Vibration Spectrum of Graphite and Boron Nitride. I. The Two-Dimensional Spectrum

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Abstract
A detailed analysis of the vibrational modes of graphite and boron nitride, which are sufficiently similar that they can conveniently be considered together, begins here with a study of the effects of interactions only between particles in the same layers of the lamellar structure. The secular equation for any lattice vibration problem can be written as a sum of contributions arising from the interactions between various groups of particles. After describing a system of notation which will eventually be used to describe all types of interactions, that part of the secular equation resulting only from nearest neighbors is derived. The frequencies and the spectral densities of all frequency branches are obtained for a system with only these forces. Because of the existence of some low‐frequency branches, it is necessary to also consider second nearest neighbor forces, even though they may be relatively much weaker, in order to obtain a complete realistic description. The frequencies of all modes of the system with both types of forces are then analyzed. Part of the spectrum can still be found in closed form; other parts can be found only approximately. This analysis gives a fairly complete description of a ``coarse grain'' frequency distribution which can, in principle, be used to describe the thermodynamic behavior of the system at all but the lowest temperatures where the interactions between layers become significant. The principle limitation on its usefulness is at present due to the sizable number of as yet unknown parameters necessary to describe the system numerically.

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