The Schoenflies notation is one of two conventions commonly used to describe crystallographic point groups. In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom This notation is used in spectroscopy. Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ The other convention is the Hermann-Mauguin notation, also known as the International notation. Hermann-Mauguin notation is used to represent the Symmetry elements in Point groups Plane groups and Space groups It is named after the German A point group in the Schoenflies convention is completely adequate to describe the symmetry of a molecule; this is sufficient for spectroscopy. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed The Hermann-Maunguin notation is able to describe the space group of a crystal lattice, while the Schoenflies notation isn't. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure Thus the Hermann-Maunguin notation is used in crystallography. Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The
Symmetry elements
Symmetry elements are denoted by i for centers of inversion, C for proper rotation axes, and σ for mirror planes, and S for improper rotation axes (rotation-reflection axes). In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or C and S are usually followed by a subscript n denoting the order of rotation possible.
By convention, the axis of proper rotation of greatest order is defined as the principle axis. All other symmetry elements are described in relation to it. Thus, mirror planes are denoted σv or σh for vertical mirror planes (parallel with the principal axis) and horizontal mirror planes (perpendicular with the principal axis).
Point groups
In three dimensions, there are 32 crystallographic point groups.
- The letter O (for octahedron) indicates that the group has the symmetry of an octahedron (or cube), with (Oh) or without (O) improper operations (those that change handedness). An octahedron (plural octahedra is a Polyhedron with eight faces A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex.
- The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. Td includes improper operations, T excludes improper operations, and Th is T with the addition of an inversion.
- Cn (for cyclic) indicates that the group has an n-fold rotation axis. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Cnh is Cn with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. Cnv is Cn with the addition of a mirror plane parallel to the axis of rotation.
- Sn (for Spiegel, German for mirror) denotes a group that contains only an n-fold rotation-reflection axis. A mirror is an object with a surface that has good Specular reflection; that is it is smooth enough to form an Image. In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or
- Dn (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus a twofold axis perpendicular to that axis. Dihedral is the upward angle from horizontal of the wings or tailpane of a Fixed-wing aircraft or the wing of a Bird. Dnh has, in addition, a mirror plane perpendicular to the n-fold axis. Dnv has, in addition to the elements of Dn, mirror planes parallel to the n-fold axis.
Due to the crystallographic restriction theorem, n is restricted to the values of 1, 2, 3, 4, or 6. The crystallographic restriction theorem in its basic form was based on the observation that the Rotational symmetries of a Crystal are usually limited to
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