LXXXI. Lattice vibrations and specific heat of graphite

Abstract

The vibrations of the graphite lattice can be treated approximately by ignoring the interaction between displacements parallel and perpendicular to the layer planes. The normal modes then separate into two types, which may be called planar and transverse. The complete frequency distribution of the transverse modes is calculated, and, with the aid of an approximate calculation of the distribution of the planar modes, the specific heat is deduced and compared with experimental results from 1·5° to 1000°K. Above 200° the atomic model gives results close to those obtained by superimposing two Debye two-dimensional continuum models, but below 100° the interaction between layers becomes important. Below 20°, the transverse modes are dominant, and the result is very sensitive to the coefficient of interaction between second neighbours in the layer This coefficient is close to a critical value which it would attain if the transverse restoring forces on the atoms of a single layer could be considered as acting only against out-of-plane bending of the bonds. It is shown that for any two-dimensional crystal the spectrum of the transverse vibrations is anomalous at low frequencies whenever the potential energy function satisfies certain conditions, and that these conditions are satisfied if the function is independent of any uniform small rotation of the crystal.