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MULTIFRAGMENTATION OF NUCLEI
C. Ngô, R. Boisgard, J. Desbois, J. Nemeth, M. Barranco
To cite this version:
C. Ngô, R. Boisgard, J. Desbois, J. Nemeth, M. Barranco. MULTIFRAGMENTATION OF NUCLEI.
Journal de Physique Colloques, 1987, 48 (C2), pp.C2-157-C2-163. �10.1051/jphyscol:1987223�. �jpa- 00226489�
JOURNAL D E PHYSIQUE
Colloque C2, supplkment au n o 6 , Tome 48, juin 1987
MULTIFRAGMENTATION OF NUCLEI
C. N G ~ , R. BOISGARD, J.DESBOIS* , J. N E M E T H * * and M. B A R R A N C O * " * Service de Physique NuclBaire-MBtrologie Fondamentale.
CEN-Saclay, F-91191 Gif-sur-Yvette Cedex, France
" ~ n s t i t u t de Physique ~ u c l B a i r e ( ~ ) , Division de Physique ThBorique, F-91406 Orsay Cedex, France
" " ~ n s t f t u t e for Theoretical Physics, E6tvos UniversitSt, Budapest, Hungary
* * * Universittit de Barcelona, Facultat de Fisica, SP-08028 Barcelona, Spain
R6sum6 - Nous p r b e n t o n s un modgle base s u r l'hydrodynamique i r r o t a t i o n e l l e e t
~ c o l a o l a n q u i permet de d e c r i r e l e ph6nomGne de mu1 t i f r a g m e n t a t i o n obser- v6 dans l e s c o l l i s i o n s e n t r e deux noyaux lourds. Nous c a l c u l o n s l e s c o n d i t i o n s q u i s o n t ngcessaires pour que ce mgcanisme a i l i e u .
A b s t r a c t - We review a model based on i r r o t a t i o n a l hydrodynamics and percola- t i o n which has been a p p l i e d t o t h e study o f h o t and e x c i t e d nucleus disassem- b l y . The d i f f e r e n t c o n d i t i o n s under which an e x c i t e d nucleus breaks up i n several pieces (mu1 ti fragmentation) are given.
When a h i g h energy p r o t o n ( 2 2 GeV) goes through a nucleus i t can break i t up. A s i n i - l a r s i t u a t i o n i s observed w i t h heavy i o n s a t lower bombarding energies per nucleon ( 3 3 0 - 5 0 MeV/u). The process i n which a nucleus breaks up i n several pieces i s c a l l e d m u l t i f r a g m e n t a t i o n . There a r e two t h i n g s one would l i k e t o kndw :
1- why do n u c l e i break up ? 2- how do they break up ?
Apart from emulsion data [l], which show d i r e c t l y p i c t u r e s where a nucleus undergoes m u l t i f r a g m e n t a t i o n , most o f t h e experiments performed so f a r a r e very i n c l u s i v e . One o f t h e i m p o r t a n t observable i s t h e mass d i s t r i b u t i o n o f t h e m u l t i f r a g m e n t a t i o n pro- d u c t s w h i c h f o l l o w s a A-% l a w [2], where A i s t h e fragment mass and 7 an exponent which takes values around 2-3. Several model S [ 3 ] t r y t o d e s c r i b e mu1 t i f ragmentation b u t a f u l l microscopic theory i s n o t y e t a v a i l a b l e . The reason i s t h a t t h e problem i s d i f f i c u l t because one has t o go beyond a mean f i e l d approach. The c o r r e l a t i o n s between several nucleons become i m p o r t a n t and one u s u a l l y r e f e r s t o these as t h e f l u c t u a t i o n s o f t h e mean f i e l d . Most o f t h e models s t a r t w i t h an i n i t i a l e x c i t e d nucleus, i n g l o b a l s t a t i s t i c a l e q u i l i b r i u m , which i s assumed t o be formed i n t h e f i r s t stage o f t h e c o l l i s i o n . These models a r e u s u a l l y a b l e t o reproduce t h e i n c l u - s i v e mass d i s t r i b u t i o n o f t h e m u l t i f r a g m e n t a t i o n products. I n t h i s c o n t r i b u t i o n we s h a l l describe a model which belongs t o t h i s category. It i s based on hydrodynamics t o describe mean f i e l d e f f e c t s , and on a p e r c o l a t i o n approach t o e v a l u a t e t h e f l u c - t u a t i o n s o f t h e mean f i e l d [4]. We have b u i l t i t as simple as p o s s i b l e i n order t o t r y t o p i n down t h e p h y s i c a l e f f e c t s which may be r e s p o n s i b l e o f m u l t i f r a g m e n t a t i o n o f n u c l e i . Indeed, i f one can d e s c r i b e t h i s process t o a l a r g e e x t e n t w i t h minimal assumptions i t would probably i n d i c a t e t h a t they have t o be i n c l u d e d i n any more s o p h i s t i c a t e d model which aim t o d e s c r i b e mu1 t i f r a g m e n t a t i o n .
(''~aboratoire associb au C.N.R.S.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987223
C2-158 JOURNAL DE PHYSIQUE
Although one can address t h i s problem by s i m i l a r methods, we s h a l l n o t consider h e r e p r o t o n induced mu1 t i f ragmentation. Indeed t h i S process i S more r e 1 a t e d t o a cascade o f t h e p r o t o n i n s i d e t h e t a r g e t nucleus, which leads t o an e j e c t i o n o f nucleons and f i n a l l y t o a break up o f t h e t a r g e t i n t o several pieces. We s h a l l r a t h e r r e s t r i c t ourselves t o t h e simple f o l l o w i n g problem which i s i m p o r t a n t f o r heavy i o n c o l l i - sions a t i n t e r m e d i a t e bombarding energies, namely : what happens t o an e x c i t e d and compressed nucleus i n g l o b a l s t a t i s t i c a l e q u i l i b r i u m ?
Central and medium impact parameters heavy i o n c o l l i s i o n a t i n t e r m e d i a t e bombarding energies can be r o u g h l y decomposed i n two main phases :
- I n t h e f i r s t one, t h e p r o j e c t i l e and t a r g e t i n t e r a c t very deeply. T h e i r r e l a t i v e v e l o c i t y suddenly decreases and a l o t o f e x c i t a t i o n energy i s deposited i n t o t h e i r common p a r t . S t a t i s t i c a l e q u i l i b r i u m i s n o t achieved i n e a r l y stages and e n e r g e t i c p a r t i c l e s a r e emitted. A p r e e q u i l i b r i u m mode developed i n [ r e f . 5 ] shows t h a t one needs about 10-22s t o reach g l o b a l thermal e q u i l i b r i u m . Furthermore, we have p e r f o r - med [6], f o r head-on c o l l i s i o n s , a hydrodynamical c a l c u l a t i o n based on a t h r e e f l u i d component model : t h e p a r t i c i p a n t s and t h e two s p e c t a t o r s o f t h e p r o j e c t i l e and t a r g e t . F o r t h e Ca + Pb system a t 52 MeV/u, f o r instance, we found t h a t i n any r e - g i o n o f t h e system, t h e d e n s i t y never exceds 20 % o f t h e c e n t r a l ground s t a t e densi- t y value. Consequently, i n heavy i o n c o l l i sions a t i n t e r m e d i a t e bombarding energies i t i s o n l y p o s s i b l e t o o b t a i n moderate compressions.
- I n the second phase t h e e x c i t e d nucleus, which i s formed previously, expands and c o o l s down. Two s i t u a t i o n s can occur depending on t h e amount o f e x c i t a t i o n energy o f t h e nucleus. There can be a succession of o s c i l l a t i o n s around a mean c o n f i g u r a t i o n w h i l e t h e system evaporates p a r t i c l e s . I n t h i s case t h e nucleus de-excite normally by p a r t i c l e emission a n d l o r f i s s i o n . T h i s occurs i f t h e e x c i t a t i o n energy i s n o t t o o large. Above a c e r t a i n e x c i t a t i o n energy value t h e nucleus expands so much t h a t t h e mean f i e l d f l u c t u a t i o n s become suddenly very large. I n such a s i t u a t i o n , which oc- c u r s a t l a r g e e x c i t a t i o n energies, t h e r e i s a m u l t i f r a g m e n t a t i o n o f t h e nucleus. It i s t h i s second phase o f a heavy i o n c o l l i s i o n t h a t we want t o describe w i t h i n a schematic p i c t u r e .
1 - F l u c t u a t i o n s o f t h e mean f i e l d
The i n t e r a c t i o n between two nucleons has a s h o r t r e p u l s i v e range and a l o n g a t t r a c - t i v e one. However, i n t o t a l , nuclear forces a r e o f s h o r t range compared t o t h e Cou- lomb forces. I n o r d e r t o understand t h e importance o f nucleon c o r r e l a t i o n s i n n u c l e i l e t us consider a volume V i n which we p u t A nucleons, neutrons and protons. I f V i s o f t h e o r d e r o f t h e v01 ume o f o r d i n a r y n u c l e i , each nucleon can i n t e r a c t by n u c l e a r f o r c e s w i t h a l l i t s neighbours. The physics o f t h i s system can be described, t o a l a r g e extent, by t h e mean f i e l d c r e a t e d by t h e whole s e t o f nucleons.
If t h e volume i s bigger, each nucleon can no l o n g e r i n t e r a c t by n u c l e a r f o r c e s w i t h a l l i t s neighbours because o f t h e s h o r t range o f t h i s i n t e r a c t i o n . C l u s t e r o f nu- cleons w i l l be formed and t h e physics o f each o f them i s dominated by t h e i r own mean f i e l d . However, t h e physics o f t h e whole system i s no l o n g e r dominated by t h e mean f i e l d o f t h e t o t a l system because t h e Coulomb f i e l d , which has a l o n g range, w i l l push t h e c l u s t e r s a p a r t from each other. T h i s w i l l l e a d t o a break-up o f t h e i n i t i a l system (mu1 i fragmentation).
I n t h e f i r s t s i t u a t i o n , where t h e physics o f t h e system i s dominated by t h e mean f i e l d o f t h e whole nucleus, one says t h a t t h e f l u c t u a t i o n s o f t h e mean f i e l d a r e small while, i n t h e second case, one says t h a t they a r e large. When t h e f l u c t u a t i o n s are large, i n s t a b i l i t i e s develops exponential l y and t h e system undergoes mu1 t i f r a g - mentation. When they a r e small, c l u s t e r s m i g h t be present i n t h e n u c l e a r medium b u t t h e i r p r o b a b i l i t y i s small. The r e s t o f t h e nucleus can then be viewed a a b i g c l u s - t e r whose mean f i e l d i s p r a c t i c a l l y i d e n t i c a l t o t h e one o f t h e t o t a l system. The p r o p o r t i o n o f small c l u s t e r s , which gives an o r d e r o f magnitude o f t h e f l u c t u a t i o n s , i s small.
The d i f f e r e n c e between small and l a r g e mean f i e l d f l u c t u a t i o n s t e l l s us about t h e c o r r e l a t i o n d i s t a n c e i n t h e n u c l e a r medium ( t h e c o r r e l a t i o n d i s t a n c e i s d e f i n e d here i n a d i f f e r e n t way from t h e c o r r e l a t i o n l e n g t h used i n p e r c o l a t i o n theory : one i n c l u d e s a1 l c l u s t e r s i n c l u d i n g t h e l a r g e s t one) : when they a r e small, t h e c o r r e l a - t i o n d i s t a n c e i s l a r g e and t h e i n f o r m a t i o n about t h e n u c l e a r i n t e r a c t i o n can propa- g a t e through t h e whole nucleus. On t h e contrary, i f they a r e l a r g e , t h e c o r r e l a t i o n d i s t a n c e suddenly decreases and t h e i n f o r m a t i o n about t h e n u c l e a r f o r c e s i s trapped w i t h i n c l u s t e r s . C l u s t e r s can " t a l k " t o o t h e r c l u s t e r s o n l y through t h e Coulomb
f i e l d which has a tendency t o p u l l them apart.
I n s t e a d o f t r e a t i n g a s i n g l e nucleus and f o l l o w i n g t h e nucleons which move i n s i d e o f it, one can a l t e r n a t i v e l y consider, as i n s t a t i s t i c a l physics, an ensemble o f seve- r a l n u c l e i w i t h f i x e d nucleons d i s t r i b u t e d i n such a way t h a t t h e i r g l o b a l proper- t i e s a r e i d e n t i c a l t o t h e one we study. Then one has t o p o s t u l a t e t h a t t h e time average o f macroscopic q u a n t i t i e s o f t h e f i r s t system a r e equal t o t h e i r ensemble average ( e r g o d i c hypothesis). I f one does t h i s , one can n o t i c e t h a t t h e e v a l u a t i o n o f t h e f l u c t u a t i o n s i s i d e n t i c a l t o a p e r c o l a t i o n problem i n which p a r t i c l e s a r e connected t o t h e i r c l o s e neighbours.
2 - P e r c o l a t i o n
P e r c o l a t i o n i s one o f t h e s i m p l e s t s t a t i s t i c a l approach which a l l o w s t o describe c r i t i c a l phenomena i n v a r i o u s areas. I t has been a p p l i e d t o several problems i n s o l i d s t a t e physics l i k e b i n a r y a l l o y s , s o l - g e l t r a n s i t i o n s , complex r e s i s t o r net- works, p o l y m e r i s a t i o n r e a c t i o n s , etc... It has a l s o been used i n a g r e a t v a r i e t y o f o t h e r f i e l d s l i k e t h e extension o f f o r e s t f i r e s o r t h e propagation o f diseases.
A p e r c o l a t i o n system i s e s s e n t i a l l y d e f i n e d by two i n g r e d i e n t s : a c o l l e c t i o n o f p o i n t s and a r u l e t o connect them. The, p e r c o l a t i o n model t h a t we have used t o evalu- a t e t h e f l u c t u a t i o n s o f t h e mean f i e l d i s a site-bond p e r c o l a t i o n model [4,7] based on a cubic l a t t i c e . The two b a s i c parameters o f t h i s approach a r e : i ) p, t h e r a t i o between t h e number o f occupied s i t e s and t h e t o t a l number o f s i t e s .
ii q, t h e r a t i o o f t h e number o f bonds l e f t , t o t h e i n i t i a l number o f bonds avai- l a b l e t o t h e nucleus i n i t s ground state.
F o r d i f f e r e n t values o f p and q t h i s model leads t o two s i t u a t i o n s which a r e i l l u s - t r a t e d i n f i g . 1 :
a r e l a r g e and t h e nucleus t o which one a p p l i e s t h i s model breaks up (mu1 t i f r a g - mentation).
0.0
l
The t r a n s i t i o n between r e g i o n s I and I1 0.0 0.5 1 .o
occurs smoothly here because we have a P
f i n i t e system. For an i n f i n i t e system t h e
t r a n s i t i o n i s very sharp. I t has been F i g u r e 1 checked t h a t t h e r e s u l t s obtained w i t h a
c u b i c l a t t i c e a r e n o t t o o d i f f e r e n t from those obtained i n a continuous medium per- c o l a t i o n . As a m a t t e r o f f a c t t h e approximation i s analogous t o t h e f i n i t e d i f f e r -
- i f p and q a r e c l o s e t o u n i t y ( r e g i o n 1.0 I ) one always g e t a b i g fragment c a l l e d p e r c o l a t i o n c l u s t e r , and small c l u s t e r s .
i s correspon S o t h e case where t h e ::uctuations ofd th: mean f i e l d a r e small.
Then, a mean f i e l d t h e o r y can describe t h e dynamical e v o l u t i o n o f t h e system.
- if p and q belong t o r e g i o n 11 t h e r e i s 0.5 no l o n g e r a p e r c o l a t i o n c l u s t e r b u t seve- r a l small o r medium s i z e c l u s t e r s . I n t h i s case t h e f l u c t u a t i o n s o f t h e mean f i e l d
-
Multifraqmentation reqion
C2-160 JOURNAL DE PHYSIQUE
ence methods used i n numerical a n a l y s i s where one a l s o works on a l a t t i c e . We s h a l l see below how one r e l a t e s t h e two parameters p and q t o t h e p r o p e r t i e s o f t h e nucle- us.
3 - The i s e n t r o p i c expansion model
Before going i n t o a more i n v o l v e d model i t i s u s e f u l t o study w i t h i n t h e framework o f an elementary model what happens t o an i n i t i a l nucleus which i s suddenly heated up w h i l e keeping i t s d e n s i t y p r o f i l e f i x e d . I n order t o do t h i s we have modeled a nucleus by a l i q u i d drop w i t h c o n s t a n t d e n s i t y and sharp edges. The energy o f t h i s drop has been c a l c u l a t e d u s i n g t h e energy d e n s i t y formal ism which a l l o w s t h e energy t o be c a l c u l a t e d even f o r non e q u i l i b r i u m c o n f i g u r a t i o n s . It i s reasonable and cascade c a l c u l a t i o n s c o n f i r m i t [8], t o assume t h a t t h e e v o l u t i o n o f t h e nucleus w i l l f o l l o w an isentrope. Then, as proposed by Cugnon [ g ] , one can f i n d t h e equation o f motion o f t h e d e n s i t y by u s i n g t h e c o n t i n u i t y and energy conservation equations o f t h e f l u i d only.
I f we heat up suddenly a nucleus w h i l e keeping t h e d e n s i t y f i x e d , and l e t evolve t h i s system, i t w i l l expand. The process we have i n mind i s s c h e m a t i c a l l y d i s p l a y e d i n fig.2 : as t h e nucleus expands t h e number o f nucleons remains p r a c t i c a l l y constant, a p a r t from those few e m i t t e d by evaporation o r p r e e q u i l i b r i u m emission.
However, t h e packing no l o n g e r remains c o n s t a n t and empty c e l l S become a v a i l able.
Therefore, t h e r a t i o between t h e number o f occupied c e l l s and t h e t o t a l number o f c e l l s decreases n o t because nucleons a r e emitted, b u t because new s i t e s a r e created.
I f t h e expansion i s s u f f i c i e n t , several small and medium s i z e c l u s t e r s a r e formed : one has a mu1 t i f r a g m e n t a t i o n o f t h e system.
F i g u r e 2
I n o r d e r t o use t h e p e r c o l a t i o n approach, one has t o d e f i n e p and q i n terms o f q u a n t i t i e s associated t o the expanding nucleus. We have made t h e f o l l o w i n g choices
( t )
1 - p ( t ) = , where p. i s t h e d e n s i t y a t t i m e t = 0 and p ( t ) i s t h e same q u a n t i t y P 0
a t t i m e t- A t t = 0 one has p = 1. As t h e nucleus expands p decreases since, accor- d i n g t o o u r p i c t u r e , new s i t e s a r e created.
2 - The parameter q ( t ) i s r e l a t e d t o t h e s t r e n g t h o f t h e bonding between nucleons.
For a nucleus w i t h zero thermal e x c i t a t i o n energy q = 1. I n c r e a s i n g 'the e x c i t a t i o n e n e r g y p e r n u c l e o n , E$, decreases t h e s t r e n g t h o f t h e bonds which vanishes comple- t e l y when E * becomes e q i a l t o t h e b i n d i n g energy. Therefore, we assume t h a t : q ( t ) =
T
E *
1- 2, where B i s t h e b i n d i n g energy per nucleon a t time t = 0.
B
p ( t ) and q ( t ) being defined, one can now e v a l u a t e t h e f l u c t u a t i o n s o f t h e mean f i e l d a t each stage o f t h e expansion [10]. As soon as they become l a r g e one stops t h e dynamical model, because the nucleus breaks up, and c a l c u l a t e t h e p r o p e r t i e s o f t h e fragments u s i n g t h e p e r c o l a t i o n model.
The above model, which i s very simple, has been s t u d i e d i n d e t a i l s i n r e f . [ l O ] . For a l l t h e r e s u l t s i t was p o s s i b l e t o o b t a i n e i t h e r a n a l y t i c a l o r f i t t e d expressions which can be used very e a s i l y . L e t us quote two i m p o r t a n t r e s u l t s :
1 - the c r i t i c a l energy value, above which a nucleus undergoes m u l t i f r a g m e n t a t i o n , i s equal t o 70 % o f t h e b i n d i n g energy o f t h e ground state.
2 - the mass d i s t r i b u t i o n o f t h e products, a t t h e p o i n t where t h e " s u s c e p t i b i l i t y "
o f t h e medium i s maximum, goes l i k e A - ~ ' ~ . I f one r e q u i r e s , as i n ref.[4], t h a t mu1 t i f r a g m e n t a t i on i s d e f i n e d when t h e mass o f t h e p e r c o l a t i o n c l u s t e r i s s m a l l e r than ha1 f o f t h e i n i t i a l nucleus, t h e exponent -C o f A ' i s a b i t l a r g e r .
4 - The t i m e dependent Thomas Fermi approach.
It i s now i n t e r e s t i n g t o study t h e same problem w i t h i n a more r e a l i s t i c approach. I n r e f . [ l l ] a time dependent Thomas-Fermi model has been proposed t o study t h e evolu- t i o n o f h o t and compressed s p h e r i c a l n u c l e i . T h i s model i s e s s e n t i a l l y an i r r o t a - t i o n a l hydrodynamical approach i n which one evaluates t h e i n t e r n a l energy w i t h t h e Thomas-Fermi approximation u s i n g a Skyrme force. Thi S d e s c r i p t i o n i S assumed t o g i v e a reasonable d e s c r i p t i o n of the expansion o f t h e nucleus and has been checked by c a l c u l a t i n g t h e p r o p e r t i e s o f t h e i s o s c a l a r g i a n t monopol e resonance [12]. Since one can vary independently t h e thermal and compressional e x c i t a t i o n energies, one can compare t h e i r i n f l u e n c e on t h e s t a b i l i t y o f n u c l e i towards mu1 t i f r a g m e n t a t i o n . As i t has been done above, one evaluates t h e f l u c t u a t i o n s o f t h e mean f i e l d a t each stage o f t h e dynamical e v o l u t i o n o f t h e system u s i n g t h e p e r c o l a t i o n model described i n s e c t i o n 2. The two parameters p ( t ) and q ( t ) a r e now d e f i n e d as :
p ( t ) = 522- and q ( t ) = l - - €7
<PO' B(t=O)
where <p> and < p > a r e t h e average d e n s i t y a t t i m e t and t = 0 r e s p e c t i v e l y ; and B ( t - 0 ) i s t h e b i n s i n g energy p e r nucleon o f t h e nucleus a t t i m e t = 0. L e t us now discuss a few r e s u l t s :
3 I I I I l I l I 1 I
I
Multifraqmentation
Normal de-excitation
F i g u r e 3
- I n f i g . 3 we d i s p l a y a k i n d o f phase diagram, f o r 208Pb, showing t h e r e g i o n of m u l t i f r a g m e n t a t i o n i n s t a b i l i t y as a f u n c t i o n o f E * ~ , t h e thermal e x c i t a t i o n energy p e r nucleon, and E * ~ , t h e compressional e x c i t a t i o n energy p e r nucleon. I f 0 one needs E * ~ = 1.5 MeV only. T h i s shows t h a t , f o r t h e same amount o f e x c i t a t i o n energy, one breaks up a nucleus more e a s i l y by compression than by thermal e x c i t a t i o n .
C2-162 JOURNAL DE PHYSIQUE
- F o r two v a l u e s o f EC we have c a l c u l a t e d t h e r a t i o between t h e e x c i t a t i o n energy per nucleon ( E * = ET + ~ e ) , and t h e b i n d i n g energy p e r nucleon ,B, o f t h e nu'clei i n t h e i r ground state, above which a nucleus undergoes m u l t i f r a g m e n t a t i o n . ( f i g . 4 ) . I f t h e r e i s no c o m p r e s s i o n one observes t h a t t h e maximum ET value t h a t a nucleus can s u s t a i n w i t h o u t b r e a k i n g up i s equal t o 70 % o f B. T h i s r e s u l t s i s t h e same as t h e one deduced i n t h e p r e v i o u s section.
F i g u r e 4
- A t t h e p o i n t where a nucleus breaks up i t i s i n t e r e s t i n g t o n o t e t h a t t h e nucleus has n o t expanded t o o much. T h i s i s i l l u s t r a t e d f o r a t y p i c a l example i n f i g . 5 which shows t h e i n i t i a l d e n s i t y p r o f i l e ( f u l l l i n e ) and t h e one where mu1 t i f r a g m e n t a t i o n takes p l a c e (dashed l i n e ) . One sees t h a t , i n t h i s l a t t e r case, t h e c e n t r a l d e n s i t y i s s t i l l 0.1 nucleon/fm3. The mean square r a d i u s <r2>"2 has changed from = 4.94 fm, f o r t h e i n i t i a l d e n s i t y t o 6.2 fm a t t h e p o i n t o f i n s t a b i l i t y . F o r t h i s p a r t i c u l a r example t h e volume o f t h e system has doubled.
F i g u r e 5
Conclusion
L e t us now summarize t h e m a j o r - r e s u l t s o f our approach t o t h e disassembly o f h o t and e x c i t e d n u c l e i .
- M u l t i f r a g m e n t a t i o n i s a r e s u l t o f t h e f l u c t u a t i o n s o f t h e mean f i e l d which become l a r g e .
- Compression i s more e f f i c i e n t t o break n u c l e i than thermal e x c i t a t i o n .
- F o r a system a t normal d e n s i t i e s one needs a thermal e x c i t a t i o n l a r g e r than 70 % o f t h e b i n d i n g energy i n o r d e r t h a t i t breaks up.
- A t t h e i n s t a b i l i t y p o i n t , t h e d e n s i t y i s o f t h e o r d e r o f 0.1 nucleon/fm3 i n t h e case o f t h e *@Pb nucleus.
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