The Born-Lande equation

Result

The Born-Lande equation

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Statement

Lattice energy from electrostatics and repulsion.

Why it matters

crystal-lattices-unit-cell describes how ions pack into a periodic lattice, but packing geometry alone does not say how strongly that lattice is held together. Lattice energy, the energy released on assembling an ionic solid from its gaseous ions, is the quantity that governs an ionic compound's melting point, hardness, and solubility, and the Born-Lande equation is the standard way to estimate it directly from electrostatics, without needing to measure it via an experimental thermodynamic cycle.

Because lattice energy is not itself directly measurable by a single experiment, this electrostatic calculation and the Born-Haber cycle's experimentally derived value serve as independent cross-checks on one another; band-theory then treats the qualitatively different case of metallic, delocalised bonding, making the Born-Lande equation and band theory the two complementary pictures of how atoms hold together in the solid state.

Hypotheses

Ions are treated as rigid point (or spherical) charges interacting purely electrostatically, with no covalent contribution to the bonding.This is a good approximation for compounds of clearly electropositive metals and electronegative non-metals with a large electronegativity difference, but becomes progressively less accurate for compounds with significant covalent character, where electron density is shared rather than purely localised on discrete ions. Short-range repulsion between ions in contact, arising from overlapping electron clouds (Pauli repulsion), is modelled as a simple power law in \(1/r^n\).Without an additional repulsive term the purely attractive Coulombic energy would diverge to \(-\infty\) as \(r\to0\), predicting no stable equilibrium separation at all; the repulsive term is what gives the total potential energy a genuine minimum at a finite equilibrium spacing \(r_0\). The Madelung constant \(M\), which sums the electrostatic interactions of one ion with every other ion in the infinite lattice (not merely its nearest neighbours), depends only on the lattice's geometric structure type, not on the specific ions' identity or charge; different structure types (rock-salt, caesium chloride, zinc blende, and so on) have different, tabulated Madelung constants.

Proof

V_{\text{Coulomb}}(r) = -\frac{M\,z_+z_-\,e^2}{4\pi\varepsilon_0\,r}

Summing the electrostatic potential energy of one reference ion with every other ion in the infinite lattice (attractive to oppositely charged neighbours, repulsive to like-charged ones at greater distance) converges to a single, structure-dependent geometric factor, the Madelung constant \(M\), multiplying the simple two-point-charge Coulomb energy for the nearest-neighbour distance \(r\). B

V_{\text{rep}}(r) = \frac{B}{r^n}

At short range, overlapping electron clouds of adjacent ions repel strongly, modelled empirically as a steep power law in \(1/r^n\) (the Born exponent \(n\), typically in the range \(5\) to \(12\) depending on the ions' electronic configuration, is obtained from independent compressibility measurements); \(B\) is a constant fixed by requiring the total potential to be minimised at the experimentally known equilibrium separation. A

V(r) = -\frac{M z_+z_- e^2}{4\pi\varepsilon_0 r} + \frac{B}{r^n}, \qquad \left.\frac{dV}{dr}\right|_{r_0} = 0

The equilibrium separation \(r_0\) is found, as for any potential energy curve, by setting the derivative of the total potential (Coulombic attraction plus short-range repulsion) to zero, giving the point at which attractive and repulsive forces exactly balance. A

B = \frac{M z_+z_- e^2\, r_0^{\,n-1}}{4\pi\varepsilon_0\, n}

Solving the equilibrium condition of Step 3 for \(B\) eliminates the otherwise-unknown repulsive-term prefactor in favour of the experimentally measurable equilibrium separation \(r_0\) and the already-known Madelung constant and Born exponent. A

U = -\frac{N_A\,M\,z_+z_-\,e^2}{4\pi\varepsilon_0\,r_0}\left(1-\frac{1}{n}\right)

Substituting \(B\) from Step 4 back into the total potential \(V(r_0)\) and scaling from a single ion pair to one mole of formula units (multiplying by Avogadro's number \(N_A\)) gives the lattice energy in its standard, final closed form; the factor \((1-1/n)\) is the correction subtracted from the pure Coulombic energy to account for the short-range repulsion's own (smaller) contribution at equilibrium. B

Result

U = -\frac{N_A\,M\,z_+z_-\,e^2}{4\pi\varepsilon_0\,r_0}\left(1-\frac{1}{n}\right)

Reading. Lattice energy is predicted directly from purely electrostatic and geometric quantities — ionic charges, equilibrium separation, and a structure-dependent Madelung constant — corrected by a small factor accounting for short-range electron-cloud repulsion.

Scope. Reliable for predominantly ionic compounds (Hypotheses); systematically under- or overestimates the true lattice energy for compounds with substantial covalent character, precisely the compounds for which comparing this predicted value against the experimental Born-Haber-cycle value is most diagnostically useful.

Corollaries & converses

Lattice energy magnitude increases with higher ionic charge (entering as the product \(z_+z_-\)) and with smaller equilibrium separation \(r_0\); this is the standard electrostatic explanation for why, for example, magnesium oxide (\(2+\), \(2-\) charges) has a dramatically larger lattice energy than sodium chloride (\(1+\), \(1-\) charges) of comparable ionic size.
close-packing-efficiency's geometric packing description and this result's Madelung-constant summation both depend on the identical underlying lattice geometry, one giving the fraction of space occupied, the other giving the electrostatic consequence of that same arrangement.
Converse: a significant discrepancy between a Born-Lande-calculated lattice energy and the experimentally derived Born-Haber-cycle value is itself standard evidence of substantial covalent character in a nominally "ionic" compound (e.g. silver halides), since the purely electrostatic, point-charge picture underestimates the true (typically more negative) lattice energy whenever appreciable covalent bonding is also present.

Fails without

Apply the equation to a compound with substantial covalent character: the purely ionic, point-charge picture (Hypotheses) systematically underestimates the true lattice-energy magnitude, since it omits the extra stabilisation genuine electron sharing provides.
Omit the short-range repulsive term of Step 2 entirely: the purely attractive Coulombic potential would diverge to \(-\infty\) as \(r\to0\), predicting no stable equilibrium separation at all, rather than the genuine finite-\(r_0\) minimum observed experimentally.

Common errors

Using the nearest-neighbour Coulomb energy alone (\(-z_+z_-e^2/4\pi\varepsilon_0 r\)) without the Madelung constant \(M\), which accounts for the full, converged sum over the entire infinite lattice, not merely the nearest ion pair.
Forgetting the \((1-1/n)\) correction factor and reporting the pure Coulombic energy alone as the lattice energy, overestimating its magnitude.
Applying the Born-Lande equation with confidence to compounds with substantial covalent character, where the purely ionic, point-charge assumption (Hypotheses) is a poor approximation to begin with.
Sign confusion: lattice energy as defined here (energy released on formation from gaseous ions) is negative; some sources define it with the opposite sign convention (energy required to separate the lattice into gaseous ions), which is numerically identical but of opposite sign.

Discussion

Max Born and Alfred Landé developed this treatment in the early twentieth century as one of the first successful quantitative applications of a combined attractive-plus-repulsive potential to a real crystalline solid, shortly after X-ray diffraction (bragg-law) first made precise ionic separations experimentally measurable, supplying the \(r_0\) values the equation itself requires as input.

A refined version, the Born-Mayer equation, replaces the power-law repulsive term with an exponential form (\(Be^{-r/\rho}\)), which more accurately reflects the true exponential decay of overlapping electron-cloud repulsion at short range; the two forms give closely similar numerical predictions for most compounds, differing mainly in how sensitively the fitted repulsive parameter responds to small errors in the assumed equilibrium separation.

Common misconception: that the Madelung constant is somehow related to, or comparable in size across, all ionic compounds regardless of structure. It is purely geometric, fixed entirely by the specific structural arrangement (rock-salt, caesium chloride, fluorite, and so on); two compounds with the identical structure type share the identical Madelung constant regardless of which ions are actually present.

Worked examples

\text{NaCl (rock-salt structure): } M=1.748,\ z_+=z_-=1,\ r_0=2.82\times10^{-10}\,\text{m},\ n=8

Sodium chloride's well-known rock-salt Madelung constant, singly charged ions, and measured equilibrium separation are substituted into the Result to estimate its lattice energy directly. A

U = -\frac{(6.022\times10^{23})(1.748)(1)(1)(1.602\times10^{-19})^2}{4\pi(8.854\times10^{-12})(2.82\times10^{-10})}\left(1-\frac18\right) \approx -756\,\text{kJ/mol}

The calculated value is close to sodium chloride's well-established experimental lattice energy (of order \(-780\,\text{kJ/mol}\) from the Born-Haber cycle), consistent with sodium chloride being a good example of a predominantly ionic, point-charge-like compound (Hypotheses). A

U(\text{NaCl}) \approx -756\,\text{kJ/mol} \quad (\text{Born-Haber experimental value} \approx -780\,\text{kJ/mol})

Reading. A purely electrostatic calculation from structural and charge data alone reproduces the experimentally derived lattice energy closely, for a compound well described by the ionic, point-charge model.

Scope. The same calculation, repeated for a compound of higher ionic charge (e.g. MgO), predicts a substantially larger lattice-energy magnitude, directly reflecting the \(z_+z_-\) dependence of the Result.

Problems

Explain, without recalculating, why MgO (\(z_+=z_-=2\)) has a substantially larger-magnitude lattice energy than NaCl (\(z_+=z_-=1\)), even though the two compounds have similar ionic separations and identical rock-salt structure (identical \(M\)).
Solution
The Result's numerator contains the product \(z_+z_-\), which is \(4\) for MgO versus \(1\) for NaCl; since \(M\), \(r_0\), and \(n\) are comparable between the two compounds, the lattice energy magnitude scales roughly fourfold, consistent with MgO's experimentally much larger lattice energy (of order \(-3800\,\text{kJ/mol}\)) compared with NaCl's roughly \(-780\,\text{kJ/mol}\).
A compound's Born-Lande-calculated lattice energy is \(-650\,\text{kJ/mol}\), but its experimental (Born-Haber-cycle) lattice energy is \(-905\,\text{kJ/mol}\), a substantially larger magnitude. What does this discrepancy suggest about the bonding in this compound?
Solution
By the Corollaries' converse, a Born-Lande value that under-predicts the experimental magnitude suggests the compound has significant covalent character not captured by the purely ionic, point-charge model (Hypotheses); the true bonding is stronger than pure electrostatics alone predicts, because some genuine electron sharing (partial covalency) contributes additional stabilisation beyond the Coulombic term.
Explain why omitting the short-range repulsion term entirely and using only the pure Coulombic energy at \(r_0\) would systematically overestimate the magnitude of the lattice energy, referencing the Result.
Solution
The full Result multiplies the pure Coulombic energy by the correction factor \((1-1/n)\), which is always less than \(1\) for any finite Born exponent \(n\); omitting this factor (equivalently, omitting the repulsive term entirely) leaves the full, uncorrected Coulombic magnitude, which is always larger than the true lattice energy magnitude, since the short-range repulsion genuinely opposes and partially cancels the attractive Coulombic contribution at the equilibrium separation.