The Geometry of Niggli Reduction
Lawrence C. Andrews, Herbert Jacob Bernstein, William Floyd Parkway · arXiv (Cornell University) · 2012
Correct identification of the Bravais lattice of a crystal is an important early step in structure solution. Niggli reduction is a commonly used technique. We investigate the boundary polytopes of the Niggli-reduced cone in the six-dimensional space G 6 by organized random probing of regions near 1-, 2-, 3-, 4-, 5-, 6-, 7- and 8-fold boundary polytope intersections. We limit our consideration of valid boundary polytopes to those avoiding the mathematically interesting but crystallographically impossible cases of zero length cell edges. Combinations of boundary polytopes without a valid intersection or with an intersection that would force a cell edge to zero or without neighboring probe points are eliminated. 216 boundary polytopes are found. There are 15 5-D boundary polytopes of the full G 6 Niggli cone, 53 4-D boundary polytopes resulting from intersections of pairs of the 15 5-D boundary polytopes, 79 3-D boundary polytopes resulting from 2-fold, 3-fold and 4-fold intersections of the 15 5-D boundary polytopes, 55 2-D boundary polytopes resulting from 2-fold, 3-fold, 4-fold and higher intersections of the 15 5-D boundary polytopes, 14 1-D boundary polytopes resulting from 3-fold and higher intersections of the 15 5-D boundary polytopes. The classification of the boundary polytopes into 5-, 4-, 3-, 2- and 1-dimensional boundary polytopes corresponds well to the Niggli classification and suggests other possible symmetries. All of the primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes. All of the non-primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes and of the 7 special-position subspaces of the 5-D boundary polytopes. This study provides a new, simpler and arguably more intuitive basis set for the classification of lattice characters and helps to illuminate some of the complexities in Bravais lattice identification. The classification is intended to help in organizing database searches and in understanding which lattice symmetries are “close” to a given experimentally determined cell.