Pair distribution function

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 $g_{ab}(\vec{r})$ is the probability of finding the particle b at the distance $\vec{r}$ from a, with a taken as the origin of coordinates.

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 $\vec{r}$:

$p(\vec{r})=1/V$,

where $V$ 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 $g(\vec{r},\vec{r'})$ is obtained by scaling the two-body probability density function by the total number of objects $N$ and the size of the container:

$g(\vec{r}, \vec{r}') = p(\vec{r},\vec{r}') V^2 \frac{N-1}{N}$.

In the common case where the number of objects in the container is large, this simplifies to give:

$g(\vec{r}, \vec{r}') \approx p(\vec{r},\vec{r}') V^2$.

Simple models and general properties

The simplest possible pair distribution function assumes that all object locations are mutually independent, giving:

$g(\vec{r})=1$,

where $\vec{r}$ 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:

$g(r) = \begin{cases} 0,&r<b,\\ 1,&r\geq{}b \end{cases},$

where $b$ 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 $r=nb$ where $n$ 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:

$g(r)=\sum\limits_i\delta(r-ib)$.

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,

$\lim\limits_{r\to\infty}g(r) = 1$.

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 $f$.

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

In Statistical Mechanics the PDF is given by the expression:

$g_{ab}(r) = \frac{1}{N_{a} N_b}\sum\limits_{i=1}^{N_a} \sum\limits_{j=1}^{N_b} \langle \delta( \vert \mathbf{r}_{ij} \vert -r)\rangle$

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

See also

• Radial distribution function
• classical-map hypernetted-chain method