Electrostatic Potentials for Semi-Infinite and Lamellar Cubic Lattices Containing Several Different Kinds of Ions Per Unit Cell

The electrostatic potential for a square planar lattice of positive, unit, point charges neutralized by a uniform-negative-background charge is developed and numerically tabulated. By stacking up such planes, potentials are constructed for infinite, semi-infinite, and lamellar-neutralized, simple-cubic lattices of positive point charges; numerical results are presented. The relation between potentials obtained by smearing out the neutralizing charge over all space and by confining the neutralizing charge to lattice planes is explicitly exhibited. Using SrTi${\mathrm{O}}_3$, a perovskite, as a specific example, it is shown how the tabulated potentials given here may be used to obtain the electrostatic potential on the surface of and within a complex cubic crystal containing several kinds of ions per unit cell. The methods for obtaining the potentials above the crystal surface or in the presence of vacancies and impurities are briefly indicated.