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).

An example of the tetragonal crystals, wulfenite

Bravais lattices
Further information: Bravais lattice

Two-dimensional
Main article: Square lattice

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 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 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

See also

References
  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
  4. "The 32 crystal classes". Retrieved 2018-06-19.
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