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

We come back to the 1979 controversy about the value of the energy per particle, 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 energy_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 relation_i=1/2_el (often accepted without discussion) does not hold and we calculate the difference, 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 of_el and in this case_i=1/2_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 equality_i=1/2_el holds also in the case of an infinite arrangement of neutral squares (system I).This paper is dedicated to our friend and colleague Nico van Kampen in honor of his lifelong dedication to science in general and to theoretical physics in particular.