Pair distribution function

The pair distribution function 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 pair distribution function of b with respect to a, denoted by $g_{ab}({\vec {r}})$ is the probability of finding the particle b at 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 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 $f$ .

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