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# Tetragonal crystal system

An example of the tetragonal crystals, wulfenite

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.

Bravais lattice Square
Pearson symbol tp
Unit cell

### 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 C4 44 [4]+ enantiomorphic polar pinnoite,
piypite
P4, P41, P42, P43 I4, I41
81–82 Tetragonal disphenoidal 4 S4 [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 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