In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions.^{[1]} 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 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. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography Hahn (2002).
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.^{[2]}
In 1879 the German mathematician Leonhard Sohncke listed the 65 space groups (called Sohncke groups) whose elements preserve the chirality.^{[3]} 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^{[4]} (whose list had two omissions (I43d and Fdd2) and one duplication (Fmm2)), and shortly afterwards in 1891 were independently enumerated by Schönflies^{[5]} (whose list had four omissions (I43d, Pc, Cc, ?) and one duplication (P42_{1}m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies.^{[6]} William Barlow (1894) later enumerated the groups with a different method, but omitted four groups (Fdd2, I42d, P42_{1}d, and P42_{1}c) 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.^{[citation needed]} Burckhardt (1967) describes the history of the discovery of the space groups in detail.
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 Fm3m, the NaCl structure.
The elements of the space group fixing a point of space are the identity element, reflections, rotations and improper rotations, including inversion points.
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.
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 b or c, a or b, a or c. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb. In 1992, it was suggested to use symbol e 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 |
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, n, 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_{1} is a twofold rotation followed by a translation of 1/2 of the lattice vector.
The general formula for the action of an element of a space group is
where M is its matrix, D is its vector, and where the element transforms point x into point y. In general, D = D (lattice) + D(M), where D(M) 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):
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.
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.
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.
The viewing directions of the 7 crystal systems are shown as follows.
Position in the symbol | Triclinic | Monoclinic | Orthorhombic | Tetragonal | Trigonal | Hexagonal | Cubic |
---|---|---|---|---|---|---|---|
1 | — | b | a | c | c | c | a |
2 | — | b | a | a | a | [111] | |
3 | — | c | [110] | [210] | [210] | [110] |
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_{1} and P4_{3}). So instead of the 54 affine space groups that preserve chirality there are 54 + 11 = 65 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 P2_{1}3 for FeSi,^{[10]} but for others, such as quartz, they 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_{n}(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.
| |
(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_{n}(Z), are conjugate in the larger group GL_{n}(Q). | These are determined by the underlying Bravais lattice type.
These correspond to conjugacy classes of lattice point groups in GL_{n}(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 (7 in three dimensions) | Lattice systems (7 in three dimensions) |
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 GL_{n}(Z)) in the larger group GL_{n}(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. |
Crystal families (6 in three dimensions) | |
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. |
Conway, Delgado Friedrichs, and Huson et al. (2001) 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 groups, and the remaining 35 irreducible groups are the same as the cubic groups and are classified separately.
In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n-dimensional Euclidean space with a compact fundamental domain. Bieberbach (1911, 1912) proved that the subgroup of translations of any such group contains n linearly independent translations, and is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert's eighteenth problem. Zassenhaus (1948) showed that conversely any group that is the extension^{[when defined as?]} of Z^{n} by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of Z^{n} by a finite group acting faithfully.
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 Z^{3}.
This table gives the number of space group types in small dimensions, including the numbers of various classes of space group. The numbers of enantiomorphic pairs are given in parentheses.
Dimensions | Crystal families, OEIS sequence A004032 | Crystal systems, OEIS sequence A004031 | Bravais lattices, OEIS sequence A256413 | Abstract crystallographic point groups, OEIS sequence A006226 | Geometric crystal classes, Q-classes, crystallographic point groups, OEIS sequence A004028 | Arithmetic crystal classes, Z-classes, OEIS sequence A004027 | Affine space group types, OEIS sequence A004029 | Crystallographic space group types, OEIS sequence A006227 |
---|---|---|---|---|---|---|---|---|
0^{[a]} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1^{[b]} | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
2^{[c]} | 4 | 4 | 5 | 9 | 10 | 13 | 17 | 17 |
3^{[d]} | 6 | 7 | 14 | 18 | 32 | 73 | 219 (+11) | 230 |
4^{[e]} | 23 (+6) | 33 (+7) | 64 (+10) | 118 | 227 (+44) | 710 (+70) | 4783 (+111) | 4894 |
5^{[f]} | 32 | 59 | 189 | 239 | 955 | 6079 | 222018 (+79) | 222097 |
6^{[g]} | 91 | 251 | 841 | 1594 | 7103 | 85308 (+?) | 28927915 (+?) | ? |
In addition to crystallographic space groups there are also magnetic space groups (also called two-color (black and white) crystallographic groups or Shubnikov groups). These symmetries contain an element known as time reversal. They treat time as an additional dimension, and the group elements can include time reversal as reflection in it. They are of importance in magnetic structures that contain ordered unpaired spins, i.e. ferro-, ferri- or antiferromagnetic structures as studied by neutron diffraction. The time reversal element flips a magnetic spin while leaving all other structure the same and it can be combined with a number of other symmetry elements. Including time reversal there are 1651 magnetic space groups in 3D (Kim 1999, p.428). It has also been possible to construct magnetic versions for other overall and lattice dimensions (Daniel Litvin's papers, (Litvin 2008), (Litvin 2005)). Frieze groups are magnetic 1D line groups and layer groups are magnetic wallpaper groups, and the axial 3D point groups are magnetic 2D point groups. Number of original and magnetic groups by (overall, lattice) dimension:(Palistrant 2012)(Souvignier 2006)
Overall dimension |
Lattice dimension |
Ordinary groups | Magnetic groups | |||
---|---|---|---|---|---|---|
Name | Symbol | Count | Symbol | Count | ||
0 | 0 | Zero-dimensional symmetry group | 1 | 2 | ||
1 | 0 | One-dimensional point groups | 2 | 5 | ||
1 | One-dimensional discrete symmetry groups | 2 | 7 | |||
2 | 0 | Two-dimensional point groups | 10 | 31 | ||
1 | Frieze groups | 7 | 31 | |||
2 | Wallpaper groups | 17 | 80 | |||
3 | 0 | Three-dimensional point groups | 32 | 122 | ||
1 | Rod groups | 75 | 394 | |||
2 | Layer groups | 80 | 528 | |||
3 | Three-dimensional space groups | 230 | 1651 | |||
4 | 0 | Four-dimensional point groups | 271 | 1202 | ||
1 | 343 | |||||
2 | 1091 | |||||
3 | 1594 | |||||
4 | Four-dimensional discrete symmetry groups | 4894 | 62227 |
Table of the wallpaper groups using the classification of the 2-dimensional space groups:
Crystal system, Bravais lattice |
Geometric class, point group | Arithmetic class |
Wallpaper groups (cell diagram) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Int'l | Schön. | Orbifold | Cox. | Ord. | ||||||
Oblique |
1 | C_{1} | (1) | [ ]^{+} | 1 | None | p1 (1) |
|||
2 | C_{2} | (22) | [2]^{+} | 2 | None | p2 (2222) |
||||
Rectangular |
m | D_{1} | (*) | [ ] | 2 | Along | pm (**) |
pg (××) |
||
2mm | D_{2} | (*22) | [2] | 4 | Along | pmm (*2222) |
pmg (22*) |
|||
Centered rectangular |
m | D_{1} | (*) | [ ] | 2 | Between | cm (*×) |
|||
2mm | D_{2} | (*22) | [2] | 4 | Between | cmm (2*22) |
pgg (22×) |
|||
Square |
4 | C_{4} | (44) | [4]^{+} | 4 | None | p4 (442) |
|||
4mm | D_{4} | (*44) | [4] | 8 | Both | p4m (*442) |
p4g (4*2) |
|||
Hexagonal |
3 | C_{3} | (33) | [3]^{+} | 3 | None | p3 (333) |
|||
3m | D_{3} | (*33) | [3] | 6 | Between | p3m1 (*333) |
p31m (3*3) |
|||
6 | C_{6} | (66) | [6]^{+} | 6 | None | p6 (632) |
||||
6mm | D_{6} | (*66) | [6] | 12 | Both | p6m (*632) |
For each geometric class, the possible arithmetic classes are
№ | Crystal system, (count), Bravais lattice |
Point group | Space groups (international short symbol) | ||||
---|---|---|---|---|---|---|---|
Int'l | Schön. | Orbifold | Cox. | Ord. | |||
1 | Triclinic (2) |
1 | C_{1} | 11 | [ ]^{+} | 1 | P1 |
2 | 1 | C_{i} | 1× | [2^{+},2^{+}] | 2 | P1 | |
3–5 | Monoclinic (13) |
2 | C_{2} | 22 | [2]^{+} | 2 | P2, P2_{1} C2 |
6–9 | m | C_{s} | *11 | [ ] | 2 | Pm, Pc Cm, Cc | |
10–15 | 2/m | C_{2h} | 2* | [2,2^{+}] | 4 | P2/m, P2_{1}/m C2/m, P2/c, P2_{1}/c C2/c | |
16–24 | Orthorhombic (59) |
222 | D_{2} | 222 | [2,2]^{+} | 4 | P222, P222_{1}, P2_{1}2_{1}2, P2_{1}2_{1}2_{1}, C222_{1}, C222, F222, I222, I2_{1}2_{1}2_{1} |
25–46 | mm2 | C_{2v} | *22 | [2] | 4 | Pmm2, Pmc2_{1}, Pcc2, Pma2, Pca2_{1}, Pnc2, Pmn2_{1}, Pba2, Pna2_{1}, Pnn2 Cmm2, Cmc2_{1}, Ccc2, Amm2, Aem2, Ama2, Aea2 Fmm2, Fdd2 Imm2, Iba2, Ima2 | |
47–74 | mmm | D_{2h} | *222 | [2,2] | 8 | Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce Fmmm, Fddd Immm, Ibam, Ibca, Imma | |
75–80 | Tetragonal (68) |
4 | C_{4} | 44 | [4]^{+} | 4 | P4, P4_{1}, P4_{2}, P4_{3}, I4, I4_{1} |
81–82 | 4 | S_{4} | 2× | [2^{+},4^{+}] | 4 | P4, I4 | |
83–88 | 4/m | C_{4h} | 4* | [2,4^{+}] | 8 | P4/m, P4_{2}/m, P4/n, P4_{2}/n I4/m, I4_{1}/a | |
89–98 | 422 | D_{4} | 224 | [2,4]^{+} | 8 | P422, P42_{1}2, P4_{1}22, P4_{1}2_{1}2, P4_{2}22, P4_{2}2_{1}2, P4_{3}22, P4_{3}2_{1}2 I422, I4_{1}22 | |
99–110 | 4mm | C_{4v} | *44 | [4] | 8 | P4mm, P4bm, P4_{2}cm, P4_{2}nm, P4cc, P4nc, P4_{2}mc, P4_{2}bc I4mm, I4cm, I4_{1}md, I4_{1}cd | |
111–122 | 42m | D_{2d} | 2*2 | [2^{+},4] | 8 | P42m, P42c, P42_{1}m, P42_{1}c, P4m2, P4c2, P4b2, P4n2 I4m2, I4c2, I42m, I42d | |
123–142 | 4/mmm | D_{4h} | *224 | [2,4] | 16 | P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4_{2}/mmc, P4_{2}/mcm, P4_{2}/nbc, P4_{2}/nnm, P4_{2}/mbc, P4_{2}/mnm, P4_{2}/nmc, P4_{2}/ncm I4/mmm, I4/mcm, I4_{1}/amd, I4_{1}/acd | |
143–146 | Trigonal (25) |
3 | C_{3} | 33 | [3]^{+} | 3 | P3, P3_{1}, P3_{2} R3 |
147–148 | 3 | S_{6} | 3× | [2^{+},6^{+}] | 6 | P3, R3 | |
149–155 | 32 | D_{3} | 223 | [2,3]^{+} | 6 | P312, P321, P3_{1}12, P3_{1}21, P3_{2}12, P3_{2}21 R32 | |
156–161 | 3m | C_{3v} | *33 | [3] | 6 | P3m1, P31m, P3c1, P31c R3m, R3c | |
162–167 | 3m | D_{3d} | 2*3 | [2^{+},6] | 12 | P31m, P31c, P3m1, P3c1 R3m, R3c | |
168–173 | Hexagonal (27) |
6 | C_{6} | 66 | [6]^{+} | 6 | P6, P6_{1}, P6_{5}, P6_{2}, P6_{4}, P6_{3} |
174 | 6 | C_{3h} | 3* | [2,3^{+}] | 6 | P6 | |
175–176 | 6/m | C_{6h} | 6* | [2,6^{+}] | 12 | P6/m, P6_{3}/m | |
177–182 | 622 | D_{6} | 226 | [2,6]^{+} | 12 | P622, P6_{1}22, P6_{5}22, P6_{2}22, P6_{4}22, P6_{3}22 | |
183–186 | 6mm | C_{6v} | *66 | [6] | 12 | P6mm, P6cc, P6_{3}cm, P6_{3}mc | |
187–190 | 6m2 | D_{3h} | *223 | [2,3] | 12 | P6m2, P6c2, P62m, P62c | |
191–194 | 6/mmm | D_{6h} | *226 | [2,6] | 24 | P6/mmm, P6/mcc, P6_{3}/mcm, P6_{3}/mmc | |
195–199 | Cubic (36) |
23 | T | 332 | [3,3]^{+} | 12 | P23, F23, I23 P2_{1}3, I2_{1}3 |
200–206 | m3 | T_{h} | 3*2 | [3^{+},4] | 24 | Pm3, Pn3, Fm3, Fd3, Im3, Pa3, Ia3 | |
207–214 | 432 | O | 432 | [3,4]^{+} | 24 | P432, P4_{2}32 F432, F4_{1}32 I432 P4_{3}32, P4_{1}32, I4_{1}32 | |
215–220 | 43m | T_{d} | *332 | [3,3] | 24 | P43m, F43m, I43m P43n, F43c, I43d | |
221–230 | m3m | O_{h} | *432 | [3,4] | 48 | Pm3m, Pn3n, Pm3n, Pn3m Fm3m, Fm3c, Fd3m, Fd3c Im3m, Ia3d |
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 Hahn (2002).
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.