Brillouin_zone

Brillouin zone

Brillouin zone

Primitive cell in the reciprocal space lattice of crystals

In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.

The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) square lattice and (b) hexagonal lattice.

The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner–Seitz cell). Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane. Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice.

k-vectors exceeding the first Brillouin zone (red) do not carry any more information than their counterparts (black) in the first Brillouin zone. k at the Brillouin zone edge is the spatial Nyquist frequency of waves in the lattice, because it corresponds to a half-wavelength equal to the inter-atomic lattice spacing a.[1] See also Aliasing § Sampling sinusoidal functions for more on the equivalence of k-vectors.
The Brillouin zone (purple) and the irreducible Brillouin zone (red) for a hexagonal lattice.

There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used less frequently. As a result, the first Brillouin zone is often called simply the Brillouin zone. In general, the n-th Brillouin zone consists of the set of points that can be reached from the origin by crossing exactly n − 1 distinct Bragg planes. A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice (point group of the crystal).

The concept of a Brillouin zone was developed by Léon Brillouin (1889–1969), a French physicist.[2]

Within the Brillouin zone, a constant-energy surface represents the loci of all the -points (that is, all the electron momentum values) that have the same energy. Fermi surface is a special constant-energy surface that separates the unfilled orbitals from the filled ones at zero kelvin.

Critical points

First Brillouin zone of FCC lattice, a truncated octahedron, showing symmetry labels for high symmetry lines and points

Several points of high symmetry are of special interest – these are called critical points.[3]

More information Symbol, Description ...

Other lattices have different types of high-symmetry points. They can be found in the illustrations below.

More information Lattice system, Bravais lattice (Abbreviation) ...

See also

Brillouin-zone construction by selected area diffraction, using 300 keV electrons.

References

  1. [1]
    "Topic 5-2: Nyquist Frequency and Group Velocity" (PDF). Solid State Physics in a Nutshell. Colorado School of Mines.
  2. [2]
    Brillouin, L. (1930). "Les électrons libres dans les métaux et le role des réflexions de Bragg" [Free electrons in metals and the role of Bragg reflections]. Journal de Physique et le Radium (in French). 1 (11). EDP Sciences: 377–400. doi:10.1051/jphysrad:01930001011037700. ISSN 0368-3842.
  3. [3]
    Ibach, Harald; Lüth, Hans (1996). Solid-State Physics, An Introduction to Principles of Materials Science (2nd ed.). Springer-Verlag. ISBN 978-3-540-58573-2.
  4. [4]
    Setyawan, Wahyu; Curtarolo, Stefano (2010). "High-throughput electronic band structure calculations: Challenges and tools". Computational Materials Science. 49 (2): 299–312. arXiv:1004.2974. Bibcode:2010arXiv1004.2974S. doi:10.1016/j.commatsci.2010.05.010. S2CID 119226326.

Bibliography
Share this article:

This article uses material from the Wikipedia article Brillouin_zone, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.