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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:
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 = π(r + dr)3 - πr3 4π r2 dr.
When more than one chemical species are present the so-called partial radial distribution functions gαβ(r) may be computed :
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.
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:
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