SEMI-CLASSICAL DESCRIPTION OF HOT ZONE DECAY

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SEMI-CLASSICAL DESCRIPTION OF HOT ZONE DECAY

J. De, C. Gregoire

To cite this version:

J. De, C. Gregoire. SEMI-CLASSICAL DESCRIPTION OF HOT ZONE DECAY. Journal de

Physique Colloques, 1987, 48 (C2), pp.C2-211-C2-214. �10.1051/jphyscol:1987230�. �jpa-00226497�

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SEMI-CLASSICAL DESCRIPTION OF HOT ZONE DECAY

J.N. DE(' ) and C. GREGOIRE

GANIL, BP. 5027, F-14021 Caen Cedex, France

ABSTRACT

The decay of a sharp-surface hot zone, i f formed i n nuclear c o l l i s i o n s , i s studied in c l a s s i c a l energy-transport model and i n the semi-classical VUU frame- work. Convection i s dominant. Global thermal ization takes a r e l a t i v e l y long time, a few ^10-21sec. Being a t variance w i t h recent experimental data, t h i s feature c a s t s doubt on sharp hot-zone formation as an intermediate s t a t e f o r excitation in f i n i t e nuclei in nuclear col1 isions.

Phenornenological hot-zone models [ l - 3 1 , where i t is assumed t h a t a locally thermalized hot source nay be created in the overlap region in the early stages of intermediate energy nuclear c o l l i s i o n s have been q u i t e successful in explaining several experimental features. What i s not so c l e a r i s how i t i s formed and even i t i s formed, how does i t share i t s energy w i t h the surrounding cold nuclear matter.

Emission spectra of l i g h t p a r t i c l e s from high1 y excited f i s s i l e nuclear systems show c h a r a c t e r i s t i c s pointing t o a very rapid thermalization i n the e n t i r e system [ 4 ] and therefore i f energy sharing in nuclear c o l l i s i o n s proceeds through an i n - termedi a t e hot-zone formation, the hot zone must decay and eouil i b r a t e i t s energ.y w i t h i t s surroundings very f a s t . To gain c l e a r insight into reaction mechanism, we therefore assume the formation of d sharp hot-zone in space and time and then study the response of the e n t i r e system t o i t s decay, i t s . energy relaxation modes w i t h t h e surrounding matter w i t h subsequent comparison with relevant experimental infor- mation. Collision dynamics i s switched off from the beginning t o define more clear- l y the energy transport from the hot zone, which i s treated f i r s t i n a crude clas- s i c a l model and then in the more sophisticated V1 asov-Uehl ing-Uhl enbeck approach [5]. The Coulomb interaction i s also neglected t o single out unambiguous nuclear e f f e c t s .

The hot zone geometry i s shown i n figure 1, w i t h a temperature To and density

p , assumed normal. \lucleons trans-

ported in and out of t h i s zone cool i t . Surrounding nuclear mat- t e r , in turn i s warmed u p due t o c o l l i s i o n s with the "hot" nucleons. The r a t e of excitation of the cold zone (a cold zone i s defined as the zone between two spherical surfaces concentric with the hot-zone surface) i s

Figure 1 : The hot zone geometry. The crossed region defines the cold zone.

(''permanent address : VEC-Centre. C a l c u t t a . I n d i a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987230

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JOURNAL DE PHYSIQUE

given by

where S2 i s t h e h o t zone s u r f a c e f a c i n g t h e c o l d zone, VF and ^EF a r e the Fermi v e l o c i t y and energy a t zero temperature, T i s t h e instantaneous temperature o f t h e h o t zone and ,H r e f e r s t o averaging w i t h f i n i t e temperature energy d i s t r i b u t i o n i n t h e h o t zone. Equation ( 1 ) i s a r r i v e d a t w i t h t h r e e assumptions, n e g l e c t o f i) evaporation from h o t zone surface ii ) r e r a d i a t i o n by adjacent m a t t e r o r by it- s e l f , i i i ) r e f l e c t i o n ' from surface. The r a t e o f energy l o s s from t h e h o t zone i s given by, l i k e i n e q n . ( l ) :

where S1 i s the o t h e r surface o f t h e lens-shaped h o t zone. Assuming E* = aHT2, from (21, one gets :

T = T e-Kt

0 ( 3 )

where K = B/(2aH).

I f t h e r e a r e

-

⁸nucleons i n t h i s zone, t h e r e l a x a t i o n time i s

-

²⁰fm/c. The heat energy i n t h e c o l d zone i s c a l c u l a t e d w i t h the assumption t h a t energy absorbed due t o removal o f h o t nucleon f l u x from t h i s zone as a r e s u l t o f two-body c o l l i s i o n s i s f u l l y thermal. It i s given as

F i g u r e 2 : The temperatures o f two c o l d zones l o c a t e d a t 1 and 2 fm from t h e h o t zone s u r f a c e as a f u n c t i o n o f time, i n a 20Me

+

20Ne system f o r two d i f f e r e n t mean f r e e paths.

where h i s t h e nucleon mean f r e e path, ~ d i s t h e thickness o f the c o l d zone, d i s t h e normal distance o f t h i s zone from t h e h o t zone surface and t' i s t h e time taken by nucleon f l u x t o cross t h i s normal distance. The temperature i n t h e c o l d zone can be c a l c u l a t e d assuming another l e v e l d e n s i t y parameter ( t a k e n t o be Ac/lO where Ac

i s the number o f nucleons i n t h i s zone). The temperatures o f two

6.0.- G L O W

c o n c e n t r i c c o l d zones each o f 1 fm thickness and s e t a p a r t from t h e h o t surface by 1 and 2 fm as a f u n c t i o n o f time are shown i n f i g u - r e 2 f o r 20Ne + 20Ne w i t h a h o t

HOT ZONE 15MeV h = 3 f m

--W

---l

h = 5 f m I .

z o n e t e m p e r a t u r e o f To = 15 MeV i n v o l v i n g

-

5 p a r t i c l e s , f o r h = 3 fm and 5 fm. The temperatures satu- r a t e a f t e r a time i n t e r v a l

-

²⁰

fm/c. Because o f t h e s i m p l i f y i n g assumptions, the s a t u r a t i o n tempe- r a t u r e s are n o t independent o f h.

The t i m e - l a g f o r t h e

'

second zone a r i s e s due t o t h e f i n i t e time i t takes f o r t h e n u c l e a r s i g n a l t o reach there.

' 20 L0 GO 00

TIME(fm/c)

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s o l v i n g t h e s e l f - c o n s i s t e n t Hartree-Fock hamil t o n i a n ; i n t h e o v e r l a p r e g i o n , t h e momentum d i s t r i b u t i o n i s m o d i f i e d a c c o r d i n g t o

f(P,

P)

+ ,,(P) l

l + exp

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w h e r e p, ( f ) i s t h e d e n s i t y i n t h e o v e r l a p r e g i o n , ^E= E(?, if) i s t h e s i n g l e - par- t i c l e e n e r g y , T i s t h e h o t zone temperature and ^pi s t h e chemical p o t e n t i a l , t a k e n c o n s t a n t , b e i n g determined f r o m t h e T = 0 s e l f - c o n s i s t e n t s o l u t i o n . The system then e v o l v e s from t h i s non e q u i l i b r i u m s i t u a t i o n a c c o r d i n g t o VUU e q u a t i o n s

Here f ( f , if) i s t h e one-body d i s t r i b u t i o n f u n c t i o n , Il = I l ( p ( f ) ) i s t h e s e l f c o n s i s - t e n t s i n g l e - p a r t i c l e p o t e n t i a l , Icoll i s t h e c o l l i s i o n t e r m i n c l u s i v e o f P a u l i - b l o c k i n g , and m and $ r e f e r t o nucleon mass and momentum. The e f f e c t i v e i n t e r a c t i o n i s t a k e n t o be a s i m p l i f i e d Skyrme i n t e r a c t i o n w i t h a c o m p r e s s i b i l i t y modulus o f 200 MeV. The c a l c u l a t i o n s a r e performed b o t h w i t h and w i t h o u t c o l l i s i o n terms t o d i s e n t a n g l e between c o n v e c t i o n and c o n d u c t i o n ( h e a t i n g ) .

I n f i g u r e 3, we examine t h e thermal energy d e n s i t y as a f u n c t i o n o f d i s t a n c e f r o m t h e h o t zone s u r f a c e i n one nucleus a f t e r a t i m e - l a g o f

-

¹⁰⁰fm/c. We f i n d t h a t i n t h e c o l 1 is i o n l e s s case, t h e energ,y d i s t r i b u t i o n i s inhomogeneous whereas w i t h c o l - l i s i o n s i n c l u d e d , i t i s n e a r l y homogeneous. I t i s however n o t a s i g n a t u r e o f t o t a l e q u i l i b r a t i o n as we f i n d f r o m t h e a n a l y s i s o f t h e quadrupole moment o f t h e g l o b a l momentum d i s t r i b u t i o n , d e f i n e d as

Ok = J ^{( 2 k z}

-

k$

-

^k;) ^f( f , 8) d f di? (7).

rxx-l

HOT ZONE 6MeV

I

^-^WITHOUT^COLLISION

I

F i g u r e 3 : The thermal e x c i t a t i o n energy d e n s i t y i n one nucleus as a f u n t i o n o f d i s t a n c e f r o m t h e c e n t r e o f t h e o t h e r nucleus.

L I

4 5 6 7 8 9

DlSTA NCE

(fml

T h i s i s d i s p l a y e d i n f i g u r e 4 as a f u n c t i o n o f t i m e f o r '+OCa

+

'+OCa. We f i n d t h a t i n b o t h cases, energy t r a n s p o r t f r o m t h e h o t zone induces a s t r o n g quadrupole v i b - r a t i o n i n t h e d i n u c l e a r system, r e f l e c t e d i n t h e quadrupole o s c i l l a t i o n s o f t h e momentum d i s t r i b u t i o n , i t b e i n g a s c a l i n g v i b r a t i o n . To r e a c h e q u i l i b r i u m , t h i s v i b r a t i o n must be damped. I n b o t h t h e cases, s t r o n g o s c i l l a t i o n s a r e seen even a f t e r

-

300 fm/c, t h e c o l l i s i o n a l case showing a s m a l l e r amplitude. The same e f f e c t i s seen f o r 2 0 ~ l e + Z0ble.

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JOURNAL DE _PHYSIQUE

HOT ZONE l5 MeV

F i g u r e 4 : The quadrupole moment o f the momentum d i s t r i b u t i o n i n 40Ca +

40Ca.

To conclude

,

o u r a n a l y s i s shows t h a t a h o t zone, i f formed, decays very quic- k l y i n t h e surrounding nuclear matter. The released energy i s however locked up i n convection mode, i n t h e induced i s o s c a l a r quadrupole v i b r a t i o n s o f t h e e n t i r e system, and because o f t h e l o n g nucleon mean f r e e path, i t takes a l a r g e time f o r t h i s v i b r a t i o n t o decay i n t o incoherent thermal mode. Consequently, e x c i t a t i o n energy d e p o s i t i o n i n a nuclear system by means o f h o t zone formation would r e q u i r e t i m e scales f o r thermal e o u i l i b r a t i o n n o t f u l l y compatible w i t h r e c e n t experimental data. A global convection f l o w w i t h two-body c o l l i s i o n s i s more l i k e l y t o achieve t h i s goal i n e n e r g e t i c nuclear c o l l i s i o n s .

REFERFWCES

1 A.O.T. Karvinen, J.V. De and B. Jakobsson, Flucl. Phys. A367, 122, (1981) 2 C.K. Gelbke, Worshop on Coincident P a r t i c l e Emission from Continuum,States,

Bad-Honnef, June (1984)

3 G. Caskey e t al., Phys. Rev. C31,

-

15 97 (1985)

4 J. Galin, I n "The many f a c e t s o f heavy i o n f u s i o n r e a c t i o n s " , Argonne N a t i o n a l Laboratory, (1986) and C. GrBgoire, B Tamain, Ann. Phys. Fr.

-

11 323, (1986) 5 L. V i n e t e t al., Nucl. Phys. A ( i n press) and C. GrPgoire e t al., Mucl. Phys.

A465, 317, (1987)