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 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 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:

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

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]

References

↑ "DiffPy – Atomic Structure Analysis in Python". Brookhaven National Laboratory. Retrieved December 14, 2016.
↑ "Zeitschrift für Kristallographie – Crystalline Materials: Volume 227, Issue 5 (May 2012)". Walter de Gruyter. Retrieved December 14, 2016.
↑ Proffen, Thomas; Neder, Reinhard B. (2012). "Preface – Special Issue on Analysis of Complex Materials". Z. Kristallogr. Cryst. Mater. 227 (5): III–IV. doi:10.1524/zkri.2012.0002.

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