Commensuration of the antiferroelectric incommensurate phase inPb(Co1/2W1/2)O3

Abstract

The antiferroelectric incommensurate phase in ${\mathrm{P}\mathrm{b}(\mathrm{C}\mathrm{o}}_{1/2}{\mathrm{W}}_{1/2}{)\mathrm{O}}_{3}$ 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 ${\mathrm{P}\mathrm{b}(\mathrm{C}\mathrm{o}}_{1/2}{\mathrm{W}}_{1/2}{)\mathrm{O}}_{3}$ 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 ${\mathbf{k}}_{\mathrm{I}}=[1\ensuremath{-}{\ensuremath{\delta}}_{\mathrm{I}}1\ensuremath{-}{\ensuremath{\delta}}_{\mathrm{I}}{\ensuremath{\Delta}}_{\mathrm{I}}]$ with ${\ensuremath{\delta}}_{\mathrm{I}}\ensuremath{\approx}0$ and ${\ensuremath{\Delta}}_{\mathrm{I}}\ensuremath{\approx}0,$ while the wave vector in the commensurate phase is given by ${\mathbf{k}}_{c}=[1/21/20].$ The important features of the incommensurate and commensurate phases are that the modulation modes with ${\mathbf{k}}_{1}$ and ${\mathbf{k}}_{c1}$ are transverse atomic displacements of the Pb ions associated with the ${\ensuremath{\Sigma}}_{3}$ irreducible representation of the ${C}_{2v}$ point group, and that another modulation with ${\mathbf{k}}_{c2}=[110]$ takes place in the commensurate phase. The ${\mathbf{k}}_{c2}$ modulation is associated with the ${X}_{10}$ representation corresponding to the antiparallel displacement of the Pb ions. The appearance of the ${\mathbf{k}}_{c2}$ modulation is accompanied by that of a pseudoperiodic array of antiphase boundaries with a phase shift of \ensuremath{\pi}, 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 ${X}_{10}$ 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.