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.
Point groups 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, that shows rotation-reflection axes. The rotoinversion axes are represented by the corresponding number with a macron, n – 1, 2, 3, 4, 5, 6, 7, 8, ... . 2 is equivalent to a mirror plane and usually notated as m. The direction of the mirror plane is defined as the direction perpendicular to it (the direction of the 2 axis). Hermann–Mauguin symbols show non-equivalent axes and planes in a symmetrical fashion. The direction of a symmetry element corresponds to its position in the Hermann–Mauguin symbol. If a rotation axis n and a mirror plane m have the same direction, then they are denoted as a fraction n/m or n /m. If two or more axes have the same direction, the axis with higher symmetry is shown. Higher symmetry means that the axis generates a pattern with more points. For example, rotation axes 3, 4, 5, 6, 7, 8 generate 3-, 4-, 5-, 6-, 7-, 8-point patterns, respectively. Improper rotation axes 3, 4, 5, 6, 7, 8 generate 6-, 4-, 10-, 6-, 14-, 8-point patterns, respectively. If a rotation and a rotoinversion axis generate the same number of points, the rotation axis should be chosen. For example, the 3/m combination is equivalent to 6. Since 6 generates 6 points, and 3 generates only 3, 6 should be written instead of 3/m (not 6/m, because 6 already contains the mirror plane m). Analogously, in the case when both 3 and 3 axes are present, 3 should be written. However we write 4/m, not 4/m, because both 4 and 4 generate four points. In the case of the 6/m combination, where 2, 3, 6, 3, and 6 axes are present, axes 3, 6, and 6 all generate 6-point patterns, as we can see on the figure in the right, but the latter should be used because it is a rotation axis – the symbol will be 6/m. Finally, the Hermann–Mauguin symbol depends on the type of the group.
Groups without higher-order axes (axes of order three or more) These groups may contain only two-fold axes, mirror planes, and/or an inversion center. These are the crystallographic point groups 1 and 1 (triclinic crystal system), 2, m, and 2/m (monoclinic), and 222, 2/m2/m2/m, and mm2 (orthorhombic). (The short form of 2/m2/m2/m is mmm.) If the symbol contains three positions, then they denote symmetry elements in the x, y, z direction, respectively.
Groups with one higher-order axis First position – primary direction – z direction, assigned to the higher-order axis. Second position – symmetrically equivalent secondary directions, which are perpendicular to the z-axis. These can be 2, m, or 2/m. Third position – symmetrically equivalent tertiary directions, passing between secondary directions. These can be 2, m, or 2/m. These are the crystallographic groups 3, 32, 3m, 3, and 32/m (trigonal crystal system), 4, 422, 4mm, 4, 42m, 4/m, and 4/m2/m2/m (tetragonal), and 6, 622, 6mm, 6, 6m2, 6/m, and 6/m2/m2/m (hexagonal). Analogously, symbols of non-crystallographic groups (with axes of order 5, 7, 8, 9, ...) can be constructed. These groups can be arranged in the following table (with the addition of the inversion group, designated S2 or 1, the two-fold rotation group, designated C2 or 2, and the reflexion group, designated C1v, C1h or m):
It can be noticed that in groups with odd-order axes n and n the third position in symbol is always absent, because all n directions, perpendicular to higher-order axis, are symmetrically equivalent. For example, in the picture of a triangle all three mirror planes (S0, S1, S2) are equivalent – all of them pass through one vertex and the center of the opposite side. For even-order axes n and n there are n/2 secondary directions and n/2 tertiary directions. For example, in the picture of a regular hexagon one can distinguish two sets of mirror planes – three planes go through two opposite vertexes, and three other planes go through the centers of opposite sides. In this case any of two sets can be chosen as secondary directions, the rest set will be tertiary directions. Hence groups 42m, 62m, 82m, ... can be written as 4m2, 6m2, 8m2, ... . For symbols of point groups this order usually doesn't matter; however, it will be important for Hermann–Mauguin symbols of corresponding space groups, where secondary directions are directions of symmetry elements along unit cell translations b and c, while the tertiary directions correspond to the direction between unit cell translations b and c. For example, symbols P6m2 and P62m denote two different space groups. This also applies to symbols of space groups with odd-order axes 3 and 3. The perpendicular symmetry elements can go along unit cell translations b and c or between them. Space groups P321 and P312 are examples of the former and the latter cases, respectively. The symbol of point group 32/m may be confusing; the corresponding Schoenflies symbol is D3d, which means that the group consists of 3-fold axis, three perpendicular 2-fold axes, and 3 vertical diagonal planes passing between these 2-fold axes, so it seems that the group can be denoted as 32m or 3m2. However, one should remember that, unlike Schoenflies notation, the direction of a plane in a Hermann–Mauguin symbol is defined as the direction perpendicular to the plane, and in the D3d group all mirror planes are perpendicular to 2-fold axes, so they should be written in the same position as 2/m. Second, these 2/m complexes generate an inversion center, which combining with the 3-fold rotation axis generates a 3 rotoinversion axis. Groups with n = ∞ are called limit groups or Curie groups.
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