0 Dynamics of relaxor ferroelectrics

We study a dynamic model of relaxor 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 susceptibili-ties χ 1 (ω) and χ 3 (ω), 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 χ 3 (T, ω), which mimics a transition into a glass-like state, but a real transition never occurs in the case of non-zero random fields. A freezing transition can be described by introducing the empirical Vogel-Fulcher (VF) behavior of the relaxation time τ in the equations of motion, with the VF temperature T 0 playing the role of the freezing temperature T f. The scaled third-order nonlinear susceptibility a ′ 3 (T, ω) = χ ′ 3 (ω)/χ ′ 1 (3ω)χ ′ 1 (ω) 3 , where the bar denotes a statistical average over T 0 , shows a crossover from paraelectric-like to glass-like behavior in the quasistatic regime above T f. The shape of χ 1 (ω) and χ 3 (ω)—and thus of a ′ 3 (T, ω)—depends crucially on the probability distribution of τ. It is shown that for a linear distribution of VF temperatures T 0 , a ′ 3 (T, ω) has a peak near T f and shows a strong frequency dispersion in the low temperature region.