Rapid Bravais-lattice determination algorithm for lattice parameters containing large observation errors

Abstract

A new Bravais-lattice determination algorithm is introduced herein. For error-stable Bravais-lattice determination, Andrews & Bernstein [Acta Cryst. (1988), A44, 1009-1018] proposed the use of operations to search for nearly Buerger-reduced cells. Although these operations play an essential role in their method, they increase the computation time, in particular when lattice parameters obtained in (powder) auto-indexing are supposed to contain large errors. The new algorithm requires only several permutation matrices in addition to the operations that are necessary when the lattice parameters have exact values. As a result, the computational efficiency of error-stable Bravais-lattice determination is improved considerably. Furthermore, the new method is proved to be error stable under a very general assumption. The detailed algorithms and the set of matrices sufficient for error-stable determination are presented.

Lattice reduction using a Euclidean algorithm

“…Such a reduced cell (in the Delaunay/ Niggli sense) can then be used for the proper crystallographic analysis of the crystal, beginning with the correct assignment of its Bravais lattice, and providing a normalized description, with a view to comparison. At this last step, the kind of supercell reduction that we have developed thus connects with established Delaunay/Niggli-like reduction theory, and we direct the reader to the literature on this subject and its relation to the reduction of ternary quadratic forms for further insight (see, for example, Burzlaff et al, 2002;Andrews & Bernstein, 1988;Gruber, 1989;Oishi-Tomiyasu, 2012;Andrews & Bernstein, 2014, and references therein).…”

“…With respect to the general context of reduction theory (see, for example, Burzlaff et al, 2002;Delaunay, 1933;Oishi-Tomiyasu, 2012, and references therein), this kind of specific supercell reduction problem and the algorithm that I will address in the next sections have a somewhat distinct and different scope. The main goal here is that we endeavour to reduce the volume and the associated content of an original (super)cell which can be (arbitrarily) many times larger than its truly primitive ones, to produce (in the sense explained above) a certain, translationally reduced, unit cell of the crystal.…”

Reversed de Wolff figure of merit and its application to powder indexing solutions

“…In order to compare the efficiencies of the figures of meritM M n , M M Wu n , M Rev n and M Sym n , 24 real powder diffraction patterns were prepared. Conograph algorithms were used to conduct peak searches, enumeration of powder indexing solutions and Bravais lattice determination [see Oishi-Tomiyasu (2012) for the method of Bravais lattice determination]. Table 3 indicates whether the obtained true solution is ranked at the top of all the enumerated solutions (x symbols) or all solutions belonging to the same Bravais type (double daggers).…”

Robust powder auto-indexing using many peaks

“…Of these stages, (d) is the most time-consuming. We have already contributed to reducing the time for this stage (see Oishi-Tomiyasu, 2012). Fig.…”