Madelung

Madelung is a surname associated with several concepts in physics and chemistry, most notably in solid-state physics and quantum mechanics. The term appears in a number of results named after German physicists such as Erwin Madelung and Otfried Madelung, who contributed to ionic crystals, quantum hydrodynamics, and atomic structure.

One prominent usage is the Madelung constant, a dimensionless lattice sum that characterizes the electrostatic energy per ion in an ionic crystal. In a given structure, the total Coulomb energy per ion can be written as a negative constant times e^2 divided by a characteristic distance, with the constant M determined by the lattice geometry. The constant depends on the arrangement of ions and converges only when the infinite lattice is summed correctly, often via techniques such as Ewald summation. For example, in the NaCl structure, the Madelung constant is approximately 1.74756. The Madelung constant provides a compact way to estimate lattice energies in ionic compounds.

Another use is the Madelung transformation, which expresses the quantum mechanical wave function in polar form, ψ = √ρ exp(iS/ħ). This representation converts the Schrödinger equation into a pair of hydrodynamic-like equations for the probability density ρ and a quantum phase S, featuring a quantum potential term. The transformation is a standard tool in quantum hydrodynamics and semiclassical analyses.

Madelung's rule, also called the diagonal rule, governs the order in which atomic orbitals are filled by electrons. It states that orbitals are occupied in order of increasing n + l, with ties broken by lower principal quantum number n. The rule is associated with Otfried Madelung and is widely taught as a practical guide to ground-state electron configurations.