32 Point Groups
- Practical Electron Microscopy and Database -
- An Online Book -

http://www.globalsino.com/EM/  



=================================================================================

Space groups represent the ways that the macroscopic and microscopic symmetry elements (operations) can be self-consistently arranged in space. There are totally 230 space groups. The space groups add the centering information and microscopic elements to the point groups. Figure 4549a schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups. The seven holohedric point groups F0 and all their subgroups (merohedral groups) form the 32 crystallographic point groups.

The relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups

Figure 4549a. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.

Nicolaus Steno first showed in 1669 that the angles between the faces of crystals are constant, independently of the regularity of a given crystal morphology. The analysis of crystal morphologies led to the formulation of a complete set of 32 symmetry classes, called “point groups” as shown in Table 4549a. The morphologies of all crystals obey the 32 point groups. Possible symmetry elements are 1-, 2-, 3-, 4-, and 6-fold rotations, mirror plane m, inversion center and a combination of rotation axis with inversion center (inversion axis). Table 4549b shows the relation between three-dimensional crystal families, crystal systems, and lattice systems. 32 point groups indicate that there are only 32 possible combinations of symmetry elements. Furthermore, Table 4549c lists the symmetry groups for the crystal systems together with the point groups with Hermann–Mauguin notation.

Table 4549a. 32 point groups (Notation: Schönflies).

32 point groups

Table 4549b. The relation between three-dimensional crystal families, crystal systems, and lattice systems.

Crystal
family
Crystal
system
Essential symmetries
of point group
Point
group
Space
group
Bravais
lattices
Lattice
system
Triclinic
1-fold axis 2 2 1 Triclinic
Monoclinic
1 two-fold axis (parallel to y) of rotation or 1 mirror plane 3
13
2
Monoclinic

Orthorhombic

3 two-fold axes of rotation or 1 two-fold axis of rotation and two mirror planes 3 59 4
Orthorhombic

Tetragonal

1 four-fold axis (parallel to z) of rotation

7

68

2

Tetragonal
Hexagonal
Trigonal
1 three-fold axis (parallel to z) of rotation 5 7 1 Rhombohedral
18 1 Hexagonal
Hexagonal
1 six-fold axis (parallel to z) of rotation 7 27
Cubic
4 three-fold axes of rotation 5 36 3 Cubic
Total: 6
7
  32 230 14 7

Table 4549c. Symmetry groups for the crystal systems.

System
Essential symmetry 
Lattice symmetry 
Diffraction (Laue) symmetry
Point Groups (Hermann–Mauguin notation)
X
Point Groups (Hermann–Mauguin notation)
X⊥m
(X/m)
X||m (Xm)
Point Groups (Hermann–Mauguin notation)
(Point Groups (Hermann–Mauguin notation) )
X⊥2
X⊥m||m
None Triclinic Triclinic 1 -1 -- -- -- -- --
Monoclinic 2/m 2/m 2 m 2/m -- -- -- --
222 or 2mm
mmm
mmm -- -- -- mm2 -- 222 mmm
Tetragonal 4/mmm
4/m
4 -4 4/m -- -- -- --
4/mmm -- -- -- 4mm 42m 422 4/mmm
Trigonal Trigonal 3 3 -3 -- -- -- -- --
-3m1 -- -- -- 3m -3m1 321 --
-31m -- -- -- -31m 312 --
Hexagonal 6/mmm
6/m 6 -6 6/m -- -- -- --
6/mmm -- -- -- 6mm -62m 622 6/mmm
23 m3m
m-3 23 m-3 -- -- -- -- --
m-3m -- -- -- -- 4-3m 432 m-3m
*  X: Major X-fold rotation axis alone.
   Point Groups (Hermann–Mauguin notation) : Inversion axis alone, namely, the major X-fold inversion axes.
    X/m: Rotation axis with a symmetry plane normal to it.
    Xm: Rotation axis with a symmetry plane that is not normal to it (usually a vertical symmetry plane).
   Point Groups (Hermann–Mauguin notation) : Inversion axis with a symmetry plane not normal to it.
    X⊥2: Rotation axis with a diad normal to it (The two-fold rotation axis is perpendicular to the major axis).
    X/mm: Rotation axis with a symmetry plane normal to it and another not so.

Of the 32 point groups, 11 crystal classes are centrosymmetric and thus possess no polar properties. Of the remaining 21 non-centrosymmetric classes, 20 classes show electrical polarity when a stress is applied.

Figure 4549b shows the schematic illustration of the different classes of crystal systems and their properties. In ferroelectric substances, the number of distinct spontaneous polarization directions depends on its point group symmetry.

32 point groups
No center of symmetry (21 point groups)
Center of symmetry (11 point groups)
 
Non-piezoelectric (1 point groups)
 
Non-pyroelectric
 
Ferroelectric (Only if polarization Is reversible)
Non-ferroelectric

Figure 4549b. Schematic illustration of the different classes of crystal systems and their properties.

 

 

=================================================================================