SEARCH TREE METHOD FOR THE DETERMINATION OF SIMILARITY OPERATORS BETWEEN ARBITRARY LATTICES

Lattice similarity operators are transformation matrices which describe alinear mapping of two crystallattices onto each other such that: (i) points ofa 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. Theseconditions 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 minimizesthe 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 theother. The triplets are limited to those uniquely definingall derivative lattices of interest. Applications of the method are envisaged in all situations where geometrical relationships between lattices or crystalstructures 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 there constructive transformation between the(Mn, …)SiO3 polymorphs Rhodonite and Pyroxmangite as examples. © by R. Oldenbourg Verlag