We present a Ginzburg-Landau theory on statics and dynamics of BaTiO$_3$-type ferroelectrics in the paraelectric phase with the cubic structure, where the order parameter is the polarization $\bi p$. Unique effects are caused by the electrostrictive (ES) coupling between ${ \bi p}$ and the elastic displacement $\bi u$. We show that the ES coupling gives rise to a central peak in the Fourier-Laplace transform of the displacement time-correlation function at small wave numbers. It emerges and grows with a narrow width as the transition is approached. Such central peaks have long been observed in a number of scattering experiments in various ferroelectrics, but their origin has not been well understood. From the acoustic part of the displacement dynamic correlation we obtain the frequency-dependent elastic moduli $C_{ 11}^*(\omega)$, $C_{ 12}^*(\omega)$, and $C_{ 44}^*(\omega)$, whose singular parts arise from the ES coupling, We then calculate the singular sound velocity and attenuation. In the central peak and the elastic moduli, the frequency $\omega$ appears in the scaled form $\omega\tau_D$, where $\tau_D$ is the Debye relaxation time in the frequency-dependent dielectric constant. { Keywords}: ferroelectric transition, central peak, acoustic anomaly, electrostrictive coupling
Categories: cond-mat.mtrl-sci