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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, 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[clarification needed] of the group.
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.
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
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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, 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[clarification needed] of the group.
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.
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