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Wikipedia / Crystal system

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:

  1. 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

References

Works cited