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An example of 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.

In crystallography, the tetragonal crystal system is one of the seven 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

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 face-centered tetragonal lattice is equivalent to the body-centered tetragonal lattice with a smaller unit cell,^[1] while the base-centered tetragonal lattice is equivalent to the primitive tetragonal lattice with a smaller unit cell.^[2]

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.^[3]^[4]

#	Point group					Type	Example	Space groups
#	Name^[5]	Intl	Schoen.	Orb.	Cox.	Type	Example	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	C_4h	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	C_4v	*44	[4]	polar	diaboleite	P4mm, P4bm, P4₂cm, P4₂nm, P4cc, P4nc, P4₂mc, P4₂bc	I4mm, I4cm, I4₁md, I4₁cd
111–122	Tetragonal scalenohedral	42m	D_2d (V_d)	2*2	[2⁺,4]		chalcopyrite, stannite	P42m, P42c, P42₁m, P42₁c, P4m2, P4c2, P4b2, P4n2	I4m2, I4c2, I42m, I42d
123–142	Ditetragonal dipyramidal	4/mmm (4/m 2/m 2/m)	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, P4₂/mmc, P4₂/mcm, P4₂/nbc, P4₂/nnm, P4₂/mbc, P4₂/mnm, P4₂/nmc, P4₂/ncm	I4/mmm, I4/mcm, I4₁/amd, I4₁/acd