Radial distribution function - AbsoluteAstronomy.com

Statistical mechanics

Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, a radial distribution function (RDF), (or pair correlation function) g(r), describes how the atomic density varies as a function of the distance from one particular atom.

More precisely, if there is an atom at the origin O, and if n = N/V is the average number density, then the local density at distance r from O is ng(r).
Given a potential energy

Potential energy

In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

function, the radial distribution function can be found either via computer simulation methods like the Monte Carlo method

Monte Carlo method

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

, or via the Ornstein-Zernike equation, using approximative closure relations like the Perckus-Yevick approximation or the Hypernetted Chain Theory

Hypernetted-chain equation

In statistical mechanics the hypernetted-chain equation is a closure relation to solve the Ornstein–Zernike equation which relates the direct correlation function to the total correlation function. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function...

Importance of g(r)

g(r) is of fundamental importance in thermodynamics because macroscopic thermodynamic quantities can be calculated using g(r). A few examples:

The virial equation for the pressure:

The energy equation:

The compressibility equation
Compressibility equation
In statistical mechanics and thermodynamics the compressibility equation refers to an equation which relates the isothermal compressibility to the structure of the liquid...

:

Gases

As long as the gas is dilute, the correlations in the positions of the atoms that g(r) takes into account are due to the potential

(r) that an atom at P feels owing to the presence of an atom at O. Using the Boltzmann distribution law:

were zero for all r – i.e., if the atoms did not exert any influence on each other, then g(r) = 1 for all r. Then the mean local density would be equal to the mean density

: the presence of an atom at O would not influence the presence or absence of any other atom and the gas would be ideal. As long as there is a

the mean local density will always be different from the mean density

due to the interactions between atoms.

When the density of the gas gets higher, the low-density limit is not applicable anymore because the atoms attracted to and repelled by the atom at O also repel and attract each other. The correction terms needed to correctly describe g(r) resemble the virial equation, which is an expansion in the density:

in which additional functions

appear which may depend on temperature

and distance

but not on density,

Experimental

It is possible to measure g(r) experimentally with neutron scattering

Neutron scattering

Neutron scattering,the scattering of free neutrons by matter,is a physical processand an experimental technique using this processfor the investigation of materials.Neutron scattering as a physical process is of primordial importance...

or x-ray scattering diffraction data. For an example of an RDF experiment see Eigen vs. Zundel structures in HCl solution, 2006

It is additionally possible to calculate g(r) experimentally by extracting particle positions from video or confocal microscopy, as detailed in:

Formal derivation

Consider a system of N particles in a volume V and at a temperature T. The probability of finding atom 1 in

, atom 2 in

, etc., is given by

where

and

is the configurational integral. To obtain the probability of finding atom 1 in

and atom n in

, irrespective of the remaining N − n atoms, one has to integrate (7) over the coordinates of atoms n + 1 through N:

Now the probability that any atom is in

and any atom in

, irrespective of the rest of the atoms, is

For n = 1 the one-particle distribution function is obtained which, for a crystal, is a periodic function with sharp maxima at the lattice sites. For a (homogeneous) liquid:

It is now time to introduce a correlation function

is called a correlation function, since if the atoms are independent from each other

would simply equal

and therefore

corrects for the correlation between atoms.

From (9) it can be shown that

In the theory of liquids

is of special importance for it can be determined experimentally using X-ray diffraction. If the liquid contains spherically symmetric atoms

depends only on the relative distance between atoms,

. People usually drop the subscripts:

. Now

is the probability of finding an atom at r given that there is an atom at the origin of r. Note that this probability is not normalized:

In fact, this equation gives us the number of atoms between r and r + dr about a central atom.

Currently, information on how to obtain the higher order distribution functions