Past exam of the quantum condensed matter course of the University of Cambridge / 2004 / Section A / 1

The radial distribution function is the conditional number density of neighbors at separation $r$, divided by the bulk number density $n$. In an isotropic three-dimensional liquid, the mean number in a thin spherical shell is $dN=4\pi n{r}^{2}g(r)dr$. Strong short-range repulsion makes $g(r)$ very small inside the effective particle core. A first maximum marks the nearest-neighbor coordination shell. More distant shells give weaker oscillations, which decay toward $g(r)=1$ as long-range positional correlations disappear. A typical simple liquid has no indefinitely repeated sharp lattice peaks.

The static structure factor measured by diffraction is related to the Fourier transform of $g(r)-1$. Thus the broad diffraction maxima of a liquid and the damped real-space coordination shells describe the same short-range organization.