Ferroelectric free energy expansion coefficients from double hysteresis loops

A direct method has been developed to determine the temperature dependence of the coefficients of the free energy expansion (A = 1/2χxP2 + 1/4ξP4 + 1/6ζP6) of a crystal spontaneously polarized in a temperature region where double hysteresis loops are observable (This implies that ξ < 0). The features of the (P)vs. (E = δA/δP) relationship are fully displayed in the double hysteresis loop, thus allowing the simultaneous determination of χ, ξ and ζ. The coefficient χ can be measured directly from the slope (δP/δE) of the straight line at the center of the double loop. We can define Pc+ and Pc- as the polarization before “switching” for decreasing and increasing field, respectively. This characterizes one of the single loops in the (P)vs(E) diagram, and hence the values of ξ and ζ can easily be computed.This approach has been used to analize experimental results on single crystals of BaTiO3. Oscillograms of the double hysteresis loops in the region immediately above Tc = 111·6°C were taken and the temperature dependence of χ, ξ and ζ determined. Simultaneous measurements of the dielectric constant on the same sample yielded a value of χ in good agreement with the above value in the same temperature region. The temperature dependence of ξ is more pronounced than that previously reported. In addition, ζ has been shown to have a marked temperature dependence, a result not accessible to other techniques.