Erschienen in:Jülich : Kernforschungsanlage Jülich Zentralbibliothek, Verlag, Berichte der Kernforschungsanlage Jülich 37, p. 569-600 (1962).
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Work already published under the above topic by W. LUDWIG and the author$^{1}$ was limited to crystal temperatures above the Debye temperature, as it treated the lattice vibrations by classical statistics. In this paper, a complete quantum-statistical treatment is presented. The essential features of the high temperature results remain unchanged: New, temperature dependent dispersion curves may be introduced which, for long waves, are determined by the isothermal elastic constants.