# 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 Atomes.

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: $\label{g2r_1} dn(r)\ =\ \frac{N}{V}\ g(r)\ 4\pi\ r^{2}\ dr$ 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: $\left(V_{\text{shell}}\ =\ \displaystyle{\frac{4}{3}} \pi (r+dr)^3\ -\ \displaystyle{\frac{4}{3}} \pi r^3 \ \simeq\ 4\pi\ r^{2}\ dr \right)$ When more than one chemical species are present the so-called partial radial distribution functions $$g_{\alpha\beta}(r)$$ may be computed : $\label{g2r_4} g_{\alpha \beta}(r)\ =\ \frac{dn_{\alpha \beta}(r)}{4\pi r^{2}\ dr\ \rho_{\alpha}} \qquad \text{with} \qquad \rho_{\alpha}\ =\ \frac{V}{N_\alpha}\ =\ \frac{V}{N\times c_\alpha}$ where $$c_\alpha$$ represents the concentration of atomic species $$\alpha$$. These functions give the density probability for an atom of the $$\alpha$$ species to have a neighbor of the $$\beta$$ species at a given distance $$r$$. The example features GeS$$_2$$ glass.

Figure 5.3 shows the partial radial distribution functions for GeS$$_2$$ glass at 300 K. The total RDF of a system is a weighted 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: ${\bf{G}}_{\alpha\beta}(r)\ =\ 4\pi r \rho_0 \left(g_{\alpha \beta}(r)\ - 1\right)$
Atomes gives access to:

• The partial $$g_{\alpha \beta}(r)$$ and $${\bf{G}}_{\alpha\beta}(r)$$ distribution functions, and more see [Eq.[s2q_9]].

• The corresponding $$dn_{\alpha \beta}(r)$$ integrated number of neighbors.

Also 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 (see section 5.3 for details).