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Tetragonal crystal system

Lattice point group

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated September 12, 2026
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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).

01Bravais lattices

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

Bravais lattice Primitive
tetragonal		 Body-centered
tetragonal
Pearson symbol tP tI
Unit cell

The face-centered tetragonal lattice is equivalent to the body-centered tetragonal lattice with a smaller unit cell, while the base-centered tetragonal lattice is equivalent to the primitive tetragonal lattice with a smaller unit cell.

Two different views (top down and from the side) of the unit cell of tP30-CrFe (σ-phase Frank-Kasper structure) that show its different side lengths, making this structure a member of the tetragonal crystal system.
Two different views (top down and from the side) of the unit cell of tP30-CrFe (σ-phase Frank-Kasper structure) that show its different side lengths, making this structure a member of the tetragonal crystal system.

02Crystal 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.

# Point group Type Example Space groups
Name Intl Schoen. Orb. Cox. Primitive Body-centered
75-80 Tetragonal pyramidal 4 C4 44 [4]+ enantiomorphic polar pinnoite,
piypite		 P4, P41, P42, P43 I4, I41
81-82 Tetragonal disphenoidal 4 S4 2× [2+,4+] cahnite, tugtupite P4 I4
83-88 Tetragonal dipyramidal 4/m C4h 4* [2,4+] centrosymmetric scheelite, wulfenite, leucite P4/m, P42/m, P4/n, P42/n I4/m, I41/a
89-98 Tetragonal trapezohedral 422 D4 224 [2,4]+ enantiomorphic cristobalite, wardite P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212 I422, I4122
99-110 Ditetragonal pyramidal 4mm C4v *44 [4] polar diaboleite P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc I4mm, I4cm, I41md, I41cd
111-122 Tetragonal scalenohedral 42m D2d (Vd) 2*2 [2+,4] chalcopyrite, stannite P42m, P42c, P421m, P421c, P4m2, P4c2, P4b2, P4n2 I4m2, I4c2, I42m, I42d
123-142 Ditetragonal dipyramidal 4/mmm (4/m 2/m 2/m) D4h *224 [2,4] centrosymmetric rutile, pyrolusite, zircon P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P42/mmc, P42/mcm, P42/nbc, P42/nnm, P42/mbc, P42/mnm, P42/nmc, P42/ncm I4/mmm, I4/mcm, I41/amd, I41/acd

03In two dimensions

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

Bravais lattice Square
Pearson symbol tp
Unit cell
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Sources and credits

This article is adapted from the Wikipedia article “Tetragonal crystal system”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Bravais lattice

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite array of discrete points generated by a set of discrete translation operations, where the lattice is exactly the same from each of the discrete lattice points. The set of translation operations is described in three dimensional space by R = n 1 a 1 + n 2 a 2 + n 3 a 3 , {\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3},} where the ni are any integers, and ai are primitive translation vectors, or primitive vectors, which lie in different directions (not necessarily mutually perpendicular) and span the lattice.

Crystal system

In crystallography, a crystal system is a set of point groups. A lattice system is a set of Bravais lattices (an infinite array of discrete points).

Crystal structure

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.