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

Abstract

A 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.

Anisotropic distortion and Lifshitz transition inα-Hf under pressure

“…Actually, the values of C 44 for a large number of hexagonal TMs have been estimated by using Eq. (1), and excellent agreement with those experimental results has been identified [7,8,12]. In this scenario, this method paves the way to investigate the structural and vibrational properties of TMs.…”

“…Verified by several studies [7,8], this E 2g is closely connected to the elastic constant C 44 by the following relation:…”

X-ray diffraction and Raman studies of beryllium: Static and elastic properties at high pressures

“…An expression for the elastic shear modulus has been derived for the hcp structure with the assumption of isotropic interaction potentials. 12,[36][37][38][39][40] The expression describing this relationship, as given by Olijnyk, is:…”

Sound velocities in solid hydrogen under pressure

“…This relation follows from lattice dynamics studies of the hcp crystal lattice [41] and is exact if the following conditions are met: (a) anharmonic effects are negligible, (b) interactions beyond the next-nearest neighbors are negligible, and (c) four-body and higher manybody interactions are negligible. If some of the conditions (a)-(c) are invalid one might expect that the simple relation between 44 C and 2 () g E breaks down.…”