Crystal structures of Ti, Zr, and Hf under compression: Theory.
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We have studied the crystal structures of Ti, Zr, and Hf under pressure by means of first-principles, total-energy calculations. The three metals are shown to exhibit a crystal structure sequence hcp[r arrow][omega][r arrow]bcc, with increasing pressure. This is in good agreement with experiment for Zr and Hf, whereas the bcc structure for Ti is a prediction. The calculated transition volumes as well as transition pressures compare fairly well with experiment. Also, the computed [ital c]/[ital a] ratio for hcp Ti, Zr, and Hf is found to be in good agreement with experiment. Similarly the calculated [ital c]/[ital a] ratio for Zr in the [omega] structure agrees well with measurement. The chemical bonding of the [omega] structure is shown to be quite different from what is normally the case in the transition metals, with a large degree of covalency. A search for the [omega] structure in Tc and Ru was fruitless yielding a stable hcp structure. At zero temperature and zero pressure the bcc crystal structure is found to be mechanically unstable for Ti, Zr, and Hf.