Theory of the Lattice Vibration of Graphite

A theory of the lattice vibration and specific heat of graphite, which has a typical lamellar structure, is worked out using the Born-von Kármán method. Four types of restoring forces are assumed; the first and second ones are associated with changes in the bond lengths and bond angles of the honey-comb net planes, respectively, the third with change in distance between neighoring atoms on adjacent net planes, and the fourth with the bending of the net planes. We have obtained modes whose polarizations are parallel and perpendicular to the planes. The former are essentially two-dimensional, but the latter are of more complex characters and play a particular role in the temperature range 45° to 100°K. As our main interest is in this and higher temperature ranges, we neglect the shearing forces between neighboring net planes which are effective at much lower temperatures. For small wave numbers, our theory naturally reduces to the semi-continuum theory developed by Komatsu and Nagamiya (1951) and Komatsu (1955). The calculated specific heat curve is in good agreement with the one observed by DeSorbo (1954) in the temperature range 45° to 300°K. We have concluded that the discrepancy found by Komatsu at temperatures higher than 60°K between his theoretical values of the specific heat and the observed ones is due to the failure of his semi-continuum treatment.