Primitive cell in the reciprocal space lattice of crystals
The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) square lattice and (b) hexagonal lattice.
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 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.
^"Topic 5-2: Nyquist Frequency and Group Velocity" (PDF). Solid State Physics in a Nutshell. Colorado School of Mines.
^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.
In mathematics and solid state physics, the first Brillouinzone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space...
outside the first Brillouinzone. If a material is periodic, it has a Brillouinzone, and any point outside the first Brillouinzone can also be expressed...
Zones, a virtualization feature of the Solaris operating system Zone (convex polytope), in geometry/algorithmic, a kind of convex polytope Brillouin zone...
than the size of the first Brillouinzone, which results in a portion of the Fermi surface lying in the second (or higher) zones. As with the band structure...
{\displaystyle \mathbf {k} } far outside the first Brillouinzone are still reflected back into the first Brillouinzone. See the external links section for sites...
this process, one can infer the atomic arrangement of a crystal. The Brillouinzone is a Wigner–Seitz cell of the reciprocal lattice. Reciprocal space (also...
integration over the whole domain of the Brillouinzone can be reduced to a 48-th part of the whole Brillouinzone. As a crystal structure periodic table...
Wavevectors outside the Brillouinzone simply correspond to states that are physically identical to those states within the Brillouinzone. Special high symmetry...
such wavevectors defines the first Brillouinzone. Additional Brillouinzones may be defined as copies of the first zone, shifted by some reciprocal lattice...
the bulk electronic wave functions, which are integrated in over the Brillouinzone, in a similar way that the genus is calculated in geometric topology...
each characterized by a certain crystal momentum (k-vector) in the Brillouinzone. If the k-vectors are different, the material has an "indirect gap"...
cells in the crystal; The sum on k includes all the values of k in the Brillouinzone (or any other primitive cell of the reciprocal lattice) that are consistent...
conduction band is vacant. The two bands touch at the zone corners (the K point in the Brillouinzone), where there is a zero density of states but no band...
singularities occur are often referred to as critical points of the Brillouinzone. For three-dimensional crystals, they take the form of kinks (where...
are six locations in momentum space, the vertices of its hexagonal Brillouinzone, divided into two non-equivalent sets of three points. The two sets...
atoms arranged on a square lattice. Energy splitting occurs at the Brillouinzone edge for one-dimensional situations because of a weak periodic potential...
gap in each direction becomes wider and the second one is to make the Brillouinzone more similar to sphere. However, the former is limited by the available...
excited, because unless the sum of q2 and q3 points outside of the Brillouinzone the momentum is conserved and the process is normal scattering (N-process)...
photon energy, corresponding to the band gaps at critical points of the Brillouinzone. The electroreflectance effect can be used to get a clearer picture...
a function whose Fourier transform is the indicator function of the Brillouinzone of that lattice. For example, the sinc function for the hexagonal lattice...
lattice and the Brillouinzone often belong to a different space group than the crystal of the solid. High-symmetry points in the Brillouinzone belong to different...
_{A}^{\mu }} . Therefore, a fermion with momentum near the center of the Brillouinzone is mapped to one of its corners while one of the corner fermions comes...
{\displaystyle \mathbf {k} } is a wavevector in the reciprocal-space (Brillouinzone), and u n k ( r ) {\displaystyle u_{n\mathbf {k} }(\mathbf {r} )} is...
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