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

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 $left(vphantom{V_{text{'ecorce}} = displaystyle{frac{4}{3}} pi (r+dr)^3 - displaystyle{frac{4}{3}} pi r^3 simeq 4pi r^{2} dr }right.$ V_shell = $displaystyle {frac{{4}}{{3}}}$ π(r + dr)³ - $displaystyle {frac{{4}}{{3}}}$ πr³ 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 g_αβ(r) may be computed :

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

(2)

where c_α 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.

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 g_αβ(r) as well as to the partial reduced distribution functions G_αβ(r) defined by:

G_αβ(r) = 4 π ρ₀ r [g_αβ(r) - 1 ]

(3)

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

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The radial distribution functions: definitions