Vibration Spectra and Specific Heats of Diamond-Type Lattices

A method is presented for calculating the frequency distribution of a crystal from measured dispersion curves without constructing a force constant model. The method is then used to compute a vibration spectrum for Ge based on elastic constants and the dispersion curves measured by slow neutron scattering. The vibration spectrum is used to calculate the lattice specific heat, the results agreeing with experiment to within a few percent. A slightly modified distribution yields results agreeing to within one percent. The results are sufficiently detailed to show the necessity for including corrections due to anharmonicity even at $T\sim \frac{\theta }{2}$. By analogy with the results for Ge, vibration spectra are constructed for silicon and gray tin that also give good agreement with the specific heat measurements; from the spectra and also by analogy with Ge the dispersion curves for Si and gray tin can be sketched. A similar discussion is also given for InSb and diamond. Some general remarks are made concerning the relationship between specific heat curves and vibration spectra.