Born–Landé equation

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:

E pair = − z 2 e 2 4 π ϵ 0 r

where

z = magnitude of charge on one ion e = elementary charge, 1.6022×10⁻¹⁹ C ϵ 0 = permittivity of free space 4 π ϵ 0 = 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:

E M = − z 2 e 2 M 4 π 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 / r n so that the repulsive energy term, E R , would be expressed:

E R = 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:

E ( r ) = − z 2 e 2 M 4 π ϵ 0 r + B r n

Minimizing this energy with respect to r yields the equilibrium separation r 0 in terms of the unknown constant B :

d E d r = z 2 e 2 M 4 π ϵ 0 r 2 − n B r n + 1 0 = z 2 e 2 M 4 π ϵ 0 r 0 2 − n B r 0 n + 1 r 0 = ( 4 π ϵ 0 n B z 2 e 2 M ) 1 n − 1 B = z 2 e 2 M 4 π ϵ 0 n r 0 n − 1

Evaluating the minimum intensive potential energy and substituting the expression for B in terms of r 0 yields the Born–Landé equation:

E ( r 0 ) = − M z 2 e 2 4 π ϵ 0 r 0 ( 1 − 1 n )

Calculated lattice energies

The Born–Landé equation gives a reasonable fit to the lattice energy

Born Exponent

The Born exponent is typically between 5 and 12. Approximate experimental values are listed below: