… 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.