Anisotropic Oxygen Polarizability and the Lorentz Correction in BaTiO3

The temperature dependence of the electronic polarizabilities of the ions in BaTi${\mathrm{O}}_3$ is incorporated in the Slater-Devonshire theory under the assumption that the dominant contribution arises from the ${\mathrm{O}}_a$ electronic polarizability due to the large Ti-${\mathrm{O}}_a$ overlap along the polar axis. The temperature dependence of the ${\mathrm{O}}_a$ polarizability as determined from optical data is parameterized in the spontaneous polarization, and a free-energy function for the clamped crystal is derived and compared with the (adjusted) experimental free energy. An internal check on this comparison yields $\left(\frac{\mathrm{dB}}{\mathrm{dT}}\right)=5.63\mathrm{}\times {10}^{-15\mathrm{}}$ cgs units, compared with the experimental value 4.5×${10}^{-15\mathrm{}}$ ($B$ is the fourth-order coefficient in the free energy). A minimum-internal-energy calculation is performed for the clamped crystal polarized along [001], corresponding to the tetragonal phase. This calculation illustrates the role of the ${\mathrm{O}}_a$ polarizability in limiting the spontaneous polarization: Using the ${\mathrm{O}}_a$ polarizability anisotropy data, a spontaneous polarization of 59 600 esu is obtained; if the isotropic oxygen polarizability in the cubic phase is used, 667 000 esu. Similar calculations are performed for the clamped crystal polarized along [011] and [111], corresponding to the orthorhombic and rhombohedral phases, respectively. The ${\mathrm{O}}_a$ polarizability anisotropy data are used, and for the [011] calculation a spontaneous polarization of 33 300 esu is obtained. It is found that the Lorentz correction for the clamped crystal corresponding to a [111] polar axis is not large enough to support a spontaneous polarization, but that a shear of the unit cell of about 27′ is required to stabilize a spontaneous polarization along this axis.