the radial distribution function

How do you describe the structure of something with no lattice, no unit cell, no tidy list of atom positions — like a liquid or a glass? You cannot point to where each atom sits, so instead you ask a statistical question: standing on any one atom and looking outward, how likely am I to find another atom at distance r? Plot that likelihood against distance and you get the radial distribution function. It is the single most important tool for disordered structure: a curve that packs the average neighbourhood of an atom into one graph.

The heart of it is the pair distribution function, written g(r). It measures the local density of atoms at distance r from a typical atom, divided by the average density of the whole material. So g(r) = 1 means 'just the average, no correlation'; g(r) greater than 1 means atoms cluster there; g(r) less than 1 means atoms avoid it. For a liquid or glass the curve tells a clear story: at very small r, g(r) is zero (two atoms cannot overlap); then it rises to a tall FIRST PEAK at the nearest-neighbour distance; then it dips, rises to a weaker second peak, oscillates a few more times, and settles down to a flat g(r) = 1 within a nanometre or so. That fade to a featureless line is the signature of NO long-range order — beyond a few atomic spacings, all memory of position is lost. The closely related radial distribution function proper, RDF(r) = 4 times pi times r^2 times rho times g(r), counts the actual NUMBER of atoms in a thin shell at radius r, and the area under its first peak gives the coordination number directly.

This one curve delivers real numbers. The position of the first peak is the average bond length; its width tells you how tightly that distance is defined; and integrating the first peak gives the coordination number — the average count of nearest neighbours. In amorphous silicon, for instance, the first peak sits at 2.35 angstrom and its area integrates to about 4, correctly telling you each silicon still has four neighbours. Crucially, g(r) is not just a picture: it is the direct Fourier transform of the static structure factor S(Q) that a diffraction experiment measures, so total-scattering experiments give you g(r) experimentally. One honest limitation: g(r) is a one-dimensional average over all directions and all atoms, so it tells you distances and neighbour counts but throws away the full three-dimensional arrangement — many different structures can share a similar g(r).