The pair correlation function
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## The pair correlation function

Of particular importance is the case n=2, or the correlation function known as the pair correlation function. The explicit expression for is

In general, for homogeneous systems in equilibrium, there are no special points in space, so that should depend only on the relative position of the particles or the difference . In this case, it proves useful to introduce the change of variables

Then, we obtain a new function , a function of and :

In general, we are only interested in the dependence on . Thus, we integrate this expression over and obtain a new correlation function defined by

For an isotropic system such as a liquid or gas, where there is no preferred direction in space, only the maginitude or , is of relevance. Thus, we seek a choice of coordinates that involves r explicitly. The spherical-polar coordinates of the vector is the most natural choice. If then the spherical polar coordinates are

where and are the polar and azimuthal angles, respectively. Also, note that

where

Thus, the function g(r) that depends only on the distance r between two particles is defined to be

Integrating g(r) over the radial dependence, one finds that

The function g(r) is important for many reasons. It tells us about the structure of complex, isotropic systems, as we will see below, it determines the thermodynamic quantities at the level of the pair potential approximation, and it can be measured in neutron and X-ray diffraction experiments. In such experiments, one observes the scattering of neutrons or X-rays from a particular sample. If a detector is placed at an angle from the wave-vector direction of an incident beam of particles, then the intensity that one observes is proportional to the structure factor

where is the vector difference in the wave vector between the incident and scattered neutrons or X-rays (since neutrons and X-rays are quantum mechanical particles, they must be represented by plane waves of the form ). By computing the ensemble average (see problem 4 of problem set #5), one finds that and S(k) is given by

Thus, if one can measure S(k), g(r) can be determined by Fourier transformation.

Next: Thermodynamic quantities in terms Up: Distribution functions in classical Previous: General correlation functions

Mark Tuckerman
Tue Feb 22 19:18:57 EST 2000