Axially Symmetric Model for Lattice Dynamics of Metals with Application to Cu, Al, and ZrH2

Abstract

A lattice dynamics model (A-S model) is proposed for metals which assumes that the restoring forces between the ions have the character of bond stretching and axially symmetric bond bending. It is shown that this type of restoring force is derivable from a sum of spherically symmetrical two-body potentials, $v({R}_{\mathrm{ij}})$, taken over all ($i, j$) pairs of lattice sites separated by a distance ${R}_{\mathrm{ij}}$. The dynamical matrix for a face-centered cubic lattice is worked out for the first three neighbor shells of atoms. The dispersion curves for aluminum and copper are derived from the elastic constant and thermal diffuse x-ray data. The A-S force constants or Cu and Al fall off rapidly with distance and give dispersion curves which are at least as good as those derived from the third-neighbor general tensor force model. Particularly noteworthy is the fact that Walker's tensor force constants for Al are nearly axially symmetric. However, Jacobsen's tensor force constants for Cu are not axially symmetric. Contrary to this observation, it is found that White's force constants derived by using Feynman's theorem are nearly axially symmetric.The A-S model is also applied to Zr${\mathrm{H}}_{2}$ which has a tetragonally deformed fluorite structure. It is shown that the six optical branches are flat to within 3% over the entire Brillouin zone and that the hydrogen-hydrogen interactions are very weak compared to the hydrogen-zirconium interactions. The dynamical matrix for the acoustic spectrum is derived.