Dispersion Relations for Hexagonal Close-Packed Crystal Lattices

Abstract

The dispersion relations for waves propagating in the [0001] and [$01\overline{1}0$] directions were calculated, using a Born---von K\'arm\'an model of the hexagonal close-packed crystal lattice. The interactions between nearest neighbors in the plane were considered to be central forces, while the interactions between nearest and next-nearest neighbors out of the plane were considered to be noncentral forces, i.e., the interactions involved both central and angular forces. The atomic force constants were evaluated (1) from the elastic constants by the method of long waves, (2) from a least-squares analysis of the eigenvalues of the secular equation for elastic waves propagating in certain symmetric crystallographic directions at critical points in the Brillouin zone, i.e., from a least-squares analysis of the dispersion relations, and (3) from a least-squares analysis of the elastic constants and the dispersion relations. The correlation between theoretical and experimental dispersion relations from the noncentral-force model is good for magnesium and zinc and is limited for beryllium. If the atomic force constants have been calculated by a least-squares analysis of certain points in the dispersion relations with or without the elastic constants, these atomic force constants greatly enhance the agreement between theory and experiment. As a result of this work it seems that better agreement between theory and experiment is obtained if angular forces are included in a model of the hcp crystal lattice rather than simply extending the sphere of interaction to more and more neighbors.