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In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of Isometry of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.
In crystallography, space groups are also called the crystallographic or Evgraf Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography .
In 1879 the German mathematician Leonhard Sohncke listed the 65 space groups (called Sohncke groups) whose elements preserve the chirality. More accurately, he listed 66 groups, but both the Russian mathematician and crystallographer Evgraf Fedorov and the German mathematician Arthur Moritz Schoenflies noticed that two of them were really the same. The space groups in three dimensions were first enumerated in 1891 by Fedorov (whose list had two omissions (I3d and Fdd2) and one duplication (Fmm2)), and shortly afterwards in 1891 were independently enumerated by Schönflies (whose list had four omissions (I3d, Pc, Cc, ?) and one duplication (P21m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies. later enumerated the groups with a different method, but omitted four groups (Fdd2, I2d, P21d, and P21c) even though he already had the correct list of 230 groups from Fedorov and Schönflies; the common claim that Barlow was unaware of their work is incorrect. describes the history of the discovery of the space groups in detail.
The number of replicates of the asymmetric unit in a unit cell is thus the number of lattice points in the cell times the order of the point group. This ranges from 1 in the case of space group P1 to 192 for a space group like Fmm, the NaCl structure.
where M is its matrix, D is its vector, and where the element transforms point x into point y. In general, D = D (lattice) + , where is a unique function of M that is zero for M being the identity. The matrices M form a point group that is a basis of the space group; the lattice must be symmetric under that point group, but the crystal structure itself may not be symmetric under that point group as applied to any particular point (that is, without a translation). For example, the diamond cubic structure does not have any point where the cubic point group applies.
The lattice dimension can be less than the overall dimension, resulting in a "subperiodic" space group. For (overall dimension, lattice dimension):
Among the 65 Sohncke groups are 22 that come in 11 enantiomorphic pairs.
| Two space groups, considered as subgroups of the group of affine transformations of space, have the same space group type if they are the same up to an affine transformation of space that preserves orientation. Thus e.g. a change of angle between translation vectors does not affect the space group type if it does not add or remove any symmetry. A more formal definition involves conjugacy (see Symmetry group). In three dimensions, for 11 of the affine space groups, there is no chirality-preserving (i.e. orientation-preserving) map from the group to its mirror image, so if one distinguishes groups from their mirror images, these each split into two cases (such as P41 and P43). So, instead of the 54 affine space groups that preserve chirality, there are 54 + 11 = 65 space group types that preserve chirality (the ). For most chiral crystals, the two belong to the same crystallographic space group, such as P23 for FeSi, but for others, such as quartz, they belong to two enantiomorphic space groups. | |
| Two space groups, considered as subgroups of the group of affine transformations of space, have the same affine space group type if they are the same up to an affine transformation, even if that inverts orientation. The affine space group type is determined by the underlying abstract group of the space group. In three dimensions, Fifty-four of the affine space group types preserve chirality and give chiral crystals. The two enantiomorphs of a chiral crystal have the same affine space group. | |
| Sometimes called Z-classes. These are determined by the point group together with the action of the point group on the subgroup of translations. In other words, the arithmetic crystal classes correspond to conjugacy classes of finite subgroup of the general linear group GLn( Z) over the integers. A space group is called symmorphic (or split) if there is a point such that all symmetries are the product of a symmetry fixing this point and a translation. Equivalently, a space group is symmorphic if it is a semidirect product of its point group with its translation subgroup. There are 73 symmorphic space groups, with exactly one in each arithmetic crystal class. There are also 157 nonsymmorphic space group types with varying numbers in the arithmetic crystal classes.
Arithmetic crystal classes may be interpreted as different orientations of the point groups in the lattice, with the group elements' matrix components being constrained to have integer coefficients in lattice space. This is rather easy to picture in the two-dimensional, wallpaper group case. Some of the point groups have reflections, and the reflection lines can be along the lattice directions, halfway in between them, or both.
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| Sometimes called Q-classes. The crystal class of a space group is determined by its point group: the quotient by the subgroup of translations, acting on the lattice. Two space groups are in the same crystal class if and only if their point groups, which are subgroups of GLn( Z), are conjugate in the larger group GLn( Q). | These are determined by the underlying Bravais lattice type. These correspond to conjugacy classes of lattice point groups in GLn( Z), where the lattice point group is the group of symmetries of the underlying lattice that fix a point of the lattice, and contains the point group. |
| Crystal systems are an ad hoc modification of the lattice systems to make them compatible with the classification according to point groups. They differ from crystal families in that the hexagonal crystal family is split into two subsets, called the trigonal and hexagonal crystal systems. The trigonal crystal system is larger than the rhombohedral lattice system, the hexagonal crystal system is smaller than the hexagonal lattice system, and the remaining crystal systems and lattice systems are the same. | The lattice system of a space group is determined by the conjugacy class of the lattice point group (a subgroup of GLn( Z)) in the larger group GLn( Q). In three dimensions the lattice point group can have one of the 7 different orders 2, 4, 8, 12, 16, 24, or 48. The hexagonal crystal family is split into two subsets, called the rhombohedral and hexagonal lattice systems. |
| The point group of a space group does not quite determine its lattice system, because occasionally two space groups with the same point group may be in different lattice systems. Crystal families are formed from lattice systems by merging the two lattice systems whenever this happens, so that the crystal family of a space group is determined by either its lattice system or its point group. In 3 dimensions the only two lattice families that get merged in this way are the hexagonal and rhombohedral lattice systems, which are combined into the hexagonal crystal family. The 6 crystal families in 3 dimensions are called triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. Crystal families are commonly used in popular books on crystals, where they are sometimes called crystal systems. | |
gave another classification of the space groups, called a fibrifold notation, according to the [[fibrifold]] structures on the corresponding [[orbifold]]. They divided the 219 affine space groups into reducible and irreducible groups. The reducible groups fall into 17 classes corresponding to the 17 [[wallpaper group]]s, and the remaining 35 irreducible groups are the same as the cubic groups and are classified separately.
It is essential in Bieberbach's theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counter-example is given by the 3-dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3-dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z3.
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| 2 | 4 | 4 | 5 | 9 | 10 | 13 | 17 | 17 |
| 3 | 6 | 7 | 14 | 18 | 32 | 73 | 219 (+11) | 230 |
| 4 | 23 (+6) | 33 (+7) | 64 (+10) | 118 | 227 (+44) | 710 (+70) | 4783 (+111) | 4894 |
| 5 | 32 | 59 | 189 | 239 | 955 | 6079 | 222018 (+79) | 222097 |
| 6 | 91 | 251 | 841 | 1594 | 7103 | 85308 (+?) | 28927915 (+?) | ? |
| Oblique lattice | 1 | None | |||
| 2 | None | ||||
| Rectangular | 2 | Along | |||
| 4 | Along | ||||
| Centered rectangular | 2 | Between | |||
| 4 | Between | ||||
| Square lattice | 4 | None | |||
| 8 | Both | ||||
| Hexagonal | 3 | None | |||
| 6 | Between | ||||
| 6 | None | ||||
| 12 | Both | ||||
Note: An e plane is a double glide plane, one having glides in two different directions. They are found in seven orthorhombic, five tetragonal and five cubic space groups, all with centered lattice. The use of the symbol e became official with .
The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the rhombohedral lattice system consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system is also sometimes used as an alternative name for the whole trigonal system.) The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R.
The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the non-rhombohedral groups is P, I, F, A or C, standing for the principal, body centered, face centered, A-face centered or C-face centered lattices. There are seven rhombohedral space groups, with initial letter R.
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