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

Further information: Bravais lattice


Main article: Square lattice

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


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
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 4C444[4]+ pinnoite,
P4, P41, P42, P43
I4, I41
81–82 Tetragonal disphenoidal 4S4[2+,4+] cahnite, tugtupite P4
83–88 Tetragonal dipyramidal 4/mC4h4*[2,4+] scheelite, wulfenite, leucite P4/m, P42/m, P4/n, P42/n
I4/m, I41/a
89–98 Tetragonal trapezohedral 422D4224[2,4]+ cristobalite, wardite P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212
I422, I4122
99–110 Ditetragonal pyramidal 4mm C4v*44[4] diaboleite P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc
I4mm, I4cm, I41md, I41cd
111–122 Tetragonal scalenohedral 42mD2d2*2[2+,4] chalcopyrite, stannite P42m, P42c, P421m, P421c, P4m2, P4c2, P4b2, P4n2
I4m2, I4c2, I42m, I42d
123–142 Ditetragonal dipyramidal 4/mmmD4h*224[2,4] 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

See also


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