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In mathematics,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
and chemistry, a space group is the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
of an object in space, usually in three dimensions. The elements of a space group (its
symmetry operation In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in spac ...
s) are the
rigid transformation In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformatio ...
s of an object that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups. 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 ...
, 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'' .


History

Space groups in 2 dimensions are the 17
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformat ...
s 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 Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
. 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.


Elements

The space groups in three dimensions are made from combinations of the 32
crystallographic point group In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal un ...
s with 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, each of the latter belonging to one of 7
lattice system In crystallography, 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. Space groups are classified into crystal systems according to their poi ...
s. 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 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 necessari ...
(including lattice centering), the point group symmetry operations of reflection,
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
and
improper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendic ...
(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.


Elements fixing a point

The elements of the space group fixing a point of space are the identity element, reflections, rotations and
improper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendic ...
s.


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 In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the space group by the Bravais lattice is a finite group which is one of the 32 possible
point group In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...
s.


Glide planes

A glide plane is a reflection in a plane, followed by a translation parallel with that plane. This is noted by a, b, or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d 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 Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, ...
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:


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, ''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, 21 is a twofold rotation followed by a translation of 1/2 of the lattice vector.


General formula

The general formula for the action of an element of a space group is : ''y'' = ''M''.''x'' + ''D'' 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 In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...
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 The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 also adopt this structure, including α-tin, the se ...
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 group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformat ...
s * (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


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. ; Number: The International Union of Crystallography publishes tables of all space group types, and assigns each a unique number from 1 to 230. The numbering is arbitrary, except that groups with the same crystal system or point group are given consecutive numbers. ; Hall notation : Space group notation with an explicit origin. Rotation, translation and axis-direction symbols are clearly separated and inversion centers are explicitly defined. The construction and format of the notation make it particularly suited to computer generation of symmetry information. For example, group number 3 has three Hall symbols: P 2y (P 1 2 1), P 2 (P 1 1 2), P 2x (P 2 1 1). ;
Schönflies notation The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe th ...
: The space groups with given point group are numbered by 1, 2, 3, ... (in the same order as their international number) and this number is added as a superscript to the Schönflies symbol for the point group. For example, groups numbers 3 to 5 whose point group is ''C''2 have Schönflies symbols ''C'', ''C'', ''C''. ; Coxeter notation: Spatial and point symmetry groups, represented as modifications of the pure reflectional
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
s. ; Geometric notation : A geometric algebra notation.


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. gave another classification of the space groups, called a fibrifold notation, according to the fibrifold structures on the corresponding
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
. 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 A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformat ...
s, and the remaining 35 irreducible groups are the same as the cubic groups and are classified separately.


In other dimensions


Bieberbach's theorems

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. proved that the subgroup of translations of any such group contains ''n'' linearly independent translations, and is a free
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
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. showed that conversely any group that is the extension of Z''n'' by a finite group acting faithfully is an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
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 Z3.


Classification in small dimensions

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.


Magnetic groups and time reversal

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 In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. ...
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 . It has also been possible to construct magnetic versions for other overall and lattice dimensions
Daniel Litvin's papers
, ). 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:


Table of space groups in 2 dimensions (wallpaper groups)

Table of the
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformat ...
s using the classification of the 2-dimensional space groups: For each geometric class, the possible arithmetic classes are * None: no reflection lines * Along: reflection lines along lattice directions * Between: reflection lines halfway in between lattice directions * Both: reflection lines both along and between lattice directions


Table of space groups in 3 dimensions

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 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 ...
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.


Derivation of the crystal class from the space group

# Leave out the Bravais type # Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry (Glide planes are converted into simple mirror planes; Screw axes are converted into simple axes of rotation) # Axes of rotation, rotoinversion axes and mirror planes remain unchanged.


References

* * * * * * * * ** English translation: * * * * * * * * * * * * * * * * *


External links


International Union of Crystallography

Point Groups and Bravais Lattices


Bilbao Crystallographic Server
Space Group Info (old)



Crystal Lattice Structures: Index by Space Group




* * ttp://www.geom.uiuc.edu/docs/reference/CRC-formulas/node9.html The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)
The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)
{{Authority control Symmetry Crystallography Finite groups Discrete groups