Tetragonal crystal system

An example of the tetragonal crystals, wulfenite

In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. 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 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 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.

Tetragonal Bravais lattices

Primitive	Body-centered

The point groups that fall under this crystal system are listed below, followed by their representations in international notation and Schoenflies notation, and mineral examples.

#	Point group					Example	Space groups
	Class	Intl	Schoenflies	Orbifold	Coxeter
75-80	Tetragonal pyramidal	4	C4	44	[4]+	pinnoite, richellite	P4, P41, P42, P43, I4, I41
81-82	Tetragonal disphenoidal	4	S4	2x	[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