Neutron Scattering

espectively. The former equation expresses the probability to find any particle at a time
t in a distance r from another at time 0 .
The latter equation denotes this probability
for the saure particle .
It therefore depends only on the particle's displacement during a
time interval Ari (t) = r i (t) - ri (0) leading to a simple expression for the intermediate
incoherent scattering function :
Sn, (Q, t) = NT E ( exp ( -iQ - Ari (t)) i
In certain cases this expression can be further simplified using the "Gaussian approximation"
7 :
(5 .63)
Sncuss (Q,t) = eXp -1 Q2 (Ar2 (t)))
(5 .62)
Here (Ar 2(t)) is
the average mean squared displacement which often follows simple laws,
. e .g . (Ar 2 (t)) = 6Dt for simple diffusion Because one of the prerequisites of the Gaussian
approximation is that all particles move statistically in the Same way (dynamic homogeneity)
the particle average and the index i vanish . An analogous expression can be
derived for the coherent scattering .
In order to decide whether the classical approximation can be used the following rule has
to be taken into account :
Quantum effects play a rôle if the distance of two particles is of
the order of the DeBroglie wavelength AB = / 2m S~kBT or if the times considered are
smaller than hlkBT .
Figure 5 .10 schematically shows on the left side the behaviour of the correlation functions
G(r, t) and G,(r, t) for a simple liquid (in classical approximation) . On the right side the
corresponding intermediate scattering functions Scon(Q, t) and Sn,(Q, t) are displayed :
a For t = 0 the self correlation function is given by a delta function at r = 0 . The pair
correlation function follows the static correlation function g(r) .
The intermediate
scattering functions are constant one for the incoherent and the static structure
factor for the coherent .
e For intermediate times the self correlation function broadens to a bell-shaped function
while the pair correlation function loses its structure .
The intermediate scat-
For solids the long-time limit of this equation is called the Lamb-Mößbauer factor .
Its coherent counterpart is the Debye-Waller factor .
5- 19