Empty lattice approximation: Difference between revisions

Revision as of 10:23, 25 March 2014

Revision as of 11:47, 24 April 2014

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[[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]]. 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= 1953-19761953–1976 |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]].

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[[Image:Free-electron DOS.svg|thumb|300px|right|Figure 3: Free-electron DOS in 3-dimensional k-space]]

Though the lattice cells are not spherically symmetric, the dispersion relation still has spherical symmetry from the point of view of a fixed central point in a reciprocal lattice cell if the dispersion relation is extended outside the central Brillouin zone. The [[density of states#Parabolic_dispersionParabolic dispersion|density of states]] in a three-dimensional lattice will be the same as in the case of the absence of a lattice. For the three-dimensional case the density of states <math>D_3\left(E\right)</math> is;

:<math>D_3\left(E\right) = 2 \pi \sqrt{\frac{E-E_0}{c_k^3}} \ .</math>

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==The crystal structures of metals==

Apart from a few exotic exceptions, [[metal]]s crystallize in three kinds of crystal structures: the BCC and FCC [[cubic crystal system|cubic crystal structures]] and the [[hexagonal crystal system|hexagonal]] close-packed [[close-packing of spheres#fcc_and_hcp_latticesfcc and hcp lattices|HCP]] crystal structure.

<gallery>

Image:Cubic-body-centered.svg|Body-centered cubic (I)