Radial distribution functions

The Radial Distribution Function, R.D.F. , g(r), also called pair distribution function or pair correlation function, is an important structural characteristic, therefore computed by Atomes.

Space discretization for the evaluation of the radial distribution function.

Considering a homogeneous distribution of the atoms/molecules in space, the g(r) represents the probability to find an atom in a shell dr at the distance r of another atom chosen as a reference point [Fig. 5.2]. By dividing the physical space/model volume into shells dr [Fig. 5.2] it is possible to compute the number of atoms dn(r) at a distance between r and r+dr from a given atom: dn(r) = NV g(r) 4π r2 dr where N represents the total number of atoms, V the model volume and where g(r) is the radial distribution function. In this notation the volume of the shell of thickness dr is approximated: (Vshell = 43π(r+dr)3  43πr3  4π r2 dr) When more than one chemical species are present the so-called partial radial distribution functions gαβ(r) may be computed : gαβ(r) = dnαβ(r)4πr2 dr ραwithρα = NαV = N×cαV where cα represents the concentration of atomic species α. These functions give the density probability for an atom of the α species to have a neighbor of the β species at a given distance r. The example features GeS2 glass.

Partial radial distribution functions of glassy GeS_2 at 300 K.

Figure 5.3 shows the partial radial distribution functions for GeS2 glass at 300 K. The total RDF of a system is a weighted sum of the respective partial RDFs, with the weights depend on the relative concentration and x-ray/neutron scattering amplitudes of the chemical species involved.
It is also possible to use the reduced Gαβ(r) partial distribution functions defined as: Gαβ(r) = 4πrρ0(gαβ(r) 1)
Atomes gives access to:

  • The partial gαβ(r) and Gαβ(r) distribution functions, and more see [Eq.[s2q_9]].

  • The corresponding dnαβ(r) integrated number of neighbors.

Also two methods are available to compute the radial distribution functions:

  • The standard real space calculation typical to analyze 3-dimensional models

  • The experiment-like calculation using the Fourier transform of the structure factor obtained using the Debye equation (see section 5.3 for details).