Editing Bravais lattice (section)

==In 3 dimensions==
[[File:diamond_lattice.stl|thumb|2&times;2&times;2 unit cells of a [[diamond cubic]] lattice]]
In three-dimensional space there are 14 Bravais lattices. These are obtained by combining one of the seven [[lattice system]]s with one of the centering types. The centering types identify the locations of the lattice points in the unit cell as follows:

*Primitive (P): lattice points on the cell corners only (sometimes called simple)
*Base-centered (S: A, B, or C): lattice points on the cell corners with one additional point at the center of each face of one pair of parallel faces of the cell (sometimes called end-centered)
*Body-centered (I): lattice points on the cell corners, with one additional point at the center of the cell
*Face-centered (F): lattice points on the cell corners, with one additional point at the center of each of the faces of the cell

Not all combinations of lattice systems and centering types are needed to describe all of the possible lattices, as 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.<ref name=Hahn>{{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>{{rp|744}} Below each diagram is the [[Pearson symbol]] for that Bravais lattice.

'''Note:''' In the unit cell diagrams in the following table all the lattice points on the cell boundary (corners and faces) are shown; however, not all of these lattice points technically belong to the given unit cell.  This can be seen by imagining moving the unit cell slightly in the negative direction of each axis while keeping the lattice points fixed. Roughly speaking, this can be thought of as moving the unit cell slightly left, slightly down, and slightly out of the screen. This shows that only one of the eight corner lattice points (specifically the front, left, bottom one) belongs to the given unit cell (the other seven lattice points belong to adjacent unit cells). In addition, only one of the two lattice points shown on the top and bottom face in the ''Base-centered'' column belongs to the given unit cell. Finally, only three of the six lattice points on the faces in the ''Face-centered'' column belong to the given unit cell.

{| class="wikitable skin-invert-image"
! rowspan=2| Crystal family
! rowspan=2| Lattice system
! rowspan=2| Point group <br />([[Schoenflies notation|Schönflies notation]])
! rowspan=2| Point group <br />([[Hermann–Mauguin_notation|Hermann-Mauguin notation]])
! colspan=4| 14 Bravais lattices
|-
! Primitive (P)
! Base-centered (S)
! Body-centered (I)
! Face-centered (F)
|- align=center
! colspan=2| [[Triclinic]] (a)
| C<sub>i</sub>
| <math>\bar{1}</math>
| [[File:Triclinic.svg|80px|Triclinic]]
aP
|
|
|
|- align=center
! colspan=2| [[Monoclinic]] (m)
| C<sub>2h</sub>
| <math>2/m</math>
| [[File:Monoclinic.svg|80px|Monoclinic, simple]]
mP
| [[File:Base-centered monoclinic.svg|80px|Monoclinic, centered]]
mS
|
|
|- align=center
! colspan=2| [[Orthorhombic]] (o)
| D<sub>2h</sub>
| <math>mmm</math>
| [[File:Orthorhombic.svg|80px|Orthorhombic, simple]]
oP
| [[File:Orthorhombic-base-centered.svg|80px|Orthorhombic, base-centered]]
oS
| [[File:Orthorhombic-body-centered.svg|80px|Orthorhombic, body-centered]]
oI
| [[File:Orthorhombic-face-centered.svg|80px|Orthorhombic, face-centered]]
oF
|- align=center
! colspan=2| [[Tetragonal]] (t)
| D<sub>4h</sub>
| <math>4/mmm</math>
| [[File:Tetragonal.svg|80px|Tetragonal, simple]]
tP
|
| [[File:Tetragonal-body-centered.svg|80px|Tetragonal, body-centered]]
tI
|
|- align=center
! rowspan=2| [[Hexagonal crystal family|Hexagonal]] (h)
! Rhombohedral
| D<sub>3d</sub>
| <math>\bar{3} m</math>
| [[File:Rhombohedral.svg|80px|Rhombohedral]]
hR
|
|
|
|- align=center
! Hexagonal
| D<sub>6h</sub>
| <math>6/mmm</math>
| [[File:Hexagonal latticeFRONT.svg|80px|Hexagonal]]
hP
|
|
|
|- align=center
! colspan=2| [[Cubic crystal system|Cubic]] (c)
| O<sub>h</sub>
| <math>m \bar{3} m</math>
| [[File:Cubic.svg|80px|Cubic, simple]]
cP
|
| [[File:Cubic-body-centered.svg|80px|Cubic, body-centered]]
cI
| [[File:Cubic-face-centered.svg|80px|Cubic, face-centered]]
cF
|}
{{Clear}}

The unit cells are specified according to six [[unit cell|lattice parameters]] which are the relative lengths of the cell edges (''a'', ''b'', ''c'') and the angles between them (''α'', ''β'', ''γ''), where ''α'' is the angle between ''b'' and ''c'', ''β'' is the angle between ''a'' and ''c'', and ''γ'' is the angle between ''a'' and ''b''. The volume of the unit cell can be calculated by evaluating the [[triple product]] {{nowrap|'''a''' · ('''b''' × '''c''')}}, where '''a''', '''b''', and '''c''' are the lattice vectors. The properties of the lattice systems are given below:

{| class="wikitable skin-invert-image"
! Crystal family
! Lattice system
! Volume
! Axial distances (edge lengths){{r|"Hahn"|p=758}}
! Axial angles<ref name="Hahn"/>
! Corresponding examples
|-
! colspan=2| [[Triclinic]]
| <math>abc \sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha \cos\beta \cos\gamma}</math>
|
|
| [[Potassium dichromate|K<sub>2</sub>Cr<sub>2</sub>O<sub>7</sub>]], [[Copper(II) sulfate|CuSO<sub>4</sub>·5H<sub>2</sub>O]], [[Boric acid|H<sub>3</sub>BO<sub>3</sub>]]
|-
! colspan=2| [[Monoclinic]]
| <math>abc \, \sin\beta</math>
| 
| ''α'' = ''γ'' = 90°
| [[Allotropes of sulphur|Monoclinic sulphur]], [[Sodium sulfate|Na<sub>2</sub>SO<sub>4</sub>·10H<sub>2</sub>O]], [[Lead(II) chromate|PbCrO<sub>3</sub>]]
|-
! colspan=2| [[Orthorhombic]]
| <math> abc </math>
| 
| ''α'' = ''β'' = ''γ'' = 90°
| [[Allotropes of sulfur|Rhombic sulphur]], [[Potassium nitrate|KNO<sub>3</sub>]], [[Barium sulfate|BaSO<sub>4</sub>]]
|-
! colspan=2| [[Tetragonal]]
| <math> a^2c </math>
| ''a'' = ''b''
| ''α'' = ''β'' = ''γ'' = 90°
| [[White tin]], [[Tin dioxide|SnO<sub>2</sub>]], [[Titanium dioxide|TiO<sub>2</sub>]], [[Calcium sulfate|CaSO<sub>4</sub>]]
|-
! rowspan=2| [[Hexagonal crystal family|Hexagonal]]
! Rhombohedral
| <math> a^3 \sqrt{1 - 3\cos^2\alpha + 2\cos^3\alpha} </math>
| ''a'' = ''b'' = ''c''
| ''α'' = ''β'' = ''γ'' 
| [[Calcite]] (CaCO<sub>3</sub>), [[cinnabar]] (HgS)
|-
! Hexagonal
| <math>\frac{\sqrt{3}}{2}\, a^2c</math>
| ''a'' = ''b''
| ''α'' = ''β'' = 90°, ''γ'' = 120°
| [[Graphite]], [[Zinc oxide|ZnO]], [[Cadmium sulphide|CdS]]
|-
! colspan=2| [[Cubic crystal system|Cubic]]
| <math> a^3</math>
| ''a'' = ''b'' = ''c''
| ''α'' = ''β'' = ''γ'' = 90°
| [[Sodium chloride|NaCl]], [[zinc blende]], [[copper|copper metal]], [[Potassium chloride|KCl]], [[Diamond]], [[Silver]]
|}
{{Clear}}

Some basic information for the lattice systems and Bravais lattices in three dimensions is summarized in the diagram at the beginning of this page. The seven sided polygon (heptagon) and the number 7 at the centre indicate the seven lattice systems. The inner heptagons indicate the lattice angles, lattice parameters, Bravais lattices and Schöenflies notations for the respective lattice systems.