Symmetry Groups for the Crystal Systems
- Practical Electron Microscopy and Database -
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Table 3545a. Symmetry groups for the crystal systems.

System
Essential symmetry 
Lattice symmetry 
Diffraction (Laue) symmetry
Point Groups (Hermann–Mauguin notation)
Triclinic
None Triclinic Triclinic 1, -1
Monoclinic
Monoclinic 2/m 2/m 2, m, 2/m
Orthorhombic
222 or 2mm
mmm
mmm 222, mm2, mmm
Tetragonal
Tetragonal 4/mmm
4/m
4, -4, 4/m
4/mmm 422, -42m, 4mm, 4/mmm
Trigonal
Trigonal Trigonal 3 3, -3
-3m1 321, 3m1, -3m1
-31m 312, 31m, -31m
Hexagonal
Hexagonal 6/mmm
6/m 6, -6, 6/m
6/mmm 622, -62m, 6mm, 6/mmm
Cubic
23 m3m
m-3 23, m-3
m-3m 432, -43m, m-3m

Table 3545b. Minimum number of symmetry elements.

System
Minimum number of symmetry elements
Triclinic
None
Monoclinic
One 2-fold rotation or roto-inversion, axis parallel to y-axis
Orthorhombic
Three 2-fold rotation or roto-inversion, axes parallel to x,y,z-axes
Tetragonal
One 4-fold rotation or roto-inversion, axis parallel to z-axis
Rhombohedral (Trigonal)
One 3-fold rotation or roto-inversion, axis parallel to z-axis
Hexagonal
One 6-fold rotation or rotoinversion, axis parallel to z-axis
Cubic
Four 3-fold rotation or roto-inversion, axes parallel to body diagonals

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 3545. Schematic illustration of the different classes of crystal systems and their properties.

 

 

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