Translational invariance of Coulomb series and symmetric potentials in crystals

TL;DR

This paper demonstrates that Coulomb series in crystals exhibit translational invariance when summed in a specific way, leading to unique bulk potentials and symmetric potential properties related to the lattice structure.

Contribution

It introduces a special summation mode for Coulomb series that ensures translational invariance and establishes the uniqueness of bulk potentials in triclinic lattices.

Findings

Coulomb series are translationally invariant under a specific summation mode.

Absolute bulk potentials with zero mean are uniquely determined for triclinic lattices.

Potential symmetry relates to the center of gravity of the potential field, applicable to non-local charges.

Abstract

It is shown that Coulomb series are to be considered within a special mode of summation so as to describe bulk properties of crystals. The translational invariance is then an explicit integral property of Coulomb series that is tantamount to the effect of invariant periodic boundary conditions discussed earlier. Absolute bulk potentials with zero mean value are then substantiated as a unique solution in the general case of triclinic lattices. An invariant treatment of the bulk Coulomb energy follows therefrom. The potential symmetry is verified for simple point-charge lattices and is connected with the centre of gravity of the potential field that is relevant to non-local charges as well.