Coulomb Interactions in the Uniform-Background Lattice Model

The uniform-background lattice model consists of a body-centered-cubic lattice of point charges of like sign embedded in a uniform background charge of the opposite sign. The method used by Fuchs to apply the Ewald sum formula to the case of static shear deformations is extended to apply to spatially periodic deformations. The dispersion relations (frequency as a function of wave number) are presented for propagation vectors in the [100], [110], [111], and [210] directions. Comparison is made with the dispersion relations for the same directions in a Born-von K\'arm\'an model of a body-centered cubic lattice in which only central interactions between nearest and next-nearest neighbors are considered. The values of the "Coulomb part" of the macroscopic elastic constants as calculated by Fuchs are used for this model. The problem of treating a model in which the background responds to the displacement of point charges is discussed.