The Effects of Symmetry in Real and Reciprocal Space

Abstract

Normally, we think of symmetry in real space, for example the mirror symmetry relating our left and right hands, or the 4-fold symmetry of ordinary bath-room tiles. A crystal also has symmetry and its symmetry in real space has effects in reciprocal space, i.e. in its diffraction pattern. Conversely, any symmetry present in a diffraction pattern provides information about the symmetry of the crystal itself in real space. In real space a symmetry manifests itself as two or more identical molecules or groups of atoms (a motif) being present within the unit cell. This symmetry may be only local, or it can be global which means all atoms in all unit cells are related by the same symmetry element(s). A global symmetry is called crystallographic symmetry, while the local symmetry is called non-crystallographic. Local symmetries are not unusual in crystals and they can be very important and useful, but they fall outside the scope of this article.