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.

Three-dimensional

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 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					Example	Space groups
	Class	Intl	Schoen.	Orb.	Cox.
75–80	Tetragonal pyramidal	4	C4	44	[4]^+	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^+]	scheelite, wulfenite, leucite	P4/m, P42/m, P4/n, P42/n I4/m, I41/a
89–98	Tetragonal trapezohedral	422	D4	224	[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	42m	D2d	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]	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

• Crystal structure
• point groups
• Bravais lattices

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