Self-Consistent Anharmonic Theory and Its Application to the Soft Mode of BaTiO3

The phase transition with soft mode caused by the anharmonic lattice vibration is studied by applying the variational principle at finite temperature. The normal coordinates are introduced in order to reflect the crystal symmetry to the softening. It is found how the appearance condition for the soft mode depends on the interactions between atoms and the crystal symmetry. An equation to determine the soft mode is derived, and is substantiated by applying to the phase transition from cubic phase to tetragonal phase in the BaTiO3 crystal.