- Overview
- Simple models and general properties
- Radial distribution function
- See also
- References
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 in a fluid, the pair distribution function of b with respect to a, denoted by 
is the probability of finding the particle b at distance 
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 :
,
where 
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 
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:
.
Simple models and general properties
The simplest possible pair distribution function assumes that all object locations are mutually independent, giving:
,
where 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:
where 
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 
where 
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:
.
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,
.
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 
.
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 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
See also
- classical-map hypernetted-chain method
References
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- Waslala
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- Wasmuth Portfolio
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- Wasp-class amphibious assault ship
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