Fourier expansions for the potentials of lattices of charge

In this note we apply the Poisson sum rule [1] to obtain formal expressions for the Fourier coefficients of the potential of a lattice of generalized charge. Each generalized charge is assumed to contribute to the potential a term which depends only on the vector displacement from the charge’s position. The coefficients are explicitly calculated for two types of individual charge potentials: Coulomb and Yukawa. For one and two-dimensional Coulomb lattices, we find that the nonzero frequency components decrease exponentially both with the distance to the distribution and with frequency. The exponential convergence of these expansions indicates that a truncated Fourier series will often provide both a simple and accurate analytic approximation for these potentials. In particular, this result may often be applied to justify use of the continuous charge distribution approximation for observation points further from the distribution than the spacing between the charges. This result also serves to explain numerical observations which have been presented previously [2, 3, 4]. The resulting expansions for lattices of Yukawa type charges are seen to be closely related to those of their Coulomb analogs. Yukawa potentials are physically realized in linearly screened Coulomb systems [5], for example, and so are of direct physical significance. We end the note with a brief study of finite and disordered lattices. The general form of the Poisson sum rule allows for a sum over a periodic lattice to be replaced by an integral over the volume of the lattice. For an arbitrary individual particle potential function f , the rule states