The unrelaxed vacancy formation energies were calculated, for group-IV elements with the hexagonal close-packed or body-centered cubic structure, within the local density approximation to density-functional theory, by using the full-potential linear muffin-tin orbital method. In the case of hexagonal close-packed Hf, the calculated value of 2.37eV was in excellent agreement with the experimental value of 2.45eV. The results for hexagonal close-packed Ti (2.14eV) and Zr (2.07eV) could therefore be considered to be reliable predictions. In the case of the more open body-centered cubic structure, after convincing validation of the present method for Mo and W, vacancy formation energies of 2.2 to 2.4eV were obtained for Ti, Zr and Hf. These energies, which were very similar to those for the hexagonal close-packed structure, were significantly larger than the experimentally measured activation energies for self-diffusion in the body-centered cubic structure. By assuming that the monovacancy mechanism predominated in β-Ti, β-Zr and β-Hf, it was demonstrated that structural relaxations with particularly large amplitudes were to be expected around the vacancy.

Unrelaxed Vacancy Formation Energies in Group-IV Elements Calculated by the Full-Potential Linear Muffin-Tin Orbital Method: Invariance with Crystal Structure. O.Le Bacq, F.Willaime, A.Pasturel: Physical Review B, 1999, 59[13], 8508-15