  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.
