Space group
The space group of a crystal is a mathematical description of the symmetry inherent in the structure. The word 'group' in the name comes from the mathematical notion of a group, which is used to build the set of space groups.
The set of all 230 space groups is made from the combination of the 32 point_groups with 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 transalational symetry elements. The basic translation is covered by the lattice type, leaving combinations of reflections and rotations with translation:
Screw axis: 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 compete 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 parrallel lattice vector. So, 21 is a two-fold rotation followed by a translation of 1/2 of the lattice vector.
Glide plane: A relfection in a plane, followed by a translation parrallel with that plane. This is noted by a, b or c, depening on which axis the glide is along. There is also the n glide, which is a glide along a face, and the d glide, which is along the body diagonal of the unit cell.
It is easily noted that not all of the possible combinations of the Bravais_lattices, crystal_systems and point_groups are apparent in the space groups ( 32 * 14 = 448 < 230). This is because a number of different combinations are isomorphic with each other (that is, they turn out to be the same thing). This was proved using group theory, and is the source of the word 'group' in the title.
There are a number of methods of identifying space groups. The International Union of Crystallography publishes a table (more corrcetly, a hefty tome of tables) of all space groups, and assigns each a unique number. Other than this numbering schemse 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 prominant symetry operation visable when projected from the a, b and c face respectivly. These symbols are the same as used in point_groups, with the addition of glide planes and screw axis, desribed above. By way of example, the space group for quartz is P3121, showing that it exhibits primative 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 expicitly contain the crystal_system, although this is unique to each space group (in the case of P3121, it is trigonal).
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(In particular, Schoenflies notation is not described, and aught to be)
Referenced By
List of Lie group topics | List of physics topics R-Z
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