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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