Theory of central peak and acoustic anomaly in cubic BaTiO3 close to ferroelectric transition

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 $\text{bi}p$. Unique effects are caused by the electrostrictive (ES) coupling between $\text{bi}p$ and the elastic displacement $\text{bi}u$. 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 $C_{11}^{\ast }(\omega )$, $C_{12}^{\ast }(\omega )$, and $C_{44}^{\ast }(\omega )$, 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