Remarks on Force-Constant Models for Lattice Dynamics

It is shown that the need to specify the equilibrium pressure may impose a restriction on the force constants used in calculating lattice vibration spectra. The existence of such a restriction depends upon the lattice under consideration and the generality of the force-constant model that is used. For a model including general interactions with sufficiently distant neighbors (third neighbors will usually be sufficient), specification of the equilibrium pressure imposes no restrictions. For the cubic crystals explicitly studied, one can conclude that models including third neighbor force constants are not restricted, whereas some second neighbor models and almost all nearest neighbor models are restricted. This is true even if long-range Coulomb forces are included in addition to the short-range force constants. We also present an argument to show that the Cauchy relations of elasticity theory depend upon the vanishing of external stresses in equilibrium, contrary to the conclusions of other published work. Finally we use a simple model to study the equilibrium atomic spacing near the surface of a crystal and show that this has no influence upon the relation between the equilibrium pressure and the force constants well within the bulk of the crystal.