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 [[geometricgeometry 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]].