Empty lattice approximation: Difference between revisions

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==Scattering and periodicity==

[[Image:1D-Empty-Lattice-Approximation.svg|thumb|400px|Free electron bands in a one dimensional lattice]]

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 [[geometric 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]]. One thing is clear for currently knownFor 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 easy to substitutecalculate the band structure 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.

</ref>

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]]. The periodicity of the dispersion relation and the division of [[Reciprocal lattice|k-space]] in Brillouin zones is the result of this scattering process. The periodic energy dispersion relation is