Noncentral force model for hexagonal close‐packed crystal lattices (ii)

Abstract

AbstractA Born‐von Karman model of the hexagonal close‐packed crystal lattice was considered using a combination of central and angular forces between various sets of neighbors. The interactions between nearest neighbors in the plane were considered to be central forces while the interaction between both 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 equations of motion were evolved and the atomic force constants were evaluated by three different methods. 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 eigenvalues of the secular equation were calculated for a mesh of seven hundred thirty‐five (735) points in that portion (one twenty‐fourth) of the Brillouin zone irreducible under symmetry operations. The frequency spectrum was obtained from these eigenvalues. The lattice specific heats as a function of temperature were calculated using the appropriate frequency distribution function. The Debye and Einstein characteristic temperatures were also calculated as a function of temperature and compared to experimental values. The insensitivity of the calculated thermodynamic properties of the hexagonal close‐packed crystal lattice on the frequency spectrum is demonstrated for zinc by calculating three different frequency spectra, one for each method of evaluating the atomic force constants, and from these spectra computing the lattice specific heats as a function of temperature. The Debye characteristic temperatures do not change significantly although the details of the frequency spectra differ appreciably. The conclusion has been drawn that better agreement between theory and experiment is reached if angular forces are included in a model of the hexagonal close‐packed crystal lattice rather than simply extending the sphere of interaction to more and more neighboring atoms.