Density fluctuations and the pair distribution function

In discrete systems the number density, defined as the number of particles per unit volume, is subject to fluctuations depending on the size and location of the sampling volume. Fluctuations in the local density or pair distribution function as determined from x-ray or neutron diffraction experiments, die out quickly in disordered materials but persist in crystals. Here we show that for a single atom at the origin, fluctuations persist out to very large distances and the pair distribution function does not decay, even in the case of a random system; only disappearing after ensemble averaging. Therefore, for a crystal the fluctuations in the pair distribution function persist to arbitrarily large distances. This is demonstrated here with experimental and modeling results for powdered crystalline nickel, where we find an undiminished fluctuation amplitude for the pair distribution function calculated up to distances of a micron. The characteristic separation between the peaks in the pair distribution function at large distances is determined by the thermal amplitude of vibration of a single atom and not by the interatomic spacing. Thermal broadening is included so the results obtained here are of direct experimental interest, and comparison is made to neutron diffraction data on nickel. Results are shown to be similar for crystals and for a single atom in a glass