Effect of strongly anharmonic longitudinal and transverse vibrations with wave vectork=2∕3(111)on the structural stability ofβ−Zrunder pressure

Abstract

The pressure and temperature dependence of the vibrational frequency of two interacting strongly anharmonic longitudinal and transverse modes with wave vector $\mathbf{k}=2∕3(111)$ in $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{Zr}$ is studied by solving a set of stochastic differential Langevin equations with a thermostat of the white noise type. The appropriate effective potential is calculated within the electron density functional theory, taking into account the contributions to the free energy from the electronic entropy depending on the atomic displacements. An analysis of the changes in the spectral density of vibrations with the pressure and temperature allows us to determine the stability region of the bcc phase of zirconium at pressures up to $35\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. A good agreement is obtained with the experimental data available. From the calculation performed it follows that the structural instability of the Zr bcc lattice with respect to the displacements characteristic of vibrations with wave vector $\mathbf{k}=2∕3(111)$ is of significant importance not only for the $\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\omega}$ transition, but also for the $\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\alpha}$ transformation observed at pressures less than $5\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$.