Off-centerTimodel of barium titanate

Abstract

It has recently been shown by NMR techniques that in the high-temperature cubic phase of ${\mathrm{BaTiO}}_{3}$ the $\mathrm{Ti}$ ions are not confined to the high-symmetry cubic sites, but rather occupy one of the eight off-center positions along the $[111]$ directions. The off-center $\mathrm{Ti}$ picture is in apparent contrast with most soft-mode-type theoretical descriptions of this classical perovskite ferroelectric. Here we apply the Girshberg-Yacoby off-center cation model of perovskite ferroelectrics assuming that the symmetrized occupation operators or ``pseudospins'' for the $\mathrm{Ti}$ off-center sites are linearly coupled to the normal coordinates for $\mathrm{TO}$ lattice vibrations. In the adiabatic limit, the coupling is eliminated by transforming to displaced phonon coordinates, and after excluding the self-interaction terms an effective $\mathrm{Ti}\text{\ensuremath{-}}\mathrm{Ti}$ interaction is obtained. Using the Langevin equations of motion and the soft-spin formalism for the $\mathrm{Ti}$ pseudospin degrees of freedom with ${T}_{1u}$ symmetry, the dynamic response of the coupled system is derived. The results are shown to be in qualitative agreement with the experimental data of Vogt et al. [Phys. Rev. B, 26, 5904 (1982)] obtained by hyper-Raman scattering. The nature of the phase transition, which is of a mixed displacive and order-disorder type, is discussed.