THE EXTENDED THOMAS-FERMI MODEL AT FINITE TEMPERATURE

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THE EXTENDED THOMAS-FERMI MODEL AT FINITE TEMPERATURE

M. Brack

To cite this version:

M. Brack. THE EXTENDED THOMAS-FERMI MODEL AT FINITE TEMPERATURE. Journal

de Physique Colloques, 1984, 45 (C6), pp.C6-15-C6-25. �10.1051/jphyscol:1984603�. �jpa-00224204�

JOURNAL DE PHYSIQUE

Colloque C6, supplement au n°6, Tome 45, juin 1984 page C6-15

THE EXTENDED THOMAS-FERMI MODEL AT F I N I T E TEMPERATURE

M. B r a c k

Institute of Theoretical Physics, University of Regenskurg, D-8400 Regenabur-g, F.R.G.

Abstract - We generalize the ETF model to finite temperature for a system of Fermions with variable effective mass. For the first time we present the corresponding functional x[p] which contains consistently all terms of order R2.

The kinetic energy density functional T ETp [ p ] , derived within the extended Thomas- Fermi (ETF) model /1,2,3/ has been widely used in several branches of physics / 4 / and recently also been successfully applied in density variational calculations for average nuclear ground-state properties and deformation energies /5,6/. Its general- ization to excited nuclear systems at temperatures T > 0, which are of growing interest both in heavy ion /!/ and astrophysics / 8 , 9 / , has so far only been realized in approximate ways. Either one used just the pure TF approximation /8,10/ or the low-temperature expansion /l1,12/, or an ad hoc combination /13,14/ of the TF func- tional plus the gradient correction terms known from the T = 0 case.

In the present note we shall derive the functionals for the kinetic energy density x[p] and the entropy density o[p] appropriate to the T > 0 case, consistently up to second order in ft2. For the sake of a simpler presentation, we restrict ourselves first to the case of a purely local potential V(r) and discuss nonlocalities at the end.

We start from the well-known expressions for the entropy, the intrinsic and the free (Helmholtic) energy of a system on noninteracting Fermions in an external potential:

(1) (2) (3) where e. are the single particle (s.p.) energies and n. the Fermi occupation numbers

with the chemical potential A. (We use k = 1 and express the temperature T in units of MeV.) We also write in the usual way the spatial density p(r) and the kinetic energy densities T(r) and T*(r) in terms of the s.p. wavefunctions <p. (r) :

(5) Résumé - Le modèle ETF est généralisé à des températures finies

pour un système de Fermions avec une masse effective variable.

Pour la première fois, la fonctionelle x[p] correspondante est présentée elle contient tous les termes de l'ordre R2 d'une façon consistente.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984603

C6-16 JOURNAL DE PHYSIQUE

For systems with time reversal symmetry, the densities .r and T, are connected by ( 8 ) Semiclassical approximations to the above quantities can most easily be obtained using the Bloch density C(;,B) which is related to the density p ( ? ) by an inverse Laplace transform:

c+im

Knowing that the transition from the T = 0 to the T > 0 case can be made by folding the T 0 spectral density with the function f,(E) / 1 5 / :

i~ is easy to realize that the Bloch density at T > 0 is given by

C(;,p)

k l .g(p),

where C (r,R) is the Bloch density at T + = 0 * )

cD(;, p'

z \ y i G l l l e - ~ c ~

i

and ? (B) is the (two-sided! ) Laplace transform of f ( E ) eq. _{(10) :}

T T

t m

Using simple rules of Laplace transforms, one finds the following expressions for the above defined densities:

(

ap(i) ^{- L:} [SC(;,P~] ^,

SO far, all equations are exact, i.e. they contain the quantum-mechanical shell effects. We now replace the "cold" Bloch density C (;,I31 by the one obtained in the Wigner-Kirkwood expansion /16/ up to second order ?n K :

crK(;,p,

^C,,

^{p) { I +}

[t f ' c ~ v ) ' - p a ~ v l ]

* ) Note that I? is only a mathematical parameter here and does not have the meaning

of an inverse temperature !

w h e r e t h e TF a p p r o x i n a t i o n h a s t h e form ^*)

'The I n v e r s e L a p l a c e t r a n s f o r m s c a n now b e d o n e a n a l y t i c a l l y a n d y i e l d t h e f o l l o w i n g d e n s i t i e s

T h e q u a n t i t y q i s g i v e n by

I - V ( ? )

Y 1 =

a n d J ( q ) a r e t h e s o - c a l l e d F e r m i i n t e g r a l s . F o r p > - 1 t h e y a r e d e f i n e d by P

R)

For p 5 - 1 , t h e i n t e g r a l ( 2 5 ) d o e s n o t e x i s t . However we c a n e x t e n d t h e d e f i n i t i o n o f J (11) t o p < - 1 Sy t t l c r e c u r r e n c e r e l a t i o n

P

* ) A l l f o r m u l a e h o l d f o r o n e k i n d o f F e r m i o n s ; a s p i n f a c t o r o f 2 i s i n c l u d e d .

C6-18 J O U R N A L DE PHYSIQUE

which l e a d s t o w e l l - b e h a v e d , c o n t i r ~ u o i l s fu n c t i o n s J , ( q ) f o r a l l v a l u e s o f p . N o t c t h a t we d o n o t e n c o u n t e r h e r e t h e t u r n i n g p o i n t p r o b l e m w h i c h i s w e l l known i n t h e T = 0 c a s e : all t h e d e n s i t i e s e q s . ( 1 9 ) - ( 2 2 ) a r e c o r ~ t i r ~ u o u s a n d f i n i t e e v e r y w h e r e i n s p a c c . O n l y i n t h e L i m i t T -> 0 w i l l . t h e s e m i c l a s s i c a l c o r r e c t i o n s w i t h J ( n ) f o r p < - 1 d i v e r g e a t t h e c l a s s i c a l t u r n i n g p o i n t s g i v e n by r: = 0. The f u n c t i o n a l s which P we s h a l l d e r i v e a r e t h u s r i g o r o u s l y d e f i n e d i n a l l s p a c e .

As i n t h e T = 0 c a s e , we now s e e k t o e l i m i n a t e t h e p o t c r ~ t i a l V ( i . c . n ) a n d i t s d e r i v - d t l v e s from t h e a b o v e equations, b e i n g c o n s i s t c n t t o o r d e r f i 2 r e l a t i v e t o t h e TF t e r m s . To t h i s p u r p o s e we w r i t e

We now s o l - v e e q . ( 2 7 ) f o r n w r i t i n g

A ,

y-y,

AT34l,(7.+y1),

where rlo i s d e f i n e d by i n v e r t i n g t h e e q u a t i o n

The f a c t t h a t p 2 , w h i c h i s o f o r d e r k i 2 r e l a t i v e t o a o , c o n t a i n s I- d o e s n o t d i s t u r b u s : s i n c e Q~ w i l l be o f o r d e r li7 r e l a t i v e t o qo, we c a n r e p l a c e I- by no everywhere i n v 3 o f e q . (191, h e r e b y n e g l e c t i n g o n l y Lerms or o r d e r h 2 . E x p a n d i n g cq. ( 2 8 ) we f i n d

The n e x t s t e p i s t o e x p r e s s t h c d e r i v a t i v e s o f t h c p o t e n t i a l by t h o s e o f t h e d e n s i t y , w h l c h i s e a s i l y d o n e a f t e r d e r i v i n g t w i c e t h e e q u a t i o n ( 2 9 ) . The r e s u l t i s

I t i s now j u s t a m a t t e r o f some a l g e b r a t o e x p a n d e q s . ( 1 9 ) - ( 2 2 ) a r r o u r l d rlo, t o c o l l e c t a l l g r a d i e n t t e r m s and e x p r e s s them i n t e r m s o f p u s i r ; g cqs. ( 3 1 ) , ( 3 2 ) . T h i s l e a d s u s t.o t h e "E'I'F" i u n c t l o r ~ i : l s

where t h e 'l'F t e r m s n a v e t h c w e 1 1-known form /10/

dTF

I ? ]

- 9.

7 AT

a n d no i s g l v e n b y s o l v i n g e q . ( 2 9 ) . S i n c e on.ly t h e i n t e g r a l s o f t h e a b o v e d e n s i t i e s a r e o f f i n a l i n t e r e s t , we g i v e b e l o w t h e e x p r e s s i o n s f o r t h e g r a d i e n t c o r r e c t i o n s w h i c h o n e o b t a i n s a f t e r some s u i t a b l e p a r t i a l i n t e g r a t i o n s . ('The c a s e o f a v a r i a b l e e f f e c t i v e m a s s , w h e r e i n t e g r a l s o v c r ^T{> o c c u r , w i l l b e d i s c u s s e d b e l o w . ) T h e s e e x p r c s s i o n s a r e

H c r c b y wc h a v e u s e d t h e a b b r e v i a t i . o n s

T ~ [ P ] i s s i m j l a r t o t t i c W e i z s a c k e r Cerm, b u t w i t h a rl0-de:xndent ( i . e . d c n s i t y

& p e n d e n t ) c o e f f i c i e n t . I f oric i s o n l y i n t e r e s t e d i n t h e t o t a l f r e e e n e r g y , o n e may u s e t h e following e x p r e s s i o n s

=

F T F l f l

F * i f l ,

we f o u r ~ d t h a t t h e f c n c t i o n 5 ( n o ) e q . ( 3 9 ) c a n b e a p p r o x i m a t e d t o w i t h i n l e s s t h a n 3 % f o r a l l v a l u e s of no by t h e c x p r c s s i o n

I t i s now i n t e r e s t i n g t o s t u d y t h e 1j.mj.t T .> 0 o f t h e a b o v e e x p r e s s i o n s . F o r t h a t + we h a v e t o d i s t i n g u i s h two r e g i o n s . I n s i d e t k e c l a s s i c a l l y a l l o w e d r e g i o n , -. X > V ( r ) a n d t h u s no wil.1 g o t o +

-.

t h e J p ( r l o ) 1'17,'

Wc f i n d t h a t i n t h i s c n s c

C6-20 J O U R N A L DE PHYSIQUE

We t h u s r e c o v e r t h e well-known ( c o r r e c t e d ) W e i z s a c k e r c o e f f i c i e n t i n T ~ [ ~ ] . The r e s u l t e q . ( 4 7 ) a l s o is n o t s u r p r i s i n g , s i n c e t h e e n t r o p y h a s t o g o t o z e r o l i k e 1'. O u t s i d e t h e c l a s s i c a l l y a l l o w e d r e g i o n , no g o e s t o - a n d we c a n u s e t h e a s y m p t o t i c s e r i e s f o r n e g a t i v e qo / 1 7 /

I n t h e l i m i t T + 0 we f i n d h e r e w i t h t h e r e s u l t

T h u s , 0 2 [ p ] g o e s t o z e r o i n a l l s p a c e , a s i t s h o u l d . The c o e f f i c i e n t i n T~ i s f o u n d t o b e d i f f e r e n t o u t s i d e t h e c l a s s i c a l l y a l l o w e d r e g i o n from t h a t i n s i d e b y a f a c t o r o f 3.

Note t h a t i n a l l d e r i v a t i o n s o f 1 2 [ p ] a t T = 0 / 1 - 3 / , t h e c l a s s i c a l l y f o r b i d d e n r e g i o n was n o t a c c e s s i b l e ; t h e f u n c t i o n a l d e r i v e d I n t h e c l a s s i c a l l y a l l o w e d r e g i o n was s i m p l y assumed t o h o l d o v e r a l l s p a c e by a n a l y t i c a l c o n t i n u a t i o n . From t h e a b o v e we now know t h a t t h e form o f t h e W e i z s d c k e r c o r r e c t i o n T~ i s a l s o c o r r e c t i n t h e o u t - s i d e r e g i o n , b u t t h e c o e f f i c i e n t jumps a t t h e t u r n i n g p o i n t a n d 1s t h u s n o t a n a l y t i - c a l :

I f t h e f u n c t i o n a l e q . ( 5 1 ) i s u s e d t o d e r i v e a n E u l e r t y p e variational e q u a t i o n ( w h e r e V w i l l be a f u n c t i o n a l of p , t o o ) , t h i s d i s c o n t i n u i t y w i l l h a v e c o n s e q u e n c e s w h i c h w i l l b e d i s c u s s e d e l s e w h e r e . F o r t h e p r e s e n t we n o t e t h a t i n t h o s e a p p l i c a - t i o n s w h e r e t h e e n e r g y i s c a l c u l a t e d from e q . ( 5 1 ) w i t h a p a r a m e t r i z e d d e n s i t y p , i t h a s v e r y l i t t l e i n f l u e n c e w h e t h e r o n e u s e s 1 / 3 6 o r t h e c o r r e c t c o e f f i c i e n t 1/12 i n t h e o u t s i d e r e g i o n , s i n c e I n a l l r e a l i s t i c c a s e s t h e t u r n i n g ? o i n t l i e s r a t h e r f o r i n t h e t a i l o f t h e d e n s i t y . ( T y p i c a l l y , p i s r e d u c e d t o a few p e r m i l l e s o f i t s c e n t r a l v a l u e a t t h e t u r n i n g p o i n t , s e e a l s o f i g . 1 . ) We h a v e c h e c k e d t h a t t h e d i f f e r e n c e a m o u n t s t o less t h a n 1 MeV o u t o f a t o t a l k i n e t i c e n e r g y o f s e v e r a l t h o u s a n d M e V i n medium a n d h e a v y n u c l e i . .

F o r a n y f i n i t e t e m p e r a t u r e T , t h e f u n c t i o n a l s r 2 [ p ] e q . ( 3 7 ) a n d a 2 [ p ] e q . ( 3 8 ) a r e w e l l - b e h a v e d w i t h a n a l y t i c a l c o e f f i c i e n t s w h i c h c a n b e r a d i l y c o n p u t e d . They d e p e n d e x p l i c i t l y on t h e t e m p e r a t u r e t h r o u g h t h e r e l a t i o n n o ( p ) e q . ( 2 9 ) . As a n i l l u s t r a - t l o n , we show i n f i g . 1 t h e c o e f f i c i e n t s v ( q O ) a n d y ( q o ) = 5 ( u 0 )

-

- -

- - ^{T= 1} MeV

- --- _{T = 4 MeV}

- -

- -

*

I l l ,

- -

- -

- -

- -

-

-

-

- -

-

l 1 1 1 l l I I 1 1 1 1 , 1

Fig. 1 - Density p(r) and the coefficients y and v of the gradient corrections eq. (37) and rZ[p] eq. (38), plotted versus radius r at two temperatures: T ⁼ 1 MeV and T = 4 MeV. The arrow above the tail of the density indicates the location of the classical turnlng point (t.p.1.

we obtain the same TF ekpressions as above except that the mass m must be replaced everywhere by

(53) The gradient corrections become, after suitable partial ~ntegrations:

C6-22 JOURNAL DE PHYSIQUE

where

The s i m p l e s t form o f t h e f r e e e n e r g y d e n s i t y c o r r e c t i o n is

Note t l i a t when c o m p u t i n g t h e t o t a l ( f r e e ) e n e r g y w i t h a Skyrme f o r c e , t h e i n t e g r a l o v e r f ~ [ ~ j c o n t a i n s n o t o n l y t h e k i n e t i c e n e r g y , b u t a u t o m a t i c a l l y a l s o t h a t p a r t o f t h e p o t e n t i a l e n e r g y w h i c h c o n t a i n s ~ p . ( I n t h i s c a s e , t h e t e r m V p i n e q . ( 4 2 ) h a s t o b e r e p l a c e d b y t h e c o m p l e t e Skyrme p o t e n t i a l e n e r g y d e n s i t y m i n u s t h e ~p t e r m s . ) I t i s r e a d i l y s e e n t h a t i n s i d e t h e c l a s s i c a l l y a l l o w e d r e g i o n , t h e c o e f f i c i e n t s o f f ~ ~ e q . ( 5 4 ) r e d u c e t o t h e o n e s known i n t h e T = 0 l i m i t / 2 / . I n c l u d i n g a l s o a s p i n - o r b i t p o t e n t i a l o f t h e form

A

\-\so

-

i n t h e H a m i l t o n i a n , we c o r r e c t i o n s u p t o o r d e r o r b i t d e n s i t y ( s e e r e f .

f o u n d t h a t o n e r e c o v e r s t h e same form o f t h e s e m i c l a s s i c a l F i 2 a s i n t h e T = 0 c a s e / 2 / . E x p l i c i t l y , we g e t f o r t h e s p i n - / 1 8 / f o r t h e d e f i n i t i o n )

a n d f o r t h e s p i n - o r b i t c o n t r i b u t i o n t o t h e k i n e t i c e n e r g y d e n s i t y

s o t h a t t h e to:al s p i n - o r b i t c o n t r i b u t i o n t o t h e e n e r g y d e n s i t y i s ( u p t o s e c o n d o r d e r )

( s e e r e f s . / 2 , 3 / f o r t h e d e t a i l e d t r e a t m e n t o f t h e s p i n - o r b i t a n d e f f e c t i v e m a s s c o n t r i b u t i o n s i n t h e T = 0 c a s e . )

I n a r e c e n t p u b l i c a t i o n / 1 4 / w e h a v e s t u d i e d an a p p r o x i m a t i o n w h e r e t h e e x a c t TF e x p r e s s i o n s a t T > 0 w e r e u s e d t o g e t h e r w i t h t h e g r a d i e n t c o r r e c t i o n t e r m s % [ p ] a n d T , , [ ~ ] known from t h e T = 0 c a s e / 3 / ( s t r i c t l y v a l i d o n l y i n s i d e t h e c l a s s i c a l l y a l l o w e d r e g i o n ) . We f o u n d t h i s a p p r o x i m a t e f u n c t i o n a l t o g i v e a f a i r l y g o o d r e p r o - d u c t i o n o f t h e r e s u l t s f o u n d i n H a r t r e e - F o c k c a l c u l a t i o n s a t T > 0 . I t f a i l e d , h o w e v e r , i n s u c c e e d i n g a t t e m p t s t o c a l c u l a t e f i s s i o n b a r r i e r s . T h i s is m a i n l y d u e t o t h e f a c t t h a t no c o r r e c t i o n 0 2 [ p ] t o t h e e n t r o p y was i n c l u d e d . ( I t s f o r m c o u l d o f c o u r s e n o t b e g u e s s e d f r o m t h e T = 0 c a s e . ) T h i s c o r r e c t i o n , a n d t h e c o r r e c t c o e f f i c i e n t s o f a l l g r a d i e n t t e r m s , t u r n o u t t o b e c r u c i a l f o r c a l c u l a t i o n s o f

* ) deformation energies

.

We have used the above TETF functionals, including the "cold" corrertion T , , [ P ] , to compute the fission barrier of 2 4 0 ~ ~ using the Skyrme force SkM* /6,19/ at. various temperatures. For the details of the variational ETF calculations with parametrized trial densities, we refer to refs. /5,6/. Fig. 2 shows the results. The free deforn- ation energy is plotted versus the elongation parameter c ; ~t is normalized to zero at sphericity (c = 1.0) for each temperature. We note that up to T 1 MeV, the deformation energy does not change appreciably. The barrier disappears at Tz4.5 MeV.

Fig. 2 - Fission barriers for 2 4 0 ~ u obtainedfroma variational calculation using the TETF functionals eqs. (54) - ⁽⁵⁷⁾ and the Skyrme force SkM* /19/. c is the elongation parameter (half diameter along symmetry axis in units of spherical radius). See ref. /6/ for the definition of deformed diffuse densities. The numbers indicate the temperature (in MeV)

.

This "critical temperature" is appreciably higher than what has been estimated before using the low-T expansion /11,12/. This expansion is only valid if T < < ( A - V ) , which however is never the case in the outer part of the nuclear surface, even at small temperatures. The fact that this approximation leads to a too strong decrease of the surface energy with increasing temperature has already been noted /14/. In Fig. 3 we plot the barrier height Ef (relative to the spherical rinimum) versustemperature T.

The dashed line is the result of a corresponding calculation with the low-T-expanded functional, in which the only temperature dependent contribution is a term proportio- nal to p 1 l 3 T' (see e.g. refs. /6,13/). We see that it leads, indeed, to a drastic overestimation of the finite-temperature effect on the fission barrier height. The variation found with the present TETF functionals seems to be in accordance with the estimates based on HF calculations by Sauer et al. /20/, although an exact comparison cannot be made since a different Skyrme force has been used by these authors.

*) That approximate functionals can work well for spherical systems but fail for deformation energies, is known also in the T = 0 case /6/.

JOURNAL DE PHYSIQUE

5

E f

4

3

F i g . 3 - B a r r i e r h e i g h t o f 2 4 0 ~ u v e r s u s t e m p e r a t u r e T. S o l i d l i n e : TETF r e s u l t s from f i g u r e 2. Dashed l i n e : c o r r e s p o n d i n g r e s u l t o b t a i n e d w i t h t h e l o w - t e m p e r a t u r e ex- p a n d e d f u n c t i o r i a l s .

I n summary, we h a v e d e r i v e d f o r t h e f i r s t t i m e t h e T W F - f u n c t i o n a l s T [ P ~ a n d o [ p ] appropriate t o t h e f i n i t e t e m p e r a t u r e c a s e u p t o s e c o n d o r d e r t e r m s , i n c l u d i n g a l s o t?.e e f f e c t s o f a v a r i a b l e e f f e c t i v e mass. The c o n s i s t e n t t r e a t m e n t o f a l l 4 t h o r d e r c o r r e c t i o n s w i l l b e h o p e l e s s l y c o m p l i c a t e d , b u t t h e p r e s e n t r e s u l t s i n w h i c h t h e

" c o l d " c o r r e c t i o n ~ , + [ p ] was u s e d ( a n d a,[p] was n e g l e c t e d ) seem e n c o u r a g i n g . I t w i l l b e t h e o b j e c t o f F u r t h e r s t u d i e s t o t e s t t h e s e f u n c t i o n a l s a g a i n s t HF a n d o t h e r model r e s u l t s .

The a u t h o r i s g r a t e f u l t o J. B a r t e l a n d C. G u e t f o r e n c o u r a g i n g d i s c u s s i o n s a n d a c k n o w l e d g e s t h e u s e o f t h e ETF c o d e s d e v e l o p e d i n c o l l a b o r a t i o n w i t h C. Guet a n d H.-B. Hdkansson / 6 / .

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