Hexagonal paracrystals

Abstract

Spatially interacting Gaussian random variables arranged on a hexagonal lattice are used to construct examples of hexagonal paracrystals. As for previously described paracrystals on a square lattice the distributions may be specified by a variance, 2 , L σ defining the magnitude of the displacement away from the underlying perfect lattice together with parameters defining transverse and longitudinal correlations (ρ T and ρ L ) between neighbouring displacements. A strong longitudinal correlation indicates a small variation from cell to cell in the length of the cell edge whereas a strong transverse correlation indicates a small variation in the cell edge direction. For values of σ L > ∼0.30 of the unit cell spacing the diffraction patterns contain diffuse peaks only with no evidence of Bragg peaks arising from the underlying regular lattice.

Luminescent Silicon Nanosheet Paracrystals from Elemental- and Hydride-Based Syntheses of CaSi2 Precursors: Implications for Photonic and Optoelectronic Applications

“…Alternatively, the diffuse hexagons could arise from disorder within the unit cell of isolated, individual nanosheets. Welberry demonstrated that deviations within the lattice vector direction within a plane of atoms can lead to these diffuse hexagons . Given the sharp (110) diffraction planes in the AI-spXRD, we conclude that the diffuse hexagons observed in Figure c,d arise from variations of the unit cell lattice vector direction within an isolated nanosheet.…”

“…Welberry demonstrated that deviations within the lattice vector direction within a plane of atoms can lead to these diffuse hexagons. 64 Given the sharp (110) diffraction planes in the AI-spXRD, we conclude that the diffuse hexagons observed in Figure 5c,d arise from variations of the unit cell lattice vector direction within an isolated nanosheet. Moreover, we find that both EM-and HS-SiNSs are very sensitive to the beam, with the ED patterns transforming into diffuse rings over the course of a few seconds (Figure S7).…”