Born–Mayer equation¶

A lattice-energy equation for ionic crystals that combines Coulomb attraction with an exponential short-range repulsion approximated through a characteristic range parameter.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3519
Domain group: Natural Sciences
Origin domain: Chemistry & Materials Science
Subdomain: Solid State Chemistry → Chemistry & Materials Science

Core Idea¶

The Born–Mayer equation refines Born–Landé lattice energy by replacing a power-law repulsion with an exponential form, often summarized by a correction proportional to 1−ρ/r₀. Crystal geometry fixes the Madelung electrostatic term, ionic charge and nearest separation set its scale, and electron-cloud overlap supplies a rapidly increasing repulsive correction. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

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Push and Pull in a Salt Crystal

In a grain of salt, tiny bits with plus and minus charges pull on each other and hold the crystal together. But if they get squeezed too close, their fuzzy outsides bump and push back really hard. The Born–Mayer equation figures out how much energy holds the crystal together by adding the pulling and the push-back, where the push-back gets stronger super fast as the bits get closer.

How Tightly Crystals Hold Together

The Born–Mayer equation estimates the lattice energy of an ionic crystal like table salt — how much energy holds its charged particles, called ions, together. It has two parts. The attraction part depends on the ions' charges, how close the nearest neighbors are, and a number called the Madelung constant that comes from the shape of the crystal. The repulsion part comes from the ions' electron clouds overlapping and pushing apart, and it grows very quickly as the ions get closer. Born–Mayer uses an exponential form for this push-back, which improves on an older equation called Born–Landé.

Exponential-Repulsion Lattice Energy

The Born–Mayer equation calculates the lattice energy of an ionic crystal. It starts from the electrostatic attraction, whose size depends on the ionic charges and nearest-neighbor distance, multiplied by the Madelung constant, which captures the crystal's geometry. It then corrects for repulsion caused by overlapping electron clouds. The earlier Born–Landé equation modeled that repulsion with a power law; Born–Mayer replaces it with an exponential that rises steeply at short distances, giving a correction factor of the form (1 − ρ/r₀), where ρ is a repulsion parameter and r₀ is the nearest-neighbor separation. Using it correctly means stating the charges, Madelung constant, separation, repulsion parameter, sign convention and whether energies are per mole.

The Born–Mayer equation gives the lattice energy of an ionic crystal by combining a Madelung electrostatic term with an exponential short-range repulsion. The Madelung constant is fixed by crystal geometry; the ionic charges and nearest-neighbor separation r₀ set the electrostatic scale; and electron-cloud overlap contributes a repulsion that rises rapidly at short range. It refines the Born–Landé equation by replacing the power-law repulsion with an exponential form, which after imposing equilibrium yields a correction proportional to 1 − ρ/r₀, with ρ the repulsion range parameter. The equation assumes an idealized ionic crystal. Applying it requires declaring ionic charges, Madelung constant, nearest-neighbor separation, repulsion parameter, sign convention, and whether energies are per mole. Substituting a different repulsion law or ignoring these conventions gives a different model.

Scope of Application¶

Born–Mayer equation belongs to solid state chemistry and is useful where the analyst can specify the typed solid state chemistry carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate ionic charges, Madelung constant, nearest-neighbor separation, repulsion parameter, sign convention, and per-mole energy convention are declared under the ionic-crystal assumptions. The scope is broad within that domain but bounded by the need for ionic charges, Madelung constant, nearest-neighbor separation, repulsion parameter, sign convention, and per-mole energy convention are declared under the ionic-crystal assumptions. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making ionic charges, Madelung constant, nearest-neighbor separation, repulsion parameter, sign convention, and per-mole energy convention are declared under the ionic-crystal assumptions the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Born–Mayer equation can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Born–Mayer equation. Born–Mayer equation compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed solid state chemistry carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express ionic charges, Madelung constant, nearest-neighbor separation, repulsion parameter, sign convention, and per-mole energy convention are declared under the ionic-crystal assumptions independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of solid state chemistry because they reuse the typed solid state chemistry carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Crystal geometry fixes the Madelung electrostatic term, ionic charge and nearest separation set its scale, and electron-cloud overlap supplies a rapidly increasing repulsive correction., and type the carrier, state every parameter and convention in the definition, test that ionic charges, Madelung constant, nearest-neighbor separation, repulsion parameter, sign convention, and per-mole energy convention are declared under the ionic-crystal assumptions, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Born–Mayer equation Domain-specific

Parents (1) — outgoing typed relations

Born–Mayer equation is a kind of Approximation Prime

What makes it its own entry: the domain-specific identity determined by ionic charges, Madelung constant, nearest-neighbor separation, repulsion parameter, sign convention, and per-mole energy convention are declared under the ionic-crystal assumptions.

Hierarchy path (1) — routes to 1 parentless root

Born–Mayer equation → is a kind of Approximation → is a decomposition of Representation → presupposes Abstraction

Neighborhood in Abstraction Space¶

Born–Mayer equation sits in a crowded region of the domain-specific corpus (36^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Molecular Structure & Chemical Bonding (35 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25