Similarity Operators Research Articles

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

Article Search tree method for the determination of similarity Operators between arbitrary lattices was published on October 1, 1990 in the journal Zeitschrift für Kristallographie - Crystalline Materials (volume 191, issue 1-4).

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

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.

The use of similarity operators for lattice superlattice relations. Part II. Application to rhombohedral fluorite related structures

The use of similarity operators for lattice superlattice relations. Part II. Application to rhombohedral fluorite related structures

The use of similarity operators for lattice superlattice relations. Part II. Application to rhombohedral fluorite related structures*

The vocable of superlattice means to physicists and chemists a structure closely related to a parent structure and having cell dimensions superior to that of the parent structure. In fact, the superlattice belongs to a subgroup of the parent structure because increasing cell dimensions means a loss of translational symmetry. Space groups which only differ by their translational symmetry are called isomorphic and are listed in the new International Tables for Crystallography. The theoretical part of the paper establishes the most general similarity matrix S which relates the well known CaF2 structure to its rhombohedral superlattices. Two versions are given, SR and SH, according to the so called rhombohedral and hexagonal descriptions of rhombohedral structures. The determinant of the similarity matrix is directly related to the chemical formula, for instance |S| = 19 in Ca13Y6F44 and |S| = 14 in Pb8Y6F32O where the cation numbers are respectively 19 and 14.

The use of similarity operators for lattice-superlattice relations: I. Applications to monoclinic and orthorhombic fluorite-related structures

Similarity operators are shown to relate the lattice vectors and origins of the structure and the superstructure, the symmetry operators of the space group G of the structure to those of the subgroup H of G of the superstructure, and finally those atomic positions which are conserved in the superstructure. In the monoclinic and orthorhombic fluorite-related structures of composition K 0.5− x Ln 0.5+ x F 2+2 x ( Ln = lanthanide and Y) considered here we find that the cation sublattice is conserved. The origin of the superstructure does not necessarily coincide with the origin of the structure, but is a point of high site symmetry.

Similarity operators and displacive transitions Application to the transition from the hexagonal NiAs to the orthorhombic MnP type1

Abstract Similarity operators are applied to the study of displacive transitions. The example considered is the transition from the hexagonal NiAs to the orthorhombic MnP type. The number of displacement modes is superior to that deduced from representation theory of the wave vector group.