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{{Short description|Primitive cell in the reciprocal space lattice of crystals}} |
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[[Image:Brillouin zone.svg|thumb|First Brillouin zones of (a) [[square lattice]] and (b) [[hexagonal lattice]].]] |
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[[File:Brillouin zone.svg|thumb|The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) [[square lattice]] and (b) [[hexagonal lattice]].]] |
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In [[mathematics]] and [[solid state physics]], the first '''Brillouin zone''' is a uniquely defined [[primitive cell]] of the [[reciprocal lattice]]. It is found by the same method as for the [[Wigner–Seitz cell]] in the [[Bravais lattice]]. The importance of the Brillouin zone stems from the [[Bloch wave]] description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone. |
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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 cell]]s 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. |
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Taking surfaces at the same distance from one element of the lattice and its neighbours, the [[volume]] included is the first Brillouin zone. Another definition is as the set of points in ''k''-space that can be reached from the origin without crossing any [[Bragg's law|Bragg plane]]. Equivalently, this is the [[Voronoi cell]] around the origin of the reciprocal lattice. |
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The first Brillouin zone is the [[locus (mathematics)|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. |
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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 more rarely. 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 ''n'' − 1 Bragg planes.) |
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[[File:Phonon k 3k.gif|thumb|''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''.<ref>{{Cite web|url=http://solidstate.mines.edu/videonotes/VN_5_2.pdf|title=Topic 5-2: Nyquist Frequency and Group Velocity|website=Solid State Physics in a Nutshell|publisher=[[Colorado School of Mines]]}}</ref> See also {{Section link|Aliasing|Sampling sinusoidal functions}} for more on the equivalence of ''k''-vectors.]] |
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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. |
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[[File:Ibz.png|thumb|The Brillouin zone (purple) and the irreducible Brillouin zone (red) for a [[hexagonal lattice]].]] |
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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). |
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The concept of a Brillouin zone was developed by [[Léon Brillouin]] (1889–1969), a French physicist.<ref>{{cite journal | last=Brillouin | first=L. | title=Les électrons libres dans les métaux et le role des réflexions de Bragg |trans-title=Free electrons in metals and the role of Bragg reflections| journal=Journal de Physique et le Radium | publisher=EDP Sciences | volume=1 | issue=11 | year=1930 | issn=0368-3842 | doi=10.1051/jphysrad:01930001011037700 | pages=377–400| url=https://hal.science/jpa-00233038/file/ajp-jphysrad_1930_1_11_377_0.pdf |language=fr}}</ref> |
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The concept of a Brillouin zone was developed by [[Leon Brillouin]] (1889-1969), a French physicist. |
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Within the Brillouin zone, a ''constant-energy surface'' represents the [[locus (mathematics)|loci]] of all the <math>\vec{k}</math>-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. |
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==Critical points== |
==Critical points== |
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[[Image:Brillouin Zone (1st, FCC).svg|thumb|360px|First Brillouin zone of [[Cubic crystal system|FCC lattice]] showing symmetry labels for high symmetry lines and points]] |
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[[File:Brillouin Zone (1st, FCC).svg|thumb|360px|First Brillouin zone of [[Cubic crystal system|FCC lattice]], a [[truncated octahedron]], showing symmetry labels for high symmetry lines and points]] |
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Several points of high symmetry are of special interest – these are called critical points.<ref>{{cite book |first=Harald |last=Ibach |coauthors=Hans Lüth |title=Solid-State Physics, An Introduction to Principles of Materials Science |edition=Second |year=1996 |publisher=Springer-Verlag |isbn=3-540-58573-7}}</ref> |
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Several points of high symmetry are of special interest – these are called critical points.<ref>{{cite book |first1=Harald |last1=Ibach |first2=Hans |last2=Lüth |title=Solid-State Physics, An Introduction to Principles of Materials Science |edition=2nd |year=1996 |publisher=Springer-Verlag |isbn=978-3-540-58573-2}}</ref> |
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{| class="wikitable" |
{| class="wikitable" |
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!Symbol!!Description |
!Symbol!!Description |
||
|- |
|- |
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|Γ||Center of the Brillouin zone |
|Γ||Center of the Brillouin zone |
||
|- |
|- |
||
!colspan="2"|Simple cube |
!colspan="2"|Simple cube |
||
|- |
|- |
||
|M||Center of an edge |
|M||Center of an edge |
||
|- |
|- |
||
|R||Corner point |
|R||Corner point |
||
|- |
|- |
||
|X||Center of a face |
|X||Center of a face |
||
|- |
|- |
||
!colspan="2"|Face-centered cubic |
!colspan="2"|Face-centered cubic |
||
|- |
|- |
||
|K||Middle of an edge joining two hexagonal faces |
|K||Middle of an edge joining two hexagonal faces |
||
|- |
|- |
||
|L||Center of a hexagonal face |
|L||Center of a hexagonal face |
||
|- |
|- |
||
|U||Middle of an edge joining a hexagonal and a square face |
|U||Middle of an edge joining a hexagonal and a square face |
||
|- |
|- |
||
|W||Corner point |
|W||Corner point |
||
|- |
|- |
||
|X||Center of a square face |
|X||Center of a square face |
||
|- |
|- |
||
!colspan="2"|Body-centered cubic |
!colspan="2"|Body-centered cubic |
||
|- |
|- |
||
|H||Corner point joining four edges |
|H||Corner point joining four edges |
||
|- |
|- |
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|N||Center of a face |
|N||Center of a face |
||
|- |
|- |
||
|P||Corner point joining three edges |
|P||Corner point joining three edges |
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|- |
|- |
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!colspan="2"|Hexagonal |
!colspan="2"|Hexagonal |
||
|- |
|- |
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|A||Center of a hexagonal face |
|A||Center of a hexagonal face |
||
|- |
|- |
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|H||Corner point |
|H||Corner point |
||
|- |
|- |
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|K||Middle of an edge joining two rectangular faces |
|K||Middle of an edge joining two rectangular faces |
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|- |
|- |
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|L||Middle of an edge joining a hexagonal and a rectangular face |
|L||Middle of an edge joining a hexagonal and a rectangular face |
||
|- |
|- |
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|M||Center of a rectangular face |
|M||Center of a rectangular face |
||
|} |
|} |
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Other lattices have different types of high-symmetry points. They can be found in the illustrations below. |
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{| class="wikitable" |
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|+ Brillouin zone types<ref>{{cite journal | first1=Wahyu | last1=Setyawan | first2=Stefano | last2=Curtarolo | title=High-throughput electronic band structure calculations: Challenges and tools | journal=[[Computational Materials Science]] | volume=49 | issue=2 | pages=299–312 | year=2010 | doi =10.1016/j.commatsci.2010.05.010 | arxiv=1004.2974 |bibcode=2010arXiv1004.2974S| s2cid=119226326 }}</ref> |
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! Lattice system |
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! Bravais lattice |
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(Abbreviation) |
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|- |
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! [[Triclinic]] |
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! Primitive triclinic |
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(TRI) |
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| Triclinic Lattice type 1a (TRI1a) |
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[[File:Triclinic Lattice type 1a (Brillouin zone).png|225px]] |
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| Triclinic Lattice type 1b (TRI1b) |
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[[File:Triclinic Lattice type 1b (Brillouin zone).png|225px]] |
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| Triclinic Lattice type 2a (TRI2a) |
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[[File:Triclinic Lattice type 2a (Brillouin zone).png|225px]] |
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| Triclinic Lattice type 2b (TRI2b) |
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[[File:Triclinic Lattice type 2b (Brillouin zone).png|200px]] |
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|- |
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! rowspan="2" |[[Monoclinic]] |
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! Primitive monoclinic |
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(MCL) |
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| Monoclinic Lattice (MCL) |
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[[File:MCL.png|225px]] |
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|- |
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! Base-centered monoclinic |
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(MCLC) |
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| Base-centered monoclinic lattice type 1 (MCLC1) |
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[[File:MCLC1.png|225px]] |
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| Base-centered monoclinic lattice type 2 (MCLC2) |
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[[File:Base-Centered Monoclinic Lattice type 2 (Brillouin zone).png|225px]] |
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| Base-centered monoclinic lattice type 3 (MCLC3) |
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[[File:MCLC3.png|225px]] |
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| Base-centered monoclinic lattice type 4 (MCLC4) |
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[[File:MCLC4.png|225px]] |
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| Base-centered monoclinic lattice type 5 (MCLC5) |
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[[File:MCLC5.png|225px]] |
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|- |
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! rowspan="4" | [[Orthorhombic]] |
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! Primitive orthorhombic |
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(ORC) |
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| Simple orthorhombic lattice (ORC) |
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[[File:Simple Orthorhombic Lattice (Brillouin zone).png|225px]] |
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|- |
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! Base-centered orthorhombic |
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(ORCC) |
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|Base-centered orthorhombic lattice (ORCC) |
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[[File:Base-Centered Orthorhombic Lattice (Brillouin zone).png|225px]] |
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|- |
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! Body-centered orthorhombic |
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(ORCI) |
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| Body-centered orthorhombic lattice (ORCI) |
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[[File:Body-Centered Orthorhombic Lattice (Brillouin zone).png|225px]] |
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|- |
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! Face-centered orthorhombic |
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(ORCF) |
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| Face-centered orthorhombic lattice type 1 (ORCF1) |
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[[File:Face-Centered Orthorhombic Lattice type 1 (Brillouin zone).png|225px]] |
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| Face-centered orthorhombic lattice type 2 (ORCF2) |
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[[File:Face-Centered Orthorhombic Lattice type 2 (Brillouin zone).png|225px]] |
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| Face-centered orthorhombic lattice type 3 (ORCF3) |
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[[File:Face-Centered Orthorhombic Lattice type 3 (Brillouin zone).png|225px]] |
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|- |
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! rowspan="2" | [[Tetragonal]] |
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! Primitive tetragonal |
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(TET) |
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| Simple tetragonal lattice (TET) |
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[[File:Simple Tetragonal Lattice (Brillouin zone).png|225px]] |
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|- |
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! Body-centered Tetragonal |
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(BCT) |
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| Body-centered tetragonal lattice type 1 (BCT1) |
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[[File:Body-Centered Tetragonal Lattice type 1 (Brillouin zone).png|225px]] |
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| Body-centered tetragonal lattice type 2 (BCT2) |
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[[File:Body-Centered Tetragonal Lattice type 2 (Brillouin zone).png|225px]] |
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|- |
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! [[Trigonal|Rhombohedral]] |
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! Primitive rhombohederal |
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(RHL) |
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| Rhombohedral lattice type 1 (RHL1) |
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[[File:RHL1.png|225px]] |
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| Rhombohedral lattice type 2 (RHL2) |
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[[File:RHL2.png|225px]] |
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|- |
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! [[Hexagonal crystal system|Hexagonal]] |
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! Primitive hexagonal |
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(HEX) |
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| Hexagonal lattice (HEX) |
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[[File:HEX.png|225px]] |
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|- |
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! rowspan="3" | [[Cubic crystal system|Cubic]] |
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! Primitive cubic |
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(CUB) |
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| Simple cubic lattice (CUB) |
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[[File:Simple Cubic Lattice (Brillouin zone).png|225px]] |
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|- |
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! Body-centered cubic |
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(BCC) |
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| Body-centered cubic lattice (BCC) |
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[[File:Body-Centered Cubic Lattice (Brillouin zone).png|225px]] |
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|- |
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! Face-centered cubic |
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(FCC) |
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| Face-centered cubic lattice (FCC) |
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[[File:Face-Centered Cubic Lattice (Brillouin zone).png|225px]] |
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|} |
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==See also== |
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[[File:Brillouin-zone construction by 300keV electrons.jpg|thumb|300px|Brillouin-zone construction by [[selected area diffraction]], using 300 keV electrons.]] |
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==See also == |
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* [[Fundamental pair of periods]] |
* [[Fundamental pair of periods]] |
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* [[Fundamental domain]] |
** [[Fundamental domain]] |
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==References== |
==References== |
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<references /> |
<references /> |
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* {{cite book |first=Charles |last=Kittel |title=Introduction to Solid State Physics |publisher=Wiley |location=New York City |year=1996}} |
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==Bibliography== |
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* {{cite book |first=Neil W. |last=Ashcroft |coauthors=N. David Mermin |title=Solid State Physics |publisher=Harcourt |location=Orlando |year=1976}} |
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* {{cite book |first=Charles |last=Kittel |author-link=Charles Kittel |title=Introduction to Solid State Physics |publisher=Wiley |location=New York |year=1996 |isbn=978-0-471-14286-7 |title-link=Solid State Physics}} |
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* {{cite journal |first=Léon |last=Brillouin |authorlink=Léon Brillouin |url=http://gallica.bnf.fr/ark:/12148/bpt6k31445 |title=Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes |journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences |volume=191 |issue=292 |year=1930}} |
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* {{cite book |first1=Neil W. |last1=Ashcroft |author-link=Neil William Ashcroft | first2=N. David | last2=Mermin | author2-link=David Mermin | title=Solid State Physics |publisher=Harcourt |location=Orlando |year=1976 |isbn=978-0-03-049346-1|title-link=Solid State Physics }} |
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* {{cite journal |first=Léon |last=Brillouin |author-link=Léon Brillouin |url=http://gallica.bnf.fr/ark:/12148/bpt6k31445 |title=Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes |journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences |volume=191 |issue=292 |year=1930}} |
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==External links== |
==External links== |
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*[http://www2.sjsu.edu/faculty/watkins/brillouin.htm Brillouin Zone simple lattice diagrams by Thayer Watkins] |
*[http://www2.sjsu.edu/faculty/watkins/brillouin.htm Brillouin Zone simple lattice diagrams by Thayer Watkins] |
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*[http://phycomp.technion.ac.il/~nika/brillouin_zones.html Brillouin Zone 3d lattice diagrams by Technion.] |
*[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 }} |
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*[http://www.doitpoms.ac.uk/tlplib/brillouin_zones/index.php DoITPoMS Teaching and Learning Package – "Brillouin Zones"] |
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*[http://www.aflowlib.org Aflowlib.org consortium database (Duke University)] |
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*[http://materials.duke.edu/awrapper.html AFLOW Standardization of VASP/QUANTUM ESPRESSO input files (Duke University)] {{Webarchive|url=https://web.archive.org/web/20211126213457/http://materials.duke.edu/awrapper.html |date=2021-11-26 }} |
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{{DEFAULTSORT:Brillouin Zone}} |
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[[Category:Crystallography]] |
[[Category:Crystallography]] |
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[[Category:Electronic band structures]] |
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[[Category:Vibrational spectroscopy]] |
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[[de:Brillouin-Zone]] |
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[[es:Zona de Brillouin]] |
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[[fr:Zone de Brillouin]] |
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[[it:Zona di Brillouin]] |
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[[ms:Zon Brillouin]] |
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[[nl:Brillouinzone]] |
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[[ja:ブリュアンゾーン]] |
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[[pl:Strefa Brillouina]] |
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[[ru:Зона Бриллюэна]] |
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[[uk:Зона Брілюена]] |
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[[zh:布里渊区]] |
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Latest revision as of 06:31, 24 September 2024
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.
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
[edit].svg)
Several points of high symmetry are of special interest – these are called critical points.[3]
| Symbol | Description |
|---|---|
| Γ | Center of the Brillouin zone |
| Simple cube | |
| M | Center of an edge |
| R | Corner point |
| X | Center of a face |
| Face-centered cubic | |
| K | Middle of an edge joining two hexagonal faces |
| L | Center of a hexagonal face |
| U | Middle of an edge joining a hexagonal and a square face |
| W | Corner point |
| X | Center of a square face |
| Body-centered cubic | |
| H | Corner point joining four edges |
| N | Center of a face |
| P | Corner point joining three edges |
| Hexagonal | |
| A | Center of a hexagonal face |
| H | Corner point |
| K | Middle of an edge joining two rectangular faces |
| L | Middle of an edge joining a hexagonal and a rectangular face |
| M | Center of a rectangular face |
Other lattices have different types of high-symmetry points. They can be found in the illustrations below.
| Lattice system | Bravais lattice
(Abbreviation) | ||||
|---|---|---|---|---|---|
| Triclinic | Primitive triclinic
(TRI) |
Triclinic Lattice type 1a (TRI1a)
|
Triclinic Lattice type 1b (TRI1b)
|
Triclinic Lattice type 2a (TRI2a)
|
Triclinic Lattice type 2b (TRI2b)
|
| Monoclinic | Primitive monoclinic
(MCL) |
Monoclinic Lattice (MCL)
| |||
| Base-centered monoclinic
(MCLC) |
Base-centered monoclinic lattice type 1 (MCLC1)
|
Base-centered monoclinic lattice type 2 (MCLC2)
|
Base-centered monoclinic lattice type 3 (MCLC3)
|
Base-centered monoclinic lattice type 4 (MCLC4)
|
Base-centered monoclinic lattice type 5 (MCLC5)
|
| Orthorhombic | Primitive orthorhombic
(ORC) |
Simple orthorhombic lattice (ORC)
| |||
| Base-centered orthorhombic
(ORCC) |
Base-centered orthorhombic lattice (ORCC)
| ||||
| Body-centered orthorhombic
(ORCI) |
Body-centered orthorhombic lattice (ORCI)
| ||||
| Face-centered orthorhombic
(ORCF) |
Face-centered orthorhombic lattice type 1 (ORCF1)
|
Face-centered orthorhombic lattice type 2 (ORCF2)
|
Face-centered orthorhombic lattice type 3 (ORCF3)
| ||
| Tetragonal | Primitive tetragonal
(TET) |
Simple tetragonal lattice (TET)
| |||
| Body-centered Tetragonal
(BCT) |
Body-centered tetragonal lattice type 1 (BCT1)
|
Body-centered tetragonal lattice type 2 (BCT2)
| |||
| Rhombohedral | Primitive rhombohederal
(RHL) |
Rhombohedral lattice type 1 (RHL1)
|
Rhombohedral lattice type 2 (RHL2)
| ||
| Hexagonal | Primitive hexagonal
(HEX) |
Hexagonal lattice (HEX)
| |||
| Cubic | Primitive cubic
(CUB) |
Simple cubic lattice (CUB)
| |||
| Body-centered cubic
(BCC) |
Body-centered cubic lattice (BCC)
| ||||
| Face-centered cubic
(FCC) |
Face-centered cubic lattice (FCC)
|
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See also
[edit]
References
[edit]- ^ "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] (PDF). Journal de Physique et le Radium (in French). 1 (11). EDP Sciences: 377–400. doi:10.1051/jphysrad:01930001011037700. ISSN 0368-3842.
- ^ 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.
- ^ 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
[edit]- Kittel, Charles (1996). Introduction to Solid State Physics. New York: Wiley. ISBN 978-0-471-14286-7.
- Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Orlando: Harcourt. ISBN 978-0-03-049346-1.
- Brillouin, Léon (1930). "Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. 191 (292).
External links
[edit]- Brillouin Zone simple lattice diagrams by Thayer Watkins
- Brillouin Zone 3d lattice diagrams by Technion. Archived 2006-12-05 at the Wayback Machine
- DoITPoMS Teaching and Learning Package – "Brillouin Zones"
- Aflowlib.org consortium database (Duke University)
- AFLOW Standardization of VASP/QUANTUM ESPRESSO input files (Duke University) Archived 2021-11-26 at the Wayback Machine