Structural characteristics of positionally-disordered lattices: relation to the first sharp diffraction peak in glasses

TL;DR

This study analytically and numerically investigates how positional disorder in crystalline lattices affects their structure factor and pair-correlation function, revealing a connection to the first sharp diffraction peak in glasses.

Contribution

It provides explicit analytical expressions for S(k) and g(r) in disordered lattices and links the FSDP in glasses to the scattering from the furthest-separated disordered lattice planes.

Findings

High-k peaks in S(k) are destroyed with increasing disorder

The last peak in S(k) matches the FSDP in glasses

Pair-correlation exhibits damped oscillations with a period related to furthest lattice planes

Abstract

Positional disorder has been introduced into the atomic structure of certain crystalline lattices, and the orientationally-averaged structure factor S(k) and pair-correlation function g(r) of these disordered lattices have been studied. Analytical expressions for S(k) and g(r) for Gaussian positional disorder in 2D and 3D are confirmed with precise numerical simulations. These analytic results also have a bearing on the unsolved Gauss circle problem in mathematics. As the positional disorder increases, high-k peaks in S(k) are destroyed first, eventually leaving a single peak, that with the lowest-k value. The pair-correlation function for lattices with such high levels of positional disorder exhibits damped oscillations, with a period equal to the separation between the furthest-separated (lowest-k) lattice planes. The last surviving peak in S(k) is, for example for silicon and silica, at a wavevector nearly identical to that of the experimentally-observed first sharp diffraction peak (FSDP) in the amorphous phases of those materials. Thus, for these amorphous materials at least, the FSDP can be regarded as arising from scattering from atomic configurations equivalent to the single family of positionally-disordered local Bragg planes having the furthest separation.