Trigonal crystal system

Example trigonal crystals (quartz)	Example trigonal rhombohedral crystals (dolomite)
Hexagonal lattice cell	Hexagonal (R-centered) unit cell

In crystallography, the trigonal crystal system is one of the seven crystal systems. A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are associated with a lattice system. The trigonal crystal system consists of those five point groups that have a single three-fold rotation axis (see table in Crystal_system#Crystal_classes). Sometimes the term rhombohedral lattice system is used as an exact synonym, whereas it is more akin to a subset. Crystals in the rhombohedral lattice system are always in the trigonal crystal system, but some crystals such as alpha-quartz are in the trigonal crystal system but not in the rhombohedral lattice system (alpha-quartz is in the hexagonal lattice system). There are 25 space groups (143-167) whose corresponding 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 associated with the hexagonal lattice system. The crystal structures of alpha-quartz in the previous example are described by two of those 18 space groups (152 and 154) associated with the hexagonal lattice system.^[1] To distinguish: The rhombohedral lattice system consists of the rhombohedral lattice, while the trigonal crystal system consists of the five point groups that have seven corresponding space groups associated with the rhombohedral lattice system (and 18 corresponding space groups associated with the hexagonal lattice system). An additional source of confusion is that all members of the trigonal crystal system with assigned rhombohedral lattice system (space groups 146, 148, 155, 160, 161, 166, and 167), can be represented with an equivalent hexagonal lattice with so called R-centering (rhombohedral-centering); there is a choice of using a R-centered hexagonal or a primitive rhombohedral setting for the lattice.^[2][3]

The trigonal point group describes the symmetry of an object produced by stretching a cube along the body diagonal. The resultant rhombohedral Bravais lattice is generated by three primitive vectors of equal length that make equal angles with one another.^[4]

"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.^[5]

Rhombohedral lattice system

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

• Hexagonal axes. The unit cell is a = b ≠ c; 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 3 lattice points per unit cell in total and the lattice is non-primitive, but a centered lattice (R-centered).
• 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.)^[6][5][7]

Space group #	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	3×	[2^+,6^+]	dolomite, ilmenite	P3 R3
149-155	trapezohedral	32	D3	223	[2,3]^+	abhurite, alpha-quartz (152, 154), 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

See also

• Crystal structure

References

1. In total there are 45 space groups associated with the hexagonal lattice system, the remaining 27 space groups being part of the hexagonal crystal system to which another modification of quartz (beta-quartz) belongs.
2. http://quantumwise.com/publications/tutorials/item/510-rhombohedral-and-hexagonal-settings-of-trigonal-crystals.
3. http://img.chem.ucl.ac.uk/sgp/medium/sgp.htm
4. Ashcroft, Neil W.; Mermin, N. David, 1976, "Solid State Physics," 1st ed., p. 119 ISBN 0-03-083993-9
5. Hurlbut, Cornelius S.; Klein, Cornelis, 1985, Manual of Mineralogy, 20th ed., pp. 78–89 ISBN 0-471-80580-7
6. Frederick H. Pough, Roger Tory Peterson (1998). A Field Guide to Rocks and Minerals. Houghton Mifflin Harcourt. p. 62. ISBN 0-395-91096-X.
7. Crystallography and Minerals Arranged by Crystal Form, Webmineral

External links

• Mineralogy Database