Influence of a frozen defect density wave on an incommensurate modulation

We report a theoretical study of the influence of an ordered periodic pattern of defects on an incommensurate modulation. We consider a one-dimensional model valid when the defect potential is much smaller than the potential of the underlying lattice, in which the discommensurations of the modulation (solitons) are handled as material objects. This model belongs to the class of Frenkel-Kontorowa problems. Between the normal-incommensurate transition and the transition of lock-in on the lattice, the soliton system undergoes a succession of transitions between two types of incommensurate phases either locked or unlocked on the defect wave. The average spacing between neighboring solitons as a function of the temperature is represented by a devil's staircase. The structure, the stability, and the Fourier spectrum of the unlocked phases are analyzed. In these phases, the soliton distribution is not uniform but modulated. As a consequence, the Fourier spectrum of the modulation splits into several components. Computed spectra account for the splitting of the incommensurate satellites observed in the scattering spectra of Ba₂{NaNb}₅O₁₅ or of Rb₂{ZnBr}₄$.