__THE CELLULAR METHOD__

Wigner and Seitz introduced the cellular method for calculating wave functions used to understand the behaviour of conduction electrons in the field of ionic cores. Assume first that the electrons in the core shells are not affected by the metallic bond (as they are tightly bound to the core) and their wave functions are the same as in the free state. This is justified since the corresponding wave functions practically vanish in half the interatomic distance. For the valence electrons however, the assumption is not valid, since the maximum of the corresponding wave function is just about half way between two atoms.

Figure 3: The amplitude of the 3d_{zz}-wavefunction
and the 4s-wavefunction of Ni. The half distances to the first,
second and third nearest neighbours (r_{1}, r_{2},
r_{3}) are shown for comparison.

The wave function must not drop to zero after the maximum but continue periodically through the whole crystal. It will therefore be much smoother than the wave function of the free atom, and the kinetic energy of the corresponding state will consequently be much smaller than that of the electron in the free atom. The potential energy, on the other hand, will be negatively larger in the lattice than in the free state because outside of the above-mentioned maximum the wave will not be under the influence of the nucleus considered originally, but under that of the next nucleus of the lattice with is nearer. The electron will have a larger negative energy than that in the free atom and this is considered to be the essence of the metallic state.

Figure 4: The calculated ground state energy E_{ws},
average Fermi energy 3/5E_{F} and total metallic electron
energy E of Sodium, as functions of the Wigner-Seitz radius r_{s}.

To find an approximation to the wave function of the conduction electron, first calculate the energy of the free electron in the lowest state by numerically solving the Schroedinger equation. It will not be necessary to solve it for the entire lattice, because it will have the same symmetry as the crystal and will merely repeat itself a great number of times. Because of the symmetry, the derivative of the wave function at every crystallographic symmetry plane will be zero perpendicular to this plane. This will be used as a boundary condition and is represented by the Wigner-Seitz cell.

The polyhedra constructed actually approximate not too badly
to spheres; Wigner and Seitz replace them by spheres, which are
referred to as s spheres. The radius r of the sphere is
determined by the condition that the volume of the sphere be
equal to the atomic volume and so for a bcc structure 4/3p r^{3 }=^{ }1/2a^{3}
therefore, r = .49a.

Using the potential of the ion V( r ) as the function to be applied inside the sphere and specifying only one electron about any given atom at a time yields the differential equation for the free electron which will be solved with the condition that the derivative must vanish at the boundary of the sphere. Because the whole problem then has radial symmetry it requires only the radial part of the Schroedinger equation. The differential equation for the radial function was determined by Wigner and Seitz to be:

ER( r ) = - (h^{2}/8p ^{2}m)
(d^{2}R/dr^{2}) + V( r )R

The Wigner-Seitz cell can be used in real space but it is more often used in reciprocal space to represent the first Brillouin zone, where the main purpose is to enable quantum states (typically electronic or vibrational) to be enumerated. For more information on these topics select ‘Electronic States’ or ‘Vibrational States’.

Prepared by Laura Malcolm -- last update : March 27, 1997