A model Hamiltonian has been constructed for the ferroelectric transition in the perovskite structure in terms of localized-strain and soft-normal-mode coordinates and temperature-independent model parameters. No anharmonic interactions higher than fourth order are considered. The free energy for tetragonal, orthorhombic, and trigonal distortions is calculated in a molecular-field approximation with the cubic structure as a reference configuration. From the free energy, the polarization and the strain distortions are determined. The soft-mode frequencies and the shifts in the acoustic-phonon frequencies are calculated from linearized equations of motion describing the fluctuations about these average values. The soft-mode frequencies in cubic structure vanish at the supercooling temperature, as usually assumed. However, in the distorted structure they remain finite at the stability limit determining the superheating temperature. The model Hamiltonian describes first- or second-order transitions depending on the strength of the coupling with the strain. For suitable choice of model parameters, the model allows for transition from the cubic to the tetragonal phase as in PhTi${\mathrm{O}}_3$, to the trigonal phase as in CsGe${\mathrm{Cl}}_3$ and various solid solutions, as well as a series of transitions from high- to low-symmetry structures, as in BaTi${\mathrm{O}}_3$.

• Received 9 December 1971

DOI:https://doi.org/10.1103/PhysRevB.5.3758