It was noted that, in the absence of external fields, ferro-electric materials would have a multi-domain configuration in the ferro-electric state. A detailed analysis showed that twinning could not be treated as random since the number of orientations of the domain walls were limited for a given symmetry change during a ferro-electric phase transition. In each finite region of a large crystal, or in small crystallites, a particular set of twins was favored under certain boundary conditions. This consisted of only 2 of the low-temperature variants. Statistical models for random distributions of domains did not apply to the calculation of the physical properties of such twin structures. However, it was possible to derive the 2 domain twin properties by using the constitutive equations plus appropriate mechanical boundary considerations. The present theoretical analysis of such a 2-domain twin system included its global symmetry and the effective material properties which resulted from various twinning configurations.

Effective Material Properties in Twinned Ferroelectric Crystals. J.Erhart, W.Cao: Journal of Applied Physics, 1999, 86[2], 1073-81