Tetragonal crystal system explained

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.

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.	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		S4	2×	[2<sup>+</sup>,4<sup>+</sup>]		cahnite, tugtupite	P	I
83–88	Tetragonal dipyramidal	4/m	C4h	4*	[2,4<sup>+</sup>]	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	2m	D2d (Vd)	2*2	[2<sup>+</sup>,4]		chalcopyrite, stannite	P2m, P2c, P21m, P21c, Pm2, Pc2, Pb2, Pn2	Im2, Ic2, I2m, I2d
123–142	Ditetragonal dipyramidal	4/mmm	D4h	224	[2,4]	centrosymmetric		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

See main article: Square lattice. There is only one tetragonal Bravais lattice in two dimensions: the square lattice.

See also

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

Notes and References

1. http://www.aem.umn.edu/people/faculty/shield/hane/tet.html Cubic-to-Tetragonal Transition
2. http://webmineral.com/crystal/Tetragonal.shtml Webmineral data
3. Hurlbut, Cornelius S.; Klein, Cornelis, 1985, Manual of Mineralogy, 20th ed., pp. 73–78,
4. Web site: The 32 crystal classes . 2018-06-19.