thumb|right|200px|An example of tetragonal crystals, [[wulfenite]]
thumb|right|300px|Two different views (top down and from the side) of the unit cell of tP30-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 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
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.
Crystal classes
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.
{| class="wikitable"
! rowspan=2 width=60|#
! colspan=5| Point group
! rowspan=2| Type
! rowspan=2| Example
! colspan=2| Space groups
|-
!Name
! 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
|
| S<sub>4</sub>
| 2×
| [2<sup>+</sup>,4<sup>+</sup>]
|
| cahnite, tugtupite
| align=left| P
| I
|- align=center
! 83–88
| Tetragonal dipyramidal
| 4/m
| C<sub>4h</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>4v</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
| 2m
| D<sub>2d</sub> (V<sub>d</sub>)
| 2*2
| [2<sup>+</sup>,4]
|
| chalcopyrite, stannite
| align=left| P2m, P2c, P2<sub>1</sub>m, P2<sub>1</sub>c, Pm2, Pc2, Pb2, Pn2
| Im2, Ic2, I2m, I2d
|- align=center
! 123–142
| Ditetragonal dipyramidal
| 4/mmm (4/m 2/m 2/m)
| D<sub>4h</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
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