__INTRODUCTION__

As a conduction electron moves through a metal it is continuously exposed to intense and rapidly varying electrostatic forces. At each atomic nucleus its electrostatic potential drops to negative infinity, rises up again in a complicated way in the field of the core electrons and then is exposed to fluctuating forces resulting from its interaction with all the other conduction electrons moving through the metal. Trying to determine chemical and structural properties of metals by calculating valence forces imposes great mathematical difficulty as the number of atoms increases.

The simple theory of metals has swept all this aside by
describing metals as containing free electrons, electrons which
have become separated from their parent atoms and are free to
wander through the entire body of the material as a kind of gas
or plasma. This theory explains high electrical conductivity and
thermal conductivity. And secondly by describing electrons in a
periodic field whereby the atoms in a metallic crystal act as
positively charged spheres and the cohesion of the material comes
from the electrical attraction of these positive charges to the
‘sea’ of negative electrons flowing among them. As long
as they keep close to the free electrons, these atoms are not
very sensitive to their positions relative to one another.
Conduction electrons are then considered independent of
interactions with other conduction electrons and assumed to be
governed by a one-electron Schroedinger equation. The wave function Y ( r ) represents running waves carrying momentum . Where k is the wave vector which specifies direction in the crystal and has units cm^{-1}.

Y ( r ) = C e^{ikr}

This does not however, explain why some elements crystallise to form good conductors of electricity while others form good insulators and still others are semiconductors , with electrical properties varying with temperature. Bloch represented in the simplest possible way the real state of affairs inside a crystal, he replaced the potential energy, considered to be a constant in the free-electron theory by a periodically varying potential energy in accordance with the electrostatic attraction of the electrons to the positive ions assumed to be arrange in a perfectly periodic lattice.

Y ( r ) = u( r ) e^{ikr}

By combining the atomic-like function (the same at every site) and an overall sinusoidal ‘free electron’ function extending throughout the crystal he determined that the periodic lattice field neither scatters nor destroys the free electron wave, but merely modulates it.

Figure 1: A partial representation of a Bloch function along a line of atomic centres.

u( r ) then, will give a good picture of the distribution of charge within a unit cell. Wigner and Seitz have developed a simple and fairly accurate method of calculating u( r ) if the ion core potential is known. This is explained in the section called ‘The Cellular Method’ and illustrated in the section called ‘Construction of Wigner-Seitz Cells’.

In a metal the electrons occupy a band of quantized kinetic
energy levels from a ground energy level of almost zero up to a
maximum of E_{F} the Fermi level. The number of unit
cells in a crystal lattice N, yields N quantum states. As per the
Pauli Exclusion Principle 2 electrons of opposite spin are
required to fill each state with the lowest energy states being
filled first. Partially filled bands allow freedom of movement of
electrons throughout the solid yielding good conductors. Filled
energy bands restrict the electron movement yielding good
insulators But this is not the whole story.

Electron waves cannot be scattered but they can be reflected and Bragg reflection is an important characteristic feature of the electron wave propagation in a periodic crystal. Bragg reflection leads to the existence of energy gaps in the distribution in energy of the states of conduction electrons. That is there are regions of energy in which solutions of the wave equation do not exist These gaps arise because the wave reflected from the neighbouring atom interferes constructively with the original wave of the home atom. Wave functions which represent the reflected wave where the electron is trapped between two crystallographic planes can be represented by y (x) = u(x)coskx and y (x) = (x)sinkx. These are standing waves and represent the fact that, when k satisfies the Bragg condition, the electron on average moves neither forward or backward through the reflecting planes, since it is as likely to have -k as +k. It is these gaps which determine whether a solid will be an insulator, conductor or semiconductor.

Figure 2: Discontinuities produced in the E, k parabola by Bragg reflection, at k = ± p/a, giving an energy gap AB.

For more on energy gaps, see 'Electronic States'

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