Experimental studies of structurally incommensurate crystal phases

The aim of this review is to present aperiodic crystals from a unifying point of view, showing why it is justified to call them crystals, despite the lack of a three-dimensional lattice periodicity, and to discuss in what sense they differ from periodic crystals in structure, symmetry and other physical properties. The extension of the concept of crystal has been based, during the last two decades, on investigation of incommensurate crystal phases. Among these, the most important ones are the modulated crystals. Their crystallographic nature, already apparent in the diffraction pattern, could be made explicit on the symmetry level by embedding in a higher-dimensional Euclidean space. The recent discovery of quasicrystal phases (representing a fairly different class of aperiodic crystals than the modulated ones) can also be approached in a similar way. Furthermore, it now appears that another class of incommensurate crystals, the so-called composite structures, represents a kind of intermediate ca...

Incommensurability in crystals

Structure–property relationship studies have been performed in the La-modified lead zirconate titanate (PLZT) solution as a function of quenched La impurity content and Zr/Ti ratio by transmission electron microscopy, lattice imaging, and dielectric spectroscopy. Investigations were performed for 65/35⩽Zr/Ti⩽90/10. These investigations have demonstrated a general trend in domain structure and polar order with increasing quenched impurity concentration. For Zr/Ti ratios of 90/10 and 85/15, a structurally incommensurate antiferroelectric (AFEin) state was found to be stabilized with increasing La. Temperature dependent investigations demonstrated that the incommensurate structure becomes pinned into long-time metastable states, rather than transforming to a commensurate phase. Also, the modulation wavelength (λ) was found to increase significantly with increasing Ti. When λ increased to ∼40 A, polar clusters began to condense from the AFEin order. For Zr/Ti ratios of 80/20 and 65/35, increasing quenched dis...

Effects of quenched disorder on La-modified lead zirconate titanate: Long- and short-range ordered structurally incommensurate phases, and glassy polar clusters

The hysteresis area as a function of frequency of the time-varying external electric field—hysteresis dispersion—for ferroelectric Pb(Zr0.52Ti0.48)O3 is measured, and the Monte-Carlo simulation on the hysteresis dispersion for a model ferroelectric lattice is performed too. We demonstrate the scaling behavior of the single-peaked hysteresis dispersion for the two ferroelectric systems, predicting a unique effective characteristic time for the domain reversal. This characteristic time shows an inversely linear dependence on the field amplitude as long as the amplitude is high enough that the reversible domain rotation response is negligible.

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Scaling on hysteresis dispersion in ferroelectric systems

Publisher Summary This chapter presents a study on excitations in incommensurate crystal phases. It presents an example of incommensurately modulated crystals, which are quasiperiodic structures for which the rank is related to the number of independent frozen modulations. In intergrowth compounds the rank is determined by the number of chemical subsystems participating in the common structure. Many incommensurate systems can be viewed as modulated crystals. This implies the existence of a basic three-dimensional (3-D) periodic structure to which the modulation is applied. The word modulation is used here in a generic sense and stands for any distortion or static perturbation characterized by its own independent periodicity. Nonlinear forces are at the very basis of the incommensurate phase transitions. The dynamics, however, have in the present contribution only been treated in the linear regime. Nonlinear excitations are expected to play a more important role in modulated systems than in ordinary crystals. The chapter discusses Debye–Waller factor, Frenkel–Kontorova model, DIFFOUR model, and other related concepts.

T. Janssen

Papers

One-dimensional model for a crystal with displacive modulation