Madelung constant
/ MAH-deh-loong /
Take a single ion deep inside a crystal and ask: what is the total electrostatic energy from all the other ions around it? Its nearest neighbours (opposite charge) pull it in. But just beyond them sit ions of the same charge that push it away, and beyond those, opposite charges again, on and on in alternating shells to infinity. The Madelung constant is the single number that sums up this endless tug-of-war for a given crystal geometry.
Concretely, picture NaCl. The reference Na+ has 6 chlorides touching it (attraction), then 12 sodiums slightly farther (repulsion), then 8 chlorides farther still (attraction), and so on. Each shell contributes its count divided by its distance, with alternating sign. Add the whole infinite series — carefully, because it only converges when summed the right way — and for the rock salt arrangement it settles on about 1.748. That pure number is the Madelung constant: it captures everything about how the geometry of the lattice amplifies (or reduces) the simple attraction of a single isolated ion pair. Crucially it depends only on the structure type, not on which ions occupy it — every rock-salt compound shares the same 1.748; cesium chloride has 1.763, zinc blende 1.638, fluorite 2.519, and so on.
The Madelung constant is the heart of the Born-Lande equation: it is exactly the factor by which the energy of one ion pair must be multiplied to get the electrostatic energy of the whole lattice. It is also a quiet lesson in why structure matters: two compounds with identical charges and ion sizes can have different lattice energies purely because they crystallize in different structures with different Madelung constants. (A subtle point worth flagging: published values differ depending on whether the charges and the distance are folded into the constant or kept separate, so always check the convention before plugging a number in.)
Summing the alternating shells around an ion in rock salt gives a Madelung constant of 1.748 — and because the constant depends only on the geometry, NaCl, KBr, MgO and every other rock-salt compound all share that exact value. Cesium chloride's different geometry gives 1.763, slightly more stabilizing.
The Madelung constant depends only on structure type — every rock-salt compound shares the same 1.748.
Published Madelung constants differ by convention (whether ionic charge is included or factored out), so a value of 1.748 versus a 'reduced' value can both be correct for NaCl — always match the constant to the equation you are using.