a Macroscopic Theory of Coupled Phasons and Sound Waves in Incommensurate Crystals.

A macroscopic description of the long-wavelength, low-frequency excitations in pure incommensurate solids is presented. This is facilitated by the introduction of the relative phase-displacement field which describes the distortions of the modulation pattern relative to the center-of-mass displacement field. Then, the general equations of motion for the sound waves and phasons are found and analyzed. The coupling with the phasons affects the sound -wave's attenuation at long wavelengths, and velocity at shorter wavelengths. Further, in the k to 0 limit, it is found that the attenuation of transverse sound waves, under interchange of their direction of propagation and polarization, is asymmetric. The phason solutions are diffusive at long wavelengths, and propagating at sufficiently short wavelengths. The question of whether or not a phason gap of 50 GHz exists in the incommensurate phase of K _2SeO_4 is then examined. It is shown that the data that was found to be consistent with a phason gap is also consistent with a gapless-phason spectrum provided that the phason is coupled to a longitudinal sound wave. An analysis of the transverse sound-wave asymmetries for systems with and without a phason gap then provides a possible existence test for the phason gap. The linear piezoelectric effect is known to influence sound waves in certain dielectric normal crystals. After the form of the relative phase-displacement field is established, and the super-space group symmetry is found (leading to the determination of the point-group symmetry), the effects of piezoelectricity on the incommensurate phase of quartz are discussed.