The Electrostatic Potential in a Finite Ionic Crystal

Abstract

An expression is obtained for the electrostatic potential in a system in which similar charges are distributed over all the points of a s.c. lattice, contained within an ellipsoidal surface. The potential consists of two components: one, the intrinsic potential, is (a) a periodic function of position, with the period of the lattice, (b) a function of the geometry of the lattice, and (c) independent of the size and shape of the ellipsoid; the other, the extrinsic potential, is, (a) independent of the geometry of the lattice, (b) a quadratic function of position, and (c) a function of the shape and size of the ellipsoid. The intrinsic potential is shown to be equal to the Ewald transformed potential at high symmetry points (cube corner, face centers, body center).The finite crystal potential results are applied to the case of an electrically neutral point charge model of a cubic ionic crystal. Both the electrostatic potential within the crystal and the electrostatic energy per charge repetition unit prove, in general, to exhibit nonzero extrinsic components. The special cases of the slab-shaped specimen with CsCl. NaCl, and BaTiO3 structure are discussed.Harris and Monkhurst's expression for the energy per ion pair in an infinite diatomic ionic crystal of point charges is shown to be equal to the intrinsic component of the energy per ion pair for the finite crystal.