Distribution rules of systematic absences and generalized de Wolff figures of merit applied to electron backscatter diffraction ab initio indexing

Ideas of lattice-basis reduction theory for error-stable Bravais lattice determination and abinitio indexing

In ab initio indexing, for a given diffraction/scattering pattern, the unit-cell parameters and the Miller indices assigned to reflections in the pattern are determined simultaneously. `Ab initio' means a process performed without any good prior information on the crystal lattice. Newly developed ab initio indexing software is frequently reported in crystallography. However, it is not widely recognized that use of a Bravais lattice determination method, which is tolerant of experimental errors, can simplify indexing algorithms and increase their success rates. One of the goals of this article is to collect information on the lattice-basis reduction theory and its applications. The main result is a Bravais lattice determination algorithm for 2D lattices, along with a mathematical proof that it works even for parameters containing large observational errors. It uses two lattice-basis reduction methods that seem to be optimal for different symmetries, similarly to the algorithm for 3D lattices implemented in the CONOGRAPH software. In indexing, a method for error-stable unit-cell identification is also required to exclude duplicate solutions. Several methods are introduced to measure the difference in unit cells known in crystallography and mathematics.

A remark on ab initio indexing of electron backscatter diffraction patterns

There is a growing interest in ab initio indexing of electron backscatter diffraction (EBSD) patterns. The methods of solving the problem are presented as innovative. The purpose of this note is to point out that ab initio EBSD indexing belongs to the field of indexing single-crystal diffraction data, and it is solved on the same principles as indexing of patterns of other types. It is shown that reasonably accurate EBSD-based data can be indexed by programs designed for X-ray data.