Closed-form expression for the potential within a face centred cubic ionic crystal

In this work we deal with the electrostatic potential generated by a periodical system of charges in the space. Specifically, we consider a three-dimensional mesh having cubic cells, whose nodes are occupied by point charges +Q or -Q, so that each positive (negative) charge is surrounded by six negative (positive) charges at distances d. This system clearly models a face centred cubic (FCC) ionic crystal that we further assume to extend to infinity in every direction. Considering ions as point particles subject only to coulombic interactions, the potential here concerned can be expressed by means of an alternating series that exhibits a very slow convergence, giving rise to computational difficulties. In the following we illustrate an alternative formula for the potential, consisting in a closed-form expression involving the elliptic Jacobi Theta functions. Such formula provides either an immediate interpretation from an analytical point of view, and a fast converging procedure for numerical purposes. Finally, we illustrate an application of our methodology concerning the calculus of the Madelung constant in the case of an ideal FCC ionic crystal. Numerical accuracy and computational costs are comparable with those of the most rapidly converging formulas previously known. (C) 2004 Elsevier B.V. All rights reserved.