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

Abstract

The crystallography of displacive/martensitic phase transformations can be described with three types of matrix: the lattice distortion matrix, the orientation relationship matrix and the correspondence matrix. Given here are some formulae to express them in crystallographic, orthonormal and reciprocal bases, and an explanation is offered of how to deduce the matrices of inverse transformation. In the case of the hard-sphere assumption, a continuous form of distortion matrix can be determined, and its derivative is identified to the velocity gradient used in continuum mechanics. The distortion, orientation and 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 and 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.

Complements to Mügge and Friedel’s Theory of Twinning

“…As the coordinate of the vectors p and w are integers or half-integers, and as p t ·w = q is integer, C is necessarily rational. This is a general property of the correspondence matrices [23]. It can also be checked that (C α→γ ) 2 = I, which also implies that C γ→α = (C α→γ ) −1 = C α→γ · With the convention that twinning transforms a right-hand basis into a left-hand one, Det(C) = −1.…”

“…Matrix notation in which the indices are not written allow lighter equations, as already commented by Christian and Mahajan [10]. Here, we will use the same matrix notations and conventions as in Reference [23]. We briefly recall them.…”

“…Three matrices play key roles in the crystallographic description of structural phase transformation in general, and for deformation twinning in particular. They are the lattice distortion matrix F, the orientation relationship matrix T, and the correspondence matrix C, as detailed in Reference [23,24]. The methods to calculate them have been progressively developed at the end of 19th and along the 20th centuries by Mügge, Kihô, Jaswon, Dove, Bilby, Bevis and Crocker.…”

“…The lower index Υ refers to the crystal Υ. Please note that in our paper [23] Υ was written as an upper index, but here we prefer to put it to the lower position to leave space for the sign "t" (transpose) in the dyadic products. When the coordinates of a vector u are expressed in the crystallographic basis of the crystal Υ, we use the notation u /γ .…”

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

“…In Appendices B4 and B5 of Cayron (2019), we confused different notations related to the point groups, Laue groups and lattice groups. We denote the point group by G and the metric tensor by M. As the symmetries preserve the norms and the angles, it is correct to write that g t Mg ¼ M for any symmetry matrix g. However, the reciprocal is not always true.…”