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).

"Tetragonal" redirects here. For other uses, see tetragon.

An example of the tetragonal crystals, wulfenite

Two different views (top down and from the side) of the unit cell of tP30-CrFe (σ-phase Frank–Kasper structure) that show its different side lengths, making this structure a member of the tetragonal crystal system.

Bravais lattices

Further information: Bravais lattice

There are two tetragonal Bravais lattices: the primitive tetragonal and the body-centered tetragonal.

Bravais lattice	Primitive tetragonal	Body-centered tetragonal
Pearson symbol	tP	tI
Unit cell

The body-centered tetragonal lattice is equivalent to the primitive tetragonal lattice with a smaller unit cell, while the face-centered tetragonal lattice is equivalent to the body-centered tetragonal lattice with a smaller unit cell.^[1]

Crystal classes

Further information: Crystallographic point group

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

In two dimensions

Main article: Square lattice

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

Bravais lattice	Square
Pearson symbol	tp
Unit cell

See also

• Bravais lattices
• Crystal system
• Crystal structure
• Point groups