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. Usually the term is applied to groups that repeat in all three dimensions, but the line groups and the wallpaper groups also apply, or can apply, to three-dimensional arrangements. The space groups that repeat in all three dimensions are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries 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 Fedorov groups, and represent a description of the symmetry of the crystal. Large collected sources of this information are available.
History
Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only given in 1891, after the much more difficult classification of space groups had largely been completed.
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 (P2m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies. William Barlow later enumerated the groups with a different method, but omitted four groups (Fdd2, I2d, P2d, and P2c). A detailed description of the history of the discovery of the space groups has been written.
Elements The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, each of the latter belonging to one of 7 lattice systems. What this means is that the action of any element of a given space group can be expressed as the action of an element of the appropriate point group followed optionally by a translation. A space group is thus some combination of the translational symmetry of a unit cell (including lattice centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries.
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.
Elements fixing a point The elements of the space group fixing a point of space are the identity element, reflections, rotations and improper rotations, including inversion points.
Translations The translations form a normal abelian subgroup of rank 3, called the Bravais lattice (so named after French physicist Auguste Bravais). There are 14 possible types of Bravais lattice. The quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups.
Glide planes A glide plane is a reflection in a plane, followed by a translation parallel with that plane. This is noted by , , or , depending on which axis the glide is along. There is also the glide, which is a glide along the half of a diagonal of a face, and the glide, which is a fourth of the way along either a face or space diagonal of the unit cell. The latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two perpendicular directions simultaneously, i.e. the same glide plane can be called or , or , or . For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb. In 1992, it was suggested to use symbol for such planes. The symbols for five space groups have been modified: Space group no. 39 41 64 67 68 New symbol Aem2 Aea2 Cmce Cmme Ccce Old Symbol Abm2 Aba2 Cmca Cmma Ccca
Screw axes A screw axis is a rotation about an axis, followed by a translation along the direction of the axis. These are noted by a number, , to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e.g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So, 2 is a twofold rotation followed by a translation of of the lattice vector.
General formula The general formula for the action of an element of a space group is
where is its matrix, is its vector, and where the element transforms point into point . In general, , where is a unique function of that is zero for being the identity. The matrices 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): (1,1): One-dimensional line groups (2,1): Two-dimensional line groups: frieze groups (2,2): Wallpaper groups (3,1): Three-dimensional line groups; with the 3D crystallographic point groups, the rod groups (3,2): Layer groups (3,3): The space groups discussed in this article
Chirality The 65 "Sohncke" space groups, not containing any mirrors, inversion points, improper rotations or glide planes, yield chiral crystals, not identical to their mirror image; whereas space groups that do include at least one of those give achiral crystals. Achiral molecules sometimes form chiral crystals, but chiral molecules always form chiral crystals, in one of the space groups that permit this.
Among the 65 Sohncke groups are 22 that come in 11 enantiomorphic pairs.
Combinations Only certain combinations of symmetry elements are possible in a space group. Translations are always present, and the space group P1 has only translations and the identity element. The presence of mirrors implies glide planes as well, and the presence of rotation axes implies screw axes as well, but the converses are not true. An inversion and a mirror implies two-fold screw axes, and so on.
Notation
There are at least ten methods of naming space groups. Some of these methods can assign several different names to the same space group, so altogether there are many thousands of different names.
Classification systems There are (at least) 10 different ways to classify space groups into classes. The relations between some of these are described in the following table. Each classification system is a refinement of the ones below it. To understand an explanation given here it may be necessary to understand the next one down.
(Crystallographic) space group types (230 in three dimensions) 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 P4 and P4). So, instead of the 54 affine space groups that preserve chirality, there are space group types that preserve chirality (the Sohncke groups). For most chiral crystals, the two enantiomorphs belong to the same crystallographic space group, such as P23 for FeSi, but others, such as quartz belong to two enantiomorphic space groups. Affine space group types (219 in three dimensions) 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. Arithmetic crystal classes (73 in three dimensions) 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 GL(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. None: C: p1; C: p2; C: p3; C: p4; C: p6 Along: D: pm, pg; D: pmm, pmg, pgg; D: p31m Between: D: cm; D: cmm; D: p3m1 Both: D: p4m, p4g; D: p6m (geometric) Crystal classes (32 in three dimensions) Bravais flocks (14 in three dimensions) 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 GL(Z), are conjugate in the larger group GL(Q). These are determined by the underlying Bravais lattice type.