We study the bcc-hcp phase transition in Ti and Zr with the use of first-principles calculations. We have determined the complete energy surface from the bcc to hcp structure. The results are used to find an appropriate Landau free energy density for describing this transformation. The proposed Landau free energy density has two relevant order parameters: shear and shuffle. Through first-principles calculations, we show that the bcc structure is unstable with respect to the shuffle of atoms (TA1 N-point phonon) rather than the shear. Therefore, we reduce the two order parameter Landau free energy to an effective one order parameter (shuffle) potential, which is a reasonable approximation. In general, the effective Landau free energy is a triple-well potential. From the phonon dispersion data and the change in entropy at the transition temperature we find the free energy coefficients for Ti and Zr.
Physical Review B
- Pub Date:
- June 2001
- Surface states band structure electron density of states;
- Electron states and collective excitations in multilayers quantum wells mesoscopic and nanoscale systems;
- Magnetic properties of monolayers and thin films