Editing Tetragonal crystal system

{{short description|Lattice point group}}
{{refimprove|date=July 2011}}
{{redirect|Tetragonal|other uses|tetragon}}
[[File:WulfeniteUSGOV.jpg|thumb|right|200px|An example of the tetragonal crystals, [[wulfenite]]]]
[[File:TP30-CrFe crystalmaker.pdf|thumb|right|300px|Two different views (top down and from the side) of the unit cell of ''tP''30-CrFe (σ-phase [[Frank–Kasper phases|Frank–Kasper structure]]) that show its different side lengths, making this structure a member of the tetragonal crystal system.]]
In [[crystallography]], the '''tetragonal crystal system'''  is one of the 7 [[crystal system]]s. Tetragonal [[crystal lattice]]s result from stretching a cubic lattice along one of its lattice vectors, so that the [[Cube (geometry)|cube]] becomes a rectangular [[Prism (geometry)|prism]] with a square base (''a'' by ''a'') and height (''c'', which is different from ''a'').

== Bravais lattices ==
{{further|Bravais lattice}}

There are two tetragonal Bravais lattices: the primitive tetragonal and the body-centered tetragonal.

{| class=wikitable
! Bravais lattice
! Primitive<br/>tetragonal
! Body-centered<br/>tetragonal
|- align=center
! [[Pearson symbol]]
| tP
| tI
|-
! [[Unit cell]]
| [[File:Tetragonal.svg|class=skin-invert-image|100px]]
| [[File:Body-centered tetragonal.svg|class=skin-invert-image|100px]]
|}

The body-centered tetragonal lattice is equivalent to the primitive tetragonal lattice with a smaller unit cell, while the face-centered tetragonal lattice is equivalent to the body-centered tetragonal lattice with a smaller unit cell.<ref>[http://www.aem.umn.edu/people/faculty/shield/hane/tet.html Cubic-to-Tetragonal Transition]</ref>

==Crystal classes==
{{further|Crystallographic point group}}
The point groups that fall under this crystal system are listed below, followed by their representations in international notation, [[Schoenflies notation]], [[orbifold notation]], [[Coxeter notation]] and [[mineral]] examples.<ref>[http://webmineral.com/crystal/Tetragonal.shtml Webmineral data]</ref><ref>Hurlbut, Cornelius S.; Klein, Cornelis, 1985, ''Manual of Mineralogy'', 20th ed., pp. 73–78, {{isbn|0-471-80580-7}} </ref>

{| class="wikitable"
! rowspan=2 width=60|#
! colspan=5| Point group
! rowspan=2| Type
! rowspan=2| Example
! colspan=2| [[Space group]]s
|-
!Name<ref>{{cite web |title=The 32 crystal classes |url=https://www.tulane.edu/~sanelson/eens211/32crystalclass.htm |access-date=2018-06-19}}</ref>
! [[Hermann–Mauguin_notation|Intl]]
! [[Schoenflies notation|Schoen.]]
! [[Orbifold|Orb.]]
! [[Coxeter notation|Cox.]]
! Primitive
! Body-centered
|- align=center
! 75–80
| Tetragonal pyramidal
| 4
| ''C''<sub>4</sub>
| 44
| [4]<sup>+</sup>
| [[enantiomorphic]] [[Polar point group|polar]]
| [[pinnoite]], <br>[[piypite]]
| align=left| P4,  P4<sub>1</sub>, P4<sub>2</sub>, P4<sub>3</sub>
| I4, I4<sub>1</sub>
|- align=center
! 81–82
| Tetragonal disphenoidal
| {{overline|4}}
| ''S''<sub>4</sub>
| 2×
| [2<sup>+</sup>,4<sup>+</sup>]
|
| [[cahnite]], [[tugtupite]]
| align=left| P{{overline|4}}
| I{{overline|4}}
|- align=center
! 83–88
| Tetragonal dipyramidal
| 4/m
| ''C''<sub>4''h''</sub>
| 4*
| [2,4<sup>+</sup>]
| [[centrosymmetric]]
| [[scheelite]], [[wulfenite]], [[leucite]]
| align=left| P4/m, P4<sub>2</sub>/m, P4/n, P4<sub>2</sub>/n
| I4/m, I4<sub>1</sub>/a
|- align=center
! 89–98
| Tetragonal trapezohedral
| 422
| ''D''<sub>4</sub>
| 224
| [2,4]<sup>+</sup>
| [[enantiomorphic]]
| [[cristobalite]], [[wardite]]
| align=left| P422, P42<sub>1</sub>2, P4<sub>1</sub>22, P4<sub>1</sub>2<sub>1</sub>2, P4<sub>2</sub>22, P4<sub>2</sub>2<sub>1</sub>2, P4<sub>3</sub>22, P4<sub>3</sub>2<sub>1</sub>2
| I422, I4<sub>1</sub>22
|- align=center
! 99–110
| Ditetragonal pyramidal
| 4mm
| ''C''<sub>4''v''</sub>
| *44
| [4]
| [[Polar point group|polar]]
| [[diaboleite]]
| align=left| P4mm, P4bm, P4<sub>2</sub>cm, P4<sub>2</sub>nm, P4cc, P4nc, P4<sub>2</sub>mc, P4<sub>2</sub>bc
| I4mm, I4cm, I4<sub>1</sub>md, I4<sub>1</sub>cd
|- align=center
! 111–122
| Tetragonal scalenohedral
| {{overline|4}}2m
| ''D''<sub>2''d''</sub> (V<sub>d</sub>)
| 2*2
| [2<sup>+</sup>,4]
|
| [[chalcopyrite]], [[stannite]]
| align=left| P{{overline|4}}2m, P{{overline|4}}2c, P{{overline|4}}2<sub>1</sub>m, P{{overline|4}}2<sub>1</sub>c, P{{overline|4}}m2, P{{overline|4}}c2, P{{overline|4}}b2, P{{overline|4}}n2
| I{{overline|4}}m2, I{{overline|4}}c2, I{{overline|4}}2m, I{{overline|4}}2d
|- align=center
! 123–142
| Ditetragonal dipyramidal
| 4/mmm
| ''D''<sub>4''h''</sub>
| *224
| [2,4]
| [[centrosymmetric]]
| [[rutile]], [[pyrolusite]], [[zircon]] 
| align=left| P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4<sub>2</sub>/mmc, P4<sub>2</sub>/mcm, P4<sub>2</sub>/nbc, P4<sub>2</sub>/nnm, P4<sub>2</sub>/mbc, P4<sub>2</sub>/mnm, P4<sub>2</sub>/nmc, P4<sub>2</sub>/ncm
| I4/mmm, I4/mcm, I4<sub>1</sub>/amd, I4<sub>1</sub>/acd
|}

== In two dimensions ==
{{main|Square lattice}}
There is only one tetragonal [[Bravais lattice]] in two dimensions: the square lattice.

{| class=wikitable
! Bravais lattice
! Square
|- align=center
! [[Pearson symbol]]
| tp
|-
! [[Unit cell]]
| [[File:2d tp.svg|class=skin-invert-image|100px]]
|}

==See also==
*[[Bravais lattices]]
*[[Crystal system]]
*[[Crystal structure]]
*[[Point groups]]

==References==
{{Reflist}}

==External links==
*{{Commonscatinline|Tetragonal lattices}}

{{Crystal systems}}
{{Authority control}}

[[Category:Crystal systems]]
[[Category:Tetragonal crystals|System]]
[[Category:Tetragonal crystal system| ]]