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,
Epair, between a pair of ions of equal and opposite charge is: :E_\text{pair} = -\frac{z^2 e^2 }{4 \pi \epsilon_0 r} where :
z = magnitude of charge on one ion :
e = elementary charge, 1.6022
C :
ε0 =
permittivity of free space ::4
ε0 = 1.112 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
EM, sometimes called the
Madelung or lattice energy: :E_\text{M} = -\frac{z^2 e^2 M}{4 \pi \epsilon_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 so that the repulsive energy term,
ER, would be expressed: :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: :E(r) = -\frac{z^2 e^2 M}{4 \pi \epsilon_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: :\begin{align} \frac{\mathrm{d}E}{\mathrm{d}r} &= \frac{z^2 e^2 M}{4 \pi \epsilon_0 r^2} - \frac{n B}{r^{n+1}} \\ 0 &= \frac{z^2 e^2 M}{4 \pi \epsilon_0 r_0^2} - \frac{n B}{r_0^{n+1}} \\ r_0 &= \left( \frac{4 \pi \epsilon_0 n B}{z^2 e^2 M}\right) ^\frac{1}{n-1} \\ B &= \frac{z^2 e^2 M}{4 \pi \epsilon_0 n} r_0^{n-1} \end{align} Evaluating the minimum intensive potential energy and substituting the expression for
B in terms of
r0 yields the Born–Landé equation: :E(r_0) = - \frac{M z^2 e^2 }{4 \pi \epsilon_0 r_0}\left(1-\frac{1}{n}\right) ==Calculated lattice energies==