On the energy per particle in three- and two-dimensional Wigner lattices

Abstract

We come back to the 1979 controversy about the value of the energy per particleФ, in an infinite Wigner lattice of electrons in a uniform compensating background. For simplicity we restrict ourselves to the simple cubic (and square) lattice. We present an accurate calculation of the energyФ _el of one electron in the field of the other electrons plus background for the case that the system (system I) is considered as an infinite arrangement of neutral cubes (Wigner-Seitz cells). The value obtained is checked by computer calculations. We confirm the conclusion of de Wette that for this system the relationФ _i=1/2Ф _el (often accepted without discussion) does not hold and we calculate the differenceΔФ, which represents the average potential in the system. On the other hand, if the system is considered as the limit of a set of spheres with increasing radii, such that the spheres are neutral (system II), we obtain a different value ofФ _el and in this caseФ _i=1/2Ф _el. We show explicitly that the Ewald method of summation, used by Fuchs and others, leads to the same analytical expression as the limit obtained for a set of neutral spheres (system II). We extend the calculations to the two-dimensional square lattice. Here the equalityФ _i=1/2Ф _el holds also in the case of an infinite arrangement of neutral squares (system I).