A MODEL FOR THE EXCHANGE IN LIQUID He3

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A MODEL FOR THE EXCHANGE IN LIQUID He3

B. Castaing

To cite this version:

B. Castaing. A MODEL FOR THE EXCHANGE IN LIQUID He3. Journal de Physique Colloques,

1980, 41 (C7), pp.C7-227-C7-229. �10.1051/jphyscol:1980736�. �jpa-00220173�

JOURNAL DE PHYSIQUE

CoZZoque

C 7 ,

suppl6ment au n07, Tome

41,

juiZZet

1980,

page

C 7 - 2 2 7

A MODEL FOR THE EXCHANGE IN LIQUID

~e~

B. C a s t a i n g

Groupe de Physique des SoZides de Z'EcoZe Normaze Supe'rieure, 24, rue Lhomond, 75231 Paris Cedex 05

~ Q s u m 6

-

On remarque que ~e~ l i q u i d e normal p e u t t t r e d g c r i t comme un ensemble de systsmes 2 deux niveaux qui s o n t l e s s p i n s des a t o n e s ( i l e s t donc suppose que, pour une r a i s o n inconnue, l e s Q t a t s

+

e t - de chaque s p i n o n t des g n e r g i e s d i f f g r e n t e s ) . Le r a p p o r t C / X T a i n s i obtenu e s t proche de s a v a l e u r expkri- mentale ( X : s u s c e p t i b i l i t 6 , C : c h a l e u r s ~ e c i f i q u e , p r o p o r t i o n n e l l e 3 l a temperature T ) .

A b s t r a c t - One remarks t h a t normal l i q u i d He3 can b e d e s c r i b e d a s a s e t of two l e v e l systems which a r e t h e s p i n s of atoms ( t h u s it i s assumed t h a t , f o r an unknown r e a s o n , + and

-

p o l a r i s a t i o n s o f each s p i n have d i f f e r e n t e n e r g i e s ) . The C /xT r a t i o t h u s o b t a i n e d i s c l o s e t o t h e experimental v a l u e (X : s u s c e p t i b i l i t y , C : s p e c i f i c h e a t , p r o p o r t i o n a l t o t h e temperature T).

INTRODUCTION

-

There e x i s t s a very r e l i a b l e des- n e t i c o r almost s o l i d . The paramagnons model / 2 / , c r i p t i o n of t h e Fermi l i q u i d s a t low temperature. which c o n s i d e r s it a s n e a r l y f e r r o m a g n e t i c , has It i s given by Landau's model

/ ? / .

It t e l l s us had many successes. It can p r e d i c t f o r example, t h a t t h e s p e c i f i c h e a t f o r example w i l l be l i n e a r w i t h o u t a d j u s t a b l e parameter, t h e temperature de- i n T a t low t e m p e r a t u r e , and w i l l depend only on pendence of t h e s u s c e p t i b i l i t y . However, a s t h e t h e d e n s i t y o f s t a t e s a t t h e Fermi l e v e l : p r e s s u r e i n c r e a s e s , l i q u i d He3 obviously t e n d s t c -

c

=

-

s2 N ( E ~ ) $ T

3 ( 1 ) ward s o l i d i f i c a t i o n . It would t h u s b e i n t e r e s t i c g

It t e l l s u s a l s o t h a t t h e s u s c e p t i b i l i t y i s con- t o understand i f t h e p i c t u r e of n e a r l y l o c a l i z e d s t a n t a t low temperature. I n t h e c a s e of an, i d e a l atoms i s compatible w i t h t h e main r e s u l t s o f t h e g a s t h e formula would b e

xid

⁼ L I ~ N ( E ~ ) where LI i s paramagnons model. A s a f i r s t s t e p , we w i l l pre- t h e magnetic momentum of t h e p a r t i c l e s . I n t h e gen- s e n t h e r e a model where we c o n s i d e r t h e ~e~ atoms e r a 1 c a s e a c o r r e c t i o n f a c t o r appears : a s n e a r l y l o c a l i z e d .

which i s c a l l e d t h e exchange f a c t o r , by r e f e r e n c e t o t h e Hartree-Fock t r e a t m e n t of t h e e l e c t r o n gas i n m e t a l s where such r e n o r m a l i s a t i o n o f t h e sus- c e p t i h i l i t y o c c u r s . I n f a c t , o b v i o u s l y , t h e whole Fermi l i q u i d behavior i s due t o exchanges between p a r t i c l e s .

The Landau model i s g e n e r a l and g i v e s no answer on t h e microscopic s i t u a t i o n . To go f u r t h e r , it i s n e c e s s a r y t o go i n t o t h e models which w i l l a l l o w us t o c a l c u l a t e t h e Landau parameters such a s F"

0 -

For example, it i s an o l d problem t o know i f l i q u i d ~e~ must b e c o n s i d e r e d a s almost ferromag-

1. The model

Let us f i r s t remark t h a t t h e l i n e a r dependence of t h e s p e c i f i c h e a t w i t h t h e t e m p e r a t u r e can be des- c r i b e d a s a g l a s s - l i k e behavior / 3 / , due t o t h e e x i s t e n c e of a s e t of two l e v e l systems. The q u e s t i o n of what t h e s e systems a r e i s answered b ~ - t h e behavior of t h e entropy. A t t h e degeneracy t e m p e r a t u r e TF, t h e l i q u i d entropy i s e q u a l t o t h e s o l i d one, which i s e n t i r e l y due t o t h e d i s o r d e r o f t h e s p i n s . Moreover, t h e c h a r a c t e r i s t i c temper- a t u r e f o r t h e non s p i n degrees of freedom ( t h a t i s t h e Debye t e m p e r a t u r e TD), i s much l a r g e r t d a n T

F' Thus it i s n a t u r a l t o assume t h a t t h e d e c r e a s e of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980736

JOURNAL DE P H Y S I Q U E

entropy a t low t e m p e r a t u r e i s e n t i r e l y due t o t h e p r o g r e s s i v e o r d e r i n g of t h e s p i n s . We w i l l t h u s assume t h a t t h e two-level systems we need a r e simply t h e two s p i n s t a t e s of each atom.

The assumptions o f t h e model a r e t h u s t h e follow- i n g ones :

a ) The

+

and

-

s p i n s t a t e s of each atom have d i f f - e r e n t e n e r g i e s

+

E and - E.

b ) I f f ( s ) d e i s t h e number of atoms f o r which E i s between E and E

+

d ~ , f ( 0 ) i s d i f f e r e n t from zero.

Without t h e second h y p o t h e s i s , a s we w i l l s e e , t h e s p e c i f i c h e a t would not be p r o p o r t i o n a l t o T. A s d i s c u s s e d i n t h e I n t r o d u c t i o n , we do not c o n s i d e r any o t h e r e x c i t a t i o n of t h e l i q u i d , n e g l e c t i n g t h e i r c o n t r i b u t i o n t o t h e entropy.

A s i m i l a r model h a s been i n t r o d u c e d by Blandin and F r i e d e l

/4/

i n o r d e r t o e x p l a i n t h e p r o p e r t i e s o f magnetic a l l o y s ( s p i n g l a s s e s ) . I n such a mean

f i e l d t h e o r y , t h e s p i n c o r r e l a t i o n s a r e n e g l e c t e d . The entropy i s t h u s t h e sum of t h e i n d i v i d u a l s p i n e n t r o p i e s i n t h e i r l o c a l " f i e l d " E :

S ( T ) = /+m f ( ~ ) S ( -5-- ) da k ~ T

where :

-m

~ ( x ) = k {en (2 cosh x )

-

x t a n h x}

B

A t low temperature :

2

=

5

f ( 0 ) k ; ~ = C ( 3 )

Within t h i s model we can a l s o c a l c u l a t e t h e magnet- i z a t i o n Y induced by a magnetic f i e l d H , and t h u s t h e s u s c e p t i b i l i t y . Neglecting i n a f i r s t s t e p t h e s h i f t of f ( e ) i n e n e r g i e s due t o t h e magnetiz- a t i o n , we can w r i t e :

E - ~ J H ) Nm = ?/I =

-

^1.1

/

de f ( ~ ) t a n h

kBT ^{( 4 )}

N i s t h e number o f atoms. A t low temperature :

x

= 2lJ2 f ( 0 )

P u t t i n g = To

,

we o b t a i n : 2kRf ( 0 )

By comparison between ( 5 ) , ( 6 ) and ( I ) , ( 2 ) we o b t a i n Fa =

-

0.75, which i s very c l o s e t o t h e ex- p e r i m e n t a l v a l u e , n e a r t h e s o l i d i f i c a t i o n p r e s s u r e . We can go f u r t h e r . Let us i n t r o d u c e a s h i f t i n energy of f ( ~ ) under magnetization. ( b ) and ( 6 ) be- come :

e-lJH+kB%m M = - l~

/

de f ( ~ ) t a n h (

kgT

⁾

and

Thus

Note t h a t % > 0 corresponds t o a s h i f t o f a n t i - ferromagnetic t y p e , a s it i s a c t u a l l y i n t h e s o l i d phase. I n t h e f i g u r e we g i v e t h e v a l u e s o f 8 we need i n o r d e r t o f i t t h e experimental d a t a / 5 / , / 6 / , v e r s u s t h e molar volume V i n l o g a r i t h m i c s c a l e s . The d i f f e r e n c e s between t h e two s e r i e s of d a t a a r e due t o t h e s m a l l number of p u b l i s h e d measurements of l i q u i d ~ e ~ s p e c i f i c h e a t / 7 / . Anyway, t h e y a r e i n agreement on t h e main p o i n t , which i s t h a t t h e '8 v a l u e a t high p r e s s u r e goes toward t h e NBel temperature o f t h e s o l i d .

Conclusion

Normal l i q u i d ~ e ~ i s t h u s very w e l l r e p r e s e n t e d by a " s p i n g l a s s " model. Obviously, such a model i s a l i t t l e f r u s t r a t i n g , because it g i v e s no account of t h e main p r o p e r t y of l i q u i d ~ ewhich ~ ,i s i t s f l u i d i t y . It s a y s only t h a t , a s f a r a s energy and s u s c e p t i b i l i t y a r e concerned, t h e q u a s i p a r t i c l e s l o o k very much l i k e l o c a l i z e d s p i n s . T h i s i s ob- v i o u s l y wrong i f we a r e i n t e r e s t e d i n momentum t r a n s f e r s .

References

1. See f o r i n s t a n c e P. NOZIERES, L e problsme 5 N c o r p s , Dunod (1963).

2. M.T. BEALMONOD

,

SHANG-KENG D. II- FIIEDKIN

,

Phys. Fev. L e t t e r s , 20, 929 (1968).

3. See f o r i n s t a n c e F. DOUSSINEAU, Rev. de Phys.

Appl.,

12,

809 (1977).

4.

A. BLANDIN and J . FRIEDEL, J . Phys. Radium, 20,

8 ,

5. H. RAMM, P. PEDBONI, J . R . THOWPSON, H. IDYET, 100

/

/

J o u r n a l of Lov Temp. Phys.,

2,

539 (1973).

6. J . C . WHEATLEY, The Helium L i q u i d s , Academic

30 P r e s s , p.310 (1975).

/ *

4

4

F i g u r e 1 - 0 dependence v e r s u s molar volume ( l o g -

LA ^,'

a r i t h m i c s c a l e s ) . These v a l u e s a r e com- /

/

puted from t h e d a t a o f r e f . ( 5 ) : and ( 6 ) : A . The N6el temperature of t h e s o l i d i s g i v e n by t h e f u l l l i n e and i t s

e x t r a p o l a t i o n by t h e dashed ?.ir,e. 25 30 3s

V~cdhle)