d u a l i n t e r a c t i o n V as s p e c i f i e d i n eq. (2.7) i n Me1 fm3. The two values f o r the i n t e r p o l a t i o n r a d i u s Ro are i n ' O Z ~ and '08pb respec, t i v e l y .

C4-284 JOURNAL DE PHYSIQUE

To discuss q u a l i t a t i v e f e a t u r e s o f t h e response f u n c t i o n SQ i t i s u s e f u l t o d e f i n e operator averaged q u a n t i t i e s l i k e t h e average ph energy

the average i n t e r a c t i o n s t r e n g t h t o f i r s t o r d e r

< v l Q =

1

Yh

P h '

and t h e frequency dependent second order term

2 -1

< v ( w ) lQ =

1

:.

Since f o r f i n i t e 7) <v2> i s complex we o b t a i n a frequency dependent energy s h i f t

and a w i d t h

I n terms o f these q u a n t i t i e s one can d e f i n e an operator averaged response as

Here

i s t h e p r o b a b i l i t y f o r f i n d i n g t r a n s i t i o n s t r e n g t h induced by Q a t frequency w. E!

i n t h e denominator denotes the unperturbed energy < E g h ) ~ p l u s t h e f i r s t order c o r - r e c t i o n <V>Q. Note t h a t i n t h e l i m i t where t h e t r a n s i t i o n s t r e n g t h i s concentrated i n a s i n g l e s t a t e In> = QI> expression (2.12) becomes exact. Comparison o f <SQ> w i t h t h e exact SQ shows t h a t the awrage gives a good r e p r e s e n t a t i o n o f t h e gross f e a t u r e s o f t h e response l i k e the c e n t r o i d energy and t h e variance i f the resonance i s l o c a l - i z e d on the l p - l h l e v e l .

I l l

-

Monopole v i b r a t i o n s

Monopole v i b r a t i o n s are compressional modes o f the nucleus and hence g i v e informa- t i o n on the bulk moduli i n symmetric nuclear m a t t e r and ( v i a N#Z n u c l e i ) also i n asymmetric nuclear matter. The damping o f t h e i s o s c a l a r and i s o v e c t o r branches i s q u i t e d i f f e r e n t as we s h a l l discuss. Bertsch /5/ f i r s t p o i n t e d o u t t h a t t h e width o f t h e i s o s c a l a r monopole v i b r a t i o n s i s very small because of a s t r o n g i n t e r f e r e n c e between ph-unlinked and p h - l i n k e d diagrams. I n i s o s p i n synanetric nu l e a r matter t h e imaginary p a r t s o f selfenergy and bubble diagrams cancel t o order q /k$ f o r s p i n

3

scalar i s o s p i n s c a l a r d e n s i t y modes as discussed i n r e f . 6. This i s a l k o t r u e t o a very l a r g e e x t e n t f o r a f i n i t e nucleus / 7 / . Fig. 2 d i s p l a y s r Q ( w ) i n t h e l o n g wave-

2 2

l e n g t h l i m i t f o r Q = r and Q = r zo. I n t h e i s o s c a l a r channel one observes remark

Fig. 2 - Operator averaged w i d h f o r i s o s c a l a r (upper p a r t ) and i s o v e c t o r (lower p a r t ) monopole e x c i t a t i o n s i n ' O ~ P ~ . The dotted 1 in e i n d i c a t e s c o n t r i b u t i o n s from selfenergy diagrams and t h e dashed and dash-double d o t t e d l i n e s give ladder and bub- b l e terms. The f u l l l i n e represents the sum.

I ' ' c I " ' ~ a

1 0 ...,

1

... - - - - - - -

_

-..

'

<

-

- 5

1

-

able c a n c e l l a t i o n o f s e l f e n e r g i e s and bubbles (dash-dotted and dash-double d o t t e d l i n e s ) . The ladders (dashed l i n e s ) only play a minor r o l e . The s i t u a t i o n i n t h e i s o - v e c t o r channel i s q u i t e d i f f e r e n t . Bubble c o n t r i b u t i o n s are s u b s t a n t i a l l y reduced and ladders i n s t e a d o f enhancing reduce t h e width somewhat a t higher frequencies.

These average features are a l s o present i n the f u l l c a l c u l a t i o n (Fig. 3).

>

s ;- ; ^10-

3

5

0

-5

The i s o s c a l a r t r a n s i t i o n s t r e n g t h (upper p a r t ) remains f a i r l y 1 ocal i zed a f t e r 2p-2h m i x i n g i s t u r n e d on and t h e increase i n "width" has t o be i n t e r p r e t e d as a r e d i s t r i - b u t i o n o f s t r e n g t h between nearby l p - l h doorway s t a t e s ( d o t t e d l i n e s ) v i a 2p-2h i n - t e r a c t i o n s . Note t h a t t h e c e n t r o i d i s s h i f t e d due t o a r e p u l s i v e c o n t r i b u t i o n from bubbles t o B E ( w ) which i s w e l l known i n symmetric nuclear mat r /8/. The second

e r s h i f t s Bre q u i t e sizable. While < A Q i s about -4 h V i n "Zr and -3.3 MeV i n q6'Pb a t the resonance energy the s h i f t from bubbles i s 2 MeV i n both n u c l e i . For i s o v e c t o r modes t h e s i t u a t i o n i s d i f f e r e n t . P a r t i c l e and h o l e s ~ r e a d i n q dominate the

....

- 1 5 . " ' L I L t L " ' l ' -

.. . r2z0

-

-

---- ^- - - - =:.- --:

-

.--

' ' ' 1 ' 1 ' 1 " 1 1 '

decay and hence r i s l a r g e . Consequently the modes are s t r o n g l y damped (middle p a r t o f Fig. 3). Q

10 2 0 3 0 Energy [NeVI

JOURNAL DE PHYSIQUE

90 Z r M o n o p o l e

2000 r 2

Fig. 3

-

L

Quadrupol e modes

4 0 0 0

2 0 0 0

I n t h e quadrupole response we see the same features: 1 i t t l e damping o f t h e i s o s c a l a r v i b r a t i o n (upper p a r t o f Fig. 4) due t o l a r g e c a n c e l l a t i o n o f s e l f e n e r q i e s by bub- b l e s and a l a r g e spreading o f i s o v e c t o r s t r e n g t h (middle p a r t o f Fig.

.

very l o n g high energy t a i l which i n extends up t o 50 MeV and i n 2''Pb up t o 40 MeV. The i s o v e c t o r FWHM i n ' O Z ~ has been r e p o r t e d by P i t t h a n /9/ as 7-8 MeV. We f i n d roughly 11 MeV somewhat l a r g e r than experiment. There are, however, t h e o r e t i c a l un- c e r t a i n t i e s i n the i n t e r a c t i o n as w e l l as experimental d i f f i c u l t i e s i n t h e back- ground s u b t r a c t i o n f o r broad resonances.

-

-

I I

2 0 3 0

E n e r g y ['lev1

20000 3000

16000

2000 12000

BOO0 1000

4000

0 0

-

=

,\ I000 10000

-

-

L

- -

-

0 0

600 3000

400 2000

200 1000

0 0

10 20 30 40 10 2 0 3 0

Energy [ M e V ] E n e r g y [MeV]

Fig. 4 - Quadrupole response i n ' O Z ~ ( l e f t p a r t ) and 2 0 8 ~ b ( r i g h t p a r t ) . The dashed l i n e s give l p - l h alone w h i l e the f u l l l i n e includes 2p-2h.

Charge exchange i s o v e c t o r monopole

Not only t h e g i a n t d i p o l e , a l s o i s o v e c t o r monopole ~ x c i t a t i o n s p r o v i d e i n f o r m a t i o n about the bulk and surface symmetry energies. The r %+-resonance i n the charge ex- change channel i s o f s p e c i a l i n t e r e s t s i n c e i t s w i d t h i s expected t o be much narrow- e r than t h e r 2 T 0 - o r t h e r 2 s - -partners /lo/. The l a t t e r i s pushed up t o h i g h e r energies by the Coulom9 displacement energy EC and t h e proton-neutron mass d i f f e r - ence Mpn r e l a t i v e t o r so. Monopole s t a t e s i n t h e (N+l,Z-1)-nucleus, however, are lowered by the same amount. The 2p-2h l e v e l d e n s i t y i s much smaller r ducing phase space f o r the decay. The c a l c u l a t e d s t r e n g t h d i s t r i b u t . ns i n "Zr and

8$Sp

a r e hown i n Fig. 5. The shape o f the i sovector monopole i n 48Y i s simi 1 a r t o t h a t i n "Zr b o t h on the l p - l h l e v e l (dashed l i n e s i n Fig. 3 and Fig. 5) and on the l e v e f u l l l i n e s i n Fig. 3 and F i g . 5 ) . But t h e r e are d i s t i n c t d i f f e r e n c e s i n and "'Pb because o f P a u l i b l o c k i n g e f f e c t s . Due t o the neutron excess the 3p and 2f neutron o r b i t s are occupied and t h e r e f o r e the analogs o f the s t r o n g 3 p 4 p and 2f+3f neutron t r a n s i t i o n s a r e missing (Fig. 5 ) . S i m i l a r l y the 2p+3p and l f + 2 f t r a n s i t i o n s are blocked. The c o u p l i n g t o 2p-2h s t a t e s a f f e c t s the l p - l h s t r e n g t h only s l i g h t l y . The reason f o r t h i s i s the r e d u c t i o n o f the l e v e l density from

-

C4-288 JOURNAL DE PHYSIQUE

exchange monopole may be small /12/ t h e r e i s evidence from our c a l c u l a t i o n t h a t the s t r e n g t h remains l o c a l i z e d and can be seen i n charge exchange r e a c t i o n s .

Energy [MeV]

Fig. 5

-

The g i a n t d i p o l e resonance

The g i a n t d i p o l e i s the experimentally best known g i a n t resonance. It s t i l l has problems i n the t h e o r e t i c a l d e s c r i p t i o n . The value of the nuclear matter bulk symme- t r y energy B between 28-36 MeV /13/ gives d i p o l e e x c i t a t i o n energies i n n u c l e i ~ h i c h are too low. I n Fermi 1 iq u i d theory he e i s a simple r e l a t i o n between B, the Landau parameter i n the nuclear i n t e r i o r fAflny and t h e q u a s i p a r t i c l e e f f e c t i v e mass

With m*=m one o b t z i n s from B = 28 MeV f b = 0.53. To f i t the d i p o l e energy systemat- i c s one needs f; = 1 f o r m*=m /14/. I n our model the Landau parameter has t o be ob- t a i n e d from order V and v2. The p h - l i n k e d v2 c o n t r i b u t i o n s t o t h e energy are small however, such t h a t the s t r e n g t h o f t h e T T ' - p i e c e o f V determines f:. We f i x e d f;

from a bulk symmetry o f 30 MeV from eq. (3 . I ) . I t has been suggested t h a t t h e r a p i d

" o f f the energy s h e l l v a r i a t i o n " o f the e f f e c t i v e mass might produce a d d i t i o n a l r e - p u l s i o n t o t h e d i p o l e energy. This e f f e c t i s included i n our model since the r e a l p a r t o f the selfenergy diagrams which gives a w-dependent c o r r e c t i o n t o the s i n g l e par c l e ene 'es /15/, i s retained. Fig. 6 d i s p l a y s t h e r e s u l t s o f the c a l c u l a t i o n i n "Zr and 'jQPb. We see almost no s h i f t i n the resonance p o s i t i o n as Zp-2h s t a t e s a r e coupled t o the l p - l h response. Decomposition i n t o the various diagrams shows n e g l i g i b l e c o n t r i b u t i o n s from t h e bubbles, a very weak r e p u l s i o n from the ladders and a weak a t t r a c t i o n from the selfenergies. The higher order c o r r e c t i o n s are too weak t o place the d i p o l e a t the r i g h t p o s i t i o n i f only nuclear matter i n f o r m a t i o n i s used. The s o l u t i o n might be provided by the surface dynamics i n a n o n t r i v i a l way:

( i ) our model neglects r e s i d u a l i n t e r a c t i o n i n the 2p-2h space. P a r t i c l e - h o l e r e - s c a t t e r i n g terms (Fig. 7 ) w i l l b u i l d up c o l l e c t i v e surface modes. I n c l u s i o n o f those

Fig. 6

-

Fig. 7 - Rescattering diagrams in the 2p-2h space neglected in the present calcula- t i o n .

rescattering terms /16/ changes the U-dependence of the e f f e c t i v e mass and therefore enhances off-shell e f f e c t s . ( i i ) the density dependence of VO1[P] i s crucial f o r the r a t i o between surface and volume symmetry energy /17/. Our density dependence i s rather weak (Table 1) giving a small surface symmetry energy.

The high energy dispersion of t r a n s i t i o n strength i s not only a feature of the iso- vector monopole and quadrupole response functions, which are in an energy region of high level density, b u t i s also seen i n the d'pole ( f u l l l i n e i n F' 6 ) . The inte- grated strength in the t a i l (U > 22.5 MeV i n ''i!r and w. 16 MeV i n ?"Pb) are 13 % and 10 % respectively. The classical Thomas-Reiche-Kuhn sum rule i s exceeded by about 20 %. This corresponds roughly t o the amount of energy weighted strength i n the t a i l . Thus the energy weighted strength under the peak exhausts the sum rlrte.

Isobaric analo-ates and isospin mixing

If e f f e c t s Prom the Coul omb interaction coul d be neglected the isobaric analogue s t&te

C4-290 J O U R N A L DE PHYSIQUE

would be an e i g e n s t a t e o f H, s i n c e then T- commutes w i t h t h e hamiltonian. This i s t h e reason why, experimentally, the IAS i s an extremely sharp resonance although i t l i e s i n a r e g i o n w i t h a h i g h d e n s i t y o f 2p-2h states.

Enerqy M e V 1 Enerov M e V 1

Fig. 8

-

We have looked a t the w i d t h o f t h i s s t a t e i n 9 0 ~ r and '08pb as a check o f our model.

Fig. 8 shows the l p - l h r e s u l t which i s given by t h e dashed l i n e . The f i n i t e w i d t h here i s a mathematical a r t i f a c t coming from t h e f i f i i t e imaginary i n i n t r o d u c e d t o a l l o w numerical i n v e r s i o n o f the s t r e n g t h function. The d o t t e d l i n e s show the e f - f e c t s o f t h e selfenergy i n s e r t i o n s which r e s u l t s i n a broadening comparable t o t h e s i n g l e p a r t i c l e width. The i n c l u s i o n o f a l l second order terms cancels t o a l a r g e e x t e n t the s i n g l e p a r t i c l e w i d t h r e s t o r i n g p a r t l y i s o s p i n synnnetry. The bottom of Fig. 8 i n d i c a t e s t h a t t h i s c a n c e l l a t i o n i s due t o the ladder diagrams and t h a t the bubbles do n o t c o n t r i b u t e . The cancel 1 a t i o n between s e l fenergy and 1 adder, however, i s n o t complete since f o r ' O Z ~ and '08pb 1 7 % o f the s t r e n g t h r e s i d e s i n t h e h i g h energy t a i l . C l e a r l y , a b e t t e r d e s c r i p t i o n would r e q u i r e c o n s i d e r a t i o n o f ground s t a t e c o r r e l a t i o n s /18/, because t h e ground s t a t e used m s t be an e i g e n s t a t e of H if the symmetry argument i s t o hold.

Fig. 9 d i s p l a y s t h e response f u n c t i o n t o the electromagnetic quadrupol e operator

5

( f u l l l i n i n comparison t o t h e sum o f i s o s c a l a r (1/2 r YZm) and i s o v e c t o r (1/2 z r Y ) response (dashed l i n e ) . Since N#Z f o r 2 0 8 ~ b the two curves need n o t c o i n c i l e and lndeed we observe a r a t h e r l a r g e d e s t r u c t i v e i n t e r f e r e n c e . This i s due 2c t o the f a c t t h a t the c o u p l i n g t o 2p-2h c o n f i g u r a t i o n s c a r r i e s i s o v e c t o r s t r e n g t h

Fig. 9 - I s o s p i n i n t e r f e r e n c e e f f e c t s i n t h e qu drupole response o f '08pb.

h

3000

-

<

-

down t o the r e g i o n where i s o s c a l a r s t r e n g t h i s concentrated (on t h e l p - l h l e v e l t h e two peaks are w e l l i s o l a t e d ) . The d e s t r u c t i v e i n t e r f e r e n c e observed here moves 18 % of the s t r e n g t h o u t o f the i s o s c a l a r peak and might provide t h e explanation t o t h e observed d i f f e r e n c e s i n E2 s t r e n g t h from (e,el) and (a,al) /19/.

,

.

.

.

I I

P b

; ^I

I '

Q u a d r u p o l e -

I '

The t w i s t mode

Energy [MeV]

One o f the more e x o t i c nuclear resonances t h a t has been speculated upon /4/ i s the nuclear t w i s t . I t s motion can be v i s u a l i z e d as a r o t a t i o n of the d i f f e r e n t l a y e r s o f a Fermi f l u i d a g a i n s t each other, t h e angle of r o t a t i o n around t h e z-axis being pro- p o r t i o n a l t o t h e z-coordinate o f the l a y e r (Fig. 10).

F i g . 10 - V e l o c i t y f i e l d f o r t h e lowest e x c i t e d 2- " t w i s t mode".

I n a f l u i d dynamical d e s c r i p t i o n the v e l o c i t y f i e l d o f the t w i s t i s p u r e l y r o t a t i o n - a l . Such a motion i s only p o s s i b l e i f t h e r e are t a n g e n t i a l r e s t o r i n g f o r c e s e f f e c - t i v e i n the f l u i d . The t w i s t propagates v i a c o l l i s i o n l e s s transverse zero sound so i t s energy i s independent o f Fo ( i .e. t h e nuclear i n c o m p r e s s i b i l i t y ) and since i t i s a spinless mode, i t i s o f course a l s o independent o f Go:

'F

/-

Etw = 2.3 - - _{m* 5} (1 + - F _{3 1} )R

-

For '08pb t h i s gives an energy o f Etw = 7.6 MeV.

M i c r o s c o p i c a l l y t h e t w i s t motion i s expressed most e a s i l y through an operator o f the form

C4-292 JOURNAL DE PHYSIQUE

a c t i n g on the corresponding ground s t a t e . The special combination o f i s o s p i n opera- t o r s chosen i n (3.3) r e s t r i c t s the t w i s t t o protons only. I n t h i s case T,, i s t h e o r b i t a l p a r t o f the electromagnetic M2 operator thus i n d i c a t i n g a way t o e x c i t e the mode. However, t h e M2 operator a l s o contains a s p i n p a r t

which c o n t r i b u t e s 80 % t o the M2 s t r e n g t h i n 2 0 8 ~ b l e a v i n g only 20 % f o r t h e t w i s t . Therefore i t seems very l i k e l y , t h a t t w i s t s t r e n g t h w i l l be obscured by s p i n f l i p s t r e n g t h i n (e,el) experiments /20/.

Fig. 11 - Low energy 2- response: ( a ) denotes t h e t w i s t p a r t w i t h o u t i n t e r a c t i o n (lower p a r t ) , V i n c l u d e d t o f i r s t order (middle p a r t ) and t o second order (upper p a r t ) , (b) d i s p l a y s t h e same f o r the spin p a r t and ( c ) gives the t o t a l electromag- n e t i c strength.

However, the s i t u a t i o n t u r n s o u t t o be much mor favourable as i s demonstrated i n Fig. 11. Each f i g u r e d i s p l a y s the response o f 298Pb t o the operators i n d i c a t e d and c o n t a i n s t h r e e cases and one smaller i n s e t . The bottom graph i n each shows the s t r e n g t h f u n c t i o n f o r t h e case o f no r e s i d u a l i n t e r a c t i o n a t a l l (V=O), i.e. the hamil t o n i a n i s j u s t made from s i n g l e p a r t i c l e energies. The middle s e c t i o n d i s p l a y s the case where the r e s i d u a l i n t e r a c t i o n between l p - l h c o n f i g u r a t i o n s only i s taken i n t o account whereas the upper curve gives the r e s u l t o f the f u l l c a l c u l a t i o n cou- p l i n g 2p-2h t o the l p - l h states. A l l t h r e e curves cover the energy range o f 0-10 MeV. The i n s e t i n t h e t o p graph shows the r e s u l t o f the f u l l c a l c u l a t i o n i n the lp-lh+2p-2h space over t h e energy range 0-30 MeV. A l l curves are normalized t o t h e t o t a l electromagnetic s t r e n g t h i n t h e range o f 0-30 MeV

The most s t r i k i n g f e a t u r e e x h i b i t e d by the r e s u l t s i s t h e sharp concentration o f the t w i s t s t r e n g t h around 7 MeV. This concentration i s already present i n the s i n g l e p a r t i c l e energies alone. Adding a r e s i d u a l i n t e r a c t i o n among t h e l p - l h s t a t e s has only l i t t l e e f f e c t on t h e s t r e n g t h d i s t r i b u t i o n , and a c t u a l l y a v a r i a t i o n o f t h e i n - t e r a c t i o n parameters over a l a r g e range does n o t move the t w i s t from i t s unperturbed p o s i t i o n . C l e a r l y t h i s r e f l e c t s the f i n d i n g s from f l u i d dynamical d e s c r i p t i o n s , t h a t t h e energy o f the t w i s t does n o t depend on the Landau parameter Fo. Coupling t h e 2p-2h c o n f i g u r a t i o n s t o the l p - l h s t a t e s f i n a l l y breaks the t w i s t i n t o two pieces, one a t 7.2 MeV and one a t 7.9 MeV.

I n c o n t r a s t t o t h e t w i s t , t h e fragmentation o f spin f l i p s t r e n g t h i s already present w i t h no i n t e r a c t i o n a t a l l and p e r t a i n s when the i n t e r a c t i o n and 2p-2h couplings are added. The i n s e t i n Fig. l l b shows t h a t 30 % o f t h e s p i n f l i p s t r e n g t h r e s i d e s above 10 MeV. 30 % however must be considered as a lower bound on the dispersion, f o r two reasons: ( i ) t h e e f f e c t i v e GA r e s u l t i n g from the a n t i symmetrized zero range i n t e r a c - t i o n used here i s too small. Any f i n i t e range i n t e r a c t i o n , which we had t o d i s c a r d because o f computational l i m i t a t i o n s , would s h i f t the spin f l i p s t a t e s t o higher energies where the density o f 2p-2h s t a t e s i s higher. This s h i f t would r e s u l t i n an even stronger d i s p e r s i o n o f the s p i n f l i p s t r e n g t h from the 2p-2h states. ( i i ) We have n o t introduced any mechanism t o quench magnetic s t r e n g t h such as isobar-hole e x c i t a t i o n s . Such mechanisms only a f f e c t t h e s p i n f l i p modes and thus enhance the r e l a t i v e importance o f t h e t w i s t t o t h e M2-sum.

Considering these two e f f e c t s , i t seems very l i k e l y t h a t t h e 25 % M2 s t r e n g t h found experimentally 1201 i n t h e energy range between 6-9 MeV i s almost e n t i r e l y due t o t h e t w i s t which c o n t r i b u t e s 20 % t o t h e M2 sum. Fig. 4 shows t h a t even w i t h t h e r a t h e r l a r g e amount o f spin f l i p s t r e n g t h l o c a t e d below 9 MeV the electromagnetic response i s dominated by t h e t w i s t peaks.

L e t us examine t h e p a r t i c l e h o l e amplitudes of the t w i s t operator. Because a, i s a s p i n l e s s 1% o p e r a t r connecting A L = ~ s t a t e s only, t h e l a r g e s t ph amplitudes a r e ( n , j , r + l ) ( n , j - l , r ) - ' g i v i n g

Therefore t h e t w i s t i s mainly b u i l t up from s t a t e s where b o t h p a r t i c l e and h o l e have high angular momenta. Table 2 shows some o f the more important ph-states p a r t i c i - p a t i n g i n the t w i s t motion where energies of s t a t e s designated by an a s t e r i s k have been measured experimentally

.

P h (ph)-energy r e l . weight

Table 2

-

C4-294 JOURNAL DE PHYSIQUE

Remembering t h a t t h e p o s i t i o n o f t h e t w i s t i s i n s e n s i t i v e t o the r e s i d u a l i n t e r a c - t i o n , t h e f a c t t h a t most s i n g l e p a r t i c l e energies are measured ones c o n t r i b u t e s s t r o n g l y t o the s i g n i f i c a n c e o f t h e r e s u l t s presented. Obviously t h e c o n c e n t r a t i o n o f the ph energies around 7 MeV r e f l e c t s the f l u i d dynamical f i n d i n g s t h a t the p o s i - t i o n o f t h e t w i s t i s governed by t h e k i n e t i c energy alone.

I n c o n t r a s t t o t h e t w i s t the s p i n f l i p mode does n o t weigh t h e high angular momentum ph-amp1 i t u d e s because one looses one power o f a by rep1 a c i n g R, i n (3.5) by the s p i n operator a,. Therefore t h e s p i n f l i p i s dispersed over a l l ( i n our b a s i s 86) ph states.

1 0 0 0

- 500

E

6

.

-

U 1-

2

E -

5 . ^-; -

U a

1000 500

Excitation Energy /MeV

Fig. 12

-

Fig. 12 shows the (e,el) spectrum from t h e Darmstadt Linac group /20/ where arrows i n d i c a t e the p o s i t i o n of M2 states. According t o t h e arguments presented the two prominent peaks a t 6.9 MeV and a t 7.9 MeV r m s t c o n t a i n predominantly t w i s t strength.

Probably a c c i d e n t a l l y these two peaks agree p e r f e c t l y w i t h the two t w i s t peaks c a l - c u l ated.

Gamow-Teller resonances

The d i s p e r s i o n o f g: s t r e n g t h due t o mu1 t i p a i r e x c i t a t i o n s i s o f special relevance f o r t h e q u a n t i t a t i v e understanding o f isobar-hole admixtures t o Gamow-Teller (GT)

t r a n s i t i o n s . It has been suggested /18/ t h a t the g i a n t GT-resonances have l o n g t a i l s due t o c o u p l i n g t o 2p-2h s t a t s. We have analyzed t h e g-c- - s t r e n g t h f u n c t i o n s w i t h i n our model i n both and 20BPb. The s i n g l e p a r t i c l e t r a n s i t i o n s i n t h e independent p a r t i c l e model are i n d i c a t e d i n Fig. 13.

The weaker 3 p 3 p and 2f+2f t r a n s i t i o n s i n 2 0 8 ~ b are n o t l i s t e d i n the f i g u r e f o r simp1 i c i t y . The antisymmetrized r e s i d u a l i n t e r a c t i o n V (Table 1) which has been f i t - t e d t o n a t u r a l p a r i t y e x c i t a t i o n s i s too weak i n t h e s p i n - i s o s p i n channel t o repro- duce t h e features o f t h e experimental fp,n) spectra /21/. We t h e r e f o r e r e a d j u s t e d Vll (eq. (1.7)) t o o p t a i n the proper resonance p o s i t i o n s on the l p - l h l e v e l ( c o n t r i - b u t i o n s o f order v2 t o t h e e x c i t a t i o n energy are very s m a l l ) . We obtained

Vll = 228 MeV fm3 c l o s e t o the value quoted by Gaarde e t a1

.

n u c l e i t h e resonances are s u b s t a n t i a l l y broadened compared t o t h e l p l h - d i s t r i b u t i o n s and they e x h i b i t l o n g t a i l s on the high nergy side. The i n t e g r a t e d t a i l strengths

(0 > 22.5 MeV i n " ~ r and u r 25 MeV i n 20&b1 are 25 % and 28 % r e s p e c t i v e l y . Since our model i s an extended TDA and P a u l i unblocking o f s t r e n g t h due t o ground s t a t e c o r r e l a t i o n s i s n o t i n c l u d e d we conserve the 3(N-Z) Ikeda sum r u l e /23/ (Sp+ = 0).

F i g . 13 - Dominant l p - l h t r a n s i t i o n s i n ' O Z ~ ( r i g h t p a r t ) and '08pb ( l e f t p a r t ) i n - c l uding 2p-2h mixing.

Fig. 15 - S t r u c t u r e o f the GT wave f u n c t i o n i n the v i c i n i t y o f t h e resonance peak.

I

. 8

.6

The h i g h - l y i n g GT-peak i n ' O Z ~ i s mainly b u i l t on the j, + j, t r a n s i t i o n , i n t h i s case nlg9/2 + ~ 1 9 ~ 1 ~ . There i s one s p e c i f i c 2p-2h t r a n s i t i o n i n which a p r o t o n i n t h e & - s h e l l drops t o the j,-shell e x c i t i n g a neutron l p - l h t r a n s i t i o n j, + j,

(Fig. 15). I f t h e one body LS p o t e n t i a l s f o r protons and neutrons are t h e same, t h i s

-

5 4 , , , , , , , , , , , , , -

m + c z

3 -

2 -

Energy [MeV]

Fig. 14 - GT-strength f u n c t i o n s i n ' O Z ~ (upper p a r t ) and '08pb (lower p a r t ) i n c l u d - i n g 2p-2h mixing.

, , , , , , , 1 , , 1 1 1 1 I , , , ,

I

-

-

90

Z

GTR

C4-296 JOURNAL DE PHYSIQUE

c o n f i g u r a t i o n i s degenerate w i t h t h e nj, + pj, t r a n s i t i o n . The coup1 i n g m a t r i x e l e - ment i s a l s o large, since t h e r a d i a l wave f u n c t i o n s m a i n t a i n maximum overlap such t h a t t h i s 2 -2h s t a t e becomes t h e dominant decay channel i n the v i c i n i t y o f t h e peak. I n ' 0 4 b t h e l p - l h wave f u n c t i o n i s somewhat more camplicated, however, f o r the nli1312 + plill/2 t r a n s i t i o n which has t h e l a r g e s t amplitude t h e same arguments hold.

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