Space group

The space group of a crystal or crystallographic group is a mathematical description of the symmetry inherent in the structure. In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The word 'group' in the name comes from the mathematical notion of a group, which is used to build the set of space groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

1 Crystallography
- 1.1 Glide planes and screw axes
- 1.2 Notation
2 Group theory
3 See also
4 Bibliography
5 External links

Crystallography

The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 crystal systems. In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation A crystal system is a category of Space groups which characterize Symmetry of structures in three dimensions with Translational symmetry in three directions This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering, and the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion). In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or Furthermore one must consider the screw axis and glide plane symmetry operations. The screw axis ( helical axis or twist axis) of an Object is a Parameter for describing Simultaneous Rotation and Translation In Crystallography, a glide plane is symmetry operation describing how a reflection in a plane followed by a translation parallel with that plane may These are called compound symmetry operations and are combinations of a rotation or reflection with a translation less than the unit cell size. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.

Glide planes and screw axes

Two of the symmetry operations involved in the space groups are not contained in the corresponding point group or Bravais lattice. These are the compound symmetry operations called the glide plane and the screw axis. In Crystallography, a glide plane is symmetry operation describing how a reflection in a plane followed by a translation parallel with that plane may The screw axis ( helical axis or twist axis) of an Object is a Parameter for describing Simultaneous Rotation and Translation

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 often called the diamond glide plane as it features in the diamond structure.

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.

Notation

There are a number of methods of identifying space groups. The International Union of Crystallography publishes a table (more correctly, a hefty tome of tables) of all space groups, and assigns each a unique number. Other than this numbering scheme there are two main forms of notation, the Hermann-Mauguin notation and Schönflies notation. Hermann-Mauguin notation is used to represent the Symmetry elements in Point groups Plane groups and Space groups It is named after the German In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom

The Hermann-Mauguin (or international) notation is the one most commonly used in crystallography, and consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, A, B, C, I, R or F). In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. These symbols are the same as used in point groups, with the addition of glide planes and screw axis, described above. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed By way of example, the space group of quartz is P3₁21, showing that it exhibits primitive centering of the motif (i. Quartz (from German) is the most abundant Mineral in the Earth 's Continental crust (although Feldspar is more common in e. , once per unit cell), with a threefold screw axis and a twofold rotation axis. Note that it does not explicitly contain the crystal system, although this is unique to each space group (in the case of P3₁21, it is trigonal). A crystal system is a category of Space groups which characterize Symmetry of structures in three dimensions with Translational symmetry in three directions

In HM notation the first symbol (3₁ in this example) denotes the symmetry along the major axis (c-axis in trigonal cases), the second (2 in this case) along axes of secondary importance (a and b) and the third symbol the symmetry in another direction. In the trigonal case there also exists a space group P3₁12. In this space group the twofold axes are not along the a and b-axes but in a direction rotated by 30^o.

Group theory

Mathematically, a space group is a symmetry group or symmetry group type of n-dimensional structures with translational symmetry in n independent directions, such as, for n = 3, a crystal. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Geometry, a translation "slides" an object by a vector a: T a (p = p + a In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating Only discrete symmetry groups are included in the categorization; i. e. , infinitely fine structure or homogeneity in one or more directions is excluded. This comes from the necessity to describe discrete sets of 'points' (i. e. atoms or ions in a crystal), as opposed to continuous media (see Symmetry in physics for the latter case). Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these See the articles Bravais lattices, Crystals, and Translation (geometry) for a fuller discussion. In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating In Euclidean geometry, a translation is moving every point a constant distance in a specified direction

Two symmetry groups are of the same crystallographic space group type if they are the same up to an affine transformation of space that preserves orientation. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which 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. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is

Two symmetry groups are of the same affine space group type if they are the same up to an affine transformation, even if that inverts orientation. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector

This can be expressed by saying that two symmetry groups which are chiral and each other's mirror image, are of different crystallographic space group type, but of the same affine space group type. In Geometry, a figure is chiral (and said to have chirality) if it is not identical to its Mirror image, or more particularly if it cannot be mapped to "Mirror Image" is an episode of the Television series The Twilight Zone.

In 1D and 2D space groups of the same affine space group type are also of the same crystallographic space group type, but in 3D this need not be the case: in 2D, the mirror image of a rotation is a reversed rotation, which is in the group anyway, and the mirror image of a mirror is still a mirror, but the mirror image of a righthand screw operation is a lefthand one, not the inverse of the righthand screw operation.

The Bieberbach theorem states that in each dimension all affine space group types are different even as abstract groups (as opposed to e. g. Frieze groups, of which two are isomorphic with Z). A frieze group is a mathematical concept to classify designs on Two-dimensional surfaces which are repetitive in one direction based on the symmetries in the pattern

The term "space group" is often used for space group type. It is often clear from the context what is meant. However, when considering subgroup relationships a specific symmetry group should not be confused with the space group type.

Various dimensions

In 1D there are two space group types: those with and without mirror image symmetry, see symmetry groups in one dimension. A one-dimensional symmetry group is a mathematical group that describe symmetries in one dimension

In 2D there are 17; these 2D space groups are also called wallpaper groups or plane groups. A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the

In 3D there are 230 crystallographic space group types, which reduces to 219 affine space group types because of some types being different from their mirror image; these are said to differ by "enantiomorphous character" (e. In Geometry, a figure is chiral (and said to have chirality) if it is not identical to its Mirror image, or more particularly if it cannot be mapped to g. P3₁12 and P3₂12). Usually "space group" refers to 3D. They are by themselves purely mathematical, but play a large role in crystallography. Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The

In 4 dimensions there are 4895 crystallographic space group types, or 4783 affine space group types. ^[1]

The number of affine space group types in $n$ dimensions is given by sequence A004029 in OEIS; the number of crystallographic space group types in $n$ dimensions is given by A006227. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

Double groups and time reversal

In addition to crystallographic space groups there are also magnetic space groups or double groups. These symmetries contain an element known as time reversal. They are of importance in magnetic structures that contain ordered unpaired spins, i. The term magnetic structure of a material pertains to the ordered arrangement of magnetic spins typically within an ordered Crystallographic lattice. e. ferro-, ferri- or antiferromagnetic structures as studied by neutron diffraction. Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it In Physics, a ferrimagnetic material is one in which the Magnetic moment of the atoms on different sublattices are opposed as in Antiferromagnetism; however In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules usuallyrelated to the spins of Electrons align in a regular pattern with neighboring Neutron diffraction is a crystallographic method for the determination of the atomic and/or magnetic structure of a material 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. ^[2]

Grouping by point group

A symmetry group consists of isometric affine transformations; each is given by an orthogonal matrix and a translation vector (which may be the zero vector). In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T Space groups can be grouped by the matrices involved, i. e. ignoring the translation vectors (see also Euclidean group). In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional This corresponds to discrete symmetry groups with a fixed point: the point groups. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed However, not all point groups are compatible with translational symmetry; those that are compatible are called the crystallographic point groups. This is expressed in the crystallographic restriction theorem. The crystallographic restriction theorem in its basic form was based on the observation that the Rotational symmetries of a Crystal are usually limited to (In spite of these names, this is a geometric limitation, not just a physical one. )

In 1D both space group types correspond to their own "crystallographic point group".

In 2D the 17 wallpaper groups are grouped according to 10 associated crystallographic point groups: 1-, 2-, 3-, 4-, and 6-fold rotational symmetry, each with or without reflections. Thus a wallpaper group with glide reflection axes is associated with the same point group as the wallpaper group with reflection axes parallel to these glide reflection axes.

In 3D this gives a grouping of the 230 space group types into 32 crystal classes, one for each associated crystallographic point group. In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom A space group with a screw axis is in the same crystal class as one with a corresponding pure axis of rotation. Similarly a space group with a glide plane is in the same crystal class as one with a corresponding pure reflection.

In addition to translations, and the point operations of reflection, rotation and improper rotation, there are combinations of reflections and rotations with translation: the screw axis and the glide plane. The screw axis ( helical axis or twist axis) of an Object is a Parameter for describing Simultaneous Rotation and Translation In Crystallography, a glide plane is symmetry operation describing how a reflection in a plane followed by a translation parallel with that plane may

The number of crystallographic point groups in n dimensions is given by OEIS:A004028.

Further categorizing

Space groups are categorized by Bravais lattice and crystal class. In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom However, for some combinations there are multiple space groups, while other combinations are not possible.

The 230 space group types can be subdivided in two categories:

73 symmorphic space group types: a space group is symmorphic if all symmetries can be described in terms of rotation axes and reflection planes all through the same point (including rotoreflections), without screw axes and glide planes). Equivalently, a space group is symmorphic if it is equivalent to a semidirect product of its point group with its translation subgroup. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can
157 nonsymmorphic space group types.

Conway and Thurston gave another classification of the space groups, where they divided the 230 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 classified separately. A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the

Bibliography

International Tables for Crystallography, Volume A, edited by Th. For the Mechanical engineering and Architecture usage see Isometric projection. Triclinic system Monoclinic system Orthorhombic system Tetragonal system Rhombohedral (trigonal system Yevgraf Stepanovich Fyodorov, sometimes spelled Evgraf Stepanovich Fedorov (Евгра́ф Степа́нович Фёдоров ( December 22, 1853 &ndash Hahn. Reidel Publishing Company, Dordrecht, Boston, 1996.
Conway, John H. ; Delgado Friedrichs, Olaf; Huson, Daniel H. ; Thurston, William P. On three-dimensional space groups. Beiträge Algebra Geom. 42 (2001), no. 2, 475--507. From the summary: "An entirely new and independent enumeration of the crystallographic space groups is given, based on obtaining the groups as fibrations over the plane crystallographic groups, when this is possible. "

^ H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978, p. 52.
^ p. 428 Group Theoretical Methods and Applications to Molecules and Crystals. By Shoon Kyung Kim. 1999. Cambridge University. Press. ISBN 0521640628

External links

Dictionary

-noun

The set of all symmetry operations that can be applied to a given crystal without changing it.

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