The lattice energy of a simple ionic crystal

In the NaCl structure an ion has six equidistant neighbours of the other kind at a distance r (Na-Q), twelve of its own kind at r /2, eight of the other kind at r s/3 d so on, so that the potential at an ion is found by summing the infinite series 

If this summation is written —A elr) then the coulomb energy is 

Since crystals are not indefinitely compressible there is evidently a repulsion force which operates when the electron clouds of the ions begin to interpenetrate (without electron-sharing between the ions). This repulsion energy is not readily calculable, and Born represented it by 5/r , a function which increases very rapidly with decreasing distance r if n is large, that is, it corresponds to the ions being hard spheres. The expression for the lattice energy is now 

The value of the exponent n can be deduced from the compressibility of the crystal values used in calculating lattice energies are 7, 9,10, and 12 for ions with Ne, Ar, Kr, and Xe configurations. The value of B is calculated in the following way. At equilibrium the energy is a minimum, that is, the attractive and repulsive forces balance one another, therefore 

The calculation of lattice energy has been refined by including terms arising from the van der Waals (London) forces and from the zero-point energy. The former is important only if both ions are readily polarizable, as may be seen from the following figures  

---