Electrostatic energy in ionic crystals

Abstract

The problem of calculating electrostatic energies in large, finite and arbitrarily shaped pieces of ionic crystal is analysed. The electrostatic energy of a unit cell of the crystal deep within the interior of the piece of crystal is shown to be composed of a shape-independent part depending on the structure of the crystal lattice concerned and the distribution of ions within a unit cell, and a shape-dependent part which depends on the shape of the piece of crystal and the dipole moment of a unit cell. The shape-dependent part is zero if this dipole moment is zero. The electrostatic energy of the whole piece of crystal is shown to be the unit cell energy multiplied by the number of unit cells in the piece of crystal, plus corrections proportional to the surface area of the piece of crystal. These surface corrections are calculated explicitly for a finite cube of simple cubic crystal. Different descriptions of the same crystal structure are shown to lead to different bulk energies. This disagreement is discussed for the CsCl lattice, and is shown to arise from the way the different descriptions of the lattice imply different surface structures on the surface of a cube of crystal. The energy of a test charge at the surface of a cube of simple cubic crystal, and then the energy of layers of charges on the surfaces of a cube of simple cubic crystal, are analysed. The analyses confirm the origin of the disagreement in bulk energies for the two descriptions of the CsCl lattice. The role of the energies of surface layers in the bulk electrostatic energy of a piece of ionic crystal, and the relation of this bulk energy to a shape-independent Madelung constant are discussed. Some conjectures on the role of bulk energies in surface reconstruction are also discussed.