Neutron Scattering

2 . The pair correlation function has the following asymptotic behaviour : For fixed
distance and t --> oc or fixed time and r --> oc the averages in equation (5 .47) can
be executed separately and in consequence
G (r, t) --> N J d 3 r' (p(r' - r, o)) (p(r', t)) = po . (5 .54)
3 . For t = 0 the operators commute and the convolution integral of equation (5 .47)
can be carried out :
G(r, 0) = N
E- (6 (r + Ei (0) - rj (0)) > . (5 .55)
i,j
For indistinguishable particles the relation to the static pair correlation function as defined
in (5 .6) can be drawn . Because of the identity of the particles we can set i = 1 in (5 .55)
and drop the average over i :
G(r, 0) = ~- (6 (r + rl(0) - rj (0)) > = S(r) + ~- (6 (r + r,(0) - r j (0)) > . (5 .56)
i
j 7~ l
We now consider the average number of particles 6N(r) in a volume SV at a vector
distance r from a given particle at rl .
It is obviously given by the integral over the second
term in the preceding expression which for small 6V can be written as
6N(r) = 6VZ (6 (r + rl (o) - rj(0))) . (5 .57)
j 7~ l
Using the definition (5 .6) and the expression for the number density in homogeneous
fluids (5 .2) one can relate 6N(r) also to the static pair correlation function :
SN(r) = po-q(r)6V . (5 .58)
Finally, by comparison of the last three equations we get a relation between the dynamic
correlation function at time zero and its static counterpart :
G(r, 0) = 6(r) + pog(r) . (5 .59)
This equation expresses the fact that the diffraction experiment (g(r)) gives an average
snapshot picture (G(r, 0)) of the sample .
In the classical approximation the operators commute always, especially also at different
times .
Then the integrals of equations (5 .45) and (5 .52) can be carried out and yield
G"(r,t) = N
E6(r- rj (t) +ri(0)) and (5 .60)
G9'(r, t) = N ~ ö (r - ri (t) + ri (0)) , (5 .61)
i
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