Fluctuation effects as a cause for the incommensurate phase formation

The role of the incommensurate phase in the opalescence of quartz

“…A global hysteresis is observed in the temperature variations of the modulation waves and of most physical properties, as shown by curve (B) of figure 1 for the birefringence [25]. In fact the coupling term F of equation (1) corresponds to an interaction between the soft mode of the α-β transition and an acoustic mode [18,26], which has indeed been observed in subsequent inelastic neutron scattering measurements in the β-phase [27]. The presence of an inc phase leads to the existence of new vibration modes (amplitudons and phasons).…”

“…In 1978, Aslanyan and Levanyuk [18] suggested, in a theoretical work, that an inc phase could exist between the classical αand β-phases of quartz: this is due to the presence, in the Landau free energy, of a coupling between elastic strains u ij and the spatial gradient of the order parameter η, given by…”

“…Using the classical theory of the inc phase of quartz [18], Saint-Grégoire et al [10] developed a model based on their TEM observations of the 2D-ELT structure reported in the same paper. By evaluating the competition between the energy of distant parallel walls and the energy of wall crossing, they predicted the existence, just above T c , of a stable ferroelastic ELT phase.…”

“…The model of Aslanyan et al [11] represents a major departure from the classical model, previously used with great success to explain the existence and the properties of the inc phase of quartz [18,26]. They also proposed that the intense opalescence of quartz is due to the domains of a ferroelastic inc phase, but with a drastically different structure.…”

Universal emergence of spatially modulated structures induced by flexoantiferrodistortive coupling in multiferroics

“…The first one consider a one-component long-range order parameter assuming that its gradient coupling coefficient in the Landau-Ginzburg-Devonshire (LGD) power expansion is negative and positively-defined higher order derivatives cause the ICP [48,49,50]. The second approach, that seems more relevant to the ICP in multiferroics description, considers at least a two-component order parameter with positive gradient coefficients, conventional LGD functional for each component, and biquadratic coupling between the order parameters and Lifshitz invariant [38,47,51,52].…”