Hermann Mauguin symbolism

The Hermann-Mauguin notation is used to describe symmetry elements and symmetry groups used. It is named after the crystallographers Carl Hermann and Charles-Victor Mauguin . Its main area of ​​application is the description of the 32 crystallographic point groups and the 230 crystallographic space groups . It is also used to describe two-dimensional planar groups , two- and three-dimensional subperiodic groups ( ribbon ornament , rod and layer groups ) and non-crystallographic groups. It is standardized in the International Tables for Crystallography . In addition to the symbolism according to Hermann-Mauguin, there is a spelling according to Arthur Moritz Schoenflies , the Schoenflies symbolism . However, it is rarely used to describe a crystalline state, but rather to describe the symmetry of molecules .

Symbols of symmetry elements

Inversion center

• ${\displaystyle {\bar {1}}}$: Center of inversion. Multiplication of a particle by point reflection . A total of two symmetry-equivalent particles are created.

Axes of rotation

A rotation around is represented by (pronounced "n-fold rotation"). Special cases are , a rotation by 360 °, according to the identity and , a rotation by any small angle. ${\displaystyle {\frac {360^{\circ }}{n}}}$ ${\displaystyle n\ }$ ${\displaystyle 1\ }$ ${\displaystyle \infty {}}$

The following rotations can occur in crystallographic space and point groups.

• ${\displaystyle 1\ }$: Identity is an element of every group.
• ${\displaystyle 2\ }$: twofold axis of rotation, i.e. rotation by 180 °. A total of two symmetry-equivalent particles are created.
• ${\displaystyle 3\ }$: threefold axis of rotation, i.e. rotation by 120 °. A total of three symmetry-equivalent particles are created.
• ${\displaystyle 4\ }$: fourfold axis of rotation, i.e. rotation by 90 °. A total of four symmetry-equivalent particles are created.
• ${\displaystyle 6\ }$: six-fold axis of rotation, i.e. rotation by 60 °. A total of six symmetry-equivalent particles are created.

Mirror plane

• ${\displaystyle m\ }$: Mirror plane. Multiplication of a particle by mirroring it on a plane . A total of two symmetry-equivalent particles are created.

Coupled symmetry operations (rotational inversion axes)

• ${\displaystyle {\bar {2}}}$: twofold rotational inversion axis, i.e. rotation by 180 ° and subsequent point reflection. A total of two symmetry-equivalent particles are created. Since this operation leads to the same result as the mirroring on a plane, this symbol is not used, but always specified as the mirror plane . ${\displaystyle m\ }$
• ${\displaystyle {\bar {3}}}$: threefold rotary inversion axis, i.e. rotation by 120 ° and subsequent point reflection. A total of six symmetry-equivalent particles are created.
• ${\displaystyle {\bar {4}}}$: four-fold rotation inversion axis, i.e. rotation by 90 ° and subsequent point reflection. A total of four symmetry-equivalent particles are created.
• ${\displaystyle {\bar {6}}}$: six-fold rotation inversion axis, i.e. rotation by 60 ° and subsequent point reflection. A total of six symmetry-equivalent particles are created.

Combined symmetry operations (axes of rotation perpendicular to mirror planes)

Both of the notations given are equivalent. The former is common in the older literature.

• ${\displaystyle {\frac {2}{m}}}$or : two-fold axis of rotation perpendicular to a mirror plane (pronounced “two over m”). A total of four symmetry-equivalent particles are created. ${\displaystyle 2/m\ }$
• ${\displaystyle {\frac {3}{m}}}$or : three-fold axis of rotation perpendicular to a mirror plane (pronounced "three over m"). A total of six symmetry-equivalent particles are created. Since this operation leads to the same result as the six-fold rotation inversion axis, this symbol is not used, but always specified as a six-fold rotation inversion axis . ${\displaystyle 3/m\ }$ ${\displaystyle {\bar {6}}}$
• ${\displaystyle {\frac {4}{m}}}$or : four-fold axis of rotation perpendicular to a mirror plane (pronounced “four over m”). A total of eight symmetry-equivalent particles are created. ${\displaystyle 4/m\ }$
• ${\displaystyle {\frac {6}{m}}}$or : six-fold axis of rotation perpendicular to a mirror plane (pronounced “six over m”). A total of twelve symmetry-equivalent particles are created. ${\displaystyle 6/m\ }$

Symbols of the point groups

The 32 point groups (crystal classes) can be described with the symbols described above , since the symmetry operations of the crystal classes do not contain any translation (see section on space groups).

In the triclinic crystal system there are point groups (absence of inversion centers) and (presence of inversion centers). For other crystal systems, the symmetry operations are given with respect to three given crystallographic directions. ${\displaystyle 1\ }$ ${\displaystyle {\bar {1}}}$

Crystal system	1 position	2nd position	3rd position
monoclinic	${\displaystyle [100]\;}$	${\displaystyle [010]\;}$	${\displaystyle [001]\;}$
orthorhombic	${\displaystyle [100]\;}$	${\displaystyle [010]\;}$	${\displaystyle [001]\;}$
tetragonal	${\displaystyle [001]\;}$	${\displaystyle \langle 100\rangle }$	${\displaystyle \langle 110\rangle }$
trigonal	${\displaystyle [00.1]\;}$	${\displaystyle \langle 10.0\rangle }$	${\displaystyle \langle 12.0\rangle }$
hexagonal	${\displaystyle [00.1]\;}$	${\displaystyle \langle 10.0\rangle }$	${\displaystyle \langle 12.0\rangle }$
trigonal , rhombohedral arrangement	${\displaystyle [111]\;}$	${\displaystyle \langle 1{\bar {1}}0\rangle }$
cubic	${\displaystyle \langle 100\rangle }$	${\displaystyle \langle 111\rangle }$	${\displaystyle \langle 110\rangle }$

(The directions highlighted in color are generally not indicated in the point group symbols, since there are never symmetry elements other than or . However, they are occasionally required for the space group symbols.) ${\displaystyle 1\ }$ ${\displaystyle {\bar {1}}}$

The rotation and rotation inversion axes are specified parallel and the mirror planes perpendicular to these directions. In the case of trigonal point groups, please note that the directions in the first line are given in relation to the hexagonal arrangement of the coordinate system. Redundant information is omitted from the abbreviated notation of the Hermann Mauguin symbols. Instead , for example, is written. ${\displaystyle 4/m\ 2/m\ 2/m}$ ${\displaystyle 4/m\ m\ m}$

Space group symbols

The designation for the room groups works in principle like that of the point groups. In addition, the Bravais grid is prepended:

• P: Primitive
• A, B or C: Face-centered
• Q: Centered on all sides
• I: Inner Centered
• R: Hexagonal grid with rhombohedral centering

There are also additional symbols:

• ${\displaystyle n_{m}\ }$: -numerous screw axis with translation around parts of a grid vector. ${\displaystyle n\ }$ ${\displaystyle {\frac {m}{n}}}$
• ${\displaystyle a\ }$, or : gliding mirror plane with translation along half a grid vector. ${\displaystyle b\ }$ ${\displaystyle c\ }$
• ${\displaystyle n\ }$: Gliding mirror plane with translation along half a surface diagonal.
• ${\displaystyle d\ }$: Gliding mirror plane with translation along a quarter surface diagonal.
• ${\displaystyle e\ }$: Two glide mirrors with the same glide mirror plane and translation along two (different) half lattice vectors.

An example of a tetragonal space group in abbreviated form is . ${\displaystyle I\ 4_{1}/a\ m\ d}$

literature

• Hahn, Theo (Ed.): International Tables for Crystallography Vol. A D. Reidel publishing Company, Dordrecht 1983, ISBN 90-277-1445-2