In statistical mechanics, the radial distribution function, (or pair correlation function in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle.

If a given particle is taken to be at the origin O from O. This simplified definition holds for a homogeneous and isotropic system. A more general case will be considered below.

away from a given reference particle, relative to that for an ideal gas away from a particle. This general theme is depicted to the right, where the red particle is our reference particle, and the blue particles are those whose centers are within the circular shell, dotted in orange.

.

Given a potential energy function, the radial distribution function can be computed either via computer simulation methods like the Monte Carlo method, or via the Ornstein–Zernike equation, using approximative closure relations like the Percus–Yevick approximation or the hypernetted-chain theory. It can also be determined experimentally, by radiation scattering techniques or by direct visualization for large enough (micrometer-sized) particles via traditional or confocal microscopy.

The radial distribution function is of fundamental importance since it can be used, using the Kirkwood–Buff solution theory, to link the microscopic details to macroscopic properties. Moreover, by the reversion of the Kirkwood–Buff theory, it is possible to attain the microscopic details of the radial distribution function from the macroscopic properties. The radial distribution function may also be inverted to predict the potential energy function using the Ornstein–Zernike equation or structure-optimized potential refinement.

Definition

(for an average number density is the Boltzmann constant. The potential energy and we do not consider the case of an externally applied field.

The appropriate averages are taken in the canonical ensemble, etc. is given by

particles. To this end, one has to integrate (:

1. .

The probability that those positions ARE occupied is found by summing over all configurations in which a particle is at each of those locations. This can be done by taking every permutation, in the symmetric groupThis quantity is called the n-particle density function. For indistinguishableIntegrating the n-particle density gives the permutation factor.

, we have the one-particle density. For a crystal it is a periodic function everywhere else. The partition function in this case is

from which the definition gives the desired result

. With this in hand, the n-point correlation is defined by factoring out the non-interacting contributionwhere it is clear that the n-point correlation function is dimensionless.

Relations involving g(r)

Structure factor

is of special importance, as it is directly related (via a Fourier transform) to the structure factor of the system and can thus be determined experimentally using X-ray diffraction or neutron diffraction.

is the average.

the ensemble average, yielding:

, since we have:

and thus:

1. , proving the Fourier relation alluded to above.

This equation is only valid in the sense of distributions, leading to a Dirac peak at the origin for the structure factor. Since this contribution is inaccessible experimentally we can subtract it from the equation above and redefine the structure factor as a regular function:

1. .

and, if the system is a liquid, we can invoke its isotropy:

Compressibility equation

Evaluating ( and using the relation between the isothermal compressibility and the structure factor at the origin yields the compressibility equation:

Potential of mean force

It can be shown that the radial distribution function is related to the two-particle potential of mean force by: .

Energy equation

, the average internal energy per particle is:

Pressure equation of state

Developing the virial equation yields the pressure equation of state:

Thermodynamic properties in 3D

The radial distribution function is an important measure because several key thermodynamic properties, such as potential energy and pressure can be calculated from it.

For a 3-D system where particles interact via pairwise potentials, the potential energy of the system can be calculated as follows:

where N is the pair potential.

The pressure of the system can also be calculated by relating the 2nd virial coefficient. The pressure can be calculated as follows:

1. .

.

Approximations

engendered by the reference particle, neglecting indirect effects. In the first approximation, it is thus simply given by the Boltzmann distribution law:

: the presence of a particle at O).

experiences not only the interaction with the particle at O, which resembles the virial equation:

This similarity is not accidental; indeed, substituting () in the relations above for the thermodynamic parameters (Equations , and ) yields the corresponding virial expansions. The auxiliary function is known as the cavity distribution function. under equilibrium is unique up to an additive constant, if it exists.

In recent years, some attention has been given to develop pair correlation functions for spatially-discrete data such as lattices or networks.

Experimental

) using neutron scattering or x-ray scattering data. The technique can be used at very short length scales (down to the atomic level) but involves significant space and time averaging (over the sample size and the acquisition time, respectively). In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals to charged colloids. is not straightforward and the analysis can be quite involved.

directly by extracting particle positions from traditional or confocal microscopy. This technique is limited to particles large enough for optical detection (in the micrometer range), but it has the advantage of being time-resolved so that, aside from the statical information, it also gives access to dynamical parameters (e.g. diffusion constants) and also space-resolved (to the level of the individual particle), allowing it to reveal the morphology and dynamics of local structures in colloidal crystals, glasses, gels, and hydrodynamic interactions.

Direct visualization of a full (distance-dependent and angle-dependent) pair correlation function was achieved by a scanning tunneling microscopy in the case of 2D molecular gases.

Higher-order correlation functions

It has been noted that radial distribution functions alone are insufficient to characterize structural information. Distinct point processes may possess identical or practically indistinguishable radial distribution functions, known as the degeneracy problem. In such cases, higher order correlation functions are needed to further describe the structure.

were less studied, since they are generally less important for the thermodynamics of the system; at the same time, they are not accessible by conventional scattering techniques. They can however be measured by coherent X-ray scattering and are interesting insofar as they can reveal local symmetries in disordered systems.

See also

• Ornstein–Zernike equation

• Structure Factor

References

• Widom, B. (2002). Statistical Mechanics: A Concise Introduction for Chemists. Cambridge University Press.

• McQuarrie, D. A. (1976). Statistical Mechanics. HarperCollins Publishers.