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Radial distribution functions

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The Radial Distribution Function, R.D.F. , g(r), also called pair distribution function or pair correlation function, is the elementary tool used to extract the structural information from numerical simulations.


Image gr
Figure 1: Space discretization for the evaluation of the radial distribution function.

Considering an homogeneous repartition of the particles in space, the g(r) represents the probability to find a particle in the shell dr at the distance r of another particle [Fig. 1].

By discretizing the space in intervals dr [Fig. 1] it is possible to compute for a given atom, the number of atoms dn(r) at a distance between r and r + dr of this atom:

dn(r)  =  $displaystyle {frac{{N}}{{V}}}$ g(r) 4π r2 dr (1)


where N represents the total number of particles, V the 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)$.
By distinguishing the chemical species it is possible to compute the partial radial distribution functions gαβ(r):

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 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.


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


Figure [Fig 2] illustrates the results of the calculation of the partial radial distribution functions for gGeS2 at 300 K, and allows to visualize the characteristics of the repartition of the chemical species.

Since version 1.1 the R.I.N.G.S. code also gives access to the reduced distribution functions Gαβ(r) defined by:

Gαβ(r) = 4 π ρ0 r [ gαβ(r) - 1 ]
(3)
Last Updated on Thursday, 08 July 2010 06:44  

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