COOPERATIVE IONIC MOTION IN LIQUID METALS

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COOPERATIVE IONIC MOTION IN LIQUID

METALS

T. Gaskell, P. Mason

To cite this version:

JOURNAL DE PHYSIQUE ColZoque C8, suppldment

au

n08, Tome 41,

aoCt

1980,

page

C8-330

COOPERATIVE I O N I C M O T I O N

I N

L I Q U I D

METALS

T. Gaskell and P.E. Mason

P h y s i c s Department, t h e U n i v e r s i t y , S h e f f i e l d , U.K. 1

.

ZNTRODUCTION Theorles of v e l o c l t y c o r r e l a t i o n s i n l i q u i d s have i n v a r i a b l y been r e s t r i c t e d t o t h e a u t o c o r r e l a t i o n f u n c t i o n of e i t h e r t h e s i n g l e p a r t i c l e v e l o c i t y o r a F o u r i e r component of t h e c u r r e n t d e n s i t y . Both a r e important f o r a number of r e a s o n s , n o t l e a s t of which i s t h a t t h e y a r e i n p r i n c i p l e a c c e s s i b l e through n e u t r o n s c a t t e r i n g experiments. However, t h e y do n o t n e c e s s a r i l y c o n t a i n a l l t h e u s e f u l i n f ormat l o n one might seek from a knowledge of t h e way i n

which t h e v e l o c i t i e s of atoms i n a l i q u i d a r e c o r r e l a t e d . We demonstrate t h i s by u s i n g a configuration-dependent v e l o c i t y f i e l d t o d i s c u s s

t h e way i n which t h e i n i t i a l momentum of an atom i n t h e l l q u i d i s t r a n s f e r r e d t o t h e surrounding medium. T h i s approach may be used t o d e r i v e an

expression f o r t h e v e l o c l t y a u t o c o r r e l a t i o n f u n c t i c n , b u t a l s o a l l o w s f o r cooperative e f f e c t s , such a s t h e coherent motion w i t h i n a c l u s t e r of atoms, t o be mvestiga-ked. The v e l o c i t y f i e l d ,

2*(x,t)

,

which t a k e s i n t o account t h e f i n i t e s l z e of t h e atoms, i s r e p r e s e n t e d by an e x p r e s s i o n of t h e form f

t

=

C?L;(t)

f(~:-:~:;(t)l)

1% i s

k ( - ,

argued t h a t t h e v e l o c i t y f i e l d should be e s s e n t i a l l y c o n s t a n t a c r o s s

an

atomic d i a m e t e r

and

f (f)

1s chosen a c c o r d i n g l y . ' I n a d d i t i o n it should s a t i s f y t h e c o n d i t i o n ,

where

n

i s t h e number d e n s i t y , which g u a r a n t e e s

t h e c o r r e c t s t r u c t u r e of t h e v e l o c i t y f i e l d i n t h e hydrodynamic l i m i t . Momentum t r a n s f e r may be c o n v e n i e n t l y c o n s i d e r e d through t h e time-dependent c o r r e l a t i o n of t h e v e l o c i t y of an atom ( l a b e l l e d 1 )

a t

t

= O

,

w i t h t h a t of t h e c e n t r e of mas: of a group of atoms w i t h i n a sphere whose c e n t r e i s t h e l o c a t i o n of t h e same atom a t some l a t e r time. This c o r r e l a t i o n f u n c t i o n c l e a r l y i n c l u d e s a

c o n t r i b u t i o n from t h e a u t o c o r r e l a t i o n term, and a s t h e l a t t e r decays with ti^, t h e c o r r e l a t i o n of t h e i n t i t i a l v e l o c i t y of atom 1 w i t h t h e vel-ocity of t h e surrounding s h e l l s of atoms b u i l d s up from z e r o w l t h t h e t r a n s f e r of t h e i n i t i a l momentum of p a r t i c l e 1 t o t h e surrounding medlum. I n a group of f i n i t e s i z e t h e l a t t e r correlation w l l l i t s e l f e v e n t u a l l y decay as t h e momentum t r a n s f e r spreads. The c h a r a c t e r i s t i c f e a t u r e s of such c o r r e l a t i o n s w i l l give some i n s i g h t i n t o any s t r o n g l y c o r r e l a t e d motlon which may e x i s t w i t h i n

a c l u s t e r .

2.

THEORY

The atomic v e l o c i t i e s a r e o b t a i n e d by e v a l u a t i n g t h e v e l o c i t y f i e l d at t h e l o c a t i o n of t h e atom. Hence, b e a r i n g i n mind ( 1

)

,

t h e sun of t h e atomic v e l o c i t i e s ,

s(b)

,

w i t h i n a sphere of r a d i u s

R

c e n t r e d a t

Ti(t)

1s o b t a i n e d by i n t e g r a t i o n of t h e v e l o c i t y f i e l d . Theref o r e

I'igure

3.

The s o l i d curve shows

(1.5

U,

t)

minus t h e a u t o c o r r e l a t i o n term, f o r rubidium. The open c i r c l e s r e p r e s e n t t h e v e l o c i t y a u t o c o r r e l a t i o n f u n c t i o n

q ( t )

.

!

t r a n s f e r r e d t o t h e i o n s i n t h e f i r s t s h e l l v a r i e s

w i t h time

in

much t h e same way as t h e

a u t o c o r r e l a t i o n f u n c t i o n . T h i s i s more c l e a r l y demonstrated when t h e a u t o c o r r e l a t i o n c o n t r i b u t i o n

i s s u b t r a c t e d from

Qd(I.50)t).

The r e s u l t i s shown i n f i g u r e

3.

It s t r o n g l y s u g g e s t s a

p h y s i c a l p i c t u r e of an i o n rebounding from t h e

f i r s t s h e l l , w i t h t h e c e n t r e of mass of t h e s h e l l i t s e l f i n o s c i l l a t o r y motion, though o u t of phase

w i t h t h e i o n a t i t s c e n t r e . It should be remembered t h a t i n t h i s approach t h e c o r r e l a t i o n of p a r t i c l e 1 i s w i t h t h a t of t h e c e n t r e of mass

of t h e n e a r e s t neighbours. Consequently, our p r e v i o u s statement should n o t be t a k e n t o imply a

s h e l l of i o n s moving a s a r i g i d u n i t . We would Deed t o see t h e a n g u l a r c o r r e l a t i o n w i t h r e s p e c t J;o c e n t r a l p a r t i c l e ' s i n i t i a l d i r e c t i o n of motion t o t e s t t h a t concept. N e v e r t h e l e s s , t h e r e Bppears t o be s t r o n g l y c o o r d i n a t e d behaviour. p g u r e

4.

The c u r v e s show r e s u l t s f o r I$

( R ,

t )

hf

!in a r i g i d

-

sphere f l u i d . The open c l r o l e s

/

r e p r e s e n t t h e v e l o c i t y a u t o c o r r e l a t i o n f u n c t i o n

Y(6

Facking f r a c t i o n 0.4628,

k

in mean collis;oi? time

.

_,

Although qd(2.5G,b)a1so d i s p l a y s an o s c i l l a t o r y decay, it i s c l e a r t h a t

$,

(2.5

u ) t )

-

$N(1.5c,t)

w i l l n o t show any s i g n i f i c a n t o s c i l l a t o r y p a t t e r n . T h i s i s presumably because of t h e c o n s i d e r a b l e combined i n e r t i a l mass of t h e second s h e l l neighbours. Contrast t h e s e r e s u l t s w i t h t h o s e i n

f i g u r e

4.

When R i n c l u d e s t h e n e a r neighbours,

H,(g,t)

shows no t r a c e of t h e n e g a t i v e r e g i o n

p r e s e n t i n

y(r(t).

T h i s emphasi s e s t h e caging e f f e c t i n Rb. There, we argue t h a t a s an i o n c o l l i d e s w i t h

a neighbour t h e momentum t r a n s f e r , because o f t h e s t r o n g cohesion, i s w i t h a s i g n i f i c a n t p a r t of t h e s h e l l , c a u s i n g t h e i o n t o rebound from t h e g r e a t e r mass. T h i s e f f e c t i s a b s e n t w i t h r i g i d s p h e r e s , c o n s i d e r a b l y reducing t h e p r o b a b i l i t y of a rebound.

REFERENCES

1 GASKELL,T and

MILLER,S

,J .F'hys. C . ~ ( I 978) 3749

2 4839

3

FlDKMAN,A,Phys.Rev.Ag (1974)

1667