LIMITING TEMPERATURES OF EXCITED NUCLEI : STATIC AND DYNAMICAL ASPECTS

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LIMITING TEMPERATURES OF EXCITED

NUCLEI : STATIC AND DYNAMICAL ASPECTS

P. Bonche, D. Vautherin, M. Veneroni

To cite this version:

P. Bonche, D. Vautherin, M. Veneroni.

LIMITING TEMPERATURES OF EXCITED

NU-CLEI : STATIC AND DYNAMICAL ASPECTS. International Conference on Heavy Ion Nuclear

Collisions in the Fermi Energy Domain Hicofed 86, 1986, Caen, France.

pp.C4-339-C4-350,

JOURNAL DE PHYSIQUE

Colloque C4, supplément au n° 8, Tome 47, août 1986 C4-339

LIMITING TEMPERATURES OF EXCITED NUCLEI : STATIC AND DYNAMICAL ASPECTS

P. BONCHE, D. VAUTHERIN* and M. VENERONI*

DPhG/SPhT, CEW-Saclay, F-91191 Gif-sur-Yvette Cedex, France "Division de Physique Theorique* , Institut de Physique Nucleaire, F-91406 Orsay Cedex, France

Résumé - Des calculs statiques et dynamiques de noyaux à haute température sont présentés et discutés dans le cadre de l'approximation de champ moyen.

A b s t r a c t - Some s t a t i c and dynamical c a l c u l a t i o n s o f hot n u c l e i are discussed w i t h i n the framework o f the mean f i e l d approximation.

The phase diagram o f nuclear matter i s believed t o e x h i b i t the t y p i c a l s t r u c t u r e o f a Van der W a a l s ' f l u i d / I , 2 / . In p a r t i c u l a r t h i s diagram shows a c r i t i c a l p o i n t , whose temperature Tc i s about 20 MeV and below which the minimum

value o f the f r e e energy corresponds t o an e q u i l i b r i u m between a l i q u i d and a vapor phase. Whether the corresponding l i q u i d - v a p o r t r a n s i t i o n i s observable i n heavy-ion c o l l i s i o n s has been the s u b j e c t o f much d i s c u s s i o n . I n c o n t r a s t w i t h the case o f s t a t i c i n f i n i t e nuclear m a t t e r , a r e a l i s t i c d e s c r i p t i o n o f h i g h l y e x c i t e d n u c l e i , as they are produced i n heavy-ion c o l l i s i o n s , should include f i n i t e s i z e , surface and Coulomb e f f e c t s , as w e l l as the dynamics o f the c o l l i s i o n . With the aim o f e x p l o r i n g some o f these e f f e c t s , we i n t e n d i n the f o l l o w i n g t o compare the r e s u l t s o f some s t a t i c and dynamical c a l c u l a t i o n s w i t h i n the mean f i e l d approximation a t f i n i t e temperature. This approximation i s c h a r a c t e r i z e d by the replacement o f the exact d e n s i t y operator by an u n c o r r e l a t e d one o f the form

(1)

where a:- and a-j are the nucleon c r e a t i o n and a n n i h i l a t i o n operators i n the o r b i t a l s

| t f . > , Z being a n o r m a l i z a t i o n f a c t o r . We w i l l t h e r e f o r e assume t h a t the s i n g l e -p a r t i c l e degrees o f freedom have been thermalized a t some stage o f the c o l l i s i o n by the r e s i d u a l i n t e r a c t i o n s .

I - STATIC MEAN FIELD CALCULATIONS OF HOT NUCLEI

For Hartree-Fock c a l c u l a t i o n s a t f i n i t e temperature, the p o p u l a t i o n o f the i n d i v i d u a l s t a t e s i n the continuum, prevents t h e d i r e c t d e t e r m i n a t i o n o f D by minimizing the f r e e energy. A way t o solve t h i s problem i s t o perform mean f i e l d

c a l c u l a t i o n s f o r an e q u i l i b r i u m between a nucleus and an e x t e r n a l vapor / 3 / . (For an a l t e r n a t i v e approach see r e f . / 4 / ) . However t h i s method makes i t necessary t o i d e n t i f y unambiguously the nucleus and vapor c o n t r i b u t i o n s .

i ) I s o l a t i n g the nucleus from i t s vapor.

I t was shown i n r e f . / 3 / t h a t the Hartree-Fock equations a t f i n i t e temperature, f o r given values o f the chemical p o t e n t i a l \i and volume V, have i n general two s o l u t i o n s . One describes the nucleus-vapor system whereas the second describes the vapor o n l y . The nucleus grand p o t e n t i a l was defined i n r e f . / 3 / as the d i f f e r e n c e

+Laboratoire associe au CNRS

C4-340

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between the grand p o t e n t i a l s o f these two s o l u t i o n s ; t h i s d i f f e r e n c e was shown t o have a w e l l d e f i n e d l i m i t when V goes t o i n f i n i t y .

For independent nucleons i n a p o t e n t i a l U ( r ) t h i s s u b t r a c t i o n procedure gives t h e f o l l o w i n g formula f o r the grand p o t e n t i a l o f t h e nucleus

where 6 = l / k T i s the i n v e r s e temperature and a/@ = p t h e chemical p o t e n t i a l , w h i l e

En and 6g(E) denote r e s p e c t i v e l y t h e energies o f the bound s t a t e s and the phase s h i f t s f o r the p o t e n t i a l U ( r ) . T h i s formula agrees w i t h t h e expression of t h e second o r d e r v i r i a l c o e f f i c i e n t d e r i v e d i n 1937 by Bethe and Uhlenbeck /5/.

The s u b t r a c t i o n procedure of r e f . /3/ a l s o c o n t a i n s as a s p e c i a l case t h e formula o f Tubbs and Koonin /6/, which was d e r i v e d i n t h e semi-classical approxima- t i o n f o r independent nucleons i n a p o t e n t i a l U ( r ) . It gives the nucleus grand p o t e n t i a l

R

as

- a 2

w i t h

where F 3 j 2 i s t h e Fermi i n t e g r a l o f o r d e r 3/2. This formula may be shown t o be t h e semi-classical l i m i t o f eq. ( 2 ) /3/.

ii ) L i m i t i n g temperature.

When a p p l i e d w i t h i n a Hartree-Fock framework the s u b t r a c t i o n procedure ( 3 ) p r e d i c t s t h e e x i s t e n c e o f a l i m i t i n g temperature Ti beyond which an e q u i l i b r i u m w i t h an e x t e r n a l vapor i s no l o n g e r p o s s i b l e . The value o f t h i s l i m i t i n g temperature has been found t o be o f the o r d e r o f 10 MeV : i n the case o f lead-208,TR = 11 MeV f o r t h e Skyrme I l l i n t e r a c t i o n w h i l e i t i s 8 MeV f o r t h e SKY i n t e r a c t i o n /3/. Because o f the Coulomb e f f e c t s , these values a r e a p p r e c i a b l y smal l e r than t h e corresponding ones i n ( n e u t r a l ) n u c l e a r m a t t e r . Systematic i n v e s t i g a t i o n s u s i n g t h e Thomas-Fermi approximation /7/ and t h e h o t l i q u i d drop model /8/ have shown t h a t TQ decreases from l i g h t t o heavy n u c l e i (see f i g . 1). This decrease i s a l s o due t o t h e Coulomb e f f e c t s .

I t can be shown t h a t t h e l i m i t i n g temperature TR i s n o t l a r g e r than the temperature To f o r which t h e nucleus b i n d i n g energy vanishes. Indeed when t h e energy o f t h e nucleus i s p o s i t i v e , i t becomes p o s s i b l e t o lower t h e f r e e energy by a scale t r a n s f o r m a t i o n on t h e o r b i t a l s , t h e occupation numbers ( i . e . t h e entropy) being k e p t fixed. I n p r a c t i c e , numerical c a l c u l a t i o n s have shown t h a t

T i s always s l i g h t l y s m a l l e r than To (see Table 2 o f r e f . /3/).

II

i i i ) L i f e t i m e s o f h o t n u c l e i .

Fig.

1

-

Limiting temperature obtained i n t h e hot l i q u i d drop model as a function of

the nucleus mass number

/ 8 / .

The r e s u l t s obtained f o r d i f f e r e n t e f f e c t i v e i n t e r -

actions (see ref.

/a/)

are compared. High values of a are associated w i t h high

incompressi bi 1 i t i e s i n nuclear matter.

f o r interaction

S I I I

and

f o r various temperatures

Table

1

-

Lifetime of lead-208, calculated i n r e f .

/3/

f o r interaction S I I I , a t

various temperatures.

I t may be noted t h a t l i f e t i m e s become comparable t o reaction times f o r temperatures

of the order of

5

MeV. The values in t a b l e

1

a r e , however, presumably underesti-

mated

131.

C4-342 JOURNAL

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I 1

-

TIME DEPENDENT HARTREE-FOCK CALCULATIONS

The e v o l u t i o n o f h o t n u c l e i has recent1 y been i n v e s t i g a t e d by several v a r i a n t s of t h e dynamical mean-field approach. L e t us quote the constrairled Hartree-Fock method /9/, the time dependent Thomas-Fermi approximation

/ l o / ,

t h e Vlasov e q u a t i o n

/11/, the Monte-Carlo Hartree-Fock method /12/, and the time dependent Hartree-Fock (TDHF) approximation /13/. Ide w i l l now d e s c r i b e some r e s u l t s which were r e c e n t l y obtained b y the l a s t approach /14/.

L e t us f i r s t r e c p l l t h a t , i n TDHF, the d e n s i t y m a t r i x i s assumed, a t each time t, t o b$ o f the form given by eq. (1). The TDHF e v o l u t i o n equations f o r the o r b i t a l s p i ( r ) and f o r the q u a n t i t i e s a i a r i s i n g i n eq. ( 1 ) preserve t h e occupation numbers, t h e s i n g l e p a r t i c l e entropy, and the Hartree-Fock energy.

The r e s u l t s described below were obtained by numerical i n t e g r a t i o n o f t h e TDHF equations a 0.2 fm l a t t i c e c o n t a i n i n g 100 p o i n t s , w i t h a t i m e s t e p A t = 0.25 x s. The zero range d e n s i t y dependent f o r c e o f Jaquaman, l l e k j i a n and Zamick (JNZ) /15/ was used because o f i t s s i m p l i c i t y and a l s o because i t s parameters were a d j u s t e d t o reproduce t h e s a t u r a t i o n p r o p e r t i e s o f n u c l e a r m a t t e r :

b i n d i n g enerqy, d e n s i t y , and compression modulus. However i t overbinds l i g h t n u c l e i . For i n s t a n c e E/A =

-

11 MeV i n calcium-40.

I n f i g , 2 we show t h e e v o l u t i o n o f a h o t calcium-40, whose i n i t i a l temperature i s T = 6 MeV. The i n i t i a l wave f u n c t i o n s are taken t o be pure o s c i l l a t o r f u n c t i o n s , w i t h a parameter b =

-

= 1.7 fm a d j u s t e d t o reproduce t h e r o o t mean square

(rms) r a d i u s o f t h e Hartree-Fock ground s t a t e (3.08 fm). The i n i t i a l occupation numbers a r e determined from t h e Hartree-Fock s i n g l e p a r t i c l e energies associated t o t h e i n i t i a l wave functions. It i s assumed, i n o t h e r words, t h a t t h e r e s i d u a l i n t e r a c t i o n s , d u r i n g a f i r s t stage o f the r e a c t i o n , have t h e r m a l i z e d the s i n g l e p a r t i c l e degrees o f freedom, and t h a t t h e r a d i u s has n o t a p p r e c i a b l y changed. Fig. 2 e x h i b i t s (see a l s o f i g . 6 below) t h e existence o f a s i g n i f i c a n t Landau damping o f the monopole o s c i l l a t i o n : t h e d e n s i t y d i s t r i b u t i o n appears t o reach a l i m i t a f t e r two o s c i l l a t i o n s . T h i s damping was already observed and discussed i n zero temperature TDHF c a l c u l a t i o n s /16/. It a r i s e s because o f the c o u p l i n g t o t h e continuum s t a t e s . I t i s t h i s same c o u p l i n g which leads t o t h e emission o f p a r t i c l e s .

F i g s . 3, 4 and 5 d i s p l a y t h e e v o l u t i o n o f a h o t calcium-40 f o r h i g h e r values o f t h e i n i t i a l temperature, namely T = 10, 12 and 15 MeV. Here also, t h e i n i t i a l r a d i u s has been k e p t approximately equal t o t h e Hartree-Fock ground s t a t e rms r a d i u s . I t can be seen on f i g s . 3-5 t h a t t h e amplitude o f t h e monopole o s c i l l a t i o n increases r a p i d l y w i t h temperature. A t o t a l v a p o r i z a t i o n o f t h e nucleus occurs between 12 and 15 MeV ; i t w i l l be discussed f u r t h e r i n connection w i t h the e v o l u t i o n o f occupation numbers ( t a b l e 2).

It i s i n t e r e s t i n g t o compare t h i s r e s u l t w i t h t h a t o f t h e s t a t i c c a l c u l a t i o n s o f s e c t . I. For t h e zero range i n t e r a c t i o n we are u s i n g and f o r the case of calcium-40 the l i m i t i n g temperature TR discussed i n sect. I i s near 12 MeV. This value i s i n good agreement w i t h t h e r e s u l t s of f i g u r e s 4 and 5.

I n c a l c u l a t i o n s using t h e BKN force (and t h e Vlasov equation) a h i g h e r value o f t h e v a p o r i z a t i o n temperature was found (about 18 MeV) /11/. T h i s d i f f e - rence should probably be a t t r i b u t e d t o t h e BKN f o r c e . Indeed t h i s f o r c e leads t o a l a r g e value o f t h e compression modulus i n n u c l e a r m a t t e r ; and t h i s value i s expected t o produce a l a r g e value o f t h e l i m i t i n g temperature TR (see f i g . 1 ) .

I n f i g . 6 we show t h e e v o l u t i o n o f t h e rms r a d i u s o f t h e " r e s i d u a l nucleus" f o r t h e i n i t i a l c o n d i t i o n s T = 6, 10 and 15 MeV and an i n i t i a l calcium-40 nucleus. The r e s i d u a l nucleus i s d e f i n e d by t r u n c a t i n g t h e s i n g l e p a r t i c l e wave f u n c t i o n s p i ( r ) i n eq. ( 1 ) a t a c u t o f f r a d i u s c = 8 fm. T h i s c u t o f f r a d i u s c i s l a r g e r than

JOURNAL DE PHYSIQUE CALCIUM 40

-

T1,15MeV R _b

_,1.48

_frn E

-

U-

-

interaction

JMZ

21 + rrns -3.06

fm

.-

V) C

8

-

-

-

081

-

-

-

t - 0

-

t = I X I O - ~ ~ S

.-.

t = 2 x 1 0 - ~ ' s

...

".

t 1 3 x 10-~'s

-

0

2

4

6

8

Fig.

6

-

Time evolution of the root mean square radius of the residual nucleus for

the i n i t i a l conditions

A

= 40,

T

=

6

MeV, b

=

1.7 fm

; T =

10 MeV,

b =

1.6 fm

;

T

=

15 MeV,

b =

1.48 fm.

T

of the monopole o s c i l l a t i o n . The corresponding energy

fi x 2 8 / ~

i s 23.7 MeV f o r

T = 6

MeV and 17.3 MeV f o r

T =

10 MeV. For

T =

15 MeV the radius shows a monotonic

increase with time, as can be expected from f i g .

5.

The p a r t i c l e emission leads of course t o a decrease i n time of the mass

number and a l s o of the energy per nucleon ( i .e. t o a cooling) of the residual

nucleus as i s i l l u s t r a t e d by f i g s .

6

and 7. (Recall t h a t the t o t a l mass and the

t o t a l Hartree-Fock energy a r e constants of the motion i n TDHF). For i n i t i a l

temperatures

up

t o 12 MeV the average mass of the residual nucleus appears t o

reach a 1

i m i

t

w i t h

time. The same remark holds f o r the energy per nucleon which

goes asymptotically t o a negative value. In c o n t r a s t , f o r an i n i t i a l temperature

T

=

15 MeV the residual nucleus eventually disappears, while the energy per nucleon

remains always positive. From figure

7

one can define schematically a l i f e t i m e

.r

of the hot nucleus as the time when the number of emitted p a r t i c l e s i s half i t s

asymptotic value. For T

= 6

MeV t h i s gives

T

-

1

x

10-z2 s i n agreement with the

value quoted i n t a b l e

2 .

For higher temperatures such a d e f i n i t i o n of T(T) i s no

longer meaningful since there i s an important cooling ( i .e. decrease i n

E/A)

over

a time s c a l e of 1 0 - ~ 2

s . The importance of t h i s cooling i n a s h o r t lapse of time

suggests t h a t the system may not have the opportunity t o reach an i n i t i a l condition

with temperatures greater than some value around 10 MeV.

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CALCIUM 40 ~nteroct~on JMZ

c = 8 f m

Fig.

7

-

Time evolution of the average mass of the residual nucleus f o r various

i n i t i a l temperatures. The i n i t i a l nucleus i s cal ci um-40 and the i n i t i a l radius

3.08

fm.

nucleus, a t time

t l =

to

+

3

x

s . Theil calculation requires some care

because the r e s t r i c t i o n s of the o r b i t a l s q ( r ) t o the sphere of radius c are no

longer orthogonal. I t

is thus necessary t o construct and diagonalize the r e s t r i c t i o n

of the single p a r t i c l e density matrix in t h i s sphere. In contrast with the global

occupation numbers (which are constants of the motion) t a b l e 2 shows t h a t the

occupation numbers in the residual nucleus are modified during the TDHF evolution

by the p a r t i c l e emission. I t i s worthwhile noting t h a t the main e f f e c t of the

p a r t i c l e emission i s t o empty unbound o r b i t s i n the i n i t i a l Hartree-Fock potential.

For T

=

12 MeV the entropy per nucleon decreases from S/A

=

2.09 t o S/A

=

1.51 because

the evaporated p a r t i c l e s are the most disordered. Notice t h a t f o r

T

greater than

10 MeV, the emitted p a r t i c l e s produce a d r a s t i c reduction of the mean f i e l d depth

;

a t T

=

15 MeV t h i s non-linear e f f e c t i s s u f f i c i e n t t o drive a l l the p a r t i c l e s i n t o

the continuum, leading t o t o t a l vaporization.

Table

2

-

Occupation numb

rs

ni of the lowest o r b i t a l s a t the i n i t i a l time

to,

and a t ti

=

to

+

3

x

s

in the residual nucleus. The i n i t i a l condition i s

A =

40, T

= 6

MeV, rms

=

3.08 fm. The Hartree-Fock single p a r t i c l e energies

Ei(MeV) a t time

to are also indicated.

CONCLUSION

AND OUTLOOK

The time dependent Hartree-Fock formalism provides, through p a r t i c l e emission,

a consistent dynamical treatment of the l i q u i d and vapor nuclear phases. While being

s t i l l numerically simple, i t allows a comparison with the r e s u l t s of e a r l i e r s t a t i c

calculations, whose conclusions are i n the overall c o n f i n e d .

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Some aspects o f n u c l e a r fragmentation have a l r e a d y been i n v e s t i g a t e d by d i f f e r e n t techniques /12, 17-19/. I n t h e TDHF framework, an approximate way t o study t h i s q u e s t i o n i s through t h e random ~ h a s e approximation (RPA) a t f i n i t e temperature /13, 20/. I n t h i s approach and f o r a density-dependent i n t e r a c t i o n such as t h e JMZ f o r c e /15/, the e f f e c t i v e " p a r t i c l e - h o l e " i n t e r a c t i o n was found t o be r e p u l s i v e a t d e n s i t i e s h i g h e r than the normal n u c l e a r d e n s i t y and a t t r a c t i v e a t lower d e n s i t i e s . T h i s e f f e c t produces imaginary RPA r o o t s which may reveal t h e onset o f fragmentation a t low n u c l e a r density.

L e t us f i n a l l y r e c a l l t h a t TDHF i s o n l y one among p o s s i b l e mean-field t h e o r i e s . Using a time dependent v a r i a t i o n a ? p r i n c i p l e , i t has been shown t o be the "best" mean-field approximation f o r p r e d i c t i n g t h e average values o f s i n g l e - p a r t i c l e observables. When the q u a n t i t i e s o f i n t e r e s t , such as f l u c t u a t i o n s o r t r a n s i t i o n p r o b a b i l i t i e s , are no more o f t h e s i n g l e - p a r t i c l e type, more s o p h i s t i - cated mean-field t h e o r i e s (which a r e s u i t e d t o t h e e v a l u a t i o n o f these q u a n t i t i e s and which i n c o r p o r a t e some c o r r e l a t i o n e f f e c t s ) may be d e r i v e d from t h i s same v a r i a t i o n a l p r i n c i p l e /21/. I n t h e zero temperature case, t h e mass d i s p e r s i o n o f the r e s i d u a l n u c l e i has been evaluated by t h i s method and t h e c o r r e c t i o n s /22-24/ found t o improve s i g n i f i c a n t l y the "naive" TDHF e v a l u a t i o n s which g i v e t o o small values.

We hope t h a t i n v e s t i g a t i o n s , now i n progress along these l i n e s , w i l l h e l p t o a b e t t e r understanding o f these e x c i t i n g n u c l e a r blobs, which our e x p e r i m e n t a l i s t f r i e n d s e x c i t e , n o t f a r from here, w i t h a t o n i c excitement.

REFERENCES

/1/ Kiipper, W.A., Wegmann, G. and H i l f , E.R., Ann. Phys. (N.Y.) 88 (1974) 454. /2/ Sauer, G., Chandra, H. and Mosel, U., Nucl. Phys. A264 (1976 221,

6

/3/ Bonche, P., L e v i t , S. and Vautherin, D., Nucl. Ph~s.-A4?7 ( 1 841 278

-

_.-

_.

; A436 (1985) 265.

/4/ E m b e r g e r , E., B a r t e l

,

J., B l i n , A., H i l l e r , 6. and Brack, M., C o n t r i b u t i o n t o t h e p r e s e n t volume, p. 66.

/5/ Landau, L.D. and L i f s c h i t z , E.

,.

S t a t i s t i c a l Mechanics, M i r , Moscow, 1967, sect. 77.

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