Geometric Theory of Lattice Vibrations and Specific Heat

TL;DR

This paper provides a geometric derivation of the classical Debye T^3 law for the specific heat of solids at low temperatures, using quantization of crystal lattices and spectral analysis.

Contribution

It introduces a geometric approach to derive the Debye law, connecting spectral geometry with solid state physics.

Findings

Rigorous derivation of Debye T^3 law

Application of discrete geometric analysis to lattice spectra

Asymptotic analysis of integrated density of states

Abstract

We discuss, from a geometric standpoint, the specific heat of a solid. This is a classical subject in solid state physics which dates back to a pioneering work by Einstein (1907) and its refinement by Debye (1912). Using a special quantization of crystal lattices and calculating the asymptotic of the integrated density of states at the bottom of the spectrum, we obtain a rigorous derivation of the classical Debye $T^3$ law on the specific heat at low temperatures. The idea and method are taken from discrete geometric analysis which has been recently developed for the spectral geometry of crystal lattices.