Fluctuations, higher anharmonisms, and the Ginzburg–Landau–Devonshire expansion for barium titanate

We show that the strong temperature dependence of the coefficients $\mathrm B_1$ and $\mathrm B_2$ of terms of the form $P_x^4$ and $P_x^2P_y^2$ in the Ginzburg–Landau–Devonshire expansion for barium titanate can result from the action of thermal polarization fluctuations if the fourth-order anharmonic constants are anomalously small, i.e., if the nonlinearity of the form $P^4$ and higher anharmonisms play comparable roles. We calculate the regular (noncritical) fluctuation contributions to $\mathrm B_1$ and $\mathrm B_2$ in the first approximation in the sixth- and eighth-order anharmonic constants. Both contributions increase monotonically with temperature, which agrees with the forms of the experimentally observed temperature dependences of these coefficients. The theory allows finding the ratio of the fluctuation components $\mathrm B_1$ and $\mathrm B_2$ without additional assumptions. This ratio also turns out to be very close to the experimental value.