Madelung constant

[1] Because the anions and cations in an ionic solid attract each other by virtue of their opposing charges, separating the ions requires a certain amount of energy.

In addition, If the distances rij are normalized to the nearest neighbor distance r0, the potential may be written with Mi being the (dimensionless) Madelung constant of the ith ion Another convention is to base the reference length on the cubic root w of the unit cell volume, which for cubic systems is equal to the lattice constant.

For example, for the ionic crystal NaCl, there arise two Madelung constants – one for Na and another for Cl.

Since both ions, however, occupy lattice sites of the same symmetry they both are of the same magnitude and differ only by sign.

The electrical charge of the Na+ and Cl− ion are assumed to be onefold positive and negative, respectively, zNa = 1 and zCl = –1.

The nearest neighbour distance amounts to half the lattice constant of the cubic unit cell

Since this sum is conditionally convergent it is not suitable as definition of Madelung's constant unless the order of summation is also specified.

[6] A fast converging formula for the Madelung constant of NaCl is The continuous reduction of M with decreasing coordination number Z for the three cubic AB compounds (when accounting for the doubled charges in ZnS) explains the observed propensity of alkali halides to crystallize in the structure with highest Z compatible with their ionic radii.

[11] These second order Madelung constants turned out to have significant effects on the lattice energy and other physical properties of heteropolar crystals.

[12] The Madelung constant is also a useful quantity in describing the lattice energy of organic salts.