What single number sets how tightly an ionic crystal like rock salt (NaCl) holds together — and why is it so easy to compute it WRONG?
▶ Launch the interactive simulationAn ionic crystal is a lattice of alternating ±1 point charges; the electrostatic energy of one ion is U = −M·e²/(4πε₀·r₀), with the Madelung constant M = −Σ_{j≠0} (±1)/r_j a pure number fixed by the geometry (r_j in nearest-neighbour units). Sum ±1/r directly — but the series is only CONDITIONALLY convergent, so the order matters: sum over expanding NEUTRAL CUBES (Evjen: face ions ½, edge ¼, corner ⅛) so every shell is charge-neutral. No closed form for M is ever written in — only ±1/r and the neutral-cell weights. The lattice grows shell by shell on screen and M(n) climbs to the limit.
M(NaCl) = 1.747565 recovered from the raw ±1/r lattice sum (to rel ≈ 2×10⁻⁸ by n = 32); the SAME machinery in reduced dimension returns 2 ln 2 = 1.386294 (1-D chain) and 1.615543 (2-D square). Decisive control: the naive expanding-SPHERE sum (full un-neutralised charges) does NOT converge — its partial sums swing with O(1) amplitude and never settle at M, because Σ ±1/r is conditionally convergent; reordering the 1-D series (two positives per negative) shifts it from 2 ln 2 to 3 ln 2 — the same terms, a different sum (Riemann).