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The periodic potential of the lattice in this free electron model must be weak because otherwise the electrons wouldn't be free. The strength of the scattering mainly depends on the geometry and topology of the system. Topologically defined parameters, like [[Scattering cross-section|scattering]] [[Cross section (physics)|cross sections]], depend on the magnitude of the potential and the size of the [[potential well]]. For 1-, 2- and 3-dimensional spaces potential wells do always scatter waves, no matter how small their potentials are, what their signs are or how limited their sizes are. For a particle in a one-dimensional lattice, like the [[Kronig-Penney model]], it is possible to calculate the band structure analytically by substituting the values for the potential, the lattice spacing and the size of potential well.<ref name=Kittel>
{{cite book |author=C. Kittel |title=Introduction to Solid State Physics |year= 1953–1976 |publisher=Wiley & Sons |isbn=0-471-49024-5 }}
</ref> For two and three-dimensional problems it is more difficult to calculate a band structure based on a similar model with a few parameters accurately. Nevertheless, the properties of the band structure can easily be approximated in most regions by [[Perturbation theory (quantum mechanics)|perturbation methods]].
In theory the lattice is infinitely large, so a weak periodic scattering potential will eventually be strong enough to reflect the wave. The scattering process results in the well known [[Bragg's law|Bragg reflections]] of electrons by the periodic potential of the [[crystal structure]]. This is the origin of the periodicity of the dispersion relation and the division of [[Reciprocal lattice|k-space]] in Brillouin zones. The periodic energy dispersion relation is expressed
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:<math>E_n(\bold{k}) = \frac{\hbar^2 (\bold{k} + \bold{G}_n)^2}{2m}</math>
The <math>\bold{G}_n</math> are the [[reciprocal lattice]] vectors to which the bands{{
The figure on the right shows the dispersion relation for three periods in reciprocal space of a one-dimensional lattice with lattice cells of length ''a''.
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:<math>U_{\bold{G}} = \frac{4 \pi Z e}{q^2 + \bold{G}^2}</math>
When the values of the off-diagonal elements <math>U_{\bold{G}}</math> between the reciprocal lattice vectors in the Hamiltonian almost go to zero. As a result, the magnitude of the band gap <math>2|U_{\bold{G}}|</math> collapses and the empty lattice approximation is obtained.
==The electron bands of common metal crystals==
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