How is the crystal class of a space group determined?
(geometric) Crystal classes (32 in three dimensions). Sometimes called Q-classes. The crystal class of a space group is determined by its point group: the quotient by the subgroup of translations, acting on the lattice.
Why is the determination of space group symmetry important?
The determination of space-group symmetry of material is an essential step in structure analysis since it minimises the amount of information needed for the complete description of the contents of the unit cell.
How are space groups studied in other dimensions?
Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space . In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal.
How is symmetry used to describe a crystal structure?
Thus symmetry can be used to shorten considerably the description of a crystal structure; more importantly, a knowledge of the crystal symmetry reduces the number of atomic coordinates that have to be found in the determination of a crystal structure.
How are space groups numbered with a point group?
The space groups with given point group are numbered by 1, 2, 3, … (in the same order as their international number) and this number is added as a superscript to the Schönflies symbol for the point group. For example, groups numbers 3 to 5 whose point group is C2 have Schönflies symbols C1
Can a space group have the same affine group type?
Two space groups, considered as subgroups of the group of affine transformations of space, have the same affine space group type if they are conjugate under an affine transformation. The affine space group type is determined by the underlying abstract group of the space group.