# Schoenflies symbolism

The schoenflies notation is a system of symbols (a symbolic ), which for the description of elements of symmetry and symmetry groups is used. The symbolism named after the German mathematician Arthur Moritz Schoenflies is next to the Hermann Mauguin symbolism one of the generally used international conventions for the description of the 32 crystallographic point groups and 230 crystallographic space groups . Nowadays, however, the Schoenflies symbolism is mainly used to describe molecular symmetries. Typical areas of application are therefore primarily found in the area of molecular spectroscopy and molecular physics .

## Symbols of symmetry elements

Symmetry elements are described using the following symbols:

• Rotation: C N describes an axis of rotation,
• Reflection: σ denotes a mirror plane,
• Inversion: i describes a center of inversion,
• Rotational inversion: S ' N (does not apply)
• Rotational mirroring: S N denotes an axis of rotation with subsequent mirroring. It describes the same facts as an inversion, whereby both can have different counts. In contrast to the Hermann Mauguin symbolism, the Schoenflies symbolism always specifies the rotating mirror axis and not the inversion axis.

The symbols C and S are usually designated with a numerical index N , which indicates the order of the possible rotations.

By convention, the axis of the greatest order of rotation is defined as the principal axis and all other symmetry elements are described in relation to it; the main axis is defined as "vertical". Correspondingly, vertical mirror planes (containing the main axis) are denoted by σ v and horizontal mirror planes (perpendicular to the main axis) are denoted by σ h .

Symmetry operations and elements are denoted by the same symbols.

## Symbols of point groups and space groups

There are 32 possible crystallographic point groups in the three spatial dimensions. According to Schoenflies, they are classified into the following subgroups:

To describe the symmetry, the symbols of the point groups are given an additional subscript:

• horizontal mirror plane: h
• vertical plane of symmetry: v
• Diagonal plane of symmetry: d (only if two-fold horizontal axes of symmetry occur simultaneously that are not on the mirror planes)
• Inversion Center: i
• Mirror plane: s

Furthermore, the number of the axis or a symbol for other symmetry elements is specified in a subscript, e.g. B. D 2h for an orthorhombic crystal structure, where the h denotes a mirror plane perpendicular to the n -fold axis (horizontal mirror plane).

## Space group symbols

With the Schoenflies symbolism it is also possible to describe room groups. A superscript numeric index is added to a point group symbol. The room groups are numbered consecutively: e.g. B. , , etc. The symbolism, however, rarely is used because it can not recognize the existing symmetry elements. ${\ displaystyle D_ {2h} ^ {1}}$${\ displaystyle D_ {2h} ^ {2}}$${\ displaystyle D_ {2h} ^ {3}}$

When describing a factor group , the superscript index is usually not included. Similarly, the symbol in Hermann-Mauguin notation (or , , etc.) are omitted. ${\ displaystyle P}$${\ displaystyle C}$${\ displaystyle F}$${\ displaystyle I}$

## literature

• Ulrich Müller: Inorganic Structural Chemistry . Vieweg + Teubner, 2008, ISBN 978-3-8348-0626-0 , p. 26–38 ( limited preview in Google Book search).
• Erhard Scholz: symmetry, group, duality. On the relationship between theoretical mathematics and applications in crystallography and structural engineering in the 19th century . Birkhäuser, 1989, ISBN 978-3-7643-1974-8 , pp. 120–148 ( limited preview in the Google book search - brief description of historical circumstances leading to the origin of the symbolism; the work also includes a larger discussion of the symbolism).

## Individual evidence

1. Arthur Schoenflies: Crystal systems and crystal structure . Berlin 1877 ( online resources [accessed on April 9, 2011] habilitation thesis, University of Göttingen).
2. Erhard Scholz: Symmetry, group, duality. On the relationship between theoretical mathematics and applications in crystallography and structural engineering in the 19th century . Birkhäuser, 1989, ISBN 978-3-7643-1974-8 , pp. 120 ( limited preview in Google Book search).
3. ^ A b Johann Weidlein, Ulrich Müller, Kurt Dehnicke: Schwingungsspektoskopie. 2nd edition ISBN 3-13-625102-4 , pp. 59-61.
4. Molecular spectroscopy of inorganic compounds. ( Memento of the original from March 16, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.
5. ^ Robert J. Naumann: Introduction to the Physics and Chemistry of Materials . CRC Press, 2011, ISBN 978-1-4200-6134-5 , pp. 71 ( limited preview in Google Book search).