Distribution rules of crystallographic systematic absences on the Conway topograph and their application to powder auto-indexing

Abstract

Powder auto-indexing is the crystallographic problem of lattice determination from an average theta series. There, in addition to all the multiplicities, the lengths of part of lattice vectors cannot be obtained owing to systematic absences. As a consequence, solutions are not always unique. We develop a new algorithm to enumerate powder auto-indexing solutions. This is a novel application of the reduction theory of positive-definite quadratic forms to a problem of crystallography. Our algorithm is proved to be effective for all types of systematic absences, using their newly obtained common properties. The properties are stated as distribution rules for lattice vectors corresponding to systematic absences on a topograph. Conway defined topographs for 2-dimensional lattices as graphs whose edges are associated with $|l_1|^2$, $|l_2|^2$, $|l_1+l_2|^2$, $|l_1-l_2|^2$. In our enumeration algorithm, topographs are utilized as a network of lattice vector lengths. As a crystal structure is a lattice of rank 3, the definition of topographs is generalized to any higher dimensional lattices using Voronoi's second reduction theory. The use of topographs allows us to speed up the algorithm. The computation time is reduced to 1/250--1/32, when it is applied to real powder diffraction patterns. Another advantage of our algorithm is its robustness to missing or false elements in the set of lengths extracted from a powder diffraction pattern. Conograph is the powder indexing software which implements the algorithm. We present results of Conograph for 30 diffraction patterns, including some very difficult cases.