Hermann-Mauguin notation

In geometry, 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 (who introduced it in 1928) and the French mineralogist Charles-Victor Mauguin (who modified it in 1931). This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.

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.

Rotation axes are denoted by a number n — 1, 2, 3, 4, 5, 6, 7, 8 ... (angle of rotation φ = 360°/n). For improper rotations, Hermann–Mauguin symbols show rotoinversion axes, unlike Schoenflies and Shubnikov notations, where the preference is given to rotation-reflection axes. The rotoinversion axes are represented by the corresponding number with a macron, n — 1, 3, 4, 5, 6, 7, 8 ... The symbol for a mirror plane (rotoinversion axis 2) is m. The direction of the mirror plane is defined as the direction of perpendicular to the face (the direction of 2 axis).

Hermann–Mauguin symbols show symmetrically non-equivalent axes and planes. The direction of a symmetry element is represented by its position in the Hermann–Mauguin symbol. If a rotation axis n and a mirror plane m have the same direction (i.e. the plane is perpendicular to axis n), then they are denoted as a fraction n/m or n/m.

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