Hermann-Mauguin symbols

Symboles d'Hermann-Mauguin　(Fr); Hermann-Mauguin-Symbolik　(Ge); Símbolos de Hermann-Mauguin　(Sp); Simboli di Hermann-Mauguin　(It);. Символика Германа — Могена　(Ru); ヘルマン・モーガン記号　(Ja).

Hermann-Mauguin symbols, also known as international symbols, are oriented symbols giving the symmetry operations or symmetry elements of a point or space group. Three types of Hermann-Mauguin are used: short, full and, for space groups, extended. The three types of symbols represent different levels of information content with respect to the symmetry elements and the related symmetry operations of the space group.

The short and full Hermann-Mauguin symbols only display information about a chosen set of generators for a space group from which all elements of a space group can in principle be deduced. Both the short and the full Hermann-Mauguin symbols consist of two parts:

• a letter indicating the centring type of the conventional cell, and
• a set of characters indicating symmetry elements of the space group parallel or perpendicular to the symmetry directions of the lattice of the space group.

In general, the symbol of symmetry elements shown in the Hermann-Mauguin symbols are selected according to the so-called priority rule which requires that rotation axes are chosen over screw axes of the same rotational order, when these coexist along the same symmetry direction, and mirror planes are chosen in the sequence m > e > a, b, c > n (where > means “has priority over”). A few exceptions occur when these rules would not allow differentiating space-group types bringing the same type of symmetry elements but differently positioned in space, like I222 (No. 23) and I2₁2₁2₁ (No. 24), where the rotation axes cross in a point for the former but not for the latter.

Short and full Hermann-Mauguin symbols coincide for most space groups with the exception of the monoclinic space groups (the full symbol explicitly states the symmetry direction: for example, Pc short symbol vs. P1c1 full symbol), the space groups of the holohedries (with the exception of P-1) and of the geometric crystal class m-3 (for example, Pa-3 short symbol vs. P2₁/a-3 full symbol). Short symbols give a reduced set of symmetry elements about the symmetry directions of the lattice, where reduced means that some of the symmetry elements along certain symmetry directions may not be explicitly shown.

In the orthorhombic or higher crystal systems when a symmetry axis of order two and a normal to a symmetry plane are parallel, in the full symbols, the symmetry axes and symmetry planes normal to them for each symmetry direction are listed while in the short symbols, only the symmetry-planes symbols are shown. For example, the full and short symbol of space-group type No. 55 are, respectively, P2₁/b2₁/a2/m and Pbam. This reduced set of symmetry elements shown in the short Hermann-Mauguin symbols is however not limited to the elements corresponding to the generators of the space group; for example, in orthorhombic merohedral space groups the operations corresponding to the symmetry elements about two of the three symmetry directions of the lattice are generators, the third is obtained as the product (composition) of these two.

In the extended Hermann-Mauguin symbols, the symmetry of the space group is listed in a rather complete fashion and provide information not only on the chosen set of space-group generators but also on the additional symmetry operations in most of the cases obtained by the compositions of the generating symmetry operations with lattice translations. These symbols are constructed following the same rules as the short symbols but give the corresponding full list of symmetry operations, including the different symmetry operation of the same nature (rotation or screw rotations of the same order; mirror or glide reflections) that may occur about geometric elements in parallel orientation. The result is a symbol composed of one (when the extended symbols coincides with the short one), two or four lines depending on the crystal system and the type of unit cell.