ON THE ROLE OF QUANTUM AND STATISTICAL EFFECTS IN THE LIQUID GAS PHASE TRANSITION OF HOT NUCLEI

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ON THE ROLE OF QUANTUM AND STATISTICAL

EFFECTS IN THE LIQUID GAS PHASE

TRANSITION OF HOT NUCLEI

A. Blin, B. Hiller, H. Reinhardt, P. Schuck

To cite this version:

C o l l o q u e C4, s u p p l 6 m e n t au

no 8 ,

T o m e

47,

a o O t

1986

ON THE ROLE OF QUANTUM AND STATISTICAL EFFECTS IN THE LIQUID GAS

PHASE TRANSITION OF HOT NUCLEI

A . H . B L I N , B . HILLER, H . REINHARDT*

and

P . SCHUCK"

I n s t i t u t fiir T h e o r e t i s c h e P h y s i k , U n i v e r s i t d t R e g e n s b u r g . 0 - 8 4 0 0 R e g e n s b u r g , F.R.G.

' ~ e n t r a l i n s t i t u t

fur K e r n f o r s c h u n g , R o s s e n d o r f ,

Postfach

19, 0 - 8 0 5 1 D r e s d e n , D.R.G.

"ISN, 53, A v e n u e d e s M a r t y r s , F - 3 8 0 2 6 G r e n o b l e C e d e x , France RCsume

-

La t r a n s i t i o n de phase Liquide-gaz dans La m a t i e r e n u c l e a i r e chaude

--

due aux f l u c t u a t i o n s quantiques e t s t a t i s t i q u e s e s t Ctudiee dans un modble microscopique de La n u c l e a t i o n . Ce modkle e s t une v e r s i o n hydrodynamique de La t h e o r i e de champs moyen dCpendant du temps i m a g i n a i r e aux temperatures f i n i e s . A b s t r a c t

- The t r i g g e r i n g o f t h e Liquid-gas phase t r a n s i t i o n i n h o t n u c l e a r

m a t t e r by quantum and s t a t i s t i c a l f l u c t u a t i o n s i s s t u d i e d i n a m i c r o s c o p i c approach t o nucleation, which i s a f l u i d - d y n a m i c a l v e r s i o n o f t h e imaginary t i m e dependent mean f i e l d t h e o r y a t f i n i t e temperature.

I

-

INTRODUCTION

The e q u a t i o n o f s t a t e o f h o t n u c l e a r m a t t e r e x h i b i t s a t y p i c a l van der Waals behavi- o u r and i n d i c a t e s t h e coexistence o f a l i q u i d phase a t normal n u c l e a r m a t t e r d e n s i t y and o f a gas phase ( a t lower d e n s i t i e s ) which c o n s i s t s o f a f r e e gas o f nucleons, deuterons, A t low temperatures, where s t a t i s t i c a l f l u c t u a t i o n s can be d i s - carded i n a c l a s s i c a l theory, t h e two homogeneous phases which correspond t o t h e li- q u i d and gas a r e s t a b l e and may have d i f f e r e n t o r equal energy d e n s i t i e s . However, i n a quantum t h e o r y t h e s t a t e o f h i g h e r energy i s rendered u n s t a b l e through b a r r i e r pe- n e t r a t i o n and a phase t r a n s i t i o n may t a k e place. Phase t r a n s i t i o n s a r e a l s o p o s s i b l e v i a thermodynamical f l u c t u a t i o n s . A new phase may be formed i n s m a l l volumes, say, bubble i n a n u c l e a r m a t t e r environment, which a r e s t a b l e i f t h e y have a c e r t a i n size, b e i n g e n e r g e t i c a l l y f a v o u r a b l e i n t h e i n t e r p l a y o f volume and s u r f a c e energies. The r o l e o f quantum and s t a t i s t i c a l f l u c t u a t i o n s has been f i r s t analyzed i n a micro- s c o p i c d e s c r i p t i o n i n t h e r e c e n t work o f reference4). There a r e l a t i v i s t i c model f i e l d t h e o r y o f n u c l e a r systems (Walecka's model has been used t o s t u d y t h e L i - q u i d vapour phase t r a n s i t i o n s a p p l y i n g e u c l i d e a n space p a t h i n t e g r a l techniques 6 ) .

I n t h e p r e s e n t work we use a s i m i l a r dynamical t r e a t m e n t t o s t u d y phase t r a n s i t i o n s i n a n o n - r e l a t i v i s t i c d e s c r i p t i o n o f t h e n u c l e a r many-body system w i t h e f f e c t i v e Skyrme f o r c e s . The q u a n t i t y which d e s c r i b e s t h e decay o f one o f t h e m e t a s t a b l e phases i n t o t h e s t a b l e one ( " n u c l e a t i o n r a t e " ) has a W.K.B. l i k e form

W =

B e x p ( - A i )

(11

N

where A i i s t h e minimized c l a s s i c a l (euclidean) a c t i o n a s s o c i a t e d w i t h bubble o r d r o p l e t f o r m a t i o n and B i s a f a c t o r which a r i s e s from t h e f u n c t i o n a l i n t e g r a l over t h e f l u c t u a t i o n s . Equation

(I)

i s a p p l i c a b l e t o bubble f o r m a t i o n ( o r n u c l e a t i o n ) t r i g g e r e d b y quantum as w e l l as s t a t i s t i c a l f l u c t u a t i o n s b u t t h e corresponding a c t i o n d i f f e r s i n t h e two cases. By m i n i m i z i n g t h e a c t i o n one o b t a i n s an e q u a t i o n o f motion f o r t h e bubble o r d r o p l e t r a d i u s as f u n c t i o n o f t h e imaginary time. T h i s e q u a t i o n has s e v e r a l s o l u t i o n s . Two o f them a r e t r i v i a l s t a t i c s o l u t i o n s and correspond t o t h e homogeneous gas and l i q u i d phase r e s p e c t i v e l y . One i s s t a t i c b u t non t r i v i a l , repre- s e n t i n g a bubble ( d r o p l e t ) on t h e background o f a constant l i q u i d (gas) phase and a r i s e s due t o s t a t i s t i c a l f l u c t u a t i o n s . The a c t i o n corresponding t o t h i s s o l u t i o n

JOURNAL

DE

PHYSIQUE

e n t e r s i n t h e c a l c u l a t i o n o f s t a t i s t i c a l f l u c t u a t i o n s . F i n a l l y t h e r e e x i s t s an (ima- g i n a r y ) t i m e dependent s o l u t i o n , which d e s c r i b e s t h e f o r m a t i o n o f bubbles ( d r o p l e t s ) through b a r r i e r p e n e t r a t i o n o r quantum f l u c t u a t i o n s .

A t f i n i t e temperatures, t h e e u c l i d e a n action, d e f i n e d as ( - i ) times t h e c o n t i n u a t i o n o f t h e r e a l t i m e s e c t i o n t o i m a g i n a r y times T i s g i v e n i n t h e f l u i d dynamical a p p r o x i - m a t i o n by

6A

~ [ g p

J d ~ ( ~ d i x ) -

y T n ( p u z ) - R

CS])

(11 .I) -h/* I I -

w i t h t h e boundary c o n d i t i o n p(l3/2) =p(-R/2) and t h e i n v e r s e temperature T o f t h e sys- tem. Here p i s t h e t i m e even p a r t o f t h e d e n s i t y m a t r i x , t h e f i e l d X i s azsumed+to be l o c a l i n space, and t h e v e l o c i t y f i e l d U i s a s s o c i a t e d t o t h e f i e l d X by u =

VX

.

The q u a n t i t y Q i n eq. (11.1) i s t h e thermodynamic p o t e n t i a l , a f u n c t i o n a l o f p

RLSI=

E

CS]

-p*>

(S)

-

S/@

(11.2)

where i s t h e chemical p o t e n t i a l , S t h e e n t r o p y and ELp] t h e mean f i e l d energy f u n c t i o n a l .

The v a r i a t i o n s o f t h e a c t i o n i n t e g r a l w i t h r e s p e c t t o X and p lead t o t h e f l u i d dy- namics equations. Since eq. (11.1) i s s u b j e c t t o t h e boundary c o n d i t i o n p(l3/2)

.=

=p(-0/2),the s o l u t i o n i s n o t as u s u a l an i n i t i a l v a l u e problem, b u t r a t h e r a boundary v a l u e problem w i t h p e r i o d i c c o n d i t i o n s . T h i s renders t h e problem q u i t e d i f f i c u l t t o handle.

I11

-

SIMPLIFICATION OF THE MODEL

To g e t a f i r s t i n s i g h t i n t o t h e non t r i v i a l m e a n - f i e l d s o l u t i o n s d e s c r i b i n g t h e n u c l e - a t i o n process (bubble o r d r o p l e t f o r m a t i o n ) we r e s o r t t o s e v e r a l approximations which we L i s t as f o l l o w s :

(i) we assume r a d i a l symmetry ( f o r t h e bubble o r d r o ~ l e t ' ) ( i t ) t h e d e n s i t y o f a bubble ; o l u t i o n i s parametrized' as

P

=

P,

+

(%-$)/{I

+

e x p l ( r - ~ ( c ) ) / a ] j

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

PI

.

ere

p6 and PL a r e t h e values o f t h e d e n s i t i e s which r e s u l t from a Maxwell construc- t i o n i n a n u c l e a r m a t t e r c a l c u l a t i o n . Then t h e c o e x i s t e n c e o f t h e gas and L i q u i d phases i s c h a r a c t e r i z e d by d e n s i t i e s o f equal chemical p o t e n t i a l /(A.

P

( F c ) = p

l % ) .

(111.2)

The imaginary t i m e T e n t e r s through t h e parameter R. The parameter a, which repre- sents t h e s u r f a c e t h i c k n e s s o f t h e droplet, i s k e p t constant f o r a f i x e d temperature. (iii) The mean f i e l d h a m i l t o n i a n i s c a l c u l a t e d i n t h e f i n i t e temperature Thomas Fermi

a p p r o x i m a t i o n w i t h a Skyrme i n t e r a c t i o n 3 ) .

( i v ) We n e g l e c t Coulomb e f f e c t s and assume symmetric n u c l e a r matter.

The c h o i c e o f p a r a m e t r i z a t i o n (ii) corresponds t o our b e l i e f t h a t h o t compressed nuc- l e a r m a t t e r may d e s i n t e g r a t e t o g i v e r i s e t o bubbles a t much Lower t h a n s a t u r a t i o n d e n s i t i e s through s t a t i s t i c a l and quantum f l u c t u a t i o n s . I n a s i m i l a r f a s h i o n we a l s o c o n s i d e r t h e p o s s i b i l i t i e s t h a t a h o t nucleon gas may m a t e r i a l i z e i n s m a l l d r o p l e t s . T h i s p i c t u r e i s s u b s t a n t i a t e d by t h e e q u a t i o n o f s t a t e o f h o t n u c l e a r matter, which e x h i b i t s a t y p i c a l Van d e r Waals behaviour.

I n heavy i o n r e a c t i o n s t h e process o f bubble f o r m a t i o n i s p r o b a b l y more r e l e v a n t t h a n t h e d r o p l e t formation. D r o p l e t f o r m a t i o n c o u l d i n p r i n c i p l e be i m p o r t a n t d u r i n g t h e s t r o n g expansion phase o f a h i g h energy n u c l e a r r e a c t i o n . However, we w i l l see below t h a t t h i s occurs i n o u r model o n l y f o r d e n s i t i e s much lower t h a n t h e u s u a l l y accepted v a l u e o f t h e f r e e z e

-

o u t d e n s i t y .

I V

-

THE VELOCITY FIELD AND THE MASS PARAMETERS

y i e l d

(IV.1) T h i s e q u a t i o n a l l o w s t o determine t h e v e l o c i t y f i e l d which corresponds t o a c e r t a i n d e n s i t y . To c a l c u l a t e t h e a c t i o n i n t e g r a l (11.1) u s i n g t h e p a r a m e t r i z a t i o n (111.11, i t i s necessary t o know how p evolves i n t h e imaginary t i m e r. Since i n (111.1) p depends on T o n l y through R ( r ) , we v a r y t h e a c t i o n A w i t h r e s ~ e c t t o R and o b t a i n t h e f o l l o w i n g e q u a t i b n o f m o t i o n

fbr-p

d n

-

PR

k 2 - 2 &

R =

0

d72I

w h e r e a f i g u r e s as mass parameter. V

-

THE THERMODYNAMICAL POTENTIAL R

The thermodynamical p o t e n t i a l corresponding t o bubble f o r m a t i o n i s d e f i n e d as (V.1) Here R [ p L ] i s t h e thermodynamic p o t e n t i a l a s s o c i a t e d w i t h t h e l i q u i d background o f eq. (111.1). I n F i g . 1, we d e p i c t

aB

f o r T

=

9 MeV. Since p -t p ~ as R +

-

m one ap-

proaches from t h e l e f t t h e p o t e n t i a l m i n i ~ u m corresponding t o t h e L i q u i d phase, which i s s u b s t r a c t e d o f f i n t h e c a l c u l a t i o n o f CLB. T h e r e f o r e RB vanishes f o r R + -W. For

l a r g e values o f R, p + p ~

,

and one approaches from t h e r i g h t t h e p o t e n t i a l minimum

o f t h e gas phase. The b a r r i e r between t h e two phases i s due t o t h e s u r f a c e energy. A f t e r reaching t h e s a t u r a t i o n value, t h e volume energy o f t h e bubble s t a r t s t o domi- n a t e over t h e s u r f a c e energy, which e x p l a i n s t h e f a l l o f f o f

GB

a f t e r a c e r t a i n c r i - t i c a l R.

V1

-

QUANTUFI FLUCTUATIONS

I t i s necessary t o s o l v e e x p l i c i t l y t h e imaginary t i m e e q u a t i o n o f m o t i o n f o r R, eq. (IV.2) w i t h t h e a p p r o p r i a t e boundary c o n d i t i o n . The s o l u t i o n o f eq. (IV.2) shows how t h e r a d i u s R o f t h e d r o p l e t o r bubble behaves i n t h e course o f t h e i m a g i n a r y t i m e

T. F i g u r e 2 shows t h e imaginary t i m e e v o l u t i o n o f a bubble r a d i u s R ( ~ ) f o r p = ' 2 0 . 5 M e V a t T

':

0.5 MeV.

I n Table 1 we showthe i n i t i a l values o f R which a r e compatible w i t h quantum f l u c t u a - t i o n s a t s e v e r a l values o f t h e chemical p o t ~ n t i a l p, and t h e r e s u l t i n g bubble forma- t i o n p r o b a b i l i t i e s a t t h e temperature T

=

0.5 MeV.

V11

- STATISTICAL FLUCTUATIONS

S t a t i s t i c a l f l u c t u a t i o n s correspond t o t h e non t r i v i a l s t a t i c s o l u t i o n s o f eq. (IV.2) t h z t i s ( R

=

0, .I?- = 0) and dWdR = 0. T h e r e f o r e t h e nEn t r i v i a l s o l u t i o n i s

ai

(R)

=

= Ri(Rcr) where RC, i s t h e c r i t i c a l r a d i u s f o r which Ri(R) a c q u i r e s i t s maximum.Fig.3 shousthe r e s u l t i n g c r i t i c a l bubble r a d i i and n u c l e a t i o n r a t e s a t T

=

9 NeV as func- t i o n of t h e d e n s i t y Q.

REFERENCES

-

1) G. Sauer, H. Chandra and U. Nosel, Nucl. Phys. A 264 (1976) 221;

2) G. B e r t s c h and P.J. Siemens, Phys. L e t t .

126

(1983) 9; 3) M. Brack, C. Guet and H.-B. Hikansson, Phys. Rep.

123

(1985) 275; 4 ) H. Reinhardt and H. Schulz, Nucl. Phys. A 432 (1985) 630; 5 ) J.D. Walecka, Ann. Phys.

83

(1974) 491; Phys. L e t t .

59

(1975) 109; 6) S. Coleman, Phys. Rev.

D

(1977) 2929; H. Reinhardt, Nucl. Phys. A 367

JOURNAL

DE

PHYSIQUE

F i g .

1

-

The thermodynamical p o t e n t i a l

8,

o f a bubble on a l i q u i d background as f u n c t i o n o f t h e r a d i u s R o f t h e bubble f o r t h e isotherm T = 9 MeV and two values o f t h e chemical p o t e n t i a l ( i n MeV).

Fig. 2

-

Time-dependent s o l u t i o n t o t h e equation o f motion IV.2. The bubble r a d i u s R i s s h ~ w n as f u n c t i o n of t h e imaginary t i m e T, corresponding t o t h e s o l u t i o n i n s i d e

t h e i n v e r t e d thermodynamical p o t e n t i a l f o r 11 = -20.7 MeV and

T

= 0.5. MeV. The minimum o f

R

l i e s on one t u r n i n g p o i n t o f t h e p o t e n t i a l .