Neutron Scattering

5 . Correlation Functions Measured By Scattering
Experiments
Reiner Zorn, Dieter Richter
In this lecture static and dynamic correlation functions will be introduced . We will start by
introducing probability densities, relate those to the scattering cross section and define the
pair correlation function . Two examples will be given, one from the physics of liquids and
one from polymer physics . The concept of dynamic correlation functions will be explained
firstly by the example of correlation spectroscopy . Then the Van Hove correlation function
will be introduced which is the basis for the calculation of the double differential scattering
cross section . Finally, the concept will be applied to the example of an ideal gas .
5 .1 Probability Densities
We start by considering a homogeneous monatomic liquid with N atoms in a volume V .
We denote the probability to find a certain atom in a volume element dar at r by P(r)d'r .
Because of the homogeneity P(r) is constant and evidently
Then the number density of atoms at r is
1
P (r) (5 .1)
V
P(r) = NP(r) = N (5 .2)
We call the probability density to find a certain atom at r i and another at r 2 P(ri, r 2 ) .
This function fulfills the basic relations
P(rl , r2) = P (r2, ri ) and (5 .3)
IV d3r2P(ri, r2) = P(rj - (5 .4)
If there is no interaction between the atoms P(rl, r2) factorizes into
P(r l ,r2 ) = P(r1)P(r2) (5 .5)