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| In [[geometry]] and [[crystallography]], a '''Bravais lattice''', studied by {{harvs|txt|first=Auguste |last=Bravais|year=1850|authorlink=Auguste Bravais}},<ref>{{cite journal |last = Aroyo|first = Mois I.|coauthors = Ulrich Müller and Hans Wondratschek|title = Historical Introduction|journal = International Tables for Crystallography|volume = A1|issue = 1.1|pages = 2–5|publisher = Springer|year = 2006|url = http://it.iucr.org/A1a/ch1o1v0001/sec1o1o1/|doi = 10.1107/97809553602060000537|accessdate =2008-04-21 |ref = harv}}</ref> is an infinite array of discrete points generated by a set of discrete [[translation (geometry)|translation]] operations described by:
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| :<math>\mathbf{R} = n_{1}\mathbf{a}_{1} + n_{2}\mathbf{a}_{2} + n_{3}\mathbf{a}_{3}</math>
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| where ''n<sub>i</sub>'' are any integers and '''a<sub>i</sub>''' are known as the primitive vectors which lie in different directions and span the lattice. This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector '''R''', the lattice looks exactly the same.
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| A crystal is made up of a periodic arrangement of one or more atoms (the ''basis'') repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, namely those separated by the translation of one unit cell (the ''motive'').
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| Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 [[space groups]].
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| ==Bravais lattices in at most 2 dimensions==
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| In each of 0-dimensional and 1-dimensional space there is just one type of Bravais lattice.
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| In two dimensions, there are five Bravais lattices. They are oblique, rectangular, centered rectangular (rhombic), [[hexagon]]al, and square.<ref>{{cite book |last=Kittel |first=Charles |title=Introduction to Solid State Physics |origyear=1953 |url= http://www.wiley.com/WileyCDA/WileyTitle/productCd-047141526X.html |accessdate=2008-04-21 |edition=Seventh |year=1996 |publisher=John Wiley & Sons |location=New York |isbn=0-471-11181-3 |pages=10 |chapter=Chapter 1}}</ref>
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| [[Image:2d-bravais.svg|650px|center|thumb|The five fundamental two-dimensional Bravais lattices: 1 oblique, 2 rectangular, 3 centered rectangular (rhombic), 4 hexagonal, and 5 square]]
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| ==Bravais lattices in 3 dimensions==
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| The 14 Bravais lattices in 3 dimensions are obtained by coupling one of the 7 [[lattice system]]s (or axial systems) with one of the lattice centerings.
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| Each Bravais lattice refers to a distinct lattice type.
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| The lattice centerings are:
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| *Primitive (P): lattice points on the cell corners only.
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| *Body (I): one additional lattice point at the center of the cell.
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| *Face (F): one additional lattice point at the center of each of the faces of the cell.
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| *Base (A, B or C): one additional lattice point at the center of each of one pair of the cell faces.
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| Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. There are in total 7 × 6 = 42 combinations, but it can be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below.
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| {| class=wikitable
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| !The 7 lattice systems
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| !colspan=4| The 14 Bravais lattices
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| |- align=center
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| |rowspan=2| [[Triclinic]]|| P
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| |-
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| || [[Image:Triclinic.svg|80px|Triclinic]]
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| |- align=center
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| |rowspan=2| [[Monoclinic]] || P|| C
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| |-
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| || [[Image:Monoclinic.svg|80px|Monoclinic, simple]]
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| || [[Image:Monoclinic-base-centered.svg|80px|Monoclinic, centred]]
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| |- align=center
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| |rowspan=2| [[Orthorhombic]] ||P||C||I||F
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| |-
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| || [[Image:Orthorhombic.svg|80px|Orthohombic, simple]]
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| || [[Image:Orthorhombic-base-centered.svg|80px|Orthohombic, base-centred]]
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| || [[Image:Orthorhombic-body-centered.svg|80px|Orthohombic, body-centred]]
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| || [[Image:Orthorhombic-face-centered.svg|80px|Orthohombic, face-centred]]
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| |- align=center
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| |rowspan=2| [[Tetragonal]] || P|| I
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| |-
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| || [[Image:Tetragonal.svg|80px|Tetragonal, simple]]
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| || [[Image:Tetragonal-body-centered.svg|80px|Tetragonal, body-centred]]
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| |- align=center
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| |rowspan=2| [[rhombohedral lattice system|Rhombohedral]]|| P
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| |-
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| | [[Image:Rhombohedral.svg|80px|Rhombohedral]]
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| |-align=center
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| |rowspan=2| [[Hexagonal lattice system|Hexagonal]] ||P
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| |-
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| | [[Image:Hexagonal lattice.svg|80px|Hexagonal]]
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| |- align=center
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| |rowspan=2| [[Cubic (crystal system)|Cubic]]|| P (pcc)|| I (bcc)|| F (fcc)
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| |-
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| || [[Image:Cubic.svg|80px|Cubic, simple]]
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| | [[Image:Cubic-body-centered.svg|80px|Cubic, body-centred]]
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| | [[Image:Cubic-face-centered.svg|80px|Cubic, face-centred]]
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| |}
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| {{-}}
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| The volume of the unit cell can be calculated by evaluating '''a · b × c''' where '''a''', '''b''', and '''c''' are the [[lattice vector]]s. The volumes of the Bravais lattices are given below:
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| {| class=wikitable
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| !Lattice system
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| !colspan=4| Volume
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| | [[Triclinic]]|| <math>abc \sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha \cos\beta \cos\gamma}</math>
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| |-
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| | [[Monoclinic]]|| <math>abc ~ \sin\beta</math>
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| | [[Orthorhombic]] || <math> abc </math>
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| |-
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| | [[Tetragonal]] || <math> a^2c </math>
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| |-
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| | [[Rhombohedral lattice system|Rhombohedral]] || <math> a^3 \sqrt{1 - 3\cos^2\alpha + 2\cos^3\alpha} </math>
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| | [[Hexagonal lattice system|Hexagonal]]|| <math>\frac{\sqrt{3\,}\, a^2c}{2}</math>
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| |-
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| | [[Cubic crystal system|Cubic]] || <math> a^3</math>
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| |}
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| {{-}}
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| Centred Unit Cells :
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| {| class="wikitable"
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| ! Lattice System || Possible Variations || Axial Distances (edge lengths) || Axial Angles || Examples
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| | Cubic || Primitive, Body-centred, Face-centred || a = b = c || α = β = γ = 90°|| [[NaCl]], [[Zinc Blende]], [[Copper|Cu]]
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| |-
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| | Tetragonal || Primitive, Body-centred || a = b ≠ c || α = β = γ = 90°|| [[White tin]], [[SnO2|SnO<sub>2</sub>]], [[TiO2|TiO]]<sub>2</sub>, [[CaSO4|CaSO]]<sub>4</sub>
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| | Orthorhombic || Primitive, Body-centred, Face-centred, Base-centred || a ≠ b ≠ c || α = β = γ = 90° || [[Allotropes of sulfur|Rhombic sulphur]], [[KNO3|KNO]]<sub>3</sub>, [[Barium sulfate|BaSO]]<sub>4</sub>
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| | Hexagonal || Primitive || a = b ≠ c || α = β = 90°, γ = 120° || [[Graphite]], [[ZnO]], [[CdS]]
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| | Rhombohedral || Primitive || a = b = c || α = β = γ ≠ 90° || [[Calcite]] (CaCO<sub>3</sub>, [[Cinnabar]] (HgS)
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| |-
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| | Monoclinic || Primitive, Base-centred || a ≠ b ≠ c || α = γ = 90°, β ≠ 90° || [[Allotropes of sulfur|Monoclinic sulphur]], Na<sub>2</sub>SO<sub>4</sub>.10H<sub>2</sub>O
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| |-
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| | Triclinic || Primitive || a ≠ b ≠ c || α ≠ β ≠ γ ≠ 90° || [[K2Cr2O7|K<sub>2</sub>Cr<sub>2</sub>O<sub>7</sub>]], CuSO<sub>4</sub>.5H<sub>2</sub>O, [[H3Bo3|H<sub>3</sub>BO<sub>3</sub>]]
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| |}
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| ==Bravais lattices in 4 dimensions==
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| In four dimensions, there are 64 Bravais lattices. Of these, 23 are primitive and 41 are centered. Ten Bravais lattices split into [[enantiomorphic]] pairs.<ref>{{Citation | last1=Brown | first1=Harold | last2=Bülow | first2=Rolf | last3=Neubüser | first3=Joachim | last4=Wondratschek | first4=Hans | last5=Zassenhaus | first5=Hans | author5-link=Hans Zassenhaus | title=Crystallographic groups of four-dimensional space | publisher=Wiley-Interscience [John Wiley & Sons] | location=New York | isbn=978-0-471-03095-9 | mr=0484179 | year=1978}}</ref>
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| ==See also==
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| {{colbegin|2}}
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| *[[Translational symmetry]]
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| *[[Lattice (group)]]
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| *[[Crystal system#Classification of lattices|classification of lattices]]
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| *[[Miller Index]]
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| {{colend}}
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| ==References==
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| {{reflist|1}}
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| ==Further reading==
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| *{{Cite journal|last=Bravais|first= A. |year=1850|title= Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans l'espace|journal= J. Ecole Polytech.|volume= 19|pages= 1–128|ref=harv|postscript=<!--None-->}} (English: Memoir 1, Crystallographic Society of America, 1949.)
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| *{{Cite book|editor1-last=Hahn|editor1-first=Theo|title=International Tables for Crystallography, Volume A: Space Group Symmetry|url=http://it.iucr.org/A/|publisher=[[Springer-Verlag]]|location=Berlin, New York|edition=5th|isbn=978-0-7923-6590-7|doi=10.1107/97809553602060000100|year=2002|volume=A|ref=harv|postscript=<!--None-->}}
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| ==External links==
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| *{{Cite web|url=http://www.haverford.edu/physics-astro/songs/bravais.htm |first=Walter Fox |last=Smith
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| |title=The Bravais Lattices Song|year=2002|ref=harv|postscript=<!--None-->}}
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| [[Category:Crystallography]]
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| [[Category:Condensed matter physics]]
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| [[Category:Lattice points]]
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| [[ja:ブラベー格子]]
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Hello, dear friend! My name is Harvey. I smile that I can unite to the entire world. I live in Germany, in the south region. I dream to check out the different countries, to obtain familiarized with fascinating people.
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