Lattice Dynamics of Copper

Abstract

A fairly extensive survey of the phonon dispersion relation in copper at room temperature has been made by means of neutron scattering. The cold-neutron time-of-flight technique was used, and the results are confined to the (001) and ($1\overline{1}0$) symmetry planes of the reciprocal lattice. An interpolation formula for the dispersion relation has been obtained by making a least-squares fit of an interatomic-force-constant model to the results. However, because of the nonconvergent behavior of the parameters with the introduction of further neighbors, the values obtained for these cannot be ascribed any physical significance beyond showing that (a) nearest-neighbor interactions dominate, and (b) the forces extend up to at least third and probably up to sixth neighbors. The frequency distribution function $g\left(\nu \right)$ has been calculated, and the values obtained from it for the lattice specific heat and the Debye-Waller factor are in excellent agreement with experiment, except at very low temperatures. The results have been compared with calculations based on Toya's treatment (with a few minor modifications) of the electron-phonon interaction in monovalent metals. The values of the three most uncertain parameters in the theory, representing the depth of the pseudo-potential well due to the ions and the two core-overlap interaction parameters, have been obtained by fitting to the measured elastic constants. Phonon-dispersion curves calculated from these parameters show reasonably good agreement with the experimental results, especially for the transverse modes. The validity of these parameters and of the "free-electron-like" approximation for the electron wave functions, used in Toya's theory, is critically discussed.