Elastic and Dielectric Anomalies and the Lattice-Dynamical Theory of Ferroelectric Phase Transitions

The lattice-dynamical theory of ferroelectric phase transitions relates the dielectric anomaly with an anomalous "softening" of a TO mode as $q\to 0$. It is shown that the elastic anomalies, which are commonly observed at ferroelectric phase transitions, are predicted by the lattice-dynamical theory of ferroelectricity, within a diatomic approximation. The softening of the TO mode is associated with variations in $\frac{d\left({\omega }_{\mathrm{TA}}\right)}{\mathrm{dq}}$ which are reflected in the behavior of the transverse and shear elastic constants. The discussions are extended to more complicated systems and are illustrated by an examination of the available neutron scattering data on the temperature dependence of the dispersion curves of the TO and TA modes of SrTi${\mathrm{O}}_3$ and KTa${\mathrm{O}}_3$.