A procedure for the calculation of the number of normal modes of a single crystal is proposed which takes an intermediate position between the methods of Debye and Born–von Karman. The method of Debye is extended to an anisotropic continuum, where the cutoff and dispersion phenomena, which are due to the lattice structure, are accounted for in a semiempirical way. It appears possible to define a finite number of characteristic temperatures (one for cubic crystals and at most three for crystals of low symmetry) independent of direction. This ensures a comparatively simple calculation from the phenomenological elastic constants of the crystal, as such retaining one of the pleasing features of Debye's theory, i.e., a straightforward correlation between thermal and elastic data.The method is applied to eight cubic monatomic crystals for which elastic data are available. The results provide some additional evidence to emphasize the significance of the dispersion of the Debye heat waves.An application to the hexagonal crystals of cadmium and zinc leads to results similar to those obtained by Grüneisen and Goens who produced with these crystals the first experimental evidence of the dispersion phenomenon using the concept of a characteristic temperature dependent on direction.In the last section the correlation between the elastic constants of single crystals and the corresponding quasi-isotropic materials is discussed and illustrated with data found in the literature. It is shown that the polycrystalline state is more "elastic" (sometimes very considerably) than the single crystal state. The consequences of this "boundary layer elasticity" for the calculation of θ values are discussed.
Explaining the specific heat of liquids based on instantaneous normal modes
