The lattice sums involved in the definition of Madelung's constant for an NaCl–type crystal lattice in two or three dimensions are investigated here. The fundamental mathematical questions of convergence and uniqueness of the sum of these series, which are not absolutely convergent, are considered. It is shown that some of the simplest direct sum methods converge and some do not converge. In particular, the very common method of expressing Madelung's constant by a series obtained from expanding spheres does not converge. The concept of the analytic continuation of a complex function to provide a basis for an unambiguous mathematical definition of Madelung's constant is introduced. By these means, the simple intuitive direct sum methods and the powerful integral transformation methods, which are based on theta function identities and the Mellin transform, are brought together. A brief analysis of a hexagonal lattice is also given.

Introduction

Lattice sums have played a role in physics for many years and have received a great deal of attention on both practical and abstract levels. The term ‘lattice sum’ is not a precisely defined concept: it refers generally to the addition of the elements of an infinite set of real numbers, which are indexed by the points of some lattice in N-dimensional space. A method of performing a lattice sum involves accumulating the contributions of all these elements in some sequential order. Unfortunately, the elements of the set are not, in general, absolutely summable so the sequential order chosen can affect the answer.