The Electrostatic Potential of Periodic Crystals

Abstract

The electrostatic potentials $\phi$ associated with neutral periodic crystals are defined by lattice sums that are never absolutely convergent. The sum depends on the order of summation. The mean zero periodic solution of Poisson's equation, denoted ${\underline \phi}$, is a natural potential. So is the potential obtained by the Mellin transform algorithm of [Borwein, Borwein, and Taylor, J. Math. Phys., 26 (1985), pp. 2999--3009]. We prove that these two are equal and are both equal to the potential obtained by Abel summation. The sum defining $\partial^\alpha \phi$ converges absolutely for $|\alpha|\ge 3$ to $\partial^\alpha{\underline \phi}$. The indeterminacy in the potential is at most a harmonic polynomial of degree 2.