Empty lattice approximation: Difference between revisions

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{{short description|A theoreticalTheoretical electronic band structure model in which the potential is periodic and weak}}

{{Electronic structure methods}}

The '''empty lattice approximation''' is a theoretical [[electronic band structure]] model in which the potential is ''periodic'' and ''weak'' (close to constant). One may also consider an empty{{clarify|is there actually some lattice that is "empty"?|date=November 2014}} irregular lattice, in which the potential is not even periodic.<ref>Physics Lecture Notes. P.Dirac, Feynman, R.,1968. Internet, Amazon,25.03.2014.</ref> The empty lattice approximation describes a number of properties of energy dispersion relations of non-interacting [[Free electron model|free electrons]] that move through a [[crystal structure|crystal lattice]]. The energy of the electrons in the "empty lattice" is the same as the energy of free electrons. The model is useful because it clearly illustrates a number of the sometimes very complex features of energy dispersion relations in solids which are fundamental to all electronic band structures.

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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;

:<math>D_3\left(E\right) = 2 \pi \sqrt{\frac{E-E_0E _0}{c_k^3}} \ .</math>

In three-dimensional space the Brillouin zone boundaries are planes. The dispersion relations show conics of the free-electron energy dispersion parabolas for all possible reciprocal lattice vectors. This results in a very complicated set intersecting of curves when the dispersion relations are calculated because there is a large number of possible angles between evaluation trajectories, first and higher order Brillouin zone boundaries and dispersion parabola intersection cones.

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==External links==

*[http://www2.sjsu.edu/faculty/watkins/brillouin.htm Brillouin Zone simple lattice diagrams by Thayer Watkins] {{Webarchive|url=https://web.archive.org/web/20060914142130/http://www2.sjsu.edu/faculty/watkins/brillouin.htm |date=2006-09-14 }}

{{commons category|Dispersion relations of electrons}}

*[http://phycomp.technion.ac.il/~nika/brillouin_zones.html Brillouin Zone 3d lattice diagrams by Technion.] {{Webarchive|url=https://web.archive.org/web/20061205220050/http://phycomp.technion.ac.il/~nika/brillouin_zones.html |date=2006-12-05 }}

*[http://www2.sjsu.edu/faculty/watkins/brillouin.htm Brillouin Zone simple lattice diagrams by Thayer Watkins]

*[http://phycomp.technion.ac.il/~nika/brillouin_zones.html Brillouin Zone 3d lattice diagrams by Technion.]

*[http://www.doitpoms.ac.uk/tlplib/brillouin_zones/index.php DoITPoMS Teaching and Learning Package- "Brillouin Zones"]

[[Category:ConceptsQuantum in physicsmodels]]

[[Category:Electronic band structures]]