The Lattice Stability of Ti, Zr and Hf in an Ultra-Soft Pseudopotential Method of the First-Principles

Abstract

The lattice constants, total energies and densities of states of Ti, Zr and Hf in ⅣB group with different crystalline structures have been calculated, and the results have been compared with the results of the projector augmented wave (PAW) method in first-principles, CALPHAD and experiments. It is found that the results of lattice stability in this paper agree completely with those of the PAW method in first-principles and agree well with those of CALPHAD. Further analyses of lattice stability from electronic structures show that the crystalline Ti,Zr and Hf with hcp structures have the obvious character of stable phases, but fcc and bcc structures have the character of metastable and unstable phases, agreeing with the results of total energy calculations. It is found from the analyses of atomic populations that the transiton numbers of electrons from the s state to the p and d states for the hcp, fcc and bcc crystals decrease with the increase in elemental period, and the transiton number of electrons from the s state to the p state decrease dramatically, but the transiton from the s state to the d state increase dramatically, leading to much more number of electrons in the d state than that in the s or p state, stronger cohesion or higher cohesive energy between atoms in heavy metals and accordingly the stabler lattice in these heavier metals.