Hermann–Mauguin notation

Summary

The Hermann–Mauguin (or international) notation describes the lattice and some generators for the group. It has a shortened form called the international short symbol, which is the one most commonly used in crystallography, and usually consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, A, C, I, R or F). The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. 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 of quartz is P3${}_{1}$21, showing that it exhibits primitive centering of the motif (i.e., once per unit cell), with a threefold screw axis and a twofold rotation axis. Note that it does not explicitly contain the crystal system, although this is unique to each space group (in the case of P3${}_{1}$21, it is trigonal).

In the international short symbol the first symbol (3${}_{1}$ in this example) denotes the symmetry along the major axis (c-axis in trigonal cases), the second (2 in this case) along axes of secondary importance (a and b) and the third symbol the symmetry in another direction. In the trigonal case there also exists a space group P3${}_{1}$12. In this space group the twofold axes are not along the a and b-axes but in a direction rotated by 30°.

The international symbols and international short symbols for some of the space groups were changed slightly between 1935 and 2002, so several space groups have 4 different international symbols in use.

The viewing directions of the 7 crystal systems are shown as follows. - ! Symbol position ! Triclinic ! Monoclinic ! Orthorhombic ! Tetragonal ! Trigonal ! Hexagonal ! Cubic