# Tetragonal crystal system

In crystallography, the **tetragonal crystal system** is one of the 7 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

### Two-dimensional

There is only one tetragonal Bravais lattice in two dimensions: the square lattice.

### Three-dimensional

There are two tetragonal Bravais latices: the simple tetragonal (from stretching the simple-cubic lattice) and the centered tetragonal (from stretching either the face-centered or the body-centered cubic lattice). One might suppose stretching face-centered cubic would result in face-centered tetragonal, but face-centered tetragonal is equivalent to body-centered tetragonal, BCT (with a smaller lattice spacing). BCT is considered more fundamental, so that is the standard terminology.^{[1]}

Bravais lattice | Primitive tetragonal |
Body-centered tetragonal |
---|---|---|

Pearson symbol | tP | tI |

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.^{[2]}^{[3]}

# | Point group | Example | Space groups | ||||
---|---|---|---|---|---|---|---|

Class | Intl | Schoen. | Orb. | Cox. | |||

75–80 | Tetragonal pyramidal | 4 | C_{4} | 44 | [4]^{+} |
pinnoite, piypite |
P4, P4_{1}, P4_{2}, P4_{3}I4, I4 _{1} |

81–82 | Tetragonal disphenoidal | 4 | S_{4} | 2× | [2^{+},4^{+}] |
cahnite, tugtupite | P4 I4 |

83–88 | Tetragonal dipyramidal | 4/m | C_{4h} | 4* | [2,4^{+}] |
scheelite, wulfenite, leucite | P4/m, P4_{2}/m, P4/n, P4_{2}/nI4/m, I4 _{1}/a |

89–98 | Tetragonal trapezohedral | 422 | D_{4} | 224 | [2,4]^{+} |
cristobalite, wardite | P422, P42_{1}2, P4_{1}22, P4_{1}2_{1}2, P4_{2}22, P4_{2}2_{1}2, P4_{3}22, P4_{3}2_{1}2I422, I4 _{1}22 |

99–110 | Ditetragonal pyramidal | 4mm | C_{4v} | *44 | [4] | diaboleite | P4mm, P4bm, P4_{2}cm, P4_{2}nm, P4cc, P4nc, P4_{2}mc, P4_{2}bcI4mm, I4cm, I4 _{1}md, I4_{1}cd |

111–122 | Tetragonal scalenohedral | 42m | D_{2d} | 2*2 | [2^{+},4] |
chalcopyrite, stannite | P42m, P42c, P42_{1}m, P42_{1}c, P4m2, P4c2, P4b2, P4n2I4m2, I4c2, I42m, I42d |

123–142 | Ditetragonal dipyramidal | 4/mmm | D_{4h} | *224 | [2,4] | rutile, pyrolusite, zircon | P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4_{2}/mmc, P4_{2}/mcm, P4_{2}/nbc, P4_{2}/nnm, P4_{2}/mbc, P4_{2}/mnm, P4_{2}/nmc, P4_{2}/ncmI4/mmm, I4/mcm, I4 _{1}/amd, I4_{1}/acd |

## See also

## References

- ↑ Cubic-to-Tetragonal Transition
- ↑ Webmineral data
- ↑ Hurlbut, Cornelius S.; Klein, Cornelis, 1985,
*Manual of Mineralogy*, 20th ed., pp. 73–78, ISBN 0-471-80580-7