The pair distribution function (PDF) describes the distribution of distances between pairs of particles contained within a given volume. Mathematically, if a and b are two particles in a fluid, the PDF of b with respect to a, denoted by is the probability of finding the particle b at the distance from a, with a taken as the origin of coordinates.

Overview

The pair distribution function is used to describe the distribution of a objects within a medium (for example, oranges in a crate or nitrogen molecules in a gas cylinder). If the medium is homogeneous (i.e. every spatial location has identical properties), then there is an equal probability density for finding an object at any position :,
where is the volume of the container. On the other hand, the likelihood of finding pairs of objects at given positions (i.e. the two-body probability density) is not uniform. For example, pairs of hard balls must be separated by at least the diameter of a ball. The pair distribution function is obtained by scaling the two-body probability density function by the total number of objects and the size of the container:.
In the common case where the number of objects in the container is large, this simplifies to give:.

Simple models and general properties

The simplest possible pair distribution function assumes that all object locations are mutually independent, giving:,
where is the separation between a pair of objects. However, this is inaccurate in the case of hard objects as discussed above, because it does not account for the minimum separation required between objects. The hole-correction (HC) approximation provides a better model:
where is the diameter of one of the objects.

Although the HC approximation gives a reasonable description of sparsely packed objects, it breaks down for dense packing. This may be illustrated by considering a box completely filled by identical hard balls so that each ball touches its neighbours. In this case, every pair of balls in the box is separated by a distance of exactly where is a positive whole number. The pair distribution for a volume completely filled by hard spheres is therefore a set of delta functions of the form:.

Finally, it may be noted that a pair of objects which are separated by a large distance have no influence on each other's position (provided that the container is not completely filled). Therefore,.

In general, a pair distribution function will take a form somewhere between the sparsely packed (HC approximation) and the densely packed (delta function) models, depending on the packing density .

Radial pair distributions

Of special practical importance is the radial pair distribution function, which is independent of orientation. It is a major descriptor for the atomic structure of amorphous materials (glasses, polymers) and liquids. The radial PDF can be calculated directly from physical measurements like light scattering

Light scattering

Light scattering is a form of scattering in which light is the form of propagating energy which is scattered. Light scattering can be thought of as the deflection of a ray from a straight path, for example by irregularities in the propagation medium, particles, or in the interface between two media...

 or x-ray powder diffraction through the use of Fourier Transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

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In Statistical Mechanics the PDF is given by the expression:



The Diffpy project is used to match crystal structures with PDF data derived from X-ray or neutron diffraction data.