The transformation matrices (distortion, orientation, correspondence), their continuous forms, and their variants

TL;DR

This paper explores the mathematical description of phase transformation matrices in crystallography, providing formulas, continuous forms, and variants, with implications for understanding transformation behaviors and symmetries.

Contribution

It introduces formulas for transformation matrices in various bases, derives continuous forms under specific assumptions, and clarifies the distinctions among different variants in phase transformations.

Findings

Continuous distortion matrix form under hard-sphere assumption

Distinction between stretch, orientation, and correspondence variants

Examples illustrating variant relationships and orientation reversibility

Abstract

The crystallography of displacive phase transformations can be described with three types of matrices: the lattice distortion matrix, the orientation relationship matrix, and the correspondence matrix. The paper gives some formula to express them in crystallographic bases, orthonormal bases, and reciprocal bases, and it explains how to use them to deduce the matrices of inverse transformation. In the case of hard-sphere assumption, a continuous form of the distortion matrix can be determined, and its derivative is identified to the velocity gradient used in continuum mechanics. The distortion, the orientation and the correspondence variants are determined by coset decomposition with intersection groups that depend on the point groups of the phases and on the type of transformation matrix. The stretch variants required in the phenomenological theory of martensitic transformation should be distinguished from the correspondence variants. The orientation variants and the correspondence variants are also different; they are defined from the geometric symmetries and algebraic symmetries, respectively. The concept of orientation (ir)reversibility during thermal cycling is briefly and partially treated by generalizing the orientation variants with n-cosets and graphs. Some simple examples are given to show that there is no general relation between the numbers of distortion, orientation and correspondence variants, and to illustrate the concept of orientation variants formed by thermal cycling.