Lattice Energies of Ionic Crystals

Lattice energy represents the energy released when gaseous cations and anions combine to form an ionic solid, reflecting the strength of electrostatic interactions within the crystal. This process is fundamentally governed by Coulombic attraction between oppositely charged ions, where the potential energy varies inversely with the interionic distance and directly with the product of ionic charges. As ions approach one another, the electrostatic energy becomes increasingly negative, indicating a more stable configuration.

At very short interionic distances, electron cloud overlap leads to strong repulsive forces. Born introduced a repulsive term proportional to B/rⁿ, where B is the Born constant and n, the Born exponent, reflecting the compressibility of the ions. This repulsion counterbalances the attractive Coulombic forces, preventing collapse of the lattice.

To determine B, the key idea is to use the equilibrium condition. At the equilibrium distance between ions, the system is stable, meaning there is no net force acting on the ions. Physically, this corresponds to the point where the attractive forces pulling ions together are exactly balanced by the repulsive forces pushing them apart.

When this condition is applied to the total energy expression, it produces a relationship between the attractive term and the repulsive term at the equilibrium distance. Solving this relationship gives an explicit expression for B as

\begin{equation*}B = \frac{M {z_+}\, z_-\, e^{2}\, r_{0}^{\,n-1}}{4\pi \varepsilon_{0} n}\end{equation*}

Substituting for B and reorganizing the total lattice energy yields the Born–Landé equation for lattice energy.