Crystal system

Incrystallography, acrystal systemis a set ofpoint groups(a group of geometric symmetries with at least one fixed point). Alattice systemis a set ofBravais 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 acrystal family.
The seven crystal systems aretriclinic,monoclinic,orthorhombic,tetragonal,trigonal,hexagonal, andcubic. Informally, two crystals are in the same crystal system if they have similar symmetries (though there are many exceptions).
Crystals can be classified in three ways: lattice systems, crystal systems and crystal families. The various classifications are often confused: in particular thetrigonal crystal systemis  often confused with therhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".
A lattice system is a group of lattices with the same set of latticepoint groups. The 14Bravais latticesare grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.
A crystal system is a set of point groups in which the point groups themselves and their correspondingspace groupsare assigned to a lattice system. Of the 32crystallographic point groupsthat 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.
A crystal family is determined by lattices and point groups. It is formed by combining crystal systems that have space groups assigned to a common lattice system. In three dimensions, the hexagonal and trigonal crystal systems are combined into one hexagonal crystal family.
Five of the crystal systems are essentially the same as five of the lattice systems. 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:
The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) 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 (x,y,z) becomes (−x,−y,−z). This is the 'inverted structure'. If the original structure and inverted structure are identical, then the structure iscentrosymmetric. Otherwise it isnon-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-centrosymmetricachiralstructure. If the inverted structure cannot be rotated to align with the original structure, then the structure ischiralorenantiomorphicand its symmetry group isenantiomorphic.
A direction (meaning a line without an arrow) is calledpolarif its two-directional senses are geometrically or physically different. A symmetry direction of a crystal that is polar is called apolar axis.Groups containing a polar axis are calledpolar. A polar crystal possesses a unique polar axis (more precisely, all polar axes are parallel). Some geometrical or physical property is different at the two ends of this axis: for example, there might develop adielectric polarizationas inpyroelectric 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.
Thecrystal structuresof chiral biological molecules (such asproteinstructures) can only occur in the 65enantiomorphicspace groups (biological molecules are usuallychiral).
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.
Ingeometryandcrystallography, aBravais latticeis a category oftranslativesymmetry groups(also known aslattices) in three directions.
Such symmetry groups consist of translations by vectors of the form
wheren1,n2, andn3areintegersanda1,a2, anda3are three non-coplanar vectors, calledprimitive vectors.
These lattices are classified by thespace groupof the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. Theyrepresent the maximum symmetry a structure with the given translational symmetry can have.
All crystalline materials (not includingquasicrystals) 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 theprimitive cell. Depending on the symmetry of a crystal or other pattern, thefundamental domainis again smaller, up to a factor 48.
The Bravais lattices were studied byMoritz Ludwig Frankenheimin 1842, who found that there were 15 Bravais lattices. This was corrected to 14 byA. Bravaisin 1848.
In two-dimensional space, there are four crystal systems (oblique, rectangular, square, hexagonal), four crystal families (oblique, rectanguar, square, hexagonal), and four lattice systems (oblique,rectangular,square, andhexagonal).
‌The four-dimensional unit cell is defined by four edge lengths (a,b,c,d) and six interaxial angles (α,β,γ,δ,ε,ζ). The following conditions for the lattice parameters define 23 crystal families
The names here are given according to Whittaker.They are almost the same as in Brownet al.,with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brownet al.are given in parentheses.
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 P31and P32, P4122 and P4322. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.