Positional disorder in lattices: a model for the structural order responsible for the firstsharp diffraction peak in glasses

Abstract

Gaussian positional disorder has been introduced into the structure of various 2D and 3Dcrystal lattices. As the degree of positional disorder is increased, the weights of thehighest-k peaks in the orientationally-averaged structure factor are decreased first, eventuallyleaving only a single peak having significant weight, that with the lowestk-value. The orientationally-averaged real-space pair-correlation function for lattices withsuch high levels of positional disorder exhibits a corresponding power-law-damped series ofoscillations, with a single period equal to the separation between the furthest-separated,lowest-k lattice planes. This last surviving peak in the structure factor is at ak-value nearly identical to that of the experimentally-observed first sharp diffractionpeak for the corresponding real amorphous phases of such lattices (e.g. Si,SiO2). These results also have a bearing on the unsolved Gauss circle problem in mathematics.