Born–Landé equation

${\displaystyle E=-\frac{N_{A}M{z}^{+}{z}^{-}{e}^2}{4\pi {\epsilon }_0{r}_0}\left(1-\frac{1}{n}\right)}$

Where:

• E = the lattice energy
• N_A = Avogadro constant
• M = Madelung constant, relating to the geometry of the crystal
• z⁺ = numeric charge number of cation
• z⁻ = numeric charge number of anion
• e = elementary charge, 1.6022×10⁻¹⁹ C
• ε₀ = permittivity of free space

  4πε₀ = 1.112×10⁻¹⁰ C²/(J·m)

• r₀ = distance between closest cation [ +ve ] and anion [ -ve ]
• n = Born exponent, typically a number between 5 and 12, determined experimentally by measuring the compressibility of the solid, or derived theoretically

Derivation

The ionic lattice is modeled as an assembly of hard elastic spheres which are compressed together by the mutual attraction of the electrostatic charges on the ions. They achieve the observed equilibrium distance apart due to a balancing short range repulsion.

Electrostatic potential

The electrostatic potential energy, E_pair, between a pair of ions of equal and opposite charge is:

${\displaystyle E_{\text{pair}}=-\frac{z^2{e}^2}{4\pi {ϵ}_{0}r}}$

where

z = magnitude of charge on one ion

e = elementary charge, 1.6022×10⁻¹⁹ C

ε₀ = permittivity of free space

4πε₀ = 1.112×10⁻¹⁰ C²/(J·m)

r = distance separating the ion centers

For a simple lattice consisting ions with equal and opposite charge in a 1:1 ratio, interactions between one ion and all other lattice ions need to be summed to calculate E_M, sometimes called the Madelung or lattice energy:

${\displaystyle E_{\text{M}}=-\frac{z^2{e}^{2}M}{4\pi {ϵ}_{0}r}}$

where

M = Madelung constant, which is related to the geometry of the crystal

r = closest distance between two ions of opposite charge

Repulsive term

Born and Lande suggested that a repulsive interaction between the lattice ions would be proportional to ⁠1/rⁿ⁠ so that the repulsive energy term, E_R, would be expressed:

${\displaystyle E_{\text{R}}=\frac{B}{r^n}}$

where

B = constant scaling the strength of the repulsive interaction

r = closest distance between two ions of opposite charge

n = Born exponent, a number between 5 and 12 expressing the steepness of the repulsive barrier

Total energy

The total intensive potential energy of an ion in the lattice can therefore be expressed as the sum of the Madelung and repulsive potentials:

${\displaystyle E(r)=-\frac{z^2{e}^{2}M}{4\pi {ϵ}_{0}r}+\frac{B}{r^n}}$

Minimizing this energy with respect to r yields the equilibrium separation r₀ in terms of the unknown constant B:

${\displaystyle \begin{array}{cc}\frac{\mathrm{d}E}{\mathrm{d}r}& =\frac{z^2{e}^{2}M}{4\pi {ϵ}_0{r}^2}-\frac{nB}{r^{n+1}}\\ 0& =\frac{z^2{e}^{2}M}{4\pi {ϵ}_0{r}_0^2}-\frac{nB}{r_0^{n+1}}\\ r_0& ={\left(\frac{4\pi {ϵ}_{0}nB}{z^2{e}^{2}M}\right)}^{\frac{1}{n-1}}\\ B& =\frac{z^2{e}^{2}M}{4\pi {ϵ}_{0}n}{r}_0^{n-1}\end{array}}$

Evaluating the minimum intensive potential energy and substituting the expression for B in terms of r₀ yields the Born–Landé equation:

${\displaystyle E(r_0)=-\frac{M{z}^2{e}^2}{4\pi {ϵ}_0{r}_0}\left(1-\frac{1}{n}\right)}$