The Madelung Constant

What the Madelung constant does

Pick one sodium ion sitting inside a crystal of table salt. It feels an attractive pull from every chloride ion in the crystal and a repulsive push from every other sodium ion — millions upon millions of them, stretching out in all directions. Adding up that infinite tug-of-war looks hopeless. The Madelung constant is the single number that does it for you.

Formally, it collapses the entire lattice's Coulomb bookkeeping into one dimensionless multiplier. The electrostatic energy of one reference ion in the crystal is written

    E_ion  =  − (M · z⁺ z⁻ e²) / (4π ε₀ r₀)

  M   = Madelung constant   (dimensionless — geometry only)
  z⁺,z⁻ = ion charge numbers (e.g. +1 and −1 for NaCl)
  e   = elementary charge     1.602 × 10⁻¹⁹ C
  ε₀  = vacuum permittivity    8.854 × 10⁻¹² F/m
  r₀  = nearest-neighbour distance (Na–Cl = 2.82 Å in NaCl)

The prefactor e²/(4π ε₀ r₀) is just the Coulomb energy of one nearest-neighbour ion pair. Everything else about the crystal — the thousands of farther ions, the alternating signs, the exact geometry — is folded into M. That is the whole trick: M answers "how much bigger is the real lattice energy than one isolated ion pair?" For NaCl the answer is 1.748× bigger, and it is bigger (not smaller) despite all the repulsions, because the nearest neighbours are always the oppositely-charged ones and they dominate.

The step-by-step summation

The Madelung constant is a lattice sum. Choose a reference ion, set the nearest-neighbour distance to 1, and add up the signed inverse distance to every other ion. Attractions (opposite charge) count positive; repulsions (like charge) count negative:

    M  =  Σ_j  (± 1) / (r_j / r₀)

  where the sign is +  if ion j has opposite charge to the reference,
                    −  if ion j has the same charge.

Walk it out in shells around a Na⁺ in rock salt. In the cubic lattice the reference ion sees ions at distances that are multiples of r₀:

1. Shell 1 — 6 nearest chlorides at distance 1. These are attractive. Contribution: +6/1 = +6.000.
2. Shell 2 — 12 sodiums at distance √2. Same charge, repulsive. Contribution: −12/√2 = −8.485.
3. Shell 3 — 8 chlorides at distance √3. Attractive again. Contribution: +8/√3 = +4.619.
4. Shell 4 — 6 sodiums at distance 2. Repulsive. Contribution: −6/2 = −3.000.
5. Shell 5 — 24 chlorides at distance √5. Attractive. Contribution: +24/√5 = +10.733… and so on.

The alternating series is the key insight — and the trap. Grouped like this the running total lurches around: 6.000, then −2.485, then +2.134, then −0.866, then +9.87… These partial sums do not settle down to 1.748. The naive shell sum is only conditionally convergent: the number of ions in a shell grows like the surface area (r²) while each ion's contribution shrinks only like 1/r, so the terms don't die fast enough. The value you get depends on how you slice the crystal into shells — which is mathematically unacceptable.

Making the sum converge: Evjen and Ewald

The fix is to make sure that every finite chunk of crystal you sum over is electrically neutral. A neutral group has no net monopole, so its far-field falls off far faster than 1/r and the series converges absolutely.

• Evjen's method (1932). Sum over expanding cubes centred on the reference ion, but fractionally weight the ions on the boundary so the cube is neutral: an ion sitting on a cube face counts as ½, on an edge as ¼, and on a corner as ⅛ (it is shared with neighbouring cubes). With this weighting the very first cube already gives M ≈ 1.75, and a handful of cubes nails 1.74756 to five figures. It is the method you can actually do by hand.
• Ewald summation (1921). Paul Ewald's trick splits the 1/r potential into a smooth part summed in reciprocal (Fourier) space and a sharp part summed in real space, with a Gaussian screen bridging the two. Both halves converge exponentially fast. Every modern crystal-energy and molecular-dynamics code uses Ewald (or its fast cousins, Particle-Mesh Ewald) precisely because the direct sum is untrustworthy.

Both routes converge to the same, geometry-fixed number. That number — 1.74756 for rock salt — is the Madelung constant.

The reference-distance gotcha

Half the confusion around published Madelung constants comes from which distance the author divided by. There are two common conventions:

• Referenced to the nearest-neighbour distance (r₀ = the shortest cation–anion separation). This is the physically natural choice and gives NaCl the familiar 1.74756.
• Referenced to the cubic unit-cell edge (a). Since a = 2r₀ in rock salt, dividing by the larger edge doubles the number to the "reduced Madelung constant" 3.49513.

Neither is wrong — they describe the same crystal — but they are not interchangeable in the Born-Landé equation. Always pair the Madelung constant with the distance it was defined against. A missing factor of two here is the single most common error in student lattice-energy calculations.

Madelung constants of common structures

The Madelung constant is a fingerprint of the geometry, independent of which ions occupy the sites. All values below are referenced to the nearest-neighbour distance.

Structure type	Example	Coordination	Madelung constant M
Rock salt	NaCl, MgO, LiF	6 : 6	1.74756
Caesium chloride	CsCl, CsBr	8 : 8	1.76267
Zinc blende (sphalerite)	ZnS, GaAs	4 : 4	1.63806
Wurtzite	ZnO, wz-ZnS	4 : 4	1.64132
Fluorite	CaF₂, UO₂	8 : 4	2.51939
Rutile	TiO₂, SnO₂	6 : 3	≈ 2.408
Corundum	Al₂O₃	6 : 4	≈ 4.172

Two patterns stand out. First, among 1:1 salts, higher coordination gives a slightly larger M — CsCl (8:8) beats NaCl (6:6) beats ZnS (4:4). Second, the fluorite, rutile and corundum values are much larger, but that is partly a book-keeping effect: their formula units carry more charge, and the Madelung energy also scales with z⁺z⁻, so raw M values across different charge types can't be compared head-to-head. What is directly comparable is the full Madelung energy per formula unit.

Worked example: the lattice energy of NaCl

Put the Madelung constant to work inside the Born-Landé equation, which adds a short-range repulsion correction (the Born exponent n) to the pure Coulomb sum:

    U  =  − (N_A · M · z⁺ z⁻ e²) / (4π ε₀ r₀) · (1 − 1/n)

  N_A = 6.022 × 10²³ /mol      M   = 1.74756   (NaCl)
  z⁺  = +1,  z⁻ = −1           r₀  = 2.82 × 10⁻¹⁰ m
  e   = 1.602 × 10⁻¹⁹ C        n   = 8   (average Born exponent for Na⁺Cl⁻)
  ε₀  = 8.854 × 10⁻¹² F/m

Evaluate the Coulomb block first:

    N_A M e² / (4π ε₀ r₀)
  = (6.022e23)(1.74756)(1.602e-19)²
    ─────────────────────────────────
      4π (8.854e-12)(2.82e-10)
  ≈  8.61 × 10⁵  J/mol  =  861 kJ/mol

Now apply the Born correction (1 − 1/8) = 0.875:

    U  =  − 861 kJ/mol × 0.875  ≈  − 753 kJ/mol

The experimental lattice energy of NaCl, extracted from a Born-Haber cycle, is about −787 kJ/mol. A model with one geometric constant and one empirical repulsion exponent lands within about 4% of reality — a striking vindication of the idea that ionic crystals are held together by nothing more exotic than Coulomb's law summed correctly over the lattice. (Nudging the Born exponent up toward n ≈ 9–10, as more careful treatments do, closes most of the remaining gap.)

Where it actually matters

• Predicting lattice energies without measurement. The Madelung constant lets you estimate U for a salt you have never made — the primary use in inorganic and materials chemistry. It underlies the Kapustinskii equation, a Madelung-free shortcut that folds an "averaged" Madelung constant (≈0.874 per ion) into a formula needing only ion charges, radii and count.
• Ranking structural stability. Because M encodes geometry, comparing Madelung energies tells you which polymorph an ionic compound should adopt. It correctly explains why small-cation salts favour the 4:4 or 6:6 structures while large caesium salts flip to the 8:8 CsCl arrangement.
• Defect and surface energies. Point-defect formation energies, surface Madelung potentials and the electrostatics at grain boundaries in ceramics all start from lattice Coulomb sums built on the same machinery.
• Battery and oxide materials. The site potentials (the Madelung potential at each ion) feed directly into models of redox voltages in cathodes like LiCoO₂ and into oxygen-vacancy chemistry in fuel-cell electrolytes.
• Simulation everywhere. Every classical molecular-dynamics simulation of an ionic or partly-ionic material — from protein solvation in salt water to molten-salt reactors — computes the long-range Coulomb energy by Ewald summation, the direct descendant of Ewald's 1921 solution to the Madelung convergence problem.

Limitations and where the picture breaks

• It assumes point charges. The Madelung sum treats each ion as a mathematical point of charge ±ze. Real ions are polarisable clouds; in less ionic solids (AgCl, the silver halides, sulfides) covalency and polarisation add significant non-Coulomb bonding that M cannot see.
• It needs integer, formal charges. For partly-covalent lattices the "true" charge is fractional and structure-dependent, so plugging z = ±1 or ±2 into the formula is already an approximation.
• Short-range repulsion is separate. M captures only the long-range Coulomb part. The Pauli repulsion of overlapping electron shells has to be bolted on empirically (the 1 − 1/n Born term, or a Born-Mayer exponential), and n itself is fitted, not predicted.
• No temperature, no zero-point motion. The classic Madelung energy is a static-lattice, 0 K quantity. Thermal expansion, phonon zero-point energy and vibrational entropy all shift the real cohesive energy.
• Reference-distance ambiguity. As noted above, a Madelung constant is meaningless without stating the distance it was normalised to — the factor-of-two trap catches people constantly.

Who worked it out, and when

The constant is named for Erwin Madelung (1881–1972), a German theoretical physicist, who introduced the lattice electrostatic sum in a 1918 paper (Physikalische Zeitschrift) on the forces binding ionic crystals — work that arrived right at the birth of X-ray crystallography, when the Braggs had only just shown in 1913–1914 that NaCl is a lattice of alternating Na⁺ and Cl⁻ ions rather than discrete molecules. Madelung's number gave the new "ionic lattice" picture its first quantitative energy.

The sum as Madelung wrote it was hard to evaluate. Paul Peter Ewald supplied the reciprocal-space transformation in 1921 that made it converge exponentially — a method now used billions of times a day inside simulation software. Harald Evjen published the neutral-expanding-cube weighting in 1932, giving chemists a hand-computable route. Together, Madelung, Ewald and Born (whose lattice-energy framework wrapped the constant in a usable thermodynamic equation) turned "why does salt hold together?" into a solved arithmetic problem.