Ionic materials form crystalline lattices like metals but the lattice structure is determined by one ion (the larger, usually the anion) and the other ion fits into the holes. Since the smaller ion may not just right, lattice expansions can occur.
What holds an ionic lattice together?
A simple model: assume each ion is represented by a point charge, then sum up all the Coulomb interactions.
q+, q– are charges on the ion, d the distance between ions, ε0 = permittivity of free space, NA is Avogadro's number, A is the Madelung constant
The Madelung constant is a purely geometric factor that has been tabulated for many lattice types. It represents the potential energy associated with different relative geometries. d is the distance between ions so accounts for the different energy associated with different compounds with the same lattice type (hence, the same Madelung constant).
ECoul is negative, thus attractive, but maximizes at d = 0, i.e. when anions and cation occupy the the same point in space. Need a repulsive term (to account for nuclear-nuclear repulsion when the ions are close enough) which is done in an ad hoc fashion.
Born-Mayer:
C, d* constants
Born-Landé: