Angular distortive matrices of phase transitions in the fcc-bcc-hcp system

This work generalizes the one-step model previously developed on fcc-bcc martensitic transformations to the larger family of phase transitions in the fcc-bcc-hcp system. The angular distortive matrices are calculated for the bcc-fcc, bcc-hcp and fcc-hcp transitions, and for fcc-fcc mechanical twinning. The analytical expressions of the continuous atomic displacements, lattice distortion and lattice correspondence matrices result directly from the orientation relationships; the unique assumption is that the atoms are hard-spheres that can’t interpenetrate each other. The displacive transformations occur in one-step by the change of the unique parameter which is the angle of distortion, without any defined intermediate phase or lattice shearing. The matrices of complete distortion form an algebra over the number field Q(√6). The habit planes are predicted on the simple criterion that they are untilted by the distortion; the results are compared to experimental observations published in literature. Shuffle is required for bcc-hcp and fcc-hcp transitions because the hcp primitive Bravais lattice contains two atoms instead of one for the fcc and bcc phases; the analytical expressions of the shuffle trajectories are determined. Different crystallographic aspects are discussed. The steric barriers on dense planes are calculated and compared for fcc-fcc mechanical twining and fcc-bcc martensitic transformation. A distinction between the orientational and distortional variants is introduced, with an example given for the fcc-hcp transformation. Some crystallographic properties that could help the understanding of the transformation reversibility are also detailed. This approach is directly applicable to mechanical twinning in bcc and hcp crystals, and probably to diffusion-limited displacive transformations. This work gives a unified approach of the crystallography of displacive phase transformations and mechanical twinning in hard-sphere packed metallic alloys.