Some results on the electrostatic energy of ionic crystals

Abstract

We define a class of ionic cyrstals, the alternating Bravais lattice ionic crystals, which has the NaCl and CsCl structures as members. We calculate the electrostatic energy of finite pieces and study the convergence to the macroscopic Madelung limit. For the one-parameter family of trigonal lattices we calculate the dependence of the electrostatic energy on the parameter. The NaCl and CsCl structures correspond to minima, the CsCl minimum being deeper. This is due to long-range effects; for small clusters the NaCl structure is favored. We also study the Madelung constant of simple cubic lattices as a function of spatial dimension, and discuss the results. We finally calculate the electrostatic repulsion of two constant unit charge distributions in the unit cube. This quantity, a six-dimensional integral, can be integrated analytically five times, leaving a simple one-dimensional integral to be done numerically. Key words: ionic crystal, Madelung constant, ionic cluster, electrostatic energy.