New method for reducing the general formula for lattice specific heat to the Einstein and Nernst-Lindemann approximations

To reduce the general formula for lattice specific heat to Einstein's formula of 1907, one traditionally models the spectrum of lattice modes-of-vibration as a set of independent oscillators all of one frequency, ν1. Not only is this a poor representation of a real solid, but no formula is provided for the frequency ν1, which has to be determined empirically. We offer a new and more compelling method for reducing the general formula to Einstein's formula. The reduction involves a simple mathematical approximation, proceeds without any reference to independent oscillators all of one frequency, and leads to a formula for the characteristic frequency, ν1, equal to the mean modal frequency. The mathematical approximation is valid at all but low temperatures, thereby providing insight into the failure of Einstein's formula at low temperatures. A simple extension of the new method leads to the Nernst–Lindemann formula for specific heat, proposed in 1911 on the basis of trial and error and currently without a sound theoretical basis. Empirical values (from the literature) of the frequencies that characterize the Einstein, the Nernst–Lindemann, and also the Debye formulae are all in support of the present theory. PACS Nos.: 65.40.Ba, 01.55.+b