the Madelung constant
/ MAH-deh-loong /
A single pair of opposite ions attracts by Coulomb's law, but in a real crystal an ion is surrounded by a whole city of neighbours: nearby opposite charges pull it in, next-nearest like charges push it away, and the tug-of-war ripples out to infinity. The Madelung constant is a single pure number that sums up all of that geometry: it tells you how much stronger the total electrostatic energy of an ion in the full crystal is, compared with just its nearest-neighbour pair.
Start with the nearest neighbours (attraction, a plus term), subtract the next shell of like-charge ions (repulsion), add the shell after that, and so on. For rock salt this alternating sum converges to M = 1.748; for the caesium chloride structure it is 1.763, for zinc blende 1.638, for fluorite 2.519. The value depends only on the geometry of the structure, not on which ions occupy it, which is why every rock-salt crystal (NaCl, MgO, NiO) shares the same 1.748. Notice M is bigger than 1: the surrounding lattice makes each ion more tightly bound than a lone ion pair would be.
The Madelung constant is the geometry factor at the heart of the lattice-energy formulas. Because it captures the whole infinite lattice in one number, it lets the Born-Lande equation turn charges and a bond length into a real, quantitative binding energy. It also carries a lesson in crystal chemistry: structures with higher Madelung constants are more electrostatically stable, one reason a given compound picks the structure it does.
For NaCl the Madelung sum is 6 (nearest, attracting) minus 12/sqrt(2) (next, repelling) plus 8/sqrt(3) (attracting) minus more terms, which converges to 1.748, the same constant that appears in the lattice energy of MgO and every other rock-salt crystal.
One number standing in for a sum over the entire infinite lattice.
The Madelung constant depends on how you define it: some books fold the ion charges into it and some keep them separate, so a value only means something once you know the convention and the reference distance it goes with.