Reduction theory for determining the Bravais type of unit-cell parameters containing observation errors.

26^th May 2022, 14:00
Ryoko Tomiyasu
Kyushu University, Japan

Abstract

In crystallography, the parameters of the unit cells or crystal structures recovered from experimental data are required to be output using their conventional cells. Namely, the lattice-basis reduction must be applied, and the software also transforms the parameters of the primitive cells into those of the conventional cells, based on their symmetries (centering types).
The influence of observational errors cannot be ignored during this algebraic calculation. In 2012, the speaker provided an algorithm for 3D lattices containing observational errors; similar results have been obtained for 2D lattices. The algorithm is new in the sense that all the candidates for nearly reduced bases were explicitly given to speed up the algorithm, and we proved based on the Venkov reduction that it works correctly for parameters with errors of various sizes, ranging from rounding errors to observation errors. The Eisenstein and Selling reductions (known as the Niggli and Delaunay reductions in crystallography) can be regarded as a special case of the Venkov reduction. The code has been used in the author's research in geometry of numbers, and in the ab-initio indexing software CONOGRAPH for PXRD and EBSD distributed by and for experimental scientists.
In the course of the research, the speaker had the opportunity to study the history of the reduction theory in crystallography, which will be mentioned as well as the background in mathematics.