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Identifying the multiplicity of crystallographically equivalent variants generated by iterative phase transformations in Ti.

This work describes phase transformations in Ti from a purely crystallographic perspective. Iterative heating and cooling above and below 1155 K induce phase transitions between a low-temperature h.c.p. (hexagonal close packed) (6/m mm) and a high-temperature b.c.c. (body centred cubic) (m3m) structure. The crystallography of the two phases has been found to be related by the Burgers Orientation Relationship (Burgers OR). The transitions are accompanied by changes in texture, as an ever-increasing number of crystallographically equivalent variants occur with every cycle. Identifying their multiplicity is important to relate the textures before and after the transformation, in order to predict the resultant one and refine its microstructure. The four-dimensional Frank space was utilized to describe both h.c.p. and b.c.c. structures within the same orthogonal framework, and thus allow for their easy numerical manipulation through matrix algebra. Crystallographic group decomposition showed that the common symmetry maintained in both groups was that of group 2/m; therefore, the symmetry operations that generated the variants were of groups 3m and 23 for cubic and hexagonal generations, respectively. The number of all potential variants was determined for the first three variant generations, and degeneracy was indeed detected, reducing the number of variants from 72 to 57 and from 432 to 180 for the second and third generations, respectively. Degeneracy was attributed on some special alignments of symmetry operators, as a result of the Burgers OR connecting the relative orientation of the two structures.