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 I.S.A.A.C.S..

Image gr
Figure 1: 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. 1].

By dividing the physical space/model volume into shells dr [Fig. 1] 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) = $displaystyle {frac{{N}}{{V}}}$ g(r) 4π r2 dr (1)

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 $ left(vphantom{V_{text{'ecorce}} = displaystyle{frac{4}{3}} pi (r+dr)^3 - displaystyle{frac{4}{3}} pi r^3  simeq 4pi r^{2} dr }right.$Vshell = $displaystyle {frac{{4}}{{3}}}$π(r + dr)3 - $displaystyle {frac{{4}}{{3}}}$πr3 $displaystyle simeq$ 4π r2 dr$ left.vphantom{V_{text{'ecorce}} = displaystyle{frac{4}{3}} pi (r+dr)^3 - displaystyle{frac{4}{3}} pi r^3  simeq 4pi r^{2} dr }right)$.
When more than one chemical species are present the so-called partial radial distribution functions gαβ(r) may be computed :

gαβ(r) = $displaystyle {frac{{dn_{alpha beta}(r)}}{{4pi r^{2} dr rho_{alpha}}}}$ with ρα = $displaystyle {frac{{V}}{{N_alpha}}}$ = $displaystyle {frac{{V}}{{Ntimes c_alpha}}}$ (2)

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 GeS2glass.

Image grp300K
Figure 2: Partial radial distribution functions of glassy GeS2 at 300 K.

Figure [Fig 2] shows the partial radial distribution functions for GeS2glass at 300 K. The total RDF of a system is a weighterd 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.


I.S.A.A.C.S. gives access to the partial distribution functions gαβ(r) as well as to the partial reduced distribution functions Gαβ(r) defined by:

Gαβ(r) = 4 π ρ0 r [gαβ(r) - 1 ]

Two methods are available to compute the radial distribution functions:

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