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 ${\displaystyle g_{ab}({\vec {r}})}$ is the probability of finding the particle b at the distance ${\displaystyle {\vec {r}}}$ 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 ${\displaystyle {\vec {r}}}$:

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

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

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

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

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

Simple models and general properties

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

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

where ${\displaystyle {\vec {r}}}$ 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:

${\displaystyle g(r)={\begin{cases}0,&r

where ${\displaystyle b}$ 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 ${\displaystyle r=nb}$ where ${\displaystyle n}$ 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:

${\displaystyle g(r)=\sum \limits _{i}\delta (r-ib)}$.

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,

${\displaystyle \lim \limits _{r\to \infty }g(r)=1}$.

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 ${\displaystyle f}$.

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:

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

Applications

The Diffpy project[1] 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[2] to the use of pair distribution function methods in crystallography.[3]