System of Two (Not One) Declension Parameters for Prototype Undulation. Related U-Ferroelectricity

Abstract

This paper is concerned with a transition from an artificially incommensurately undulated prototypic phase to a ferroic phase, and also with U-ferroelectricity (which means ferroelectricity that should vanish if the prototypic phase is brought from undulated to homogeneous). Prototype undulation can be represented by the undulation of a set of declension parameters (denoted by δ, e,…). All previous papers that dealt with prototype undulation assumed only one declension parameter. In contrast the present paper assumes two. One case is studied, whose definition includes assuming \(\delta =U^{\delta}\,\cos\,(\hat{h}x+\hat{\theta}^{\delta})\) and \(\epsilon =U^{\epsilon}\,\cos\,(\hat{h}x+\hat{\theta}^{\epsilon})\) with all high harmonics ignored. The expressions of spontaneous polarization and dielectric susceptibility are deduced, the former of which indicates that, if and only if U δ ≠0, U e ≠0 and \(\sin\,(\hat{\theta}^{\delta}-\hat{\theta}^{\epsilon})\neq 0\), the ferroic phase is ferroelectric.