Empty lattice approximation: Difference between revisions

Revision as of 19:12, 6 September 2012

Revision as of 22:24, 31 March 2013

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==Scattering and periodicity==

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

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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 <math>E_n(\bold{k})</math> belong. Electrons with wave vectors outside the first [[Brillouin zone]] are scattered by the lattice and mapped back into the first Brillouin zone by a so -called [[Umklapp scattering|Umklapp process]].

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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In a one-dimensional lattice the number of reciprocal lattice vectors <math>\bold{G}_n</math> that determine the bands in an energy interval is limited to two when the energy rises. In two and three dimensional lattices the number of reciprocal lattice vectors that determine the free electron bands <math>E_n(\bold{k})</math> increases more rapidly when the length of the wave vector increases and the energy rises. This is because the number of reciprocal lattice vectors <math>\bold{G}_n</math> that lie in an interval <math>[\bold{k},\bold{k} + d\bold{k}]</math> increases. The [[density of states]] in an energy interval <math>[E,E + dE]</math> depends on the number of states in an interval <math>[\bold{k},\bold{k} + d\bold{k}]</math> in reciprocal space and the slope of the dispersion relation <math>E_n(\bold{k})</math>.

[[Image:Free-electron_DOSelectron 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_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;

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==Second, third and higher Brillouin zones==

[[Image:Brillouin_Zone_Brillouin Zone (1st,_FCC FCC).svg|thumb|300px|right|FCC Brillouin zone]]

"Free electrons" that move through the lattice of a solid with wave vectors <math>\bold{k}</math> far outside the first Brillouin zone are still reflected back into the first Brillouin zone. See the [[#External links|external links]] section for sites with examples and figures.

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==The nearly free electron model==

{{main|Nearly free electron model}}

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==References==

{{reflist}}