13.6: Scattering of free electrons in metals

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If we describe an electron bound in a solid or liquid as a free electron, we see that scattering occurs only for those electrons with wavenumbers close to periodic repeats. For simple metals (Li, Na etc) the highest occupied free-electron level has wavelength greater than any crystal spacing, so it only sees the average of the ionic potential.

To first order, only electrons with the periodicity of the lattice are scattered. To second order in perturbation theory, the potential can mix states:

\[\Delta E = \frac{|V_{ij} |^2}{(E_j − E_i)} ; \quad V_{ij} = \int e^{\pm i(a/2+\delta )x} V \cos axe^{\pm i(a/2−\delta )x} \neq 0 \nonumber\]

which gives significant energy shifts for states \pm \delta from the lattice periodicity \((E_j−E_i = −\hbar^2 a\delta /m)\). Thus free-electron levels with \(k \approx a/2\) are split by periodic potentials giving a bandgap in the density of allowed states. At first glance, this may seem to be totally different physics from the LCAO band gaps we saw earlier. In fact, its simply another manifestation of using two different mathematical basis sets to describe the same physical phenomenon.