Commensuration of the antiferroelectric incommensurate phase in Pb(Co_1/2W_1/2)O₃

The antiferroelectric incommensurate phase in Pb(Co_1/2W_1/2)O₃ 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 Pb(Co_1/2W_1/2)O₃ 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 k₁=[1-δ_I 1-δ_I Δ₁] with δ_I≈0 and Δ_I≈0, while the wave vector in the commensurate phase is given by k_c=[1/2 1/2 0]. The important features of the incommensurate and commensurate phases are that the modulation modes with k₁ and k_c1 are transverse atomic displacements of the Pb ions associated with the Σ₃ irreducible representation of the C_2v point group, and that another modulation with k_c2=[110] takes place in the commensurate phase. The k_c2 modulation is associated with the X₁₀ representation corresponding to the antiparallel displacement of the Pb ions. The appearance of the k_c2 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 X₁₀ 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.