… Born decided to investigate the simple ionic crystal – rock salt (sodium chloride) – using a ring model. He asked Landé to collaborate with him in calculating the forces between the lattice points that would determine the structure and stability of the crystal. Try as they might, the mathematical expression that Born and Landé derived contained a summation of terms that would not converge. Sitting across from Born and watching his frustration, Madelung offered a solution. His interest in the problem stemmed from his own research in Goettingen on lattice energies that, six years earlier, had been a catalyst for Born and von Karman's article on specific heat. The new mathematical method he provided for convergence allowed Born and Landé to calculate the electrostatic energy between neighboring atoms (a value now known as the Madelung constant). Their result for lattice constants of ionic solids made up of light metal halides (such as sodium and potassium chloride), and the compressibility of these crystals agreed with experimental results.

The study of lattice sums is an important topic in mathematics, physics, and other areas of science. It is not a new field, dating back at least to the work of Appell in 1884, and has attracted contributions from some of the most eminent practitioners of science (Born and Landé [1], Rayleigh, Bethe, Hardy, …). Despite this, it has not been widely recognized as an area with its own important tradition, results, and techniques.

We discuss here the topic of angular lattice sums, i.e., those which depend on the angle or angles between the vector linking the origin to lattice points and the coordinate axes. This topic is an old one, dating back to an 1892 paper by Lord Rayleigh [26], but is curiously disconnected from the main thread of investigations into lattice sums, as surveyed by Glasser and Zucker [9]. We begin with a brief account of the history of the sums and go on to give an account of some of their more recently discovered properties. We use the latter topic to discuss how properties and formulae for lattice sums may be discovered with the aid of modern symbolic algebra packages such as Mathematica or Maple. Chief among the properties of the angular lattice sums in two dimensions that we describe is their relationship to the Riemann zeta function; selected sums obey the celebrated Riemann hypothesis.

Optical properties of coloured glass and lattice sums

The technology of colouring glass by adding to the melt appropriate metals is an old one, dating back to the time of the ancient Greeks and Romans and predating the modern field known as plasmonics. Michael Faraday proposed a model of atoms as being like tiny metal particles and so opened up the question as to what could be inferred about the properties of atoms from the interaction of light with various solids.

Introduction

It has been more than 100 years since Appell [2] introduced lattice sums into physics, yet the article on which this chapter is based is apparently the first devoted entirely to the subject. We are, of course, aware that parts of other reviews (such as those by Born and Göppert-Mayer [21], Waddington [135], Tosi [133], and Sherman [125]) have dealt with Coulomb sums in ionic crystals, as a casual reading of this chapter will demonstrate. In the perusal of 100 years of literature, we will inevitably have missed or ignored relevant papers and their authors are urged to communicate with us directly.

The organization of this review is as follows. In Section 1.2 we present a historical survey, picking out and describing in detail some of the more important methods for calculation. Section 1.3 deals with the representation of lattice sums as Mellin-transformed products of theta functions. In Section 1.4 we discuss the evaluation of two-dimensional lattice sums by number-theoretic means, and in Section 1.5 we examine a promising new application of contour integration.

A modular equation of order n is essentially some algebraic relation between theta functions of arguments q and qn respectively. In his notebooks Ramanujan gave many such relations involving Lambert series, and Berndt [1] painstakingly collected these results, proved or verified them, and if necessary made corrections. However, these relations as given by Ramanujan appear in a haphazard way, and there seems to be no systematic way of ordering them or for that matter knowing whether the formulae are independent or complete. Here an attempt is made to arrange these results in a systematic fashion. It will also be demonstrated how new modular relations may be derived from those previously established. Indeed it will be shown how, using sign and Poisson transformations to be described in Chapter 6, each modular equation is essentially a set of either four or eight relations which can be generated from any one of the set by a group of simple transformations. Further it will also be shown how the use of character notation provides a shorthand for the lengthy Lambert series involved. A connection between binary quadratic forms and modular equations will be demonstrated. This will allow very simple proofs of certain modular and mixed modular equations to be executed. Finally it will be exhibited how the Mellin transforms of modular equations, in which q is replaced by e−t, lead to the evaluation of lattice sums.

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.