Long-wavelength normal mode vibrations of infinite, ionic crystal lattices

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Abstract
This paper is an extension of an earlier article [J. Math. Phys. 13, 1207 (1972)] in which a study is made of the long-wavelength normal mode vibrations of infinite, ionic crystal lattices. The work is applicable to lattices of all symmetries but is restricted to the rigid ion model without retardation. This article completes the mathematical formalism introduced in the earlier article. In addition, two theorems are proved. The first states that all branches of the dispersion relations for an ionic lattice will approach definite frequencies as the propagation vector approaches zero if the point group of the space group of the lattice belongs to the regular system. The second states that, for all other lattices, at least two branches of the dispersion relations will approach frequencies which depend upon the direction of the propagation vector as the propagation vector approaches zero. A number of useful techniques are introduced for determining both the qualitative and quantitative behavior of the long-wavelength normal modes. Finally it is shown that the work in this paper is easily extended to include some but not all lattice models with

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