Extension of Debye's theory of specific heats of solids

Van Hove's theory of singularities tells us that for three-dimensional lattices the phonon frequency distribution function has square-root singularities and hence is continuous. Using this fact and Weierstrass's theorem, we extend Debye's theory of specific heats of three-dimensional solids to arbitrary phonon frequency spectra. It is found that in the low-temperature limit both the specific heat and thermal expansion coefficient exhibit the T3 law and the Grueneisen's law is valid. In the high-temperature limit, the thermal expansion coefficient approaches a volume-dependent constant and Grueneisen's law is valid. At intermediate temperatures Grueneisen's law is invalid