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 |
---|---|---|
, | ||
, | ||
, | ||
, | ||
, |
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 | |||
Full | Short | |||||||||||
16 | trigonal | trigonal-pyramidal | C 3 | 3 | 3 | 3 | 143-146 | + | + [001] | + |
Carlinite gratonite |
|
17th | rhombohedral | C 3 i ( S 6 ) | 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 |
|||
|
For further trigonal crystallizing chemical substances see category: Trigonal crystal system
Crystal forms of the trigonal crystal system
Ditrigonal scalenohedron
Trigonal trapezoid
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).