Types of perfect solids

As early as 1933, Wigner and Seitz gave a quantum-mechanical explanation of metallic bonding, and made a reasonable estimate, to within 10%, of the bonding energy. The calculation was carried out in the case of sodium, and we'll use this example to give some qualitative indications. Let's start by describing the crystal lattice, which is cubic-centered, and use, in the direct lattice, to determine the volume assigned to each atom, the procedure employed to define the first Brillouin zone in the reciprocal lattice. This is how we define the Wigner-Seitz cell, described in the article Geometrical Crystallography . Then, for simplicity's sake, we'll assimilate this cell to a sphere of identical volume.

Our aim is to calculate the wave function of the low-band electron, whose wave vector k is zero. To a first approximation, in the Wigner-Seitz sphere,...