Trigonal crystal system

The trigonal crystal system is one of the seven crystal systems in crystallography . It includes all point groups with a threefold axis of rotation or rotation inversion. The trigonal crystal system is closely related to the hexagonal crystal system and together with it forms the hexagonal crystal family .

The trigonal point groups

The trigonal crystal system comprises the point groups 3, 3 , 32, 3 m and 3 m . These are all the point groups of the hexagonal crystal family in which there is a space group with rhombohedral centering - the space groups of the hexagonal crystal system can all be described with the hexagonal primitive axis system. The trigonal crystal system thus includes all subgroups of the point group 3 m , which have a 3-fold axis. In contrast to the hexagonal point groups, these point groups all have a cubic parent group. The following table provides an overview of the space groups of the trigonal crystal system.

Point group	Primitive space groups	Centered room groups
${\ displaystyle C_ {3}}$ , ${\ displaystyle \ 3}$	${\ displaystyle P3, \, P3_ {1}, \, P3_ {2}}$	${\ displaystyle R3}$
${\ displaystyle C_ {3i} (\ equiv S_ {6})}$ , ${\ displaystyle {\ bar {3}}}$	${\ displaystyle P {\ bar {3}}}$	${\ displaystyle R {\ bar {3}}}$
${\ displaystyle D_ {3}}$ , ${\ displaystyle \ 32}$	${\ displaystyle P312, \, P321, \, P3_ {1} 12, \, P3_ {1} 21 \, P3_ {2} 12, \, P3_ {2} 21}$	${\ displaystyle R32}$
${\ displaystyle C_ {3v}}$ , ${\ displaystyle \ 3m}$	${\ displaystyle P3m1, \, P31m, \, P3c1, \, P31c}$	${\ displaystyle R3m, \, R3c}$
${\ displaystyle D_ {3d}}$ , ${\ displaystyle {\ bar {3}} m}$	${\ displaystyle P {\ bar {3}} 1m, \, P {\ bar {3}} 1c, \, P {\ bar {3}} 1c, \, P {\ bar {3}} c1}$	${\ displaystyle R {\ bar {3}} m, \, R {\ bar {3}} c}$

The axis systems of the trigonal crystal system

Two different lattice systems are used to describe trigonal space groups : the hexagonal and the rhombohedral. These are described in detail in the article hexagonal crystal system . The terms trigonal and rhombohedral are clearly delimited in modern usage:

Trigonal is the name for a set of symmetry groups.
Rhombohedral is the name of a grid system.

Point groups in the trigonal crystal system and their physical properties

To describe the trigonal crystal classes in Hermann-Mauguin symbology , the symmetry operations with respect to given directions in the lattice system are given.

In the hexagonal axis system: 1. Symbol in the direction of the c-axis (<001>). 2. Symbol in the direction of an a-axis (<100>). 3. Symbol in a direction perpendicular to an a- and the c-axis (<120>). The generally non-equivalent direction <210> is also often specified for the 3rd direction. Even if this does not play a role in specifying the position of the symmetry elements, this specification does not correspond to the conventions.

In the rhombohedral axis system: 1. Symbol in the direction of the space diagonal (<111>). 2. Symbol in the direction of a surface diagonal (<1 1 0>).

Characteristic for all space groups of the trigonal crystal system is the 3 (or 3 ) in the 1st position of the space group symbol.

Point group (crystal class)								Physical Properties			Examples
No.	Crystal system	Surname	Schoenflies icon	International symbol ( Hermann-Mauguin )		Tepid class	Associated room groups ( no.)	Optical activity ( enantiomorphism )	Pyroelectricity	Piezoelectricity ; SHG effect
No.	Crystal system	Surname	Schoenflies icon	Full	Short	Tepid class	Associated room groups ( no.)	Optical activity ( enantiomorphism )	Pyroelectricity	Piezoelectricity ; SHG effect
16	trigonal	trigonal-pyramidal	C ₃	3	3	3	143-146	+	+ [001]	+	Carlinite gratonite
17th		rhombohedral	C _{3 i} ( S ₆ )	3	3	3	147-148	-	-	-	Dolomite Dioptas
18th		trigonal-trapezoidal	D ₃	321 or 312	32	3 m	149-155	+	-	+	Quartz tellurium
19th		ditrigonal-pyramidal	C _{3 v}	3 m 1 or 31 m	3 m		156-161	-	+ [001]	+	Tourmaline pyrargyrite
20th		ditrigonal-scalenohedral	D _{3 d}	3 2 / m 1 or 3 12 / m	3 m		162-167	-	-	-	Calcite corundum
↑ In the information on the physical properties, " - " means forbidden due to the symmetry and " + " means allowed. No statement can be made about the magnitude of the optical activity, pyro- and piezoelectricity and the SHG effect purely due to the symmetry. However, one can assume that there is always at least a weak expression of the property. For the pyroelectricity, the direction of the pyroelectric vector is given, if available.

For further trigonal crystallizing chemical substances see category: Trigonal crystal system

Crystal forms of the trigonal crystal system

Ditrigonal scalenohedron
Trigonal trapezoid
Quartz crystal

literature

W. Borchardt-Ott: Crystallography. 6th edition. Springer, Berlin 2002, ISBN 3-540-43964-1 .
W. Massa: Crystal structure determination. 3. Edition. Teubner, Stuttgart 2002, ISBN 3-519-23527-7 .
M. Okrusch, S. Matthes: Mineralogy. 7th edition. Springer, Berlin 2005, ISBN 3-540-23812-3 .
Hahn, Theo (Ed.): International Tables for Crystallography Vol. A D. Reidel publishing Company, Dordrecht 1983, ISBN 90-277-1445-2

Web links

Short script Algebra I - Crystallography . Uni Dortmund, p. 11 (PDF, 412 kB).

[Hinweise-1] In the information on the physical properties, " - " means forbidden due to the symmetry and " + " means allowed. No statement can be made about the magnitude of the optical activity, pyro- and piezoelectricity and the SHG effect purely due to the symmetry. However, one can assume that there is always at least a weak expression of the property. For the pyroelectricity, the direction of the pyroelectric vector is given, if available.