In
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wo ...
, 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
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of Translation operator (quantum mechanics)#Discrete Translational Symmetry, discrete translation operations described in t ...
.
Space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it ...
s 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.
The seven crystal systems are
triclinic
180px, Triclinic (a ≠ b ≠ c and α ≠ β ≠ γ )
In crystallography, the triclinic (or anorthic) crystal system is one of the 7 crystal systems. A crystal system is described by three basis vectors. In the triclinic system, the crystal i ...
,
monoclinic
In crystallography, the monoclinic crystal system is one of the seven crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic ...
,
orthorhombic
In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with ...
,
tetragonal
In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a squar ...
,
trigonal, hexagonal, and
cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
. Informally, two crystals are in the same crystal system if they have similar symmetries (albeit there are many exceptions).
Classifications
Crystals can be classified in three ways: lattice systems, crystal systems and crystal families. The various classifications are often confused: 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".
Lattice system
A lattice system is a group of lattices with the same set of lattice
point groups. The 14
Bravais lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
s are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.
Crystal system
A crystal system is a set of point groups in which the point groups themselves and their corresponding
space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it ...
s are assigned to a lattice system. Of the 32
crystallographic 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.
Crystal family
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.
Comparison
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:
:''Note: 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:
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 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 (meaning a line without an arrow) 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 (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 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 structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns t ...
s of chiral biological molecules (such as
protein
Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respon ...
structures) can only occur in the 65
enantiomorphic space groups (biological molecules are usually
chiral).
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
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wo ...
, a Bravais lattice is a category of
translative symmetry groups (also known as
lattices) in three directions.
Such symmetry groups consist of translations by vectors of the form
:R = ''n''
1a
1 + ''n''
2a
2 + ''n''
3a
3,
where ''n''
1, ''n''
2, and ''n''
3 are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s and a
1, a
2, and a
3 are three non-coplanar vectors, called ''primitive vectors''.
These lattices are classified by the
space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it ...
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 (not including
quasicrystals) 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
In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
. 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 other dimensions
Two-dimensional space
Two dimensional space has the same number of crystal systems, crystal families, and lattice systems. In 2D space, there are four crystal systems:
oblique,
rectangular,
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
, and
hexagonal.
Four-dimensional space
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 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 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 P3
1 and P3
2, P4
122 and P4
322. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.
See also
*
*
*
*
References
Works cited
*
External links
Overview of the 32 groupsall cubic crystal classes, forms, and stereographic projections (interactive java applet)Crystal systemat th
Online Dictionary of CrystallographyCrystal familyat th
Online Dictionary of CrystallographyLattice systemat th
Online Dictionary of Crystallography*
ttp://www.xtal.iqfr.csic.es/Cristalografia/index-en.html Learning Crystallography
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