The Evaluation of Lattice Sums for Cubic Crystals

Abstract

A method is presented of evaluating the sums which arise in calculating the total long-range interaction energy between a semi-infinite crystal and a particle placed above its surface. For example, if the particle is an ion and the crystal is polarizable, this method facilitates the calculation of the polarization and Van der Waal's energies. It is pointed out that many calculations within a solid can be reduced to a combination of cases each of which is of the above type. As an illustration, we obtain the total interaction at an interior lattice point for a simple-cubic structure with an individual atom-pair interaction of $\lambda r^{-n}$.