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).
There is only one tetragonal Bravais lattice in two dimensions: the square lattice.
Bravais lattice | Square |
---|---|
Pearson symbol | tp |
Unit cell |
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 |
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 |