Relationship between Two-Body Interatomic Potentials in a Lattice Model and Elastic Constants

General equations for the elastic constants associated with a crystal model in which the energy is given by two-body atom-atom interactions and volume-dependent terms are derived for monatomic solids. It is shown that noncentral interactions which require a choice of initial coordinate system do not necessarily give rise to an energy function which is invariant to homogeneous rotations. For central two-body forces, generalized Cauchy relations are defined. Explicit applications of these equations are given for short-ranged central forces in an hcp lattice and noncentral forces in the fcc lattice.