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 would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in a primitive cubic crystal system, a rotation of the unit cell by 90 degree around an axis that is perpendicular to two parallel faces of the cube, intersecting at its center, is a symmetry operation that projects each atom to the location of one of its neighbor leaving the overall structure of the crystal unaffected. In the classification of crystals, each point group defines a so-called crystal class. There are infinitely many three-dimensional point groups. However, the crystallographic restriction on the general point groups results in there being only 32 crystallographic point groups. These 32 point groups are one-and-the-same as the 32 types of morphological crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms. The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including optical properties such as birefringency, or electro-optical features such as the Pockels effect. For a periodic crystal, the group must maintain the three-dimensional translational symmetry that defines crystallinity.
Notation
The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, mineralogists, and physicists. For the correspondence of the two systems below, see crystal system.
In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:
Cn indicates that the group has an n-fold rotation axis. Cnh is Cn with the addition of a mirror plane perpendicular to the axis of rotation. Cnv is Cn with the addition of n mirror planes parallel to the axis of rotation.
Dn indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. Dnh has, in addition, a mirror plane perpendicular to the n-fold axis. Dnd has, in addition to the elements of Dn, mirror planes parallel to the n-fold axis.
The letter T indicates that the group has the symmetry of a tetrahedron. Td includes improper rotation operations, T excludes improper rotation operations, and Th is T with the addition of an inversion.
The letter O indicates that the group has the symmetry of an octahedron, with or without improper operations.
D4d and D6d are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plusT, Td, Th, O and Oh constitute 32 crystallographic point groups.
An abbreviated form of the Hermann–Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are
The correspondence between different notations
Deriving the crystallographic point group (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
Axes of rotation, rotoinversion axes and mirror planes remain unchanged.
