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, , for Bravais lattice determination. Results from 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 in the six-dimensional space. This article describes an investigation of the boundary polytopes of the Niggli-reduced cone in the six-dimensional space 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 Niggli cone .

A fingerprint based metric for measuring similarities of crystalline structures

“…Unique and preferably standardized cell parameters are required for comparison and analysis of different crystals. 22 Algorithms to transform unit cells to a reduced form are frequently used in crystallography, such as the Niggli-reduction [23][24][25][26][27] which produces cells with shortest possible vectors (|a + b + c| = minimal). Unfortunately, in the presence of noisy lattice vectors, cells can change discontinuously within the Niggli-reduction algorithm.…”

The geometry of Niggli reduction:SAUC– search of alternative unit cells

“…The approach of Andrews et al (1980) using seven parameters (three reduced cell edges, three reduced reciprocal cell edges and the volume), 'V7', helps, but has difficulty distinguishing cells with angles near 90 . The NCDist approach used here, derived from the work of Andrews & Bernstein (2012, 2014, both fills in the gaps and handles angles near 90 . Consider, for example, the unit cells of phospholipase A 2 discussed by Le Trong & Stenkamp (2007).…”

“…This scaling of L 1 is suggested by the fact that in general jjxjj 1 6 1=2 jjxjj 2 . (b) The square root of the BGAOL (Andrews & Bernstein, 2014) Niggli cone embedding distance NCDist based on…”

“…The embedding of Andrews & Bernstein (2012, 2014 can be used to show that the distance between these two cells is quite small in G 6 (0.004 Å 2 in G 6 ). …”

“…In the latter article by Andrews & Bernstein (2014), the embedding was used to determine likely Bravais lattices for a unit cell. Here we apply the embedding to searching within a database for lattices 'close' to the lattice of a given probe cell.…”