The Madelung Constant in $N$ Dimensions

TL;DR

This paper develops new convergent series expansions for calculating N-dimensional Madelung constants using Bessel functions and sum representations, enabling efficient computation up to high dimensions.

Contribution

Introduces two convergent series expansions and recursive methods for evaluating N-dimensional Madelung constants, extending analysis to higher dimensions and providing detailed convergence properties.

Findings

Derived explicit formulas for Madelung constants in multiple dimensions.

Computed values of Madelung constants up to N=100 for specific exponents.

Analyzed convergence and analytical continuation of the series expansions.

Abstract

We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and representations of sums $r_N(m)$ of squares for $N$-dimensional Madelung constants, $M_N(s)$, where $s$ is the exponent of the Madelung series (usually chosen as $s=1/2$). The functional behavior including analytical continuation, and the convergence of the Bessel function expansion is discussed in detail. Recursive definitions are used to evaluate $r_N(m)$. Values for $M_N(s)$ for $s=\tfrac{1}{2}, \tfrac{3}{2}, 3$ and 6 for dimension up to $N=20$ and for $M_N(1/2)$ up to $N=100$ are presented. Zucker's original analysis on $N$-dimensional Madelung constants for even dimensions up to $N=8$ and their possible continuation into higher dimensions is briefly analyzed.