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

Positional disorder has been introduced into the atomic structure of certain crystalline lattices, and the orientationally-averaged structure factor $S\left(k\right)$ and pair-correlation function $g\left(r\right)$ of these disordered lattices have been studied. Analytical expressions for $S\left(k\right)$ and $g\left(r\right)$ 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 $S\left(k\right)$ 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 $S\left(k\right)$ is, for example for silicon and silica, at a wave vector 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.

Figure 1

(Color online) Gaussian positional disorder in the diamond lattice with values of disorder parameter $\sigma$: (a) 0, (b) $0.1r_1$, (c) $0.2r_1$, (d) $0.3r_1$, where $r_1$ is the crystalline nearest-neighbor distance. All models have the same orientation. Atoms in three consecutive {111} lattice planes perpendicular to the page are darkened to show the gradual destruction of the lattice planes with increasing positional disorder.

Figure 2

(Color online) Orientationally-averaged structure factor per atom, for Gaussian-positionally-disordered diamond lattices (with 97336 atoms) with different values of disorder parameter, $\sigma$, where $r_1$ is the crystalline nearest-neighbor distance. Only the lowest-$k$ peak, at $k{r}_1=4.71$, contributes substantially for $\sigma \gtrsim 0.3r_1$.

Figure 3

(a) $g\left(r\right)$ for a Gaussian-positionally-disordered diamond lattice with disorder parameter $\sigma =0.3r_1$ (solid), and the best-fit curve (dashed). (b) The same data plotted double-logarithmically with the best-fit $r^{-1}$ decay envelope. (c) $g\left(r\right)$ for a Gaussian-positionally-disordered square lattice with $\sigma =0.25r_1$ (solid), and the best-fit curve (dashed). (d) The same data plotted double-logarithmically with the best-fit $r^{-1∕2}$ decay envelope. Both best-fit curves exclude the non-oscillatory data at low $r$.