Long-wavelength phonons in incommensurate systems

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Abstract
The dynamic structure factors of crystals with sublattices which are mutually incommensurate are examined. The sublattices may be individually ordered, but incommensurate, or only one may be ordered with an interpenetrating fluid of the second atomic species. At high frequencies there are two propagating longitudinal-acoustic modes arising from the separate subsystems but in the long-wavelength limit there is only one propagating longitudinal-acoustic mode. The form of the crossover from high- to low-frequency behavior and also the detailed form of the low-frequency-mode structure depends on the relative importance of impurities and the intrinsic processes which relax the relative linear momentum of the two subsystems.