Hermann-Mauguin notation

Hermann-Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French minerologist Charles-Victor Mauguin. This notation is sometimes called international notation.

The Hermann-Mauguin notation, compared with the Schoenflies notation, is preferred in crystallography because it can easily be used to include translational symmetry elements, and it specifies the directions of the symmetry axes.^[1]

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Nomenclature

Rotational symmetries are denoted by a number n, given by φ=360/n, where φ is the angle of rotation. A rotation of 180° would be denoted by the n = 2, and is called a two-fold rotation. Mirror planes are given by the letter m.

Point groups

Point groups can exist in both two and three dimensions. They are defined by their symmetry elements, such as the axes of proper and improper rotation and mirror planes. Translational symmetry elements which are present in plane groups and space groups are omitted. Where certain symmetry elements can be deduced, they may be omitted, allowing simplification.

In three dimensions, there are 32 crystallographic point groups:

• 1, 1
• 2, m, ²⁄_m
• 222, mm2, mmm
• 4,4, ⁴⁄_m, 422, 4mm, 42m, ⁴⁄_mmm
• 3, 3, 32, 3m, 3m
• 6, 6, ⁶⁄_m, 622, 6mm, 62m, ⁶⁄_mmm
• 23, m3, 432, 43m, m3m

Plane groups

Plane groups can be depicted using the Hermann-Mauguin system. The first letter is either lowercase p or c to represent primitive or centered unit cells. The next number is the rotational symmetry, as given above. The presence of mirror planes are denoted m, while glide reflections are denoted g.

Space groups

Space groups can be defined by combining the point group identifier with the uppercase P, C, I, or F for primitive, side-centered, body-centered, or face-centered lattices. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.

The screw axis is noted by a number, n, where the angle of rotation is 360°/n. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2₁ is a 180° (two-fold) rotation followed by a translation of ½ of the lattice vector. 3₁ is a 120° (three-fold) rotation followed by a translation of ⅓ of the lattice vector.

Glide planes are noted by a, b, or c depending on depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The latter is often called the diamond glide plane as it features in the diamond structure.

References

1. ^ Sands, Donald E. [1993]. "Crystal Systems and Geometry", Introduction to Crystallography (in English). Mineola, New York: Dover Publications, Inc., 165. ISBN 0-486-67839-3.