Lattice dynamics of crystals with tetragonalBaTiO3structure

A lattice-dynamical formalism using the rigid-ion model due to Born and Huang is applied to the ferroelectric crystals ${\mathrm{PbTiO}}_3$ and ${\mathrm{BaTiO}}_3$, in the tetragonal phase. The model includes short-range interactions of axially symmetric type between various ions in the primitive cell and long-range Coulomb interactions. The stability conditions are worked out in the manner described by Katiyar and are used to determine several first-order derivative potential constants for the crystals. The number of potential constants was further reduced by considering the variation of radial force constants with the ion-ion distance, as given by the exponential formalism of Born and Mayer. Zone-center phonons and a few of the low-frequency zone-boundary phonons were used for the nonlinear least-squares fitting. In general, we obtained excellent agreement between the calculated and observed frequencies. The resulting parameters showed that the short-range interaction between the nearest titanium and oxygen is approximately 1 order of magnitude stronger than the interactions between the lead and oxygen or between the oxygens.