Theoretical Study of the Soft Optic Mode Dynamics in a Relaxor Ferroelectric. The Effect of Polar Nanoregions

Abstract

We propose a simple and solvable mean-field model of the scattering of transverse optic modes by Polarized Nano Regions (PNR) in the paraelectric phase of relaxor ferroelectrics. The PNR is assumed to be a ferroelectric sphere embedded in the host isotropic medium. The Lagrangian parameters are taken to be the same inside and outside the PNR, with exception of the soft-mode gap temperature dependence. The interaction of the Transverse (TO) with the longitudinal (LO) optic modes is taken into account but the latter is found to be important only at the surface of the PNR. Elementary excitations of the system are found to be of two types - Vortex (V) and Quasi Polar (QP). V excitations correspond to closed polarization lines or closed TO displacements while QP excitations contain open TO polarization lines, with an electric dipole if the net angular momentum j=1. LO waves are virtually excited only in the thin layer near the PNR surface and can be excluded from the boundary conditions without any consequence. The final boundary conditions include only V and QP TO wave amplitudes. Dynamical equations are solved and a phase diagram is predicted. The phase transition temperature is found to be lower for Vortex than for QP excitations. Therefore, the QP condensation (local phase transition) occurs before the Vortex one upon cooling. TO scattering by the PNR reveals strong long-wave resonances that can be shown to result from shallow localized and quasi-localized states and may be essential to understanding the waterfall observed in TO studies by inelastic neutron scattering.