Electronic Theory of Phase Transitions in Ca, Sr, and Ba under Pressure

Abstract

The transitions induced by temperature and pressure between the face-centered cubic (fcc) and the body-centered cubic (bcc) phases in the alkaline earth metals---Ca, Sr, and Ba---are analyzed by computing the differences in Gibbs free energy between the two phases in the framework of the nearly-free-electron and harmonic approximations. At the absolute zero of temperature and pressure, the observed fcc in Sr and bcc in Ba were found to have the lower internal energy; in Ca, however, identical analysis led to lower energy in bcc rather than the observed fcc. An fcc-bcc transition characterized by a change in the sign of the difference in free energies in Sr at 0\ifmmode^\circ\else\textdegree\fi{}K was found at a critical pressure ${P}_{c}\ensuremath{\sim}10$ kbar; at zero pressure, it was found at a critical temperature, ${T}_{c}\ensuremath{\sim}150\ifmmode^\circ\else\textdegree\fi{}$K. Both these results agree only qualitatively with the observed ${P}_{c}\ensuremath{\sim}36$ kbar and ${T}_{c}\ensuremath{\sim}830\ifmmode^\circ\else\textdegree\fi{}$K. Ca and Ba, already bcc at absolute zero, showed no phase transition.