Some aspects of pressure-induced ω → β transformation in group IVB elements

The ω (hexagonal) to β (bcc) transformation in Zr and Hf occurs at 30 and 71 GPa under static pressures. This transition has not been found in Ti up to 87 GPa. On the basis of full-potential linearized augmented plane wave calculations aided with thermal and entropy correction we predict an ω → β transition in Ti at around 102 GPa along the 300 K isotherm. In addition to this, we calculate the ω → β transitions in Zr and Hf at around 27 and 65 GPa respectively, which are in excellent agreement with the experimental values. The ω → β transition pressures, 102, 27 and 65 GPa for Ti, Zr and Hf respectively, do not follow the trend implied by the principle of corresponding states. We have analysed the causes for this anomalous trend. We observe that the ω → β transition depends on how the increased d-population due to s–d transfer under pressure is distributed in the different d-substates. For example, at ambient conditions, the bcc phase is unstable and the difference between the average charges in the bcc stabilizing d-t_2g state and the destabilizing d-e_g state are 0.008, 0.082 and 0.013 for Ti, Zr and Hf respectively. Compression increases this difference and stabilizes the bcc structure when it becomes about 0.1. The charge transfer needed for stabilizing the β structure is highest for Ti followed by Hf and then Zr, in line with the trend in transition pressures.