In Sections 12.2 and 12.3 we discussed the classical and quantum models of
metals, making the simplifying assumption that the metal consists of a gas of
free electrons. Since good insulators have enormous resistivity compared to
metals (a factor of 1024 greater), their electronic configurations must be quite
different. In fact, most outer-shell electrons in an insulator are not free but are
involved in ionic or covalent bonds, as discussed earlier. Furthermore, in
order to more fully understand the electronic properties of solids, one must
consider the effect of the lattice ions.
In the general case, there are two approaches to the determination of electronic energies in a solid. One is to follow the behavior of energy levels of isolated atoms as they are brought closer and closer to form a solid. The other is
to show that energy bands arise when the Schrödinger equation is solved for
electrons subject to a periodic potential representing the lattice ions. We shall
follow the isolated-atom approach here first, since it is simpler and immediately leads to an explanation of the differences between conductors, insulators, and semiconductors. Later in this section we shall explicitly consider the
effect of a periodic potential on electron waves, to show how energy bands
arise from another viewpoint.