The geometry of Niggli reduction: BGAOL -embedding Niggli reduction and analysis of boundaries

Abstract

Niggli reduction can be viewed as a series of operations in a six-dimensional space derived from the metric tensor. An implicit embedding of the space of Niggli-reduced cells in a higher-dimensional space to facilitate calculation of distances between cells is described. This distance metric is used to create a program, BGAOL, for Bravais lattice determination. Results from BGAOL are compared with results from other metric based Bravais lattice determination algorithms. This embedding depends on understanding the boundary polytopes of the Niggli-reduced cone N in the six-dimensional space G₆. This article describes an investigation of the boundary polytopes of the Niggli-reduced cone N in the six-dimensional space G⁶ by algebraic analysis and organized random probing of regions near one-, two-, three-, four-, five-, six-, seven- and eightfold boundary polytope intersections. The discussion of valid boundary polytopes is limited to those avoiding the mathematically interesting but crystallographically impossible cases of zero-length cell edges. Combinations of boundary polytopes without a valid intersection in the closure of the Niggli cone or with an intersection that would force a cell edge to zero or without neighboring probe points are eliminated. In all, 216 boundary polytopes are found. There are 15 five-dimensional boundary polytopes of the full G⁶ Niggli cone N .

Keywords: Bravais lattice determination; Niggli reduction; computer programs.

Figure 1

Example of the difficulty of finding the shortest distance in a lattice. The distance within the asymmetric unit of the chosen cell is shown as a light dotted line. However, the shortest distance, shown as a heavy solid line, crosses multiple unit cells.

Figure 2

To illustrate distance calculations between arbitrary points, a line was drawn in between an unreduced point close to cF and its reduced image. Each curve shows the distances at 100 points along each line to the corresponding reduced cell of the starting point. The upper curve starts from , ,  Å, , , ° and ends at , ,  Å, , , °. The lower curve starts from , ,  Å, , , ° and ends at , ,  Å, , , °.