Crystal system
In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals. Informally, two crystals are in the same crystal system if they have similar symmetries, although there are many exceptions to this.
Crystal systems, crystal families and lattice systems are similar but slightly different, and there is widespread confusion between them: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".
Space groups and crystals are divided into seven crystal systems according to their point groups, and into seven lattice systems according to their Bravais lattices. Five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.
Overview
A lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.In a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.
A crystal family is determined by lattices and point groups. It is formed by combining crystal systems which have space groups assigned to a common lattice system. In three dimensions, the crystal families and systems are identical, except the hexagonal and trigonal crystal systems, which are combined into one hexagonal crystal family. In total there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic.
Spaces with less than three dimensions have the same number of crystal systems, crystal families and lattice systems. In one-dimensional space, there is one crystal system. In 2D space, there are four crystal systems: oblique, rectangular, square, and hexagonal.
The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the following table:
Crystal classes
The 7 crystal systems consist of 32 crystal classes as shown in the following table below:The point symmetry of a structure can be further described as follows. Consider the points that make up the structure, and reflect them all through a single point, so that becomes. This is the 'inverted structure'. If the original structure and inverted structure are identical, then the structure is centrosymmetric. Otherwise it is non-centrosymmetric. Still, even in the non-centrosymmetric case, the inverted structure can in some cases be rotated to align with the original structure. This is a non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral or enantiomorphic and its symmetry group is enantiomorphic.
A direction is called polar if its two directional senses are geometrically or physically different. A symmetry direction of a crystal that is polar is called a polar axis. Groups containing a polar axis are called polar. A polar crystal possesses a unique polar axis. Some geometrical or physical property is different at the two ends of this axis: for example, there might develop a dielectric polarization as in pyroelectric crystals. A polar axis can occur only in non-centrosymmetric structures. There cannot be a mirror plane or twofold axis perpendicular to the polar axis, because they would make the two directions of the axis equivalent.
The crystal structures of chiral biological molecules can only occur in the 65 enantiomorphic space groups.
Bravais lattices
There are seven different kinds of crystal systems, and each kind of crystal system has four different kinds of centerings. 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 lattice systems and crystal families is given in the following table.
In geometry and crystallography, a Bravais lattice is a category of translative symmetry groups in three directions.
Such symmetry groups consist of translations by vectors of the form
where n1, n2, and n3 are integers and a1, a2, and a3 are three non-coplanar vectors, called primitive vectors.
These lattices are classified by the space group of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. They represent the maximum symmetry a structure with the given translational symmetry can have.
All crystalline materials must, by definition, fit into one of these arrangements.
For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3 or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48.
The Bravais lattices were studied by Moritz Ludwig Frankenheim in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.
In four-dimensional space
The four-dimensional unit cell is defined by four edge lengths and six interaxial angles. The following conditions for the lattice parameters define 23 crystal families| No. | Family | Edge lengths | Interaxial angles |
| 1 | Hexaclinic | a ≠ b ≠ c ≠ d | α ≠ β ≠ γ ≠ δ ≠ ε ≠ ζ ≠ 90° |
| 2 | Triclinic | a ≠ b ≠ c ≠ d | α ≠ β ≠ γ ≠ 90° δ = ε = ζ = 90° |
| 3 | Diclinic | a ≠ b ≠ c ≠ d | α ≠ 90° β = γ = δ = ε = 90° ζ ≠ 90° |
| 4 | Monoclinic | a ≠ b ≠ c ≠ d | α ≠ 90° β = γ = δ = ε = ζ = 90° |
| 5 | Orthogonal | a ≠ b ≠ c ≠ d | α = β = γ = δ = ε = ζ = 90° |
| 6 | Tetragonal monoclinic | a ≠ b = c ≠ d | α ≠ 90° β = γ = δ = ε = ζ = 90° |
| 7 | Hexagonal monoclinic | a ≠ b = c ≠ d | α ≠ 90° β = γ = δ = ε = 90° ζ = 120° |
| 8 | Ditetragonal diclinic | a = d ≠ b = c | α = ζ = 90° β = ε ≠ 90° γ ≠ 90° δ = 180° − γ |
| 9 | Ditrigonal diclinic | a = d ≠ b = c | α = ζ = 120° β = ε ≠ 90° γ ≠ δ ≠ 90° cos δ = cos β − cos γ |
| 10 | Tetragonal orthogonal | a ≠ b = c ≠ d | α = β = γ = δ = ε = ζ = 90° |
| 11 | Hexagonal orthogonal | a ≠ b = c ≠ d | α = β = γ = δ = ε = 90°, ζ = 120° |
| 12 | Ditetragonal monoclinic | a = d ≠ b = c | α = γ = δ = ζ = 90° β = ε ≠ 90° |
| 13 | Ditrigonal monoclinic | a = d ≠ b = c | α = ζ = 120° β = ε ≠ 90° γ = δ ≠ 90° cos γ = −cos β |
| 14 | Ditetragonal orthogonal | a = d ≠ b = c | α = β = γ = δ = ε = ζ = 90° |
| 15 | Hexagonal tetragonal | a = d ≠ b = c | α = β = γ = δ = ε = 90° ζ = 120° |
| 16 | Dihexagonal orthogonal | a = d ≠ b = c | α = ζ = 120° β = γ = δ = ε = 90° |
| 17 | Cubic orthogonal | a = b = c ≠ d | α = β = γ = δ = ε = ζ = 90° |
| 18 | Octagonal | a = b = c = d | α = γ = ζ ≠ 90° β = ε = 90° δ = 180° − α |
| 19 | Decagonal | a = b = c = d | α = γ = ζ ≠ β = δ = ε cos β = − − cos α |
| 20 | Dodecagonal | a = b = c = d | α = ζ = 90° β = ε = 120° γ = δ ≠ 90° |
| 21 | Diisohexagonal orthogonal | a = b = c = d | α = ζ = 120° β = γ = δ = ε = 90° |
| 22 | Icosagonal | a = b = c = d | α = β = γ = δ = ε = ζ cos α = − |
| 23 | Hypercubic | a = b = c = d | α = β = γ = δ = ε = ζ = 90° |
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 parenthesis.
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 are 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 structures. In the current table, "enantiomorphic" means that a group itself 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.