Vibration Spectra and Specific Heats of Cubic Metals. I. Theory and Application to Sodium

A model for monovalent metals is proposed, on the basis of which the secular equations determining the frequencies of the normal modes of vibration are derived for body-centered and face-centered cubic metals. In contrast to the two force constants occurring in the usual treatments based on the Born-von K\'arm\'an model, each of these equations---one for each type of lattice---contains three force constants which may be identified with the three independent elastic constants of a cubic metal. The limitations of the model are discussed.The frequency spectrum of sodium, and hence its specific heat ${C}_{v}$ at constant volume, are calculated. It is found that the agreement between the observed and calculated values of ${C}_{v}$ is more satisfactory in the entire relevant temperature range, and especially at low temperatures, than that obtained by Bauer using the customary two force-constant model.