Ideas of lattice-basis reduction theory for error-stable Bravais lattice determination and ab initio 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.