Radial distribution functions fundamentals

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 5.2: 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:

$${\frac{{N}}{{V}}}$$ (5.2)

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:

$$\left(\vphantom{V_{\text{shell}} = \displaystyle{\frac{4}{3}} \... ...dr)^3 - \displaystyle{\frac{4}{3}} \pi r^3 \simeq 4\pi r^{2} dr }\right. {\frac{{4}}{{3}}} {\frac{{4}}{{3}}} \simeq \left.\vphantom{V_{\text{shell}} = \displaystyle{\frac{4}{3}} \... ...dr)^3 - \displaystyle{\frac{4}{3}} \pi r^3 \simeq 4\pi r^{2} dr }\right)$$ (5.3)

When more than one chemical species are present the so-called partial radial distribution functions g_αβ(r) may be computed :

$${\frac{{dn_{\alpha \beta}(r)}}{{4\pi r^{2} dr \rho_{\alpha}}}} {\frac{{V}}{{N_\alpha}}} {\frac{{V}}{{N\times c_\alpha}}}$$ (5.4)

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 GeS₂ glass.

Figure 5.3: Partial radial distribution functions of glassy GeS₂ at 300 K.

Figure [Fig 5.3] shows the partial radial distribution functions for GeS₂ glass 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.
It is also possible to use the reduced $\bf {G}_{{\alpha\beta}}^{}$(r) partial distribution functions defined as:

$$\bf {G}_{{\alpha\beta}}^{} \left(\vphantom{g_{\alpha \beta}(r) - 1}\right. \left.\vphantom{g_{\alpha \beta}(r) - 1}\right)$$ (5.5)

I.S.A.A.C.S. gives access to the partial g_αβ(r) and $\bf {G}_{{\alpha\beta}}^{}$(r) distribution functions. Two methods are available to compute the radial distribution functions:

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

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