An Invertible Seven-Dimensional Dirichlet Cell Characterization of Lattices

TL;DR

This paper introduces a novel seven-dimensional Dirichlet cell characterization for lattices, improving the understanding and computation of crystallographic lattice structures through a minimal set of parameters.

Contribution

It presents a new invertible characterization of lattices using seven key parameters derived from Dirichlet cells, enhancing lattice analysis methods.

Findings

Seven key parameters suffice to characterize Dirichlet cells

The method allows reconstruction of the Niggli-reduced cell from these parameters

Improves lattice classification and structure solution techniques

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 edge vectors) or by Delaunay-reduced cells (based on four edge vectors summing to zero and all meeting at obtuse or right angles) are commonly used. 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 than they are to any other lattice point. Starting from a Niggli-reduced cell, the Dirichlet cell is characterized by the planes determined by thirteen 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 edge lengths are sufficient: three edge lengths, the three shorter of each pair of face-diagonal lengths and the shortest body-diagonal length, from which the Niggli-reduced cell may be recovered.