Schoenflies notation
Encyclopedia
The Schoenflies notation or Schönflies notation, named after the German
mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point group
s. This notation is used in spectroscopy
. The other convention is the Hermann–Mauguin notation, also known as the International notation. A point group
in the Schoenflies convention is completely adequate to describe the symmetry of a molecule; this is sufficient for spectroscopy. The Hermann–Maunguin notation is able to describe the space group
of a crystal lattice, while the Schoenflies notation isn't. Thus the Hermann–Mauguin notation is used in crystallography
.
By convention, the axis of proper rotation of greatest order is defined as the principal axis. All other symmetry elements are described in relation to it. Thus, mirror planes are denoted σ_{v} or σ_{h} for vertical mirror planes (containing the principal axis) and horizontal mirror planes (perpendicular to the principal axis), respectively.
All groups that do not contain several higher-order axes (order 3 or more) can be arranged in the table:
The symbols that shouldn't be used are marked with maroon color.
In crystallography, due to the crystallographic restriction theorem
, n is restricted to the values of 1, 2, 3, 4, or 6. The noncrystallographic groups are shown with grayed backgrounds. D_{4d} and D_{6d} are also forbidden because they contain improper rotation
s with n = 8 and 12 respectively. The 27 point groups in the table plus T, T_{d}, T_{h}, O and O_{h} constitute 32 crystallographic point groups.
Groups with n = ∞ are called limit groups or Curie groups. There are two more limit groups, not listed in the table - K (for Kugel, German for ball, sphere) - group of all possible rotations in 3-dimesional space and K_{h} - group of all possible rotations and reflections. In mathematics and theoretical physics they are usually denoted as SO(3) and O(3) (special orthogonal group in 3-dimensional space and orthogonal group in 3-dimensional space).
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....
mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point group
Point group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
s. This notation is used in spectroscopy
Spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
. The other convention is the Hermann–Mauguin notation, also known as the International notation. A point group
Point group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
in the Schoenflies convention is completely adequate to describe the symmetry of a molecule; this is sufficient for spectroscopy. The Hermann–Maunguin notation is able to describe the space group
Space group
In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
of a crystal lattice, while the Schoenflies notation isn't. Thus the Hermann–Mauguin notation is used in crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
.
Symmetry elements
Symmetry elements are denoted by i for centers of inversion, C for proper rotation axes, σ for mirror planes, and S for improper rotation axes (rotation-reflection axes). C and S are usually followed by a subscript number (abstractly denoted n) denoting the order of rotation possible.By convention, the axis of proper rotation of greatest order is defined as the principal axis. All other symmetry elements are described in relation to it. Thus, mirror planes are denoted σ_{v} or σ_{h} for vertical mirror planes (containing the principal axis) and horizontal mirror planes (perpendicular to the principal axis), respectively.
Point groups
In three dimensions, there are infinite number of point groups, but all of them can be classified by several families.- C_{n} (for cyclicCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
) indicates that the group has an n-fold rotation axis. - C_{nh} is C_{n} with the addition of a mirror (reflection) plane perpendicular to the axis of rotation (horizontal plane).
- C_{nv} is C_{n} with the addition of n mirror planes parallel to the axis of rotation (vertical planes).
- S_{2n} (for Spiegel, German for mirrorMirrorA mirror is an object that reflects light or sound in a way that preserves much of its original quality prior to its contact with the mirror. Some mirrors also filter out some wavelengths, while preserving other wavelengths in the reflection...
) denotes a group that contains only a 2n-fold rotation-reflection axis. The index should be even because for odd n, n-fold rotation-reflection axis is equivalent to combination of n-fold rotation axis and perpendicular plane, hence S_{n} = C_{nh} for odd n. - C_{ni} indicates the group containing only rotoinversion axisImproper rotationIn 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.Equivalently it is the...
C_{ni}. These symbols are redundant, because any rotoinversion axis can be expressed as rotation-reflection axis, hence for odd n C_{ni} = S_{2n} and C_{2ni} = S_{n} = C_{nh}, and for even n C_{2ni} = S_{2n}. Only symbol C_{i} is conventionally used, nevertheless, in some texts you can see symbols like C_{3i}, C_{5i}. - D_{n} (for dihedral groupDihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. - D_{nh} has, in addition, a horizontal mirror plane and, as a consequence, also n vertical mirror planes containing the n-fold axis.
- D_{nd} has, in addition to the elements of D_{n}, n vertical mirror planes which pass between twofold axes (diagonal planes).
- The letter T (for tetrahedronTetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
) indicates that the group has the rotation axes of a tetrahedron (three 2-fold axes and four 3-fold axes). - T_{d} includes diagonal mirror planes (each diagonal plane contains only one twofold axis and passes between two other twofold axes, like in D_{2d} group). This addition of diagonal planes results in three improper rotation operations S_{4} in T_{d} group.
- T_{h} includes horizontal mirror plane and, as a consequence, vertical mirror planes. Each plane passes two twofold axes and is perpendicular to the third twofold axis, which results in inversion center i.
- The letter O (for octahedronOctahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
) indicates that the group has the rotation axes of an octahedron or cubeCubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
(three 4-fold axes, four 3-fold axes, and 6 diagonal 2-fold axes). - O_{h} includes horizontal mirror plane and, as a consequence, vertical mirror planes. It contains also inversion center and improper rotation operations.
- The letter I (for icosahedronIcosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
) indicates that the group has the rotation axes of an icosahedron or dodecahedron (six 5-fold axes, ten 3-fold axes, and 15 2-fold axes). - I_{h} includes horizontal mirror plane and contains also inversion center and improper rotation operations.
All groups that do not contain several higher-order axes (order 3 or more) can be arranged in the table:
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ... | ∞ |
---|---|---|---|---|---|---|---|---|---|---|
C_{n} | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | |
C_{∞} |
C_{nv} | C_{1v} = C_{1h} | C_{2v} | C_{3v} | C_{4v} | C_{5v} | C_{6v} | C_{7v} | C_{8v} | |
C_{∞v} |
C_{nh} | C_{1h} = C_{s} | C_{2h} | C_{3h} | C_{4h} | C_{5h} | C_{6h} | C_{7h} | C_{8h} | |
C_{∞h} |
S_{n} | S_{1} = C_{s} | S_{2} = C_{i} | S_{3} = C_{3h} | S_{4} | S_{5} = C_{5h} | S_{6} | S_{7} = C_{7h} | S_{8} | |
S_{∞} = C_{∞h} |
C_{ni} | C_{1i} = C_{i} | C_{2i} = C_{s} | C_{3i} = S_{6} | C_{4i} = S_{4} | C_{5i} = S_{10} | C_{6i} = C_{3h} | C_{7i} = S_{14} | C_{8i} = S_{8} | |
C_{∞i} = C_{∞h} |
D_{n} | D_{1} = C_{2} | D_{2} | D_{3} | D_{4} | D_{5} | D_{6} | D_{7} | D_{8} | |
D_{∞} |
D_{nh} | D_{1h} = C_{2v} | D_{2h} | D_{3h} | D_{4h} | D_{5h} | D_{6h} | D_{7h} | D_{8h} | |
D_{∞h} |
D_{nd} | D_{1d} = C_{2h} | D_{2d} | D_{3d} | D_{4d} | D_{5d} | D_{6d} | D_{7d} | D_{8d} | |
D_{∞d} = D_{∞h} |
The symbols that shouldn't be used are marked with maroon color.
In crystallography, due to the crystallographic restriction theorem
Crystallographic restriction theorem
The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold...
, n is restricted to the values of 1, 2, 3, 4, or 6. The noncrystallographic groups are shown with grayed backgrounds. D_{4d} and D_{6d} are also forbidden because they contain improper rotation
Improper rotation
In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.Equivalently it is the...
s with n = 8 and 12 respectively. The 27 point groups in the table plus T, T_{d}, T_{h}, O and O_{h} constitute 32 crystallographic point groups.
Groups with n = ∞ are called limit groups or Curie groups. There are two more limit groups, not listed in the table - K (for Kugel, German for ball, sphere) - group of all possible rotations in 3-dimesional space and K_{h} - group of all possible rotations and reflections. In mathematics and theoretical physics they are usually denoted as SO(3) and O(3) (special orthogonal group in 3-dimensional space and orthogonal group in 3-dimensional space).