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

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

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.	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	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

In two dimensions

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

Bravais lattice	Square
Pearson symbol	tp
Unit cell

References

^ Cubic-to-Tetragonal Transition
^ Webmineral data
^ Hurlbut, Cornelius S.; Klein, Cornelis, 1985, Manual of Mineralogy, 20th ed., pp. 73–78, ISBN 0-471-80580-7
^ "The 32 crystal classes". Retrieved 2018-06-19.

External links

Media related to Tetragonal lattices at Wikimedia Commons

[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

[4] "The 32 crystal classes". Retrieved 2018-06-19.

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v t e Crystal systems
Bravais lattice Crystallographic point group
Seven 3D systems	triclinic (anorthic) monoclinic orthorhombic tetragonal trigonal & hexagonal cubic (isometric)
Four 2D systems	oblique rectangular square hexagonal