# 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 lattices: 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 the face-centered tetragonal is equivalent to the body-centered tetragonal, BCT (with a smaller lattice spacing). BCT is considered more fundamental, and therefore this 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 | Type | Example | Space groups | |||||
---|---|---|---|---|---|---|---|---|---|

Name[4] | Intl | Schoen. | Orb. | Cox. | Primitive | Body-centered | |||

75–80 | Tetragonal pyramidal | 4 | C_{4} |
44 | [4]^{+} |
enantiomorphic polar | 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^{+}] |
centrosymmetric | scheelite, wulfenite, leucite | P4/m, P4_{2}/m, P4/n, P4_{2}/n |
I4/m, I4_{1}/a |

89–98 | Tetragonal trapezohedral | 422 | D_{4} |
224 | [2,4]^{+} |
enantiomorphic | 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}2 |
I422, I4_{1}22 |

99–110 | Ditetragonal pyramidal | 4mm | C_{4v} |
*44 | [4] | polar | diaboleite | P4mm, P4bm, P4_{2}cm, P4_{2}nm, P4cc, P4nc, P4_{2}mc, P4_{2}bc |
I4mm, I4cm, I4_{1}md, I4_{1}cd |

111–122 | Tetragonal scalenohedral | 42m | D_{2d} (V_{d}) |
2*2 | [2^{+},4] |
chalcopyrite, stannite | P42m, P42c, P42_{1}m, P42_{1}c, P4m2, P4c2, P4b2, P4n2 |
I4m2, I4c2, I42m, I42d | |

123–142 | Ditetragonal dipyramidal | 4/mmm | D_{4h} |
*224 | [2,4] | centrosymmetric | 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}/ncm |
I4/mmm, I4/mcm, I4_{1}/amd, I4_{1}/acd |

## 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 - "The 32 crystal classes". Retrieved 2018-06-19.