# Trigonal crystal system

(Redirected from Trigonal)
Clockwise from upper left: An example of trigonal crystals, quartz; An example of trigonal rhombohedral crystals, dolomite; Hexagonal lattice cell; Rhombohedral lattice cell.

In crystallography, the trigonal crystal system is one of the seven crystal systems. Sometimes the term rhombohedral lattice system is used as an exact synonym, whereas it is a subset. 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.

The trigonal point group describes the symmetry of an object produced by stretching a cube along the body diagonal. The resultant rhombohedral (or trigonal) Bravais lattice is generated by three primitive vectors of equal length that make equal angles with one another.[1]

"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.[2]

## Rhombohedral lattice system

There are two descriptions (settings) of the rhombohedral lattice system.

• Hexagonal axes. The unit cell is a = bc; α = β = 90°, γ = 120°. Two additional lattice points occupy space diagonal of the unit cell and have coordinates 2/3 1/3 1/3 and 1/3 2/3 2/3. Hence, there are totally 3 lattice points per unit cell.
• Rhombohedral axes. The unit cell is a rhombohedron (which gives the name for rhombohedral lattice system). This is a primitive unit cell (no additional lattice points inside unit cell) with parameters a = b = c; α = β = γ ≠ 90°.

In practice, the hexagonal description is more commonly used because it is easier to deal with coordinate system with two 90° angles. However, the rhombohedral axes are often shown in textbooks because this cell reveals 3m symmetry of crystal lattice. The relation between two settings is given below.

The rhombohedral lattice system is combined with the hexagonal lattice system and grouped into a larger hexagonal family. These lattice systems belong to hexagonal family because the same (hexagonal) unit cell can be used for both of them.

## Crystal classes

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, with their international number and notation, their space groups in name and example crystals. (All these point groups are also associated with some space groups not in the rhombohedral lattice system.)[3][2][4]

# Point group Examples Space group
Class Intl Schoen. Orb. Cox.
143-146 Pyramidal
rhombohedral tetartohedral
3 C3 33 [3]+ carlinite, jarosite P3, P31, P32
R3
147-148 Rhombohedral
rhombohedral tetartohedral
3 S6 [2+,6+] dolomite, ilmenite P3
R3
149-155 trapezohedral 32 D3 223 [2,3]+ abhurite, quartz, cinnabar P312, P321, P3112, P3121, P3212, P3221
R32
156-161 Ditrigonal Pyramidal
rhombohedral hemimorphic
3m C3v *33 [3] schorl, cerite, tourmaline, alunite, lithium tantalate P3m1, P31m, P3c1, P31c
R3m, R3c
162-167 Hexagonal Scalenohedral
rhombohedral holohedral
3m D3d 2*3 [2+,6] antimony, hematite, corundum, calcite P31m, P31c, P3m1, P3c1
R3m, R3c