Schoenflies notation

The Schoenflies symbolism is a system of symbols ( symbolism ), which is used to describe symmetry elements and symmetry groups. Named after the German mathematician Arthur Moritz Schoenflies symbolism is next to the Hermann- Mauguin symbolism one of the international conventions generally used to describe the 32 crystallographic point groups and 230 crystallographic space groups. Nowadays Schoenflies symbolism but is used mainly to describe molecular symmetries. Typical areas of application are therefore found mainly in the field of molecular spectroscopy and molecular physics.

Symbols of symmetry elements

The description of symmetry elements through the following symbols:

  • Rotation: CN discloses an axis of rotation,
  • Reflection: σ denotes a mirror plane,
  • Inversion: i describes an inversion center,
  • Rotary inversion: S'N ( does not apply )
  • Rotation-reflection: SN denotes an axis of rotation followed by reflection; It describes the same facts as an inversion, both of which may have different Zähligkeiten. Unlike the Hermann- Mauguin symbols are added in the Schoenflies symbolism always adhere to the rotating mirror axis and not the inversion axis.

The symbols C and S are this is usually indicated by a numeric index N, which indicates the order of the possible rotations.

By convention, the axis of rotation largest order is defined as the main axis of symmetry, and all other elements described with respect to them. Therefore, vertical mirror planes are called ( with the major axis ) σv and horizontal mirror planes ( perpendicular to the main axis) with σh.

Symbols of the point groups and space groups

In the three spatial dimensions result 32 possible crystallographic point groups. They are arranged according Schoenflies into the following subgroups:

  • Rotation groups: C
  • Rotating Mirror Group: S
  • Dihedral: D
  • Tetrahedral groups: T
  • Oktaedergruppen: O
  • Ikosaedergruppen: I
  • Ball Group: K

The description of the symmetry provided the symbols of the point groups with an additional subscript:

  • Horizontal mirror plane: h
  • Vertical symmetry plane: V
  • Diagonal plane of symmetry: d (only in the simultaneous presence of two-fold symmetry, horizontal axes that are not located on the mirror planes )
  • Inversion center i
  • Mirror plane: s

Furthermore, the multiplicity of the axis or a symbol for other symmetry elements in a subscript will indicate as required, eg for a D2h

Symbols of the space groups

With the Schoenflies symbolism also the description of space groups is possible. For this purpose, a point group symbol superscript numeric index is appended. The space groups are thereby numbered: eg, , etc. The symbolism is, however, seldom use because they can not be identified, the existing symmetry elements.

In the description of a factor group of the superscript is not mitangegeben generally; analogy, is omitted in the Hermann- Mauguin symbols, the symbol ( or, etc.).

Rotation group SO(3) Dihedral group Octahedral symmetry Icosahedral symmetry Quotient group
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