{{short description|Lattice point group}} {{refimprove|date=July 2011}} {{redirect|Tetragonal|other uses|tetragon}} [[File:WulfeniteUSGOV.jpg|thumb|right|200px|An example of 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 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 seven crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular 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 | class=skin-invert-image|100px | class=skin-invert-image|100px |}
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 groups |- !Name<ref>{{cite web |title=The 32 crystal classes |url=https://www.tulane.edu/~sanelson/eens211/32crystalclass.htm |access-date=2018-06-19}}</ref> ! Intl ! Schoen. ! Orb. ! Cox. ! Primitive ! Body-centered |- align=center ! 75–80 | Tetragonal pyramidal | 4 | ''C''<sub>4</sub> | 44 | [4]<sup>+</sup> | enantiomorphic 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 | 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 (4/m 2/m 2/m) | ''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 | 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 System Category:Tetragonal crystal system