Electronic structure of hcp transition metals

Using the linear muffin-tin-orbital method described in the previous paper, we have calculated the electronic structures of the hcp transition metals, Zr, Hf, Ru, and Os. We show how the band structures of these metals may be synthesized from the $\mathrm{sp}$ and $d$ bands, and illustrate the effects of hybridization, relativistic band shifts, and spin-orbit coupling by the example of Os. By making use of parameters derived from the muffin-tin potential, we discuss trends in the positions and widths of the energy bands, especially the $d$ bands, as a function of the location in the periodic table. The densities of states of the four metals are presented, and the calculated heat capacities compared with experiment. The Fermi surfaces of both Ru and Os are found to be in excellent quantitative agreement with de Haas-van Alphen measurements, indicating that the calculated $d$-band position is misplaced by less than 10 mRy. Very small pieces of Fermi surface, which have not yet been observed experimentally, are predicted for Os. The limited amount of experimental information available for Zr can be fairly satisfactorily interpreted if the calculated $d$ bands are raised by about 10-20 mRy relative to the $\mathrm{sp}$ bands. This gives rise to a Fermi surface which is topologically equivalent to that recently found in Ti, and which does not support open orbits when the magnetic field is sufficiently great that breakdown is complete. It is suggested that the Fermi surface of Hf is probably similar, although very little experimental evidence is available for this metal. Some comments are made about the calculational method, which has proved to be physically transparent, accurate and extremely fast, and the adequacy of the standard potential, which has now been successfully employed in calculations on the great majority of the transition metals.