Born Repulsion and the Total Lattice Potential

Born repulsion is not a historical curiosity — it is load-bearing machinery in modern computational chemistry and materials science. Every classical molecular dynamics or lattice-energy-minimization force field used to model ionic solids, ceramics, minerals, or battery electrolyte materials includes a short-range repulsive term of exactly this Born or Born–Mayer form (often combined with a van der Waals attractive dispersion term into what is called a Buckingham potential, $U = -A/r^6 + b\,e^{-r/\rho}$, widely used in mineralogy and solid-state ionics simulations). Predicting whether a candidate solid-state battery electrolyte (e.g. Li-ion conducting garnets or NASICON-type materials) will have the right lattice parameters and ionic mobility relies on getting these repulsive parameters right, since they set the size of the "bottleneck" channels through which ions must hop. In geochemistry and mineral physics, Born-type repulsive potentials underlie predictions of how silicate and oxide minerals respond to the immense pressures inside planetary interiors — the same curvature-at-the-minimum argument developed here (Step 2 of the second derivation) is the conceptual ancestor of the equations of state used to model Earth's mantle and cores of other rocky planets. In pharmaceutical crystal engineering, repulsive-term parametrization feeds into polymorph prediction software that determines which crystal packing of a drug molecule is thermodynamically favored — a problem of enormous practical importance since different polymorphs can have different solubility and bioavailability. Finally, in the classroom and in the Born-Haber cycle analyses used throughout inorganic chemistry to rationalize thermochemical stability, reactivity trends, and the existence (or non-existence) of compounds like "MgCl" or "NaCl₂," the Born repulsion term is the quiet, unglamorous partner to the Madelung sum that makes the whole quantitative lattice-energy framework work at all.

Ask the tutor: How does adding the Madelung constant turn this single ion-pair potential into the full lattice energy used in Born–Haber cycles?

Picture a single Na⁺–Cl⁻ ion pair as the two ions are pushed together from a large separation. At large $r$, each ion looks to the other like a point charge; the energy is the familiar Coulomb term,

which is negative (attractive, since $z_+$ and $z_-$ have opposite sign) and becomes more negative — i.e., the system gets more stable — as $r$ decreases, all the way to $r \to 0$. If this were the only term in the potential, the ions would collapse into each other; there is no finite equilibrium distance predicted by Coulomb attraction alone. Physically, that collapse cannot happen because each ion is not a point charge — it is a nucleus surrounded by a full complement of core electrons (Cl⁻ is $[\text{Ar}]$-like, Na⁺ is $[\text{Ne}]$-like, both closed-shell configurations with electron density spread over a finite ionic radius). As the ions approach, their electron clouds begin to overlap once $r$ becomes comparable to the sum of ionic radii $r_+ + r_-$. Overlapping closed shells cannot simply superpose their electrons into the same states: the Pauli exclusion principle forces some electrons into higher-energy orbitals (formally, into antibonding combinations of the atomic orbitals, or equivalently the electron density must be "pushed out" of the overlap region and raised in kinetic energy by the uncertainty principle). This costs energy, and the cost rises extremely steeply as the clouds are forced to interpenetrate further — far more steeply than the Coulomb term falls. The net potential is the sum of a gentle, long-range attractive term and a violent, short-range repulsive term. At large $r$ the attractive term dominates (repulsion is exponentially or power-law suppressed at large distances, decaying to zero faster than $1/r$), so pulling the ions together lowers the energy. At small $r$ the repulsive term dominates and blows up, so pushing the ions together further costs enormous energy. Somewhere in between, the two effects exactly balance: this is the equilibrium separation $r_0$, and it is a true minimum (not just a balance point) precisely because repulsion is more steeply varying than attraction — the second derivative of the total potential is positive there.

The steepness contrast is the entire story. In the historical Born model the repulsive energy is written as a power law $B/r^n$ with $n$ typically between 5 and 12 (larger for more compressible, more polarizable ions with more diffuse outer shells; smaller — meaning steeper in relative terms at the relevant $r$ — for "harder," less compressible ions like small, highly-charged cations). Because $n \gg 1$, this term is essentially negligible until $r$ drops within roughly the sum of ionic radii, at which point it rises almost vertically. Later refinement, due to Born and Mayer, replaced the power law with an exponential form $b\,e^{-r/\rho}$, which is physically preferable because atomic and ionic electron density genuinely falls off exponentially with distance from the nucleus (a feature of the radial solutions of the Schrödinger equation for many-electron atoms), so the overlap of two such clouds — and hence the repulsive energy it generates — also decays exponentially.

The total potential energy of an ion pair separated by distance $r$ is the sum of the attractive Coulomb term and a short-range repulsive term:

where:

• $z_+ e,\ z_- e$ are the ionic charges (with $z_-$ negative, so the product $z_+z_-$ is negative and the first term is attractive),
• $e$ is the elementary charge, $\varepsilon_0$ the vacuum permittivity,
• $r$ is the center-to-center ion separation,
• $U_{\text{rep}}(r)$ is a positive, steeply decreasing function of $r$ representing electron-cloud overlap repulsion.

Reading: the first term alone has no stationary point in $r>0$; it is monotonic. Stability requires $U_{\text{rep}}$, and the shape of $U_{\text{rep}}$ (how fast it falls with $r$) sets both the equilibrium distance and the curvature of the well there. Two standard choices for $U_{\text{rep}}$ are used throughout ionic-bonding theory:

where:

• $B > 0$ is an empirical repulsion coefficient (units chosen so $U$ comes out in energy),
• $n$ is the Born exponent, typically 5–12, obtained from ionic electronic configuration or from compressibility data (larger $n$ for ions with more electron shells, i.e. more "incompressible" cores such as noble-gas-like heavy ions).

where:

• $b > 0$ is an empirical pre-exponential repulsion constant,
• $\rho$ (units of length, typically ≈ 0.3 Å) is a "softness" parameter describing how rapidly the repulsion decays — smaller $\rho$ means a harder, steeper repulsive wall.

Reading: both functions share the essential qualitative behavior — positive, rapidly vanishing at large $r$, diverging as $r \to 0$ — but the exponential form is the physically motivated one, since it mirrors the actual exponential decay of atomic orbital wavefunctions at large distance from the nucleus, $\psi \sim e^{-r/a_0}$-type behavior, whose overlap integral between two centers separated by $r$ also decays exponentially. The full pair potential (Born model) is therefore

and this is the expression whose minimum defines the observed equilibrium bond length and whose depth (once summed over the whole lattice with the Madelung constant) becomes the lattice energy — the subject of the The Born-Lande Equation.

Example 1 — Foundational: locate the minimum for a model ion pair with round-number $n$. Take a hypothetical 1:1 ion pair with $|z_+ z_-| = 1$, equilibrium separation $r_0 = 2.50$ Å (chosen off the real NaCl value of 2.82 Å to keep the arithmetic transparent and non-suggestive of a "special" real salt), and Born exponent $n = 8$ (a mid-range value typical of ions with argon-like cores). Using Step 4 of the first derivation,

Compute $r_0^7 = (2.50\times10^{-10})^7$. First $2.50^7 = 610.35$ (since $2.5^2=6.25$, $2.5^4 = 39.0625$, $2.5^6 = 244.14$, $2.5^7=610.35$), and $10^{-10\times7} = 10^{-70}$, so $r_0^7 = 6.1035\times10^{-68}\ \text{m}^7$. Numerator: $(1.602\times10^{-19})^2 = 2.566\times10^{-38}\ \text{C}^2$; multiplied by $6.1035\times10^{-68}$ gives $1.567\times10^{-105}$. Denominator: $4\pi(8.854\times10^{-12})(8) = 4(3.1416)(8.854\times10^{-12})(8) = 8.905\times10^{-10}$.

Check the well depth via the Step 5 result:

The bare Coulomb term at $r_0$: numerator $2.566\times10^{-38}$, denominator $4\pi(8.854\times10^{-12})(2.50\times10^{-10}) = 2.783\times10^{-20}$, giving $-9.222\times10^{-19}$ J. Multiplying by $(7/8) = 0.875$:

Converting to a molar quantity (×Avogadro's number) gives $-486$ kJ/mol for this single ion-pair interaction — smaller in magnitude than a real Madelung-summed lattice energy (which includes attraction/repulsion from all neighboring ions, not just the nearest pair), exactly as expected, since this toy calculation deliberately omits the lattice sum that the The Born-Lande Equation adds on top.

Example 2 — Applied: comparing $n=6$ vs $n=10$ for the same $r_0$, and why softer ions bind slightly less tightly at fixed bond length. Suppose two different hypothetical salts happen to share exactly the same equilibrium separation $r_0 = 2.80$ Å and the same $|z_+z_-|=1$, but one has Born exponent $n=6$ (softer, more polarizable ions, e.g. involving larger anions like I⁻) and the other $n=10$ (harder, more incompressible ions, e.g. small F⁻ paired with a small, high-charge-density cation). The bare Coulomb energy at $r_0=2.80$ Å is the same for both (it doesn't depend on $n$):

(dividing the earlier denominator-scaling: $2.783\times10^{-20}\times(2.80/2.50) = 3.117\times10^{-20}$, so $U_{\text{Coul}} = -2.566\times10^{-38}/3.117\times10^{-20} = -8.233\times10^{-19}$ J.) For $n=6$: correction factor $(1-1/6) = 0.8333$, giving

For $n=10$: correction factor $(1-1/10)=0.9000$, giving

The harder-ion salt ($n=10$) binds about 8% more strongly at the same bond length than the softer one ($n=6$), because a steeper (larger-$n$) repulsive wall "gives back" less of the Coulomb attraction — repulsion only switches on right at the wall, leaving more of the pure Coulomb well depth intact. This is exactly why, in real Born-Landé calculations, ions built from more electrons (bigger, softer, more diffuse clouds — smaller effective $n$ is not generally true; conventionally larger ions get larger $n$, and the sign of this trend is worth checking against tables rather than memorized as "always"). The essential physical lesson survives regardless: the correction factor $(1-1/n)$ is always a fraction less than 1, always shaves some magnitude off the bare Coulomb energy, and the amount shaved depends on how steep the repulsive wall is.

Example 3 — Exam/competition level: extracting the Born exponent from a measured compressibility-derived curvature. A hypothetical rock-salt-structure crystal MX (1:1 charges, $|z_+z_-|=1$) has measured equilibrium nearest-neighbor separation $r_0 = 3.10$ Å. An independent compressibility experiment yields a second-derivative (curvature) of the pair potential at $r_0$ of

Find the Born exponent $n$ implied by this measurement, and compare with the Coulomb curvature term alone. Using the inverted relation from the second derivation,

First compute $r_0^3 = (3.10\times10^{-10})^3\ \text{m}^3$. $3.10^3 = 29.79$, so $r_0^3 = 2.979\times10^{-29}\ \text{m}^3$. Numerator of the fraction: $4\pi\varepsilon_0\, r_0^3 \times (d^2U/dr^2) = (1.1127\times10^{-10}\ \text{C}^2\text{N}^{-1}\text{m}^{-2})(2.979\times10^{-29}\ \text{m}^3)(2.95\times10^{-9}\ \text{J/m}^2)$ (using $4\pi\varepsilon_0 = 4\pi(8.854\times10^{-12}) = 1.1127\times10^{-10}$). Multiply stepwise: $1.1127\times10^{-10}\times 2.979\times10^{-29} = 3.315\times10^{-39}$. Then $3.315\times10^{-39}\times 2.95\times10^{-9} = 9.780\times10^{-48}$. Divide by $e^2 = (1.602\times10^{-19})^2 = 2.566\times10^{-38}$:

This is puzzlingly small — the numbers imply $n \approx 1.0000000004$, essentially $n=1$, which would be unphysical (no true minimum). This flags an important dimensional/scale check worth doing explicitly: real ionic-crystal curvature values, when properly extracted from bulk moduli (which involve the full lattice sum, the Madelung constant, and the number of nearest neighbors, not a bare single-pair potential), are enormously larger in magnitude than a bare single-pair estimate — typically of order $10^{2}$–$10^{3}$ N/m in force-constant terms once lattice geometry is folded in. The arithmetic pipeline above is correct as a method; the lesson of this worked example is that $d^2U/dr^2$ must be the correctly scaled single-pair curvature (stripped of lattice geometry factors such as the coordination number and Madelung constant) before this formula can be applied — exactly the bookkeeping problem that the The Born-Lande Equation and its lattice sums are built to handle properly. A student who plugs in a macroscopic compressibility number without first converting it to a single-pair curvature will get exactly this kind of nonsensical near-unity $n$, which is itself a useful diagnostic that a unit or lattice-factor conversion has been skipped.

Max Born (1882–1970) was a German theoretical physicist whose name is more famous today for the Born interpretation of the quantum wavefunction (probability density $|\psi|^2$) than for ionic crystals, but his early career, in Göttingen before quantum mechanics existed in its later form, was deeply concerned with the mechanics of crystal lattices. Working alongside Alfred Landé around 1918, Born sought a classical-mechanical (pre-quantum-mechanical, in the modern sense) model that could reproduce measured lattice energies and, crucially, the compressibility of ionic crystals — a property utterly unexplained by Coulomb attraction alone. Introducing a short-range repulsive term of the form $B/r^n$, calibrated against measured compressibilities, gave the first quantitatively successful model of an ionic crystal's cohesive energy, decades before the full quantum-mechanical justification (Pauli exclusion, exchange interactions) was available. Joseph E. Mayer, working with Born considerably later (the Born–Mayer refinement dates to the early 1930s, after the Pauli principle and modern quantum mechanics were established), replaced the power-law repulsion with the physically motivated exponential form, better matching the true exponential decay of atomic electron density predicted by quantum theory.