The **pair distribution function** describes the distribution of distances between pairs of particles contained within a given volume.^{[1]} Mathematically, if *a* and *b* are two particles in a fluid, the pair distribution function of *b* with respect to *a*, denoted by is the probability of finding the particle *b* at distance from *a*, with *a* taken as the origin of coordinates.

## Contents

## Overview

The pair distribution function is used to describe the distribution of 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 Dirac 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 distribution function

Of special practical importance is the radial 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 distribution function can be calculated directly from physical measurements like light scattering or x-ray powder diffraction by performing a Fourier Transform.

In Statistical Mechanics the PDF is given by the expression

## See also

## References

**^**"Pair Distribution Function (PDF) Analysis". Retrieved 2018-10-26.