The pair distribution function describes the distribution of distances between pairs of particles contained within a given volume.[1] Mathematically, if a and b are two particles, the pair distribution function of b with respect to a, denoted by
gab(\vec{r})
\vec{r}
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
V
g(\vec{r},\vec{r}')
N
g(\vec{r},\vec{r}')=p(\vec{r},\vec{r}')V2
| N-1 | |
| N |
g(\vec{r},\vec{r}') ≈ p(\vec{r},\vec{r}')V2
The simplest possible pair distribution function assumes that all object locations are mutually independent, giving:
g(\vec{r})=1
\vec{r}
g(r)=\begin{cases} 0,&r<b,\\ 1,&r\geq{}b \end{cases},
b
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
n
g(r)=\sum\limitsi\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\limitsr\toinftyg(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
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 distribution function can be calculated directly from physical measurements like light scattering or x-ray powder diffraction by performing a Fourier Transform.
In Statistical Mechanics the PDF is given by the expression
gab(r)=
| 1 | |
| NaNb |
| Na | |
| \sum\limits | |
| i=1 |
| Nb | |
| \sum\limits | |
| j=1 |
\langle\delta(\vertrij\vert-r)\rangle
When thin films are disordered, as they are in electronic devices, pair distribution is used to view the strain and structure-properties of that material or composition. They have these properties that cannot be exploited in the bulk or crystalline form. There is a method with the radial distribution that is able to view the local structure of a disordered thin film of
Fischer-Colbrie, Bienenstock, Fuoss, Marcus. Phys. Rev. B (1988) 38, 12388
Jensen, K. M., Billinge, S. J. (2015). IUCrJ, 2(5), 481-489.