Dynamics of interacting clusters and dielectric response in relaxor ferroelectrics

The dielectric response in relaxor ferroelectrics is analyzed in the framework a model for the polarization dynamics in the presence of polar clusters. We associate the origin of polar clusters with the atoms displaced from their centro-symmerical positions even above $T_c$. Their collective hopping in multi-well potentials induced by disorder is analogous with the situation in glasses. The theory explicitly takes into account the distribution of cluster reorientation frequencies and the effect of cluster-cluster interactions in highly polarizable crystals, which we describe in terms of the local field distribution function. The dielectric constant is obtained from an integral master equation for the polarization dynamics in the presence of a time dependent electric field. The theory is applied for the analysis of the shape of the frequency dependent permittivity in the typical relaxor ferrolectrics $PST$ as a function of temperature. The comparison of the theory with experiment shows that in contrast to earlier assumptions, the observed Vogel-Fulcher dependence of the permittivity maximum is a consequence of the Vogel-Fulcher temperature dependence of the cluster reorientation frequency.