Dynamics of relaxor ferroelectrics

We study a dynamic model of relaxer ferroelectrics based on the spherical random-bond-random-field model and the Langevin equations of motion written in the representation of eigenstates of the random interaction matrix. The solution to these equations is obtained in the long-time limit where the system reaches an equilibrium state in the presence of random local electric fields. The complex dynamic linear and third-order nonlinear susceptibilities chi (1)(omega) and chi (3)(omega), respectively, are calculated as functions of frequency and temperature. In analogy with the static case, the dynamic model predicts a narrow frequency dependent peak in chi (3)(T,omega), which mimics a transition into a glasslike state, but a real transition never occurs in the case of nonzero random fields. A freezing transition can be described by introducing the empirical Vogel-Fulcher (VF) behavior of the relaxation time tau in the equations of motion, with the VF temperature To playing the role of the freezing temperature T-f. The scaled third-order nonlinear susceptibility a(3)(')(T,omega) = <(<chi>)over bar>(')(3)(omega)/<(<chi>)over bar>(')(1);(3 omega)<(<chi>)over bar>(')(1)(omega)(3), where the bar denotes a statistical average over T-0, shows a crossover from paraelectriclike to glasslike behavior in the quasistatic regime above T-f. The shape of <(<chi>)over bar>(1)(omega) and <(<chi>)over bar>(3)(omega)-and thus of a(3)(')(T,omega)-depends crucially on the probability distribution of tau. It is shown that for a linear distribution of VF temperatures T-0, a(3)(')(T,omega) has a peak near T-f and shows a strong frequency dispersion in the low-temperature region.