Symmetry in Crystals

In addition to their inherent translational symmetry, most crystals display other kinds of symmetry. Since a crystal is periodic by translation, a symmetry element such as a rotation axis, mirror plane or inversion centre within a crystal must be repeated at all positions that are equivalent by translation. Hence, these symmetry elements are present as sets distributed regularly through the crystal. Describing the positions of the symmetry elements within the unit cell is an efficient description of the symmetry elements through the entire crystal. It is shown in this chapter how the symmetry operations produce relationships between atomic coordinates, and how this leads to further efficiency in the description of a crystal structure. In addition to point symmetry operations (rotations, mirrors and inversions), space symmetry operations (screws and glides) are introduced.