An invertible seven-dimensional Dirichlet cell characterization of lattices

Abstract

Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli-reduced cells (based on the three shortest non-coplanar lattice vectors) or by Delaunay-reduced cells (based on four non-coplanar vectors summing to zero and all meeting at obtuse or right angles) is commonly performed. The Niggli cell derives from Minkowski reduction. The Delaunay cell derives from Selling reduction. All are related to the Wigner-Seitz (or Dirichlet, or Voronoi) cell of the lattice, which consists of the points at least as close to a chosen lattice point as they are to any other lattice point. The three non-coplanar lattice vectors chosen are here called the Niggli-reduced cell edges. Starting from a Niggli-reduced cell, the Dirichlet cell is characterized by the planes determined by 13 lattice half-edges: the midpoints of the three Niggli cell edges, the six Niggli cell face-diagonals and the four body-diagonals, but seven of the lengths are sufficient: three edge lengths, the three shorter of each pair of face-diagonal lengths, and the shortest body-diagonal length. These seven are sufficient to recover the Niggli-reduced cell.

Keywords: Dirichlet cell; Niggli reduction; Voronoi cell; Wigner–Seitz cell.

Figure 1

Truncated octahedron (image created by https://en.wikipedia.org/wiki/User:Cyp using POV-Ray, see https://en.wikipedia.org/wiki/User:Cyp/Poly.pov for source).

Figure 2

Historical timeline of studies of crystallographic lattice characterization. Figure drawn by E. Kincaid. Used with permission of the artist.