[1] Mathematically, if a and b are two particles, the pair distribution function of b with respect to a, denoted by
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
On the other hand, the likelihood of finding pairs of objects at given positions (i.e. the two-body probability density) is not uniform.
is obtained by scaling the two-body probability density function by the total number of objects
and the size of the container: In the common case where the number of objects in the container is large, this simplifies to give: The simplest possible pair distribution function assumes that all object locations are mutually independent, giving: where
The hole-correction (HC) approximation provides a better model: where
In this case, every pair of balls in the box is separated by a distance of exactly
The pair distribution for a volume completely filled by hard spheres is therefore a set of Dirac delta functions of the form: 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).
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.
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
The creation of thin-film Pair Distribution Function (tfPDF) uses a statistical distribution of a material’s mid-range order that enables viewing important details like the disorder.
TfPDF works best when in conjunction with other characterization methods like transmission electron microscopy.
Although a developing methodology, tfPDF can give complete structure-property relationships through a reliable characterization technique.