The electronic structure of the early transition metals Ti, Zr, and Hf has been investigated for the hexagonal (\ensuremath{\omega}), hcp (\ensuremath{\alpha}), bcc (\ensuremath{\beta}), and fcc phases using the linear-muffin-tin-orbital method of band-structure calculation. The results of these investigations are discussed in the light of previous band-structure calculations and experimental data closely related to the electronic structure (low-temperature specific heat, superconductive properties and magnetic susceptibility). It is found from the theoretical calculations that the electronic density of states at the Fermi level increases in the phase sequence hexagonal (\ensuremath{\omega})\ensuremath{\rightarrow}hcp (\ensuremath{\alpha})\ensuremath{\rightarrow}fcc\ensuremath{\rightarrow}bcc (\ensuremath{\beta}) and this behavior is unambiguously reflected in all the experimental data as well. A separation of the magnetic susceptibility into its components is performed and from this, the temperature dependence of the Pauli susceptibility for Ti, Zr, and Hf is deduced.

Electronic structure and magnetic susceptibility of the different structural modifications of Ti, Zr, and Hf metals.

Theoretical study of the structural properties and thermodynamic stability of the omega phase in the 4d-transition series

Among the methods available for the study of the electronic structure of solids, the investigation of the electron momentum density with the aid of low-energy positron annihilation takes a place of increasing importance. During the 35 years that have elapsed since the first observation of the angular correlation between the two annihilation quanta by BERINGER and MONTGOMERY [2.1] the positron annihilation technique has developed into a method capable of yielding relevant information about various aspects of the electronic structure such as Fermi surface dimensions and wave functions. So far, the technique has been applied mainly to pure metals which have served as a practicing ground. For metals, however, the dimensions of the Fermi surface may be obtained by other methods such as the de Haas-van Alphen (dHvA) effect, the magnetoresistance effect, the rf size effect, etc., that are often capable of a greater precision than positron annihilation. Lately, nondilute disordered alloys and high-temperature phases of ordered materials have been gaining interest. In these systems excessive electron scattering due to the short mean free path of the electrons precludes the use of the above-mentioned methods at solute concentrations ≧1%. At higher concentrations, other methods like the measurement of Kohn anomalies in X-ray diffuse scattering [2.2] or inelastic neutron scattering [2.3], and the Faraday effect [2.4,5] can still be used for investigations of the Fermi surface geometry. The Compton effect [2.6] is able, in principle, to provide direct information about the entire electron momentum distribution, but cannot compete in resolution with positron annihilation.

Electron Momentum Densities in Metals and Alloys

The electronic structures of hexagonal-close-packed divalent titanium (3-d) and zirconium (4-d) transition metals are studied by using a non-local model potential method. From the present calculation of energy bands, Fermi energy, density of states and the electronic heat capacity of these two metals are determined and compared with the existing results in the literature.

Electronic structure and equilibrium properties of hcp titanium and zirconium

The electronic bandstructures of rare earth metal Praseodymium (Pr) have been calculated for a wide range of static pressures using RAPW method. The density of states and Fermi surface cross sections on the principal planes corresponding to various pressures are obtained. It is seen that, there is, 4f shell breakdown in Pr under pressure and formation of 4f bands. The calculation of the superconducting transition temperaturesT

 c
 corresponding to various pressures shows that Pr may become a high pressure superconductor like Cerium eventhough it is not observed experimentally to be a superconductor because of its magnetic character at normal pressure.

Bandstructure, fermi surface and superconductivity of Pr under high pressure

The electronic bandstructure of zirconium, which has a HCP lattice structure, has been calculated using crystal potentials which include the effect of correlation as well as exchange. Two methods, one due to Kohn and Sham (1965) and the other to Overhauser (1971) are used for calculating the exchange and correlation contributions to the crystal potential. With the inclusion of correlation the levels around the symmetry point K become well separated, and the entire set of bands is shifted by about 0.25 Ryd. The density of states has been obtained using one of the potentials. The bandstructure results are used to calculate the momentum density distribution of positron annihilation along the c axis.

C K Majumdar

Papers

Effect of correlation on the bandstructure of zirconium