Higher Order Electrostrictlon in Ferroelectric Tungsten Bronzes

The Landau:Ginsburg:Devonshire phenomenological theory for proper ferroelectrics has been extended to include sixth order electrostrictive coupling in the prototype (paraelectric) structure. These higher orderp terms lead to contributions to the stiffened elastic compliances s,.,,ijkl which are quadratic in polarization P, so that in the ferroelectric phase the strong temperature dependence of Ps the spontaneous polarization is reflected into the elastic response. The analysis has been applied to the tungsten bronze ferroelectric Ba,, TnSr_,, Nb,,0. and sixth order electrostriction constants measured in the prototype phase used to derive elastic behavior in the ferroelectric form. A feature which makes the phenomenology valuable is that the higher order stiffnesses and coupling terms do not change markedly with temperature or with cationic makeup, thus in principle it is possible to predict from a single family of prototypic constants quantitative trends for the dielectric, piezoelectric, elastic, and thermal constants for a very wide range of bronze ferroelectric compounds.