Groupoid of orientational variants

Abstract

Daughter crystals in orientation relationship with a parent crystal are called variants. They can be created by a structural phase transition (Landau or reconstructive), by twinning or by precipitation. Internal and external classes of transformations defined from the point groups of the parent and daughter phases and from a transformation matrix allow the orientations of the distinct variants to be determined. These are algebraically identified with left cosets and their number is given by the Lagrange formula. A simple equation links the numbers of variants of the direct and inverse transitions. The equivalence classes on the transformations between variants are isomorphic to the double cosets (operators) and their number is given by the Burnside formula. The orientational variants and the operators constitute a groupoid whose composition table acts as a crystallographic signature of the transition. A general method that determines if two daughter variants can be inherited from more than one parent crystal is also described. A computer program has been written to calculate all these properties for any structural transition; some results are given for Burgers transitions and for martensitic transitions in steels. The complexity, irreversibility and entropy of fractal systems constituted by orientational variants generated by thermal cycling are briefly discussed.

Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3ngrain boundaries

“…A similar problem has been encountered for phasetransformed materials during the identification of the operators and the reconstruction of parent grains from the EBSD data obtained from the daughter grains. A solution has been proposed by Cayron et al (2006) based on groupoid composition tables introduced in Cayron (2006). However, a phase transition corresponds to only half a cycle and we need to generalize the theory to cycled-transformed materials.…”

“…For example, H is a normal subgroup when there are only two cosets in the set (3). However, we stress here that the condition of normality of H is not fulfilled in the case of AE3 twinning as discussed in Cayron (2006). Consequently, although one matrix (or one quaternion) can be arbitrarily chosen in each coset for numerical calculations (Reed et al, 2004), a method based on arbitrary choices of representatives cannot be used to justify the algebraic structure of multiple twinning.…”

“…Consequently, although one matrix (or one quaternion) can be arbitrarily chosen in each coset for numerical calculations (Reed et al, 2004), a method based on arbitrary choices of representatives cannot be used to justify the algebraic structure of multiple twinning. The misorientations from a variant represented by the coset g i H to a variant represented by the coset g j H are isomorphic to the double cosets Hg ij H, where g ij = g À1 i g j Cayron, 2006). Therefore, the distinct types of misorientations between ordered pairs of variants are given by the partition of G 0 into double cosets,…”

“…The number of operators, N 0 , is given by the Burnside formula or it can be obtained by a class equation (Cayron, 2006). In general, it is possible to distinguish the operator transforming the variant i into the variant j from the operator transforming the variant j into the variant i because in general Hg ij H and Hg ij H, the operators are called ambivalent (Janovec & Přívratská , 2003).…”

Early Stages of Microstructure and Texture Evolution during Beta Annealing of Ti-6Al-4V

“…The reconstruction algorithm used a tolerance of 2 deg when comparing the angle/axis pairs of observed colonies to the theoretical operators for two alpha colonies exhibiting the Burgers orientation relationship with the same parent beta grain. Moreover, a minimum of 4 unique alpha variants were considered when identifying the prior beta grain orientation in order to overcome the deficiency in reconstruction noted by Cayron [27] associated with a {111} mirror plane that gives rise to an alternate, twin-related prior beta grain orientation. This is a subtle but important point as a minority texture component identified later is indeed twin related to the principal texture and so extreme care was taken to ensure this was not an artifact of the reconstruction process.…”