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
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
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
where
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
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 PDF can be calculated directly from physical measurements like light scattering or x-ray powder diffraction through the use of Fourier Transform.
In Statistical Mechanics the PDF is given by the expression:
Applications
The Diffpy project is used to match crystal structures with PDF data derived from X-ray or neutron diffraction data. The scientific journal Zeitschrift für Kristallographie – Crystalline Materials devoted a special issue in 2012 to the use of pair distribution function methods in crystallography.