Noncentral Force Model for Hexagonal Close‐Packed Crystal Lattices

A Born‐von Karman model of the hexagonal close‐packed 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 regarded as central forces while the interactions between nearest neighbors out of the plane and next nearest neighbors out of the plane were considered to be noncentral, i.e. both central and angular. Three materials with different axial (c/a) ratios were considered: beryllium (1.568), magnesium (1.624), and zinc (1.855). The equations of motions were evolved and the force constants were related to the elastic constants of the material. The eigenvalues of the secular equation were solved for one hundred twenty one (121) points in that portion of the Brillouin zone irreducible under symmetry operations. The frequency spectrum for each material was obtained from these eigenvalues. The temperature dependence of the specific heat of each material was calculated using the appropriate frequency distribution function. The Debye characteristic temperature and the Einstein characteristic temperature were also obtained as a function of temperature.