Fluctuations, higher order anharmonicities, and Landau expansion for barium titanate

The correct description of the ferroelectric phase transition in barium titanate requires that, in the expansion of the free energy, not only the sixth-order terms but also the eighth-order terms should be retained. One more unusual feature of the BaTiO3 compound is that the coefficients B1 and B2 of the terms Px4 and Px2Py2 in the Landau expansion depend on the temperature. It is demonstrated that the temperature dependence of the coefficients B1 and B2 can be associated with the thermal fluctuations of polarization under conditions where the fourth-order anharmonic constants are anomalously small; i.e., a nonlinearity of the type P4 and higher order anharmonicities make comparable contributions. Regular (noncritical) fluctuation contributions to the coefficients B1 and B2 are calculated to the first order in the sixth- and eighth-order anharmonic constants. Both contributions increase monotonically with increasing temperature, which is in agreement with the experimental dependences of these coefficients. The proposed theory without additional assumptions makes it possible to determine the ratio of the fluctuation components of the coefficients B1 and B2. This ratio also appears to be very close to that observed in the experiment.