Hermann–Mauguin notation
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).
Hermann–Mauguin symbols show non-equivalent axes and planes in a symmetrical fashion.
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.
Higher symmetry means that the axis generates a pattern with more points.
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).
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 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 many groups they can be simplified by omitting n-fold rotation axes in n/m positions.
This can be done if the rotation axis can be unambiguously obtained from the combination of symmetry elements presented in the symbol.
The first letter is either lowercase p or c to represent primitive or centered unit cells.
The symbols for symmetry elements are more diverse, because in addition to rotations axes and mirror planes, space group may contain more complex symmetry elements – screw axes (combination of rotation and translation) and glide planes (combination of mirror reflection and translation).
For example, choosing different lattice types and glide planes one can generate 28 different space groups from point group mmm, e.g. Pmmm, Pnnn, Pccm, Pban, Cmcm, Ibam, Fmmm, Fddd, and so on.
[9] These are the Bravais lattice types in three dimensions: The screw axis is noted by a number, n, where the angle of rotation is 360°/n.
