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Positional disorder has been introduced into the atomic structure of certaincrystalline lattices, and the orientationally-averaged structure factor S(k)and pair-correlation function g(r) of these disordered lattices have beenstudied. Analytical expressions for S(k) and g(r) for Gaussian positionaldisorder in 2D and 3D are confirmed with precise numerical simulations. Theseanalytic results also have a bearing on the unsolved Gauss circle problem inmathematics. As the positional disorder increases, high-k peaks in S(k) aredestroyed first, eventually leaving a single peak, that with the lowest-kvalue. The pair-correlation function for lattices with such high levels ofpositional disorder exhibits damped oscillations, with a period equal to theseparation between the furthest-separated (lowest-k) lattice planes. The lastsurviving peak in S(k) is, for example for silicon and silica, at a wavevectornearly identical to that of the experimentally-observed first sharp diffractionpeak (FSDP) in the amorphous phases of those materials. Thus, for theseamorphous materials at least, the FSDP can be regarded as arising fromscattering from atomic configurations equivalent to the single family ofpositionally-disordered local Bragg planes having the furthest separation.Comment: v2: changes in response to referees' comments: Figure 2 made more  readable, improved discussion of height of peaks in S(k), other minor changes  4 pages, 3 figures, submitted to Physical Review

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