High Harmonics in Prototype Undulation. Related U-Ferroelectricity and U-Ferroelasticity

Abstract

This paper performs the first investigation of high harmonics in prototype undulation. It also deals with U-ferroelectricity and U-ferroelasticity. [U- (or H-) ferroelectricity is ferroelectricity that should (or should not) vanish if the prototypic phase is brought from undulated to homogeneous.] For simplicity, only the second harmonic is taken into account, and no higher ones. Two cases are studied. They are the same in that the point group of the prototypic phase when homogeneous is mmm, the number of declension parameters is one and the expression of undulation of this parameter is δ =U₁ cos (ĥx +\hat{θ}₁)+U₂ cos (2ĥx+\hat{θ}₂). The two cases are different with regard to order-parameter symmetry. It is found that, if and only if U₁≠ 0, U₂≠ 0 and sin (\hat{θ}₂-2\hat{θ}₁)≠ 0, U-ferroelasticity occurs in one of the two cases, and U-ferroelectricity occurs (and is fast coupled with H-ferroelasticity) in the other case.