Space group

From Freepedia

A space group is a mathematical symmetry group or symmetry group type of n-dimensional structures with translational symmetry in n independent directions, such as, for n = 3, a crystal. Only discrete symmetry groups are included in the categorization; i.e., infinitely fine structure or homogeneity in one or more directions is excluded.

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. 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.

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.

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 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 theorems state 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).

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.

In 1D there are two space group types: those with and without mirror image symmetry, see symmetry groups in one dimension.

In 2D there are 17; these 2D space groups are also called wallpaper groups or plane groups.

In 3D there are 230 crystallographic space group types, which reduces to 219 affine space group types because of some types differing only for the enantiomorphous character (e.g. P3112 and P3212). Usually "space group" refers to 3D. They are by themselves purely mathematical, but play a large role in crystallography.

The number of affine space group types in <math>n</math> dimensions is given by (sequence A004029 in OEIS); the number of crystallographic space group types in <math>n</math> dimensions is given by A006227.

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). Space groups can be grouped by the matrices involved, i.e. ignoring the translation vectors (see also Euclidean group). This corresponds to discrete symmetry groups with a fixed point: the point groups. However, only part of the point groups are compatible with translational symmetry: the crystallographic point groups. This is expressed in the crystallographic restriction theorem. (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. 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.

The space groups within each crystal class are characterized by the 14 Bravais lattices which belong to one of 7 crystal systems. This results in a space group being a combination of a unit cell with some form of motif centering, along with the point operations of reflection, rotation and improper-rotation. In addition, there are the translational symmetry elements. The basic translation is covered by the lattice type, leaving combinations of reflections and rotations with translation:

Contents

Screw axis

A rotation of 360°/n about an axis, combined with a translation along the axis. A subscript indicates the translation distance, as a multiple of the distance obtained by dividing that of the translational symmetry by n. So, 63 is a rotation of 60° combined with a translation of 1/2 of the lattice vector, implying that there is also 3-fold rotational symmetry about this axis. The possibilities are 21, 31, 41, 42, 61, 62, and 63, and the enantiomorphous 32, 43, 64, and 65.

Glide plane

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 along the fourth of a diagonal of a face of the unit cell.

Space groups are categorized by Bravais lattice and crystal class. 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.
  • 157 nonsymmorphic space group types.

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 schemes there are two main forms of notation, Paterson notation and Schoenflies.

Paterson notation consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, C, I or F). The next three describe the most prominent symmetry operation visible when projected from the a, b and c face respectively. These symbols are the same as used in point groups, with the addition of glide planes and screw axis, described above. By way of example, the space group for quartz is P3121, showing that it exhibits primitive centering of the motif (i.e. once per unit cell), with a threefold screw axis projecting on one face, and two fold rotation axis another. 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).

See also

External links

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