Trigonal crystal system

In crystallography, the trigonal crystal system is one of the seven crystal systems, and the rhombohedral lattice system is one of the seven lattice systems. They are often confused with each other: crystals in the rhombohedral lattice system are always in the trigonal crystal system, but some crystals such as quartz are in the trigonal crystal system but not in the rhombohedral lattice system. The rhombohedral lattice system consists of the rhombohedral lattice, while the trigonal crystal system consists of the five point groups of the seven space groups with a rhombohedral lattice. There are 25 space groups whose point groups are one of the five in the trigonal crystal system, consisting of the seven space groups associated with the rhombohedral lattice system together with 18 of the 45 space groups associated with the hexagonal lattice system.

"Rhombohedral crystal system" is an ambiguous term that confuses the trigonal crystal system with the rhombohedral lattice system and may mean either of them (or even the hexagonal crystal family).

In the classification into 6 crystal families, the trigonal crystal system is combined with the hexagonal crystal system and grouped into a larger hexagonal family.

Rhombohedral lattice system

An example of trigonal rhombohedral crystals, dolomite

A lattice system is described by three basis vectors. In the rhombohedral system, the crystal is described by vectors of equal length, no two of which are orthogonal. The rhombohedral system can be thought of as the cubic system stretched along a body diagonal. a = b = c; α = β = γ ≠ 90°. In some classification schemes, the rhombohedral lattice system is combined with the hexagonal lattice system and grouped into a larger hexagonal family.

There is only one rhombohedral Bravais lattice.

List of rhombohedral space groups

Rhombohedral unit cell

The seven space groups associated with the rhombohedral lattice system are listed below, with their international number and notation, followed by their point groups in name and international notation (Hermann-Mauguin notation) and Schoenflies notation, and example crystals. (All these point groups are also associated with some space groups not in the rhombohedral lattice system.)

Number	Space group	Point group	International	Schoenflies	Examples
146	R3	rhombohedral tetartohedral	3	C3	carlinite
148	R3	rhombohedral tetartohedral	3	S6	dolomite
155	R32	trapezohedral	32	D3	abhurite
160	R3m	rhombohedral hemimorphic	3m	C3v	schorl
161	R3c	rhombohedral hemimorphic	3m	C3v	cerite
166	R3m	rhombohedral holohedral	3m	D3d	antimony
167	R3c	rhombohedral holohedral	3m	D3d	hematite, corundum

Trigonal crystal system

File:Kwarc z kalcytem.jpg

An example of trigonal crystals, quartz

Hexagonal unit cell

The trigonal crystal system is the only crystal system whose point groups have more than one lattice system associated with their space groups: the hexagonal and rhombohedral lattices both appear.

The 5 point groups in this crystal system are listed below, followed by their representations in international notation (Hermann-Mauguin notation) and Schoenflies notation, and example crystals.

Class name	international	Schoenflies	examples
Hexagonal Scalenohedral	${\overline {3}}2/m$	D3d	calcite, corundum, hematite
Ditrigonal Pyramidal	$3m$	C3v	tourmaline, alunite
Rhombohedral	${\overline {3}}$	S6	dolomite, ilmenite
Trapezohedral	$32$	D3	quartz, cinnabar
Pyramidal	$3$	C3	jarosite

See also

• Crystal structure

External links

• Mineralogy Database