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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
71:
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
146:, 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
215:
903:
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].
106:-vectors exceeding the first Brillouin zone (red) do not carry any more information than their counterparts (black) in the first Brillouin zone.
1107:
1063:
1035:
947:
963:
Setyawan, Wahyu; Curtarolo, Stefano (2010). "High-throughput electronic band structure calculations: Challenges and tools".
1153:
76:, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.
1158:
1129:
965:
221:
is a special constant-energy surface that separates the unfilled orbitals from the filled ones at zero kelvin.
114:
of waves in the lattice, because it corresponds to a half-wavelength equal to the inter-atomic lattice spacing
658:
648:
638:
709:
699:
531:
416:
Other lattices have different types of high-symmetry points. They can be found in the illustrations below.
852:
158:-th Brillouin zone consists of the set of points that can be reached from the origin by crossing exactly
620:
602:
844:
1148:
889:
751:
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119:
191:
824:
806:
584:
1116:
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480:
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460:
450:
64:
245:
Several points of high symmetry are of special interest – these are called critical points.
1104:
984:
774:
238:
234:
8:
1053:
1025:
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185:
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40:
1078:"Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes"
988:
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1000:
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857:
73:
68:
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135:
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56:
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1111:
60:
217:-points (that is, all the electron momentum values) that have the same energy.
1021:
52:
24:
1077:
23:
The reciprocal lattices (dots) and corresponding first
Brillouin zones of (a)
1142:
924:
218:
162: − 1 distinct Bragg planes. A related concept is that of the
1127:
AFLOW Standardization of VASP/QUANTUM ESPRESSO input files (Duke
University)
229:
166:, which is the first Brillouin zone reduced by all of the symmetries in the
1049:
570:
134:
The
Brillouin zone (purple) and the irreducible Brillouin zone (red) for a
93:
1126:
167:
89:
36:
940:
Solid-State
Physics, An Introduction to Principles of Materials Science
667:
489:
436:
1082:
Comptes Rendus
Hebdomadaires des Séances de l'Académie des Sciences
718:
979:
19:
88:-space that can be reached from the origin without crossing any
99:
402:
Middle of an edge joining a hexagonal and a rectangular face
241:, showing symmetry labels for high symmetry lines and points
1121:
1117:
DoITPoMS Teaching and
Learning Package – "Brillouin Zones"
16:
Primitive cell in the reciprocal space lattice of crystals
1100:
Brillouin Zone simple lattice diagrams by Thayer
Watkins
561:
551:
541:
521:
320:
Middle of an edge joining a hexagonal and a square face
742:
732:
765:
503:
130:
194:
1122:Aflowlib.org consortium database (Duke University)
655:Face-centered orthorhombic lattice type 3 (ORCF3)
645:Face-centered orthorhombic lattice type 2 (ORCF2)
635:Face-centered orthorhombic lattice type 1 (ORCF1)
209:
962:
879:"Topic 5-2: Nyquist Frequency and Group Velocity"
173:The concept of a Brillouin zone was developed by
1140:
558:Base-centered monoclinic lattice type 5 (MCLC5)
548:Base-centered monoclinic lattice type 4 (MCLC4)
538:Base-centered monoclinic lattice type 3 (MCLC3)
528:Base-centered monoclinic lattice type 2 (MCLC2)
518:Base-centered monoclinic lattice type 1 (MCLC1)
394:Middle of an edge joining two rectangular faces
1105:Brillouin Zone 3d lattice diagrams by Technion.
706:Body-centered tetragonal lattice type 2 (BCT2)
696:Body-centered tetragonal lattice type 1 (BCT1)
1044:
304:Middle of an edge joining two hexagonal faces
170:of the lattice (point group of the crystal).
120:Aliasing § Sampling sinusoidal functions
96:around the origin of the reciprocal lattice.
617:Body-centered orthorhombic lattice (ORCI)
599:Base-centered orthorhombic lattice (ORCC)
150:Brillouin zone is often called simply the
110:at the Brillouin zone edge is the spatial
1072:
978:
937:
902:
838:
228:
129:
98:
18:
1141:
1020:
1027:Introduction to Solid State Physics
739:Rhombohedral lattice type 2 (RHL2)
729:Rhombohedral lattice type 1 (RHL1)
13:
938:Ibach, Harald; LĂĽth, Hans (1996).
917:10.1051/jphysrad:01930001011037700
821:Face-centered cubic lattice (FCC)
803:Body-centered cubic lattice (BCC)
581:Simple orthorhombic lattice (ORC)
477:Triclinic Lattice type 2b (TRI2b)
467:Triclinic Lattice type 2a (TRI2a)
457:Triclinic Lattice type 1b (TRI1b)
447:Triclinic Lattice type 1a (TRI1a)
224:
14:
1170:
1093:
942:(2nd ed.). Springer-Verlag.
886:Solid State Physics in a Nutshell
365:Corner point joining three edges
177:(1889–1969), a French physicist.
905:Journal de Physique et le Radium
823:
805:
787:
764:
741:
731:
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698:
680:
678:Simple tetragonal lattice (TET)
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619:
601:
583:
560:
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540:
530:
520:
502:
479:
469:
459:
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349:Corner point joining four edges
79:The first Brillouin zone is the
1014:
997:10.1016/j.commatsci.2010.05.010
966:Computational Materials Science
847:, using 300 keV electrons.
843:Brillouin-zone construction by
122:for more on the equivalence of
956:
931:
896:
871:
201:
142:There are also second, third,
1:
911:(11). EDP Sciences: 377–400.
865:
410:Center of a rectangular face
262:Center of the Brillouin zone
180:Within the Brillouin zone, a
92:. Equivalently, this is the
7:
853:Fundamental pair of periods
834:
785:Simple cubic lattice (CUB)
629:Face-centered orthorhombic
611:Body-centered orthorhombic
593:Base-centered orthorhombic
378:Center of a hexagonal face
312:Center of a hexagonal face
10:
1175:
1154:Electronic band structures
210:{\displaystyle {\vec {k}}}
164:irreducible Brillouin zone
845:selected area diffraction
773:
690:Body-centered Tetragonal
666:
569:
512:Base-centered monoclinic
500:Monoclinic Lattice (MCL)
488:
369:
340:
295:
266:
67:in the real lattice, the
1159:Vibrational spectroscopy
890:Colorado School of Mines
762:Hexagonal lattice (HEX)
723:Primitive rhombohederal
336:Center of a square face
233:First Brillouin zone of
51:) is a uniquely defined
575:Primitive orthorhombic
182:constant-energy surface
59:. In the same way the
848:
242:
211:
139:
127:
32:
1058:. Orlando: Harcourt.
842:
672:Primitive tetragonal
494:Primitive monoclinic
420:Brillouin zone types
232:
212:
133:
102:
22:
815:Face-centered cubic
797:Body-centered cubic
756:Primitive hexagonal
441:Primitive triclinic
341:Body-centered cubic
296:Face-centered cubic
239:truncated octahedron
192:
1055:Solid State Physics
1030:. New York: Wiley.
989:2010arXiv1004.2974S
421:
154:. In general, the
63:is divided up into
41:solid state physics
1132:2021-11-26 at the
1110:2006-12-05 at the
858:Fundamental domain
849:
419:
275:Center of an edge
243:
207:
140:
128:
69:reciprocal lattice
65:Wigner–Seitz cells
33:
1065:978-0-03-049346-1
1046:Ashcroft, Neil W.
1037:978-0-471-14286-7
949:978-3-540-58573-2
832:
831:
414:
413:
357:Center of a face
291:Center of a face
204:
136:hexagonal lattice
112:Nyquist frequency
29:hexagonal lattice
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1050:Mermin, N. David
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779:Primitive cubic
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428:Bravais lattice
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57:reciprocal space
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1149:Crystallography
1139:
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1134:Wayback Machine
1112:Wayback Machine
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1074:Brillouin, LĂ©on
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1022:Kittel, Charles
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430:(Abbreviation)
425:Lattice system
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225:Critical points
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184:represents the
74:Bloch's theorem
61:Bravais lattice
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1094:External links
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973:(2): 299–312.
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175:LĂ©on Brillouin
152:Brillouin zone
53:primitive cell
49:LĂ©on Brillouin
45:Brillouin zone
25:square lattice
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386:Corner point
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328:Corner point
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283:Corner point
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219:Fermi surface
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47:(named after
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1015:Bibliography
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719:Rhombohedral
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571:Orthorhombic
559:
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539:
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429:
415:
267:Simple cube
254:Description
244:
181:
179:
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159:
155:
151:
147:
143:
141:
123:
115:
107:
103:
94:Voronoi cell
85:
78:
44:
43:, the first
34:
235:FCC lattice
188:of all the
168:point group
118:. See also
90:Bragg plane
37:mathematics
1143:Categories
866:References
668:Tetragonal
490:Monoclinic
370:Hexagonal
1005:119226326
980:1004.2974
925:0368-3842
752:Hexagonal
437:Triclinic
202:→
126:-vectors.
1130:Archived
1108:Archived
1076:(1930).
1052:(1976).
1024:(1996).
835:See also
27:and (b)
985:Bibcode
631:(ORCF)
613:(ORCI)
595:(ORCC)
514:(MCLC)
1088:(292).
1062:
1034:
1003:
946:
923:
817:(FCC)
799:(BCC)
781:(CUB)
758:(HEX)
725:(RHL)
692:(BCT)
674:(TET)
577:(ORC)
496:(MCL)
443:(TRI)
251:Symbol
1001:S2CID
975:arXiv
882:(PDF)
775:Cubic
148:first
81:locus
1060:ISBN
1032:ISBN
944:ISBN
921:ISSN
237:, a
186:loci
144:etc.
39:and
1086:191
993:doi
913:doi
55:in
35:In
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909:1
892:.
407:M
399:L
391:K
383:H
375:A
362:P
354:N
346:H
333:X
325:W
317:U
309:L
301:K
288:X
280:R
272:M
259:Γ
199:k
160:n
156:n
138:.
124:k
116:a
108:k
104:k
86:k
31:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
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