Approximating lattice similarity

Abstract

A method is proposed for choosing unit cells for a group of crystals so that they all appear as nearly similar as possible to a selected cell. Related unit cells with varying cell parameters or indexed with different lattice centering can be accommodated.

Keywords: Delaunay; Delone; Niggli; Selling; lattice matching.

Figure 1

(a) The case where the transformed point T has been scaled to be at the same distance from the origin as the reference point R. (b) Point T has not been scaled, and some areas are incorrectly assigned to point R. In each panel, the straight line between points R and T separates the regions closer to each of the points.

Figure 2

A two-dimensional example of the geometry for determining similarity. Each transformed copy of the reference cell is normalized to a constant length in the chosen space (here S ⁶). Each transformed and normalized cell then defines a zone in which every point in that zone is closer to the transformed and normalized cell defining that zone than it is to the transformed and normalized cell defining any other zone. In this example, each point within the textured zone (which extends to infinity) is closer to the gray-centered point than it is to any of the black points.