= Madelung constant
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{title2=$\alpha_M$}
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The <Madelung constant>, named for <Erwin Madelung>, is a dimensionless geometrical factor in the electrostatic binding of an <ionic crystal>. For a neutral lattice of <electric charges> $\pm e$ with nearest unlike-ion distance $r_0$, <Coulomb's law> gives the <potential energy> per ion pair as $-\alpha_M e^2/(4\pi\epsilon_0r_0)$. At a positive reference site, $\alpha_M=-\sum_{j\ne0}z_jr_0/r_j$, with neighboring <electric charges> $z_je$. Both attraction and repulsion enter this lattice sum; summing with a neutral-bulk prescription is essential. Short-range overlap repulsion and other contributions must be added before inferring a stable spacing or a thermodynamic <lattice energy>.
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