The calculation of the Madelung constant

A previous article showed that the electrostatic term of the lattice energy of a crystal contains a factor (A) that depends on the type of crystalline reticulum. This article will look more in detail how to calculate this term and its value for a simple ionic system.

The total Coulomb interaction energy of a crystal is given by the sum of the single pair interaction terms:

\displaystyle V_{AB} = \frac{e^2}{4\pi\epsilon_0} \frac{Z_AZ_B}{r_{AB}} \hfill (1)

for ions with charges qA and qB and distance rAB.  The sum extends to all pairs of ions present in the solid for any crystalline structure.

The sum converges very slowly because the first neighbours give a first substantial contribution with a negative summation term. Second neighbour atoms produce a slightly weaker positive term. It then continues to infinity with terms of alternating signs by smaller and smaller values. In this way, the resulting effect will be that the attraction between cations and anions predominates and provides a favourable negative contribution to the solid’s energy.

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