A crystal system is a class of point groups. Two point groups are placed in the same crystal system if the sets of possible lattice systems of their space groups are the same. For many point groups there is only one possible lattice system, and in these cases the crystal system corresponds to a lattice system and is given the same name. However, for the five point groups in the trigonal crystal class there are two possible lattice systems for their point groups: rhombohedral or hexagonal. In three dimensions there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystal system of a crystal or space group is determined by its point group but not always by its lattice.
A crystal family also consists of point groups and is formed by combining crystal systems whenever two crystal systems have space groups with the same lattice. In three dimensions a crystal family is almost the same as a crystal system (or lattice system), except that the hexagonal and trigonal crystal systems are combined into one hexagonal family. In three dimensions there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. The crystal family of a crystal or space group is determined by either its point group or its lattice, and crystal families are the smallest collections of point groups with this property.
In dimensions less than three there is no essential difference between crystal systems, crystal families, and lattice systems. There are 1 in dimension 0, 1 in dimension 1, and 4 in dimension 2, called oblique, rectangular, square, and hexagonal.
The relation between three-dimensional crystal families, crystal systems, and lattice systems is shown in the following table:
| Crystal family | Crystal system | Required symmetries of point group | Point groups | Space groups | Bravais lattices | Lattice system |
|---|---|---|---|---|---|---|
| Triclinic | None | 2 | 2 | 1 | Triclinic | |
| Monoclinic | 1 twofold axis of rotation or 1 mirror plane | 3 | 13 | 2 | Monoclinic | |
| Orthorhombic | 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes. | 3 | 59 | 4 | Orthorhombic | |
| Tetragonal | 1 fourfold axis of rotation | 7 | 68 | 2 | Tetragonal | |
| Hexagonal | Trigonal | 1 threefold axis of rotation | 5 | 7 | 1 | Rhombohedral |
| 18 | 1 | Hexagonal | ||||
| Hexagonal | 1 sixfold axis of rotation | 7 | 27 | |||
| Cubic | 4 threefold axes of rotation | 5 | 36 | 3 | Cubic | |
| Total: 6 | 7 | 32 | 230 | 14 | 7 | |
Read more about Crystal System: Crystal Classes, Lattice Systems
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