# 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 possible space groups in 3D is made from the combination of the 32 crystallographic 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 translational symmetry 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 parallel lattice vector. So, 21 is a two-fold rotation followed by a translation of 1/2 of the lattice vector.

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.

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 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).

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