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SMALL VIBRATIONS ABOUT EXCITED STATES
W. Besold, C. Toepffer, P.-G. Reinhard
To cite this version:
W. Besold, C. Toepffer, P.-G. Reinhard. SMALL VIBRATIONS ABOUT EXCITED STATES.
Journal de Physique Colloques, 1984, 45 (C6), pp.C6-321-C6-331. �10.1051/jphyscol:1984639�. �jpa-
00224241�
JOURNAL DE PHYSIQUE
Colloque C6, suppl6ment au n06, T o m e 45, juin 1984 page C6-321
SMALL VIBRATIONS ABOUT EXCITED STATES
W. Besold, C. Toepffer and P.-G. Reinhard
I n s t i t u t fiir Theoretische Physik 11, Universitat ErZangen, 0-8520 E r Zangen I , F. R. G.
RCsum6
-
Les 6quations de l'approximation de phase aldatoire pour de petites vibrations autour des Qtats excitQs sont dtablies. La ddpendence de la tempdrature des excitations collectives sera examinde. Nous appliquons ce formalisme ii l'Qtat de base ainsi qu'au premier dtat excitE du noyau 90~r. Par cela nous allons confirmer lihypothDse de Brink.Abstract
-
W e d e r i v e Modified RPA-equations for s m a l l vibra- t i o n s around excited states. The temperature d e p e n d e n c e o f col- l e c t i v e e x c i t a t i o n s i s examined. T h e formalism i s a p p l i e d t o the ground s t a t e and the first excited s t a t e o f 9 0 ~ r i n o r d e r t o c o n f i r m the Brink hypothesis.U s u a l l y , the nuclear g i a n t resonances a r e described by the Random- P h a s e - A p p r o x i m a t i o n (RPA) which i s a linearized theory o f s m a l l vibra- t i o n s a b o u t t h e Hartree-Fock ground-state. Recently, t h e r e h a s c o m e u p e x p e r i m e n t a l e v i d e n c e for g i a n t r e s o n a n c e s w h i c h a r e built upon e x c i t e d s t a t e s [l]. T h i s has motivated u s t o i n v e s t i g a t e a modified RPA (MRPA) w h i c h a l l o w s for s m a l l vibrations about a r b i t r a r y states.
Thereby w e have b e e n concerned mainly with thermally e x c i t e d "ground- states"; but the formalism o f the MRPA i s o p e n for any k i n d o f exci- t a t i o n , e.g. particular reoccupations i n the shell-model.
There a r e many w a y s t o d e r i v e the RPA-equations. W e c h o o s e t h e Green's function formalism and extended i t to d e r i v e the MRPA, the e x t e n s i o n c o n s i s t s in allowing for arbitrary occupation n u m b e r s o f t h e s i n g l e p a r t i c l e s t a t e s (in t h e Green's function). S i m i l a r d e r i v a t i o n s h a v e b e e n g i v e n by Vautherin and Vinh M a u [ 2 ] and by D e s C l o i z e a u x [3] for t h e special c a s e o f thermalized "ground-states".
T h i s contribution i s outlined a s follows: In s e c t i o n 2 w e d e r i v e the MRPA e q u a t i o n s by m e a n s o f the Green's function formalism. In o r d e r t o o b t a i n s e m i - q u a n t i t a t i v e predictions f o r t h e g i a n t r e s o n a n c e s w e invoke a s u m - r u l e a p p r o x i m a t i o n t o the MRPA; i t i s i n t r o d u c e d a n d dis- cussed i n s e c t i o n 3. T h e r e s u l t s for giant r e s o n a n c e s built upon ther- mally excited ground-states a r e given i n s e c t i o n 4. An e x a m p l e for t h e c a s e o f a particular shell-model e x c i t a t i o n i s s h o w n in s e c t i o n 5.
2. D E R I V A T I O N OF M O D I F I E D HF- A N 0 RPA-EQUATIONS
W e derive the MRPA-equations s t a r t i n g from the e q u a t i o n s o f m o t i o n for the s i n g l e p a r t i c l e Green's function. T h P s i n g l e p a r t i c l e Green's function q(l.1') - . i s defined a s follows
g(l,l") = - i <i(G(l)
i + ( l ' ) ) >
(1..
w h e r e T i s the time ordering o p e r a t o r , the @ and $+ a r e field o p e r a -
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984639
C6-322 JOURNAL
DE
PHYSIQUEtors and
<. ..>
m e a n s the expectation value over a n ensemble. We u s e the abbreviation (1,2) for (r 19tl,r29t2).The equation o f motion for g ( l , l l ) i s coupled t o the two p a r t i c l e Green's function. We employ the Gorkov-factorization o f t h e two-parti- c l e Green's f u n c t i o n and o b t a i n
Here 2+ m e a n s , t h a t t2+ = lim(t2 + E) and 2 - i s defined analogously.
E'O
The notation g < i n d i c a t e s t h a t the eqs. (2) d e f i n e g < f o r t l < t l r ; h o w e v e r , the s o l u t i o n c a n be continued analytically to t l
>
tl'. O n e can e x p r e s s the g ln terms o f s i n g l e particle w a v e f u n c t i o n s ,The f),~, therein i s an occupation matrix which a l l o w s t o d e s c r i b e correlated s t a t e s or ensembles.
In o r d e r t o d e s c r i b e the stationary s t a t e o f the system w e c h o o s e
In particular, i f w e set f ~ to be the F e r m i - D i r a c distribution w e have the occupation n u m b e r s for s i n g l e particle s t a t e s with energy EX o f a fermi-system at f i n i t e temperature T and c h e m i c a l potential
p . W e insert eqs. (3) and (4) into eqs. (2); then w e add eq. (2a)
t o (2b), t a k e t h e l i m i t t l = tl' and project o n t o s i n g l e p a r t i c l e states. T h i s y i e l d s finally the modified H F - e q u a t i o n s
w h e r e Irk> = rV Irk> - Ikr>. The HF-eqs. a r e contained in (5) f o r t h e special choice fk = 1 for k 5 Fermisurface and fk = 0 for k
>
Fermi surface. These e q u a t i o n s are known for l o n g and have b e e n a p p l i e d by many a u t h o r s , s e e e.g. refs. [4,5].Fig. 1 s h o w s the level s c h e m e for n e u t r o n s in 4 0 ~ a calculated with the modified HF-equations and the F e r m i - D i r a c - d i s t r i b u t i o n .
To g e t the modified RPA-equations w e have to c h o o s e the o c c u p a t i o n number matrix f ~ h t more general a s in eq. (4). In analogy with the conventional derivation o f R P A w e have t o add particle-hole-contribu- tions to the HF-state. But in our c a s e w e have t o take i n t o a c c o u n t , that P a u l i - b l o c k i n g factors ( 1 - f ~ ) and f ~ wlll a p p e a r , because f ~
$
0 and f A4
1 i s possible. T h u s the generalized a n s a t z has t o read_ - -
1%- - -
lP%!F i
.
1: n t r o n s i n g l e p a r t i c l e 1 e ; e l s o f e ' c a a t t e m p e r a t u r e ~ T = O a n d k T = 5 M e V , c a l c u l a t e d w i t h f o r c e S k y r m e M*. T h e b o l d p a r t o f t h e l i n e s i n d i c a t e s t h e o c c u p a t i o n p r o b a b i l i t y o f t h e l e v e l s . A t kT=O t h e s i x l o w e s t l e v e l s a r e o c c u p i e d w i t h p r o b a b i l i t y 1, a t kT=5MeV t h e h i g h e r l e v e l s a l s o h a v e a f i n i t e o c c u p a t i o n p r o b a b i - l i t y , .T h e f i r s t t e r m t h e r e i n d e s c r i b e s t h e s t a t i o n a r y s t a t e a s b e f o r e . T h e s e c o n d a n d t h i r d t e r m d e s c r i b e s m a l l v i b r a t i o n s a b o u t t h e s t a t i o n a r y s t a t e . T h e s y s t e m i s a s s u m e d t o v i b r a t e i n o n e e i g e n m o d e w l t h f r e q u e n -
c y w"). We i n s e r t t h e e q s . ( 6 ) + ( 3 ) i n t o e q s . ( 2 ) ; t h e n we s u b t r a c t e q . ( 2 b ) f r o m ( Z a ) , t a k e t h e l i m i t t = 0 a n d p r o j e c t o n t o s i n g l e p a r t i c l e s t a t e s .
I n l o w e s t o r d e r we o b t a i n a g a i n t h e M H F - e q u a t i o n s . C o l l e c t i n g a l l t e r m s l i n e a r i n x a n d y we o b t a i n t h e M R P A - e q u a t i o n s ,
C6-324 JOURNAL DE PHYSIQUE
w h e r e A s r k k , = *(us-Ur)(fk,-fk) !sk b r k , +
k
>
k t , s>
r ( 8 )The n o r m a l i z a t i o n o f the e i g e n v e c t o r s (F F t ) i s obtained from identi- fying Ifiwl with t h e excitation-energy o f the first excited state.
It r e a d s
2 2
c (fr-fs)
I I F ~ , I - 1 ~ 1
=~
1~ 1
( 9 )s > r
3. SUM-RULE A P P R O X I M A T I O N
A full s o l u t i o n o f t h e MRPA-equations i s a formidable task. T h e r e f o r e w e e s t i m a t e the p r o p e r t i e s o f giant-resonances i n the s u m - r u l e appro- ximation to the MRPA-equations (An application o f s u m - r u l e t e c h n i q u e s in t h e T = O - c a s e c a n b e found i n the a r t i c l e of Reinhard and F r i e d r i c h i n these p r o c e e d i n g s ; s u m - r u l e s with f i n i t e temperature have a l s o been investigated by Meyer et. al. [ 6 ] ) . Assuming that t h e giant r e s o n a n c e e x h a u s t s t h e s u m - r u l e for i t s corresponding multipole o p e r a t o r , w e c a n "solve" the MRPA-equations simply by the a n s a t z
w h e r e Q i s the m u l t i p o l e o p e r a t o r ,
P
a i [;(,PIand
X
i s t h e oscillator width. Inserting t h i s Fsr i n t o t h e MRPA-equa- tions y i e l d s finally s i m p l e traces for the widthX
and t h e energy fiu o f t h e g i a n t resonance.These t r a c e s a r e no more complicated t o e v a l u a t e than in the k T = 0 - case.
T h e exhaustion h y p o t h e s i s i s n o t perfectly satisfied. Usually t h e giant r e s o n a n c e s a r e distributed o v e r t h e s t a t e s i n an energy r a n g e A E , w h e r e w e mean with AE the fragmentation width ( n o t t h e s p r e a d i n g width and n o t the decay width). Within o u r sum-rule a p p r o a c h we will
a l s o g i v e a n e s t i m a t e f o r t h i s f r a g m e n t a t i o n w i d t h A € . To t h a t e n d w e e x p l o i t t h e f a c t , t h a t t h e s u m - r u l e a p p r o x i m a t i o n c a n a l s o b e v i e - w e d i n terms o f a s c h g m a t i c m o d e l [ 7 ] w i t h E r b e i n g t h e s i n g l e p a r t i - c l e e n e r g i e s a n d w - Q - Q b e i n g t h e r e s i d u a l i n t e r a c t i o n ; t h e a c t u a l s i z e o f w c a n b e e v a l u a t e d b y Q - a v e r a g i n g o f t h e t r u e r e s i d u a l i n t e r - a c t i o n . I n f a c t , t h e s u m - r u l e a p p r o x i m a t i o n c o r r e s p o n d s t o t h e e x - t r e m e s c h e m a t i c m o d e l w h e r e a l l p a r t i c l e - h o l e e n e r g i e s a r e t h e s a m e . I n t h e f o l l o w i n g we w i l l d e r i v e t h e e x p r e s s i o n f o r A E o n l y f o r t h e c a s e k T = O .
A t k T = O t h e M R P A - e q u a t i o n s r e d u c e t o t h e w e l l k n o w n R P A - e q u a t i o n s
I n t h e e x t r e m e s c h e m a t i c m o d e l ( e q u i v a l e n t t o t h e s u m - r u l e a p p r o x i m a - t i o n ) we i n s e r t
T h u s w e h a v e t h e s o l u t i o n f o r t h e c o l l e c t i v e m o d e
w h e r e n l a b e l s s t a t e s a b o v e t h e F e r m i s u r f a c e a n d i b e l o w . T h e o t h e r s t a t e s o f t h e e n e r g y s p e c t r u m , v d l , a r e d e g e n e r a t e a t e n e r g y E .
I n g e n e r a l t h e p a r t i c l e - h o l e e n e r g i e s w i l l n o t b e d e g e n e r a t e . We now c o n s i d e r t h e f l u c t u a t i o n a b o u t t h e m e a n v a l u e E a s p e r t u r b a t i o n ; i d e n - t i f y i n g
A
=
A. + A1,
( A l ) m i n j = ( E ~ - E ~ - E ) 6 m n 6 . .1.l ( 1 5 )
a n d B E Bo r e m a i n s a s b e f o r e . I n f i r s t o r d e r p e r t u r b a t i o n t h e o r y we o b t a i n , d e n o t i n g t h e i m p r o v e d s o l u t i o n b y '? a n d
y,
T h e f i r s t a n d s e c o n d o r d e r c o r r e c t i o n s t o t h e e n e r g y a r e 6 ~ " ) = ( V J A ~ J V )
C6-326 JOURNAL DE PHYSIQUE
N o t e t h a t 6 ~ i ' ) = 0 , b e c a u s e t h e a v e r a g e E h a s b e e n d e f i n e d s u c h t h a t C lQmi12 ( E m
-
Ei - E ) = 0 .m . i ,.
The m o m e n t s o f Q a r e d e f i n e d a s u s u a l b y
I n c a s e o f c o m p l e t e e x h a u s t i o n , o n l y v = 1 c o n t r i b u t e s a n d a l l mo- m e n t s a r e g i v e n b y k n o w i n g t w o o f t h e m . I n c a s e o f i n c o m p l e t e e x h a u s - t i o n v a r i o u s i n t e r r e l a t i o n s b e t w e e n t h e m o m e n t s c a r r y some 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 h e g i a n t r e s o n a n c e . We d e f i n e t h e f r a g - m e n t a t i o n w i d t h AE f r o m t h e moments M
.
M. a n d M,( N o t e t h a t i n c a s e o f c o m p l e t e e x h a u s t i o n AE = 0 ) . I n s e r t i n g t h e a b o v e r e s u l t s f o r t h e s c h e m a t i c m o d e l i n p e r t u r b a t i o n t h e o r y , we o b t a i n t h e r e m a r k a b l e r e s u l t
2 2 2 2
w h e r e A E = C
l ~
E~
m - Ei~
- )l /
\ Q m i \m i m i
T h i s r e l a t i o n i s t r u e u p t o s e c o n d o r d e r i n A 1 .
The e x h a u s t i o n o f t h e M - s u m - r u l e b y t h e s t a t e 1 1
>
c a n b e d e f i n e d a sI n s e r t i n g a g a i n t h e s c h e m a t i c m o d e l i n p e r t u r b a t i o n t h e o r y ( u p t o s e c o n d o r d e r ) we o b t a i n
The p e r t u r b a t i o n t h e o r y i s v a l i d i f t h e c o l l e c t i v e e n e r g y i s l a r g e c o m p a r e d t o t h e f r a g m e n t a t i o n , i . e . AE
< < IE
- E l . We w i l l u s e t h e a b o v e f o r m u l a f o r AE a n d q a l s o i n t h e c a s eA T S O ;
i n e a c h a v e r a g i n g p r o c e d u r e , o f c o u r s e t h e t h e r m a l occupation p r o b a b i l i t i e s a r e t a k e n i n t o a c c o u n t .4 . R E S U L T S FOR TEMPERATURE DEPENDENT G I A N T RESONANCES
4 . 1 G r o u n d - s t a t e e n e r g i e s
We f i r s t w a n t t o l o o k a t t h e r e l a t i o n b e t w e e n t h e r m a l e x c i t a t i o n - e n e r g y a n d t e m p e r a t u r e . I n t h e f e r m i - g a s m o d e l we h a v e [ 8 ]
( k n 2 E * = a . -
MeV
'
a = 0 . 2 5.
A ( 2 4 )w h e r e a = n u c l e a r l e v e l d e n s i t y p a r a m e t e r , A = m a s s - n u m b e r , k = B o l t z m a n n - c o n s t a n t a n d T z t e m p e r a t u r e . W i t h o u r M H F - p r o g r a m we c a l c u l a t e d t h e t h e r m a l e x c i t a t i o n - e n e r g y a s a f u n c t i o n o f t e m p e r a t u r f o r 4 0 ~ a . The r e s u l t s a r e s h o w n i n f i g . 2 t o g e t h e r w i t h t h o s e o f t h e f e r m i - g a s mo- d e l .
We s e e that the c u r v e o f the MHF-calculation has a smaller slope.
T h u s the nuclear level-density parameter a , which i s t h e s l o p e o f t h e c u r v e s in fig. 2 , i s smaller for MHF than for t h e fermi-gas model.
That i s i n good agreement with measurements [9] w h e r e the n u c l e a r l e v e l density parameter w a s found t o be particularly s m a l l for d o u b l e magic nuclei.
Fig. 2:
Thermal e x c i t a t i o n - e n e r g y E * ( T ) = E M H ~ ( T ) - EMHF(O) a s a func- tion o f ( k ~ ) ~ (dotted) c o m p a r e d t o t h e fermi- g a s f o r m u l a (24) (full) for 4 0 ~ a .
4.2 Collective enerqies:
With the aid o f eqs. (11) w e calculated the e n e r g i e s o f multipol-giant resonances in 4 0 ~ a for different temperatures and multipolarities.
-
kT (MeV)-
-
kT (MeV)-
F i.
3: Energy o f the d i f f e r e n t (LyT)-giant r e s o n a n c e s i n 4 0 ~ a a s a function o f temperature.C6-328 JOURNAL DE PHYSIQUE
In fig. 3 are s h o w n the e n e r g i e s for the isovector-monopole, dipole- and quadrupole-resonance, and the isoscalar monopole- and quadrupole- resonance. The c u r v e s for the lsovector modes r e m a i n nearly c o n s t a n t u p t o 2 MeV and then start to decrease. T h i s c a n be understood by the f a c t , that the phase-space factors i n eqs. ( 8 ) r e d u c e the residu- a l interaction with increasing temperature ( m i n d that t h e residual interaction i s repulsive for isovector modes). F o r the i s o s c a l a r mo- d e s w e s e e the expected increase i n e n e r g y ( t h e r e s i d u a l i n t e r a c t i o n i s attractive i n that case) up t o k T = 2 MeV. But then the e n e r g i e s decrease also. T h i s h a p p e n s because Ofiw-transitions become p o s s i b l e due to heating the ground-state. The w h o l e model w o r k s reasonably in 4 0 ~ a up to k T
"
5 MeV. Above that value s u b s t a n t i a l f r a c t i o n s ( 5 % for the n e u t r o n s and 12% for the protons) o f c o n t i n u u m s t a t e s b e c o m e occupied in the heated ground-state.4 . 3 F r a g m e n t a t i o n and exhaustion
In fig. 4 we s h o w A E according t o eq. (21) for the L = 0 , 1 , 2 - r e s o n a n c e in 4 O ~ a . There i s n o difference between isoscalar and i s o v e c t o r re- s o n a n c e s because A 2 € does not depend on isospin for NzZ-nuclpi. The c u r v e s a r e nearly constant up to k T = 0.5 MeV and then i n c r e a s e with t e m p e r a t u r e , u p to nearly double the value a t k T = 5 MeV. Neverthe- l e s s , the fragmentation r e m a i n s surprisingly s m a l l , in particular for the L = l mode. Above k T = 5 MeV w e o b s e r v e a decreasing AE. There are two possible explanations for that effect:
On the o n e hand one could say that at temperatures higher t h a n kTZ5MeV the c u r v e s are not significant anyway because w e d o not t r e a t the continuum s t a t e s properly. O n the other h a n d , w e k n o w t h a t w e c a n have a strong collectivity in continuum, for i n s t a n c e z e r o sound.
It i s possible that the d e c r e a s e o f the fragmentation i s a physical effect a s the r e s u l t o f a new ap- pearence o f collectivity by l o w lying t r a n s i t i o n s becoming possi- ble. But w e c a n n o t b e s u r e to s e e this effect by o u r methods.
10
fig. 4 : fragmentation w i d t h a s a function o f tem e r a t u e for t h e L=0,1 ,*-mode i n ( A E Z ~ )
fig. 5: L o s s o f exhaus- tion o f tQg c o l l e c t i v e s t a t e i n Ca a s a func- tion o f t e m p e r a t u r e
In fig. 5 w e have plotted the l o s s o f e x h a u s t i o n , 1 - v , o f t h e isovec- tor r e s o n a n c e s calculated with eq. (23). The l o s s o f e x h a u s t i o n o f the collective s t a t e increases for the ( L = l , T=1)-resonance from about 6 % to 18%, for the ( L - 0 , T-1)-resonance from about 1 3 % t o 2 7 % and for the ( L - 2 , T=l)-resonance from 19% t o 39% a s the t e m p e r a t u r e i n - c r e a s e s from 0 to k T = 5 MeV. We s e e t h e r e f o r e from A E a s w e l l a s from 1 - q , that at high temperatures w e s t i l l have c o l l e c t i v i t y for the giant resonance m o d e s (except perhaps t h e i s o s c a l a r L = O mode).
5. THE DIPOLE-GIANT-RESONANCE IN 9 0 ~ r
T h e a p p l i c a t i o n o f t h e MRPA i s by n o m e a n s restricted t o s y s t e m s i n thermal equilibrium. Indeed, w e c a n use the MRPA-equations with a r b i - trary o c c u p a t i o n - n u m b e r s for the c a l c u l a t i o n o f any e x c i t e d state.
A s a n e x a m p l e w e study 9 0 z r , where Szeflinski et. al. [lo] determined experimentally the energy dependence o f the y - s t r e n g t h f u n c t i o n o f the first excited s t a t e (0+, 1.762 MeV).
The result confirmed the Brink-Hypothesis w h i c h s t a t e s that i d e n t i c a l giant r e s o n a n c e s c a n be built upon each excited state. W i t h i n o u r approach w e calculated t h e energy o f the dipole g i a n t r e s o n a n c e built upon the ground-state and the first 0+-excited s t a t e i n 9 0 ~ r a s w e l l a s the total e n e r g i e s EMHF(O+; g.s.1 and EMHF(O+; 1.762 MeV).
D u e to the residual interaction the true ground s t a t e o f 9 0 ~ r i s no longer a pure shell model state with t w o protons i n c o n f i g u r a t i o n (2p1/2)2 on top o f t h e Sr-core; the proton c o n f i g u r a t i o n i s a linear combination o f ( 2 p 1 / 2 ) ~ and (lg9/2)2. T h e o r t h o g o n a l c o m b i n a t i o n o f these two c o n f i g u r a t i o n s describes the 1.762 MeV-state:
The c o e f f i c i e n t s w e r e estimated t o be: a 2 : 0.72, b2 = 0.28 [lo].
Inserting the corresponding f~ i n the MHF and MRPA e q u a t i o n s w e get the excitation energy o f the dipole giant resonance ( i n s u m - r u l e
C6-330 JOURNAL DE PHYSIQUE
approximation) and the t o t a l e n e r g i e s o f the ground-state and the excited state. The results are given in the table 1:
Table 1: Comparison o f the MHF- and MRPA-results with experimental r e s u l t s from [lo]
I I
I t E(L=l,T=l)/MeV 1
total energy/MeV
theory lexperimentl theorylexperiment
For these c a l c u l a t i o n s w e used a newly fitted S k y r m e f o r c e which a i m s at a good description o f the isovector dipole modes.
(1) o+-ground-state (2) first excited state difference (2) - ( 1 )
From table 1 w e s e e that the resonance energy r e p r o d u c e s w e l l t h e experimental value and that the r e s o n a n c e energy on top o f the exci- ted s t a t e is shifted upwards just by the excitation energy o f t h i s state. F o r that c a s e , w e s e e the Brink h y p o t h e s i s being confirmed very well both experimentally and theoretically.
C O N C L U S I O N S
16.94 17.03 0.09
In order to study t h e c o l l e c t i v e e x c i t a t i o n s o f a n u c l e u s w h i c h i s not necessarily i n i t s ground-state, w e derived t h e modified Hartree- Fock (MHF) and modified RPA '(MRPA) equations. T h e emerging e q u a t i o n s are an o b v i o u s generalisation o f t h e HF-and RPA-equations, modified by occupation probabilities and phase-space factors. From t h e tem- perature-dependence o f the MHF e n e r g i e s one c a n d e d u c e a l e v e l d e n s i - ty parameter; i t c o m e s o u t to be lower than that o f t h e F e r m i - g a s model and t h u s r e p r o d u c e s better the experimental values. Within a sum-rule approximation t o MRPA w e calculated properties o f giant re- s o n a n c e s in 4 0 ~ a . T h e resonance e n e r g i e s for the isovector modes de- c r e a s e a s function o f the temperature w h e r e a s for the isoscalar m o d e s i t first increases, up t o kT = Z M e V , and then d e c r e a s e s also. These features can be understood by the fact that for the i s o s c a l a r m o d e s there a r e two counteracting effects: the residual interaction d e - c r e a s e s with temperature (thus increasing the energy) w h e r e a s the average lph-energy decreases because the phase s p a c e o p e n s for l o w - energy excitations. The fragmentation w i d t h s increase with tempera- ture up to kT = 5 M e V , but they remain small t h r o u g h o u t , i n particu- lar for the i s o v e c t o r L = 1 mode. The percentage o f e x h a u s t i o n of the sum-rule d e c r e a s e s with temperature, but r e m a i n s fairly high even at k T = 5 MeV. All the r e s u l t s show the remarkable fact t h a t c o l l e c - tivity d o e s not disappear i f a nucleus i s excited t h e r m a l l y ; a t l e a s t not up t o kT = 5 MeV. Above k T = 5 MeV the r e s u l t s are n o t r e l i a b l e , s i n c e then a s u b s t a n t i a l fraction o f particles i s i n c o n t i n u u m s t a t e s (and these are not handled properly in our treatment). For a more
16.710.1 17.010.3
0.3 1.23
c o m p l e t e d e s c r i p t i o n o n e h a s t o s t u d y t h e t e m p e r a t u r e d e p e n d e n c e o f t h e s p r e a d i n g - a n d t h e e s c a p e - w i d t h .
To show t h e u n i v e r s a l i t y o f t h e MRPA, we c a l c u l a t e d t h e L r l - r e s o n a n c e s o n t o p o f t h e g r o u n d - s t a t e a n d o f t h e f i r s t e x c i t e d 0 + - s t a t e i n 9 0 ~ r . We f o u n d g o o d a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s .
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