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TWO PARTICLE VERSUS THREE PARTICLE
CONTRIBUTIONS TO A DYNAMICAL
CORRELATION FUNCTION IN LIQUIDS
R. Brinkmann, M. Elwenspoek, P. Maxim, C. Paulick, D. Quitmann
To cite this version:
R. Brinkmann, M. Elwenspoek, P. Maxim, C. Paulick, D. Quitmann.
TWO
JOURNAL DE
PHYSIQUE
Colloque C9, supplément a u n012, Tome 46, décembre 1985 page c9-135
T W O PARTICLE VERSUS THREE PARTICLE CONTRIBUTIONS T O A DYNAMICAL
CORRELATION FUNCTION IN LIQUIDS
R. ~rinkmann*, M. ~ l w e n s ~ o e k * * , P. Maxim, C.A. Paulick and D. Quitmann
I n s t i t u t für Atom
- und Festkorperphysik, Freie Universitut Berlin,
ArnimaZZee 14, D-1000 BerLin 33, F.R.G.A b s t r a c t
-
That p a r t o f nuclear s p i n r e l a x a t i o n r a t e i n a l i q u i d which i s caused by e l e c t r i c i n t e r a c t i o n w i t h neighbour p a r t i c l e s , RQ, i s determined by t h e sum o f a two p a r t i c l e and a t h r e e p a r t i c l e c o r r e l a t i o n f u n c t i o n . I n order t o e x p l a i n t h e weak temperature dependence g e n e r a l l y found i n pure l i q u i d metals, we present and discuss a simple mode1 d e s c r i p t i o n f o r t h e t i m e development o f t h e f i r s t neighbour s h e l l . Two- and t h r e e - p a r t i c l e terms cancel i n l i q u i d metals f o r i n t e r m e d i a t e and l o n g times, l e a d i n g t o a s h o r t c o r r e - l a t i o n time. T h i s i s i n aqreement w i t h t h e general t r e n d o f RQ data and w i t h r e c e n t thorouqh t h e o r e t i c a l treatments o f a p a r t i c u l a r case.1
-
INTRODUCTIONWhile bondino i n t e r a c t i o n s o f each c o n s t i t u e n t atom w i t h i t s neighbours are strong, and thus m u l t i p l e c o r r e l a t i o n s a r e e s s e n t i a l , i n any l i q u i d o r amorphous system, i t i s r a t h e r f o r weak i n t e r a c t i o n s t h a t one can s i n g l e o u t one process and separate t h e c o n t r i b u t i o n s made by t w o - p a r t i c l e and by higher terms. One such case i s nuclear s p i n r e l a x a t i o n i n l i q u i d s /1/ by quadrupolar ( i . e . e l e c t r i c ) i n t e r a c t i o n w i t h neigh- bour p a r t i c l e s . I t i s caused by t h e f l u c t u a t i o n s o f t h e e l e c t r i c f i e l d g r a d i e n t ( e f ç ) a t t h e probe ( p ) n u c l e a r s i t e . The e f g leads t o a s h o r t ranged i n t e r a c t i o n between t h e probe's nuclear quadrupole moment Q and t h e neighbour i, c e r t a i n l y so i n m e t a l l i c , i.e. e l e c t r o n i c a l l y s h i e l d e d systems which w i l l concern us as examples. For t h e s i t u a t i o n i n i o n i c l i q u i d s , see e.g. /2/. One may then r e s t r i c t oneself t o t h e nea- r e s t neighbours. The e f g component
V;
i s a sum o f c o n t r i b u t i o n s over a l 1 (nearest) neighbours, and the nuclear s p i n r e l a x a t i o n r a t e R i s p r o p o r t i o n a l t o the time i n t e - g r a l o f t h e e f g c o r r e l a t i o n f u n c t i o n cefg:Q
**Wow at Kato,lieke Universiteit, Nijmegen *Now at DESY, Hamburg
JOURNAL DE PHYSIQUE
The sum over i runs over the neighbours, k i s the m u l t i p o l e order observed ( k = l i n NMR, k=2 i n TDPAD). We s h a l l assume t h a t t h e e f g f u n c t i o n v i ( r ) can be d e r i v e d
'P
from an e f f e c t i v e i n t e r a t o m i c p o t e n t i a l 2i(r. ) /3/, /4/. It i s t h e d i s t a n c e
r.
=1 P 1 P
R
-8. which through ( 1 )-
(3) occurs t w i c e i n R thus i n t r o d u c i n g f o u r p o s i t i o nP l
Q '
vectors
R
(O),Rp(t),
Ri(0), and Ri(t) o r R.(t). S i m i l a r expressions occur f o r broa-P J
denino of 4 f i o n i c l e v e l s , o r i n l i o h t s c a t t e r i n g from i n t e r a c t i o n p o l a r i z e d atoms, e.g. l i q u i d Ar.
I t i s unfavourable t h a t t h e d i r e c t i n f o r m a t i o n about 8 ' s o r r ' s i s averaged o u t i n t h e observable q u a n t i t y R t h i s beino a time and thereby a space i n t e g r a l . The ad-
Q'
vantage o f R i s however t h a t i t can be measured by NMR i n a u i t e a few l i a u i d s , v i z .
Q
i f they have a s p i n I > 1
-
nucleus, n o t a b l y l i q u i d metals / 5 / , /6/ l i k e Cu, Ga, Ge, Rb, I n , Sb, Hg, B i and a l l o y s o f these. Even more systems become accessible by u s i n g e x c i t e d nuclear States and TDPAD technique (see e.g. / 7 / ) . An i m p o r t a n t gene- r a l t r e n d p o i n t e d o u t by Titman e t a l . /6/ i s t h a t R % 1 / d i n a l 1 pure l i q u i d metalsQ
s t u d i e d so f a r .
For one model case, Rb, a thorough t h e o r e t i c a l treatment has r e c e n t l y been given by Bosse e t a l . /4/. Also, t h e e a r l i e r t h e o r i e s f o r R /8/ by S h o l l , and by Warren
Q
have very r e c e n t l y been improved by Gaskell and Woolfson /9/ i n such a way as t o r e c o n c i l e s h o r t - t i m e and long-time behaviour o f c ( t ) w h i l e s t i l l s t a r t i n g from
efg
t h e Kirkwood s u p e r p o s i t i o n approximation f o r t h e t h r e e p a r t i c l e c o r r e l a t i o n f u n c t i o n . Since e x a c t c a l c u l a t i o n s are r a t h e r involved, i t had appeared t o us t h a t f o r a p p l i - c a t i o n s and extensions i t m i g h t be h e l p f u l t o develop a simple, c l e a r and e a s i l y c a l c u l a b l e model d e s c r i p t i o n / I O / . Because o f t h e i s o t r o p y i n the l i q u i d , (2) becomes
where a ( t ) i s t h e angle subtended by (:
f .
(O),f .
( t )1.
( 5 ) i s a sum o f atm-
i p j 1 P J P
and t h r e e p a r t i c l e term. They compensate t o a l a r g e e x t e n t i f vi=v.=v as i n pure
J
K i s the number of nearest neiohbours. The d i s t r i b u t i o n function W(ro;r,t) gives the probability t o f i n d a neighbour atom a t time
t
between r and r+dr, given t h a t i t was betweenro
and r +dr a t t = O . go i s the f i r s t peak of the s t a t i c pair d i s t r i b u t i o nO O
function. Equs. ( 6 ) contain the often stressed point /8/, /9/ t h a t the power a t which g ( r ) appears i s d i f f e r e n t in
T2
andT3.
The model chosen ensures t h a tR
>O.Q
Apart from the anpular r e s t r i c t i o n mentioned, we t r e a t the dynamics of the neighbours independent from each other as the r e l a t i v e motion p, i in one dimension. According t o Haan / I l / a good approximation f o r the r e l a t i v e motion of two neighbour atoms a s described by U ( r o ; r , t ) i s obtained from Smoluchowski's diffusion equation
with an e f f e c t i v e potential $ ( r ) = -kBT l n I g ( r ) l , and a nonlinear time scale / I l / which we approximated by
~(t)=t(l-(l-exp(-~~t))/(6~t))
with ~ ~ ' 0 . 4 kBT/mD according t o t h e r e s u l t s of / I l / . The Smoluchowski equation was carefully integrated numerical- l y , and ( 6 ) was evaluated using various assumptions f o rG(r)
of the formn,
~ ( r ) = Z/r
.
exp(-xr) or ? ( r ) = Z { ( p / r I m-
m/n
.
( p / r ) n l . For p , the position of the maximum of g o ( r ) was taken, i n order t o s t r e s s compatibility of g ( r ) and ? ( r ) . The r e s u l t s show a c h a r a c t e r i s t i c difference in the time development of the two par- t i c l e and three p a r t i c l e terni, T 2 ( t ) and -T3(t) respectively, see f i g . 1 : Because the efg function v ( r ) decreases but $(r) increases steeply withr ,
and they appear e s s e n t i a l l y as rgv2
W> i n T 2 , but as <gv>.rgvW> in T g , T2 pives stronger weight t o small r ( < O ) . In t h i s region, r e l a t i v e motion i s f a s t separation, so t h a t T 2 ( t ) acquires an additional f a s t i n i t i a l decay which essential l y destroys the unevitable i n i t i a l excess of T2(0) over T3(0). Apart from t h a t , both T 2 ( t ) and T 3 ( t ) have a similar time dependence. Equality of the potential $ ( r ) andv(r)
f o r i and j makes T 2 ( t ) and -T3(t) coincide f o r larget
where they a r e diffusion controlled. The cha- r a c t e r i s t i c decay times T ~ , ~f T2,3(t)dt/T2,3(0) d i f f e r by an order of magnitude = from T = / f T 2 ( t ) + T3(t)1dt/
CT2(0) + T3(0) 1 , independent of ? ( r ) .Since R % /(T2
+
T3)dt, i t i s dominated by the short correlation time of the com-Q
JOURNAL D E PHYSIQUE
F i g . 1
-
Normalized e f g c o r r e l a t i o n functions : two p a r t i c l e c o n t r i b u t i o n T 2 ( t ) / T2(0), magnitude of t h r e e p a r t i c l e c o n t r i b u t i o n -T ( t ) / T ( 0 ) and c ( t ) /T (O) (dashes, dash dot, and f u l l l i n e , respectiveqy).
grame et ers
%f8
f o r l i q u i dG$
a t i t s m e l t i n g p o i n t T = 323 K, w i t h S ( r ) = Z / r exp(-2.0 r / f i ) . Note t h a t h e r eT ( 0 ) cancels more than 6M % of T2(0).
-
One o b t a i n s r3 = 1.98 ps, r = 0.36 psfar t h e c o r r e l a t i o n times.
mode c o u p l i n g theory and molecular dynamics, and w i t h a s i m i l a r c a l c u l a t i o n f o r Ga /12/. They a l s o agree w i t h t h e r e c e n t c a l c u l a t i o n s by Gaskell and Woolfson /9/ where i t t u r n s o u t again t h a t i n t h e pure l i q u i d s considered (Rb, Ne) t h e small s h o r t - t i m e s u r p l u s o f the t w o - p a r t i c l e over t h r e e - p a r t i c l e c o n t r i b u t i o n i s decisive. When spea- k i n g o f two p a r t i c l e term i t i s t o be k e p t i n mind t h a t here t h e d i s t a n c e vector, taken a t two times, enters, n o t the p o s i t i o n vectors.
-
The present curves f o r T 2 ( t ) , T 3 ( t ) a r e d e f i n i t e l y smoother than those o f Bosse e t a l . o r Gaskell and Woolfson probably because o f the use o f t h e Smoluchowski equation.F u r t h e r development o f t h e use o f R f o r s t u d y i n g dynamics i n 1 iq u i d s may s t a r t from
Q
the f a c t t h a t the r e l a t i v e importance o f T2 and T j can be varied. T h i s i s achieved by choosing p a r t n e r s A, B f o r which t h e e f o f u n c t i o n s vA,vB a r e d i f f e r e n t . Then an increase o f R over cA-RQA
+
cBSRQB develops which i s p r o p o r t i o n a l t o t h e t h r e e p a r -Q
t i c l e term
T3 /IO/.
T h i s p o i n t w i l l be d i scussed i n a subsequent paper (R. B r i n k - mann e t a l . t o be published). However, when A-B bonding becomes much s t r o n g e r than A-A, B-B bonding, R i s determined by the A-8, i.e. two p a r t i c l e c o r r e l a t i o n s /13/-Q
Fig. 2
-
Temperature dependence o f t h e c o r r e l a t i o n t i m e o f t h e e f c o r r e l a t i o n f u n c t i o n , w i t h parameters f o r l i q u i d Ga w i t h $ ( r ) = Z / r exp(-1.5 r d ) . For compari- son, T s ~ - 1 / 2 ( r e f . / 6 / ) and T s D-I (see r e f . / 8 / ) a r e included. (To complete the d i s - cussion of t h e temperature dependence o f RQ, t h e moderate increase o f cefg(0) w i t h T has t o be dncluded, see e.g. /4/.)This work was supported by Deutsche Forschungsgemeinschaft through Sfb 161.
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O.:
Phys. Rev. A28 (1983) 2459/5/ Carter, G.C., Bennett, L.H., and Kahan, D.J.: ~ e t a l l i c h i f t s i n NKR, Progr. Mat. S c i 20 (1977)
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R.L., and Titman, J.M.: J. Phys. F8 (1978) 1321 F I 0 (1980) 1589 -/7/ H a r t r o t t , M., Hohne, J., Quitmann, D., RoBbach, J., Weihreter, E., and Willeke, F.: Phys. Rev. BI7 (1979) 3449
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(1984) 5087/ I O / Brinkmann, R., Elwenspoek, M., Maxim, P., and Quitmann, D. : Hyperfine I n t e r . 15/16 (1983) 581; Brinkmann, R.: Thesis F r e i e U n i v e r s i t a t B e r l i n (1983),(unpu- 6TisTied)
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fl
(1980) C8-378; Hyper- f i n e I n t e r . 10 (1981) 1035/13/ Elwenspoek,
