HIGH HARMONICS IN PROTOTYPE UNDULATION - RELATED U-FERROELECTRICITY AND U-FERROELASTICITY

This paper performs the first investigation of high harmonics in proto type undulation. It also deals with U-ferroelectricity and U-ferroelas ticity. [U- (or H-) ferroelectricity is ferroelectricity that should ( or should not) vanish if the prototypic phase is brought from undulate d to homogeneous.] For simplicity, only the second harmonic is taken i nto account, and no higher ones. Two cases are studied. They are the s ame in that the point group of the prototypic phase when homogeneous i s mmm, the number of declension parameters is one and the expression o f undulation of this parameter is delta = U-1 cos(<(h)over cap x> + <( theta)over cap (1)>) + U-2 cos(<2(h)over cap (x)> + <(theta)over cap ( 2)>). The two cases are different with regard to order-parameter symme try. It is found that, if and only if U-1 not equal 0, U-2 not equal 0 and sin(<(theta)over cap 2> - <2(theta)over cap (1)>) not equal 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 .