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Theory of First, Second and Zero Sound in One‐Dimensional Crystals
Author(s) -
Niklasson Göran
Publication year - 1969
Publication title -
fortschritte der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.19690170402
Subject(s) - boltzmann equation , phonon , physics , speed of sound , collision , perturbation (astronomy) , boltzmann constant , perturbation theory (quantum mechanics) , zero (linguistics) , relaxation (psychology) , dispersion (optics) , space (punctuation) , second sound , wave equation , classical mechanics , mathematical analysis , quantum mechanics , sound (geography) , acoustics , mathematics , psychology , social psychology , linguistics , philosophy , computer security , computer science
The propagation of thermal and mechanical disturbances in a non‐conducting crystal is studied on the basis of a generalized transport equation for phonons, derived in a previous paper. This equation agrees with the Peierls‐Boltzmann equation for slowly varying inhomogeneities in space and time and is consistent with the perturbation treatment of the phonon self energy for high frequencies and short wave‐lengths. A formal solution of the transport equation is obtained, which allows us to study in detail the transition between the low‐frequency, collision‐dominated regime and the high‐frequency, essentially collision‐less regime. Using a single relaxation time approximation we discuss the propagation of “sound” waves and obtain explicit expressions for the dispersion of the sound velocity and the damping, valid for arbitrary frequencies. In order to elucidate more clearly the technique used we consider in this paper an one‐dimensional crystal, but the essential results can easily be taken over to the threedimensional case. A separate report on this will be published elsewhere.