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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. Depending on their geometry, crystals are commonly classified into seven systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. Figure 4548a schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups. Table 4548a also lists the relation between three-dimensional crystal families, crystal systems, and lattice systems. Table 4548b also shows the seven crystal systems and the restrictions on their cell dimensions (cell edges and cell angles).
Figure 4548a. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.
Table 4548a. The relation between three-dimensional crystal families, crystal systems, and lattice systems.
Crystal
family |
Crystal
system |
Required
symmetries
of point group |
Point
group |
Space
group |
Bravais
lattices |
Lattice
system |
Triclinic |
None |
2 |
2 |
1 |
Triclinic |
Monoclinic
|
1 two-fold axis 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 of rotation |
7 |
68 |
2 |
Tetragonal |
Hexagonal |
Trigonal |
1 three-fold axis of rotation |
5 |
7 |
1 |
Rhombohedral |
18 |
1 |
Hexagonal |
Hexagonal |
1 six-fold axis of rotation |
7 |
27 |
Cubic
|
4 three-fold axes of rotation |
5 |
36 |
3 |
Cubic |
Total: 6 |
7 |
|
32 |
230 |
14 |
7 |
As indicated in Table 4548a, the 14 basic Bravais lattice types give rise to a total of 230 possible crystal types when they are combined with other symmetry operators such as mirrors planes, glide planes, and screw axes.
Table 4548b. The seven crystal systems (families) and the restrictions on their cell dimensions.
Crystal family |
Cell edges |
Cell angles |
Triclinic |
None |
None |
Monoclinic |
None |
α = γ = 90° |
Orthorhombic |
None |
α = β = γ = 90° |
Tetragonal |
a = b |
α = β = γ = 90° |
Trigonal, hexagonal |
a = b |
α = β = 90°, γ = 120° |
Cubic |
a = b = c |
α = β = γ = 90° |
Figure 4548b shows the schematic illustration of the different classes of crystal systems and their properties.
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Non-pyroelectric |
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Non-ferroelectric |
Figure 4548b. Schematic illustration of the different classes of crystal systems and their properties.
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