In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices (an infinite array of discrete points). Space groups (symmetry groups of a configuration in space) are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family. Informally, two crystals are in the same crystal system if they have similar symmetries (though there are many exceptions).
Overview
Crystals can be classified in three ways: lattice systems, crystal systems and crystal families. The various classifications are often confused as the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".
In 2 dimensions In two-dimensional space, the crystal families, crystal systems, and lattice systems are the same. There are four crystal families (oblique, rectangular, square, and hexagonal).
In 3 dimensions In three-dimensional space, there are 6 crystal families, 7 crystal systems and 7 lattice systems. Five of the crystal systems are essentially the same as five of the lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, and cubic. Of the 32 crystallographic point groups that exist in three dimensions, most are assigned to only one lattice system. However, 5 point groups have 7 corresponding space groups assigned to the rhombohedral lattice system and 18 corresponding space groups assigned to the hexagonal lattice system. These point groups are assigned to the trigonal crystal system. The hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. These are combined into the hexagonal crystal family. The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the following table:
Note: The trigonal crystal system is often confused with the rhombohedral lattice system. There is no "trigonal" lattice system. To avoid confusion of terminology, the term "trigonal lattice" is not used.
Crystal classes
The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) as shown in the following table below:
Bravais lattices
There are seven different kinds of lattice systems, and each kind of lattice system has four different kinds of centerings (primitive, base-centered, body-centered, face-centered). However, not all of the combinations are unique; some of the combinations are equivalent while other combinations are not possible due to symmetry reasons. This reduces the number of unique lattices to the 14 Bravais lattices. The distribution of the 14 Bravais lattices into 7 lattice systems is given in the following table.
In 4 dimensions In four-dimensional space there are 23 crystal families, 33 crystal systems and 33 lattice systems. The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table. Enantiomorphic systems are marked with an asterisk. The number of enantiomorphic pairs is given in parentheses. Here the term "enantiomorphic" has a different meaning than in the table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means that a group itself (considered as a geometric object) is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P31 and P32, P4122 and P4322. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.
The names here are given according to Whittaker. They are almost the same as in Brown et al., with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brown et al. are given in parentheses.
See also Crystal cluster – Group of crystals formed in an open space Crystal structure – Ordered arrangement of atoms, ions, or molecules in a crystalline material Geometrical crystallography before X-rays – History of geometrical crystallography to 1895 List of space groups Polar point group
References
Works cited Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry. Vol. A (5th ed.). Berlin, New York: Springer-Verlag. doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7.
External links
Overview of the 32 groups Mineral galleries – Symmetry all cubic crystal classes, forms, and stereographic projections (interactive java applet) Crystal system at the Online Dictionary of Crystallography Crystal family at the Online Dictionary of Crystallography Lattice system at the Online Dictionary of Crystallography Conversion Primitive to Standard Conventional for VASP input files Archived 2021-11-26 at the Wayback Machine Learning Crystallography