Born–Landé Equation - Derivation - Total Energy

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:

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

\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 in terms of yields the Born–Landé equation:

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