Commensuration of the antiferroelectric incommensurate phase in (formula presented)

The antiferroelectric incommensurate phase in (formula presented) is incommensurate in both wavelength and direction of the wave vector characterizing the modulation. In this study, the crystallographic features of the incommensurate and commensurate phases in (formula presented) have been investigated by transmission electron microscopy to elucidate the details of commensuration to the commensurate phase. The wave vector of the modulation in the incommensurate phase was found to be (formula presented) with (formula presented) and (formula presented) while the wave vector in the commensurate phase is given by (formula presented) The important features of the incommensurate and commensurate phases are that the modulation modes with (formula presented) and (formula presented) are transverse atomic displacements of the Pb ions associated with the (formula presented) irreducible representation of the (formula presented) point group, and that another modulation with (formula presented) takes place in the commensurate phase. The (formula presented) modulation is associated with the (formula presented) representation corresponding to the antiparallel displacement of the Pb ions. The appearance of the (formula presented) modulation is accompanied by that of a pseudoperiodic array of antiphase boundaries with a phase shift of Π, and thus the commensurate phase is a nearly commensurate phase. That is, commensuration of both wavelength and direction of the incommensurate modulation takes place via the introduction of the (formula presented) displacement with the antiphase boundaries. If the antiphase boundary is regarded as a discommensuration, the commensuration process of the antiferroelectric incommensurate phase to the commensurate phase is therefore characterized by the appearance of a discommensuration, not by its annihilation.