Search tree method for the determination of similarity operators between arbitrary lattices

Abstract

Lattice similarity operators are transformation matrices which describe a linear mapping of two crystal lattices on to each other such that: (i) points of a common derivative lattice of reasonably low index in both lattices form a coincidence site lattice ; (ii) the deformation occurring with the mapping is tolerably small; (iii) the determinant of the transformation matrix is fixed by the ratio of the formula units in the unit cells of both crystals. These conditions are formulated in terms of matrix algebra. To tackle the discrete optimization problem of searching for lattice similarity operators between two given arbitrary lattices, a search tree algorithm is proposed. It minimizes the number of trial matrices to be investigated by systematic use of geometrical criteria which are derived from the above conditions. The search is based on the identification of triplets of lattice vectors of one lattice in the other. The triplets are limited to those uniquely defining all derivative lattices of interest. Applications of the method are envisaged in all situations where geometrical relationships between lattices or crystal structures have to be described, in particular for the real space description of structural transitions if the transformation matrix cannot be deduced clearly from experimental observations. The method is demonstrated using the martensitic transformation in zirconia and the reconstructive transformation between the (Mn, …)SiO 3 polymorphs Rhodonite and Pyroxmangite as examples.