Crystallographic international notation

Underlines in the International notation for G show which operators are complementary ones. Alternatively, these may be identified from the classes of G H by multiplying each operator by 0 G is the ordinary crystallographic point group from which G was constructed by eq. (14.1.2) H is given first in International notation and then in Schonflies notation, in square brackets. Subscript a denotes the unit vector along [1 1 0]. 

Table 7.3 shows a few point groups of interest to molecules (and to crystals). The Schonflies notation is being replaced in the crystallographic literature by the Herrmann11-Mauguin12 or international notation. 

Table 7.8 The 32 Crystallographic Point Groups, Listed by Main Symmetry Axes or Plane, Using Both the Schoenflies Notation (S, e.g., C2v) and the Hermann-Mauguin or International Notation (HM, e.g., mm2)3...

Table 2.2. Crystallographic point groups Schoenflies and International notations...

Crystallography is an advanced discipline [318], Modern crystallography has been developed since the discovery of X-ray diffraction in 1912 from the original basis laid down by classical crystallographers. One of the beauties of this modern discipline, while it can be somewhat mathematical, is the universal use of standardised notations and conventions, as developed through the International Union of Crystallography (IUCr). 

A report by the International Union of Crystallography Commission on Crystallographic Nomenclature (Lima de Faria et al. 1990) presents a concise description of similar alternative notations, a summary of which is in Table 3.4. 

For symmetry determinations, the choice of the pertinent technique among the available techniques greatly depends on the inferred crystallographic feature. A diffraction pattern is a 2D finite figure. Therefore, the symmetry elements displayed on such a pattern are the mirrors m, the 2, 3, 4 and 6 fold rotation axes and the combinations of these symmetry elements. The notations given here are those of the International Tables for Crystallography [1]. 

Note 5 From a crystallographic point of view, the uniaxial nematic structure is characterised by the symbol Dooh in the Schoenflies notation (Wmm in the International System). 

Table 2.9. The thirty-two crystallographic point groups in both International and Schonflies notation.

The point group, which is D3 in the Schonflies notation used for example in molecular spectroscopy, is called 32 in the International (or Hermann-iMauguin) notation used by crystallographers. 

It is often said that group 432 is too symmetric to allow piezoelectricity, in spite of the fact that it lacks a center of inversion. It is instructive to see how this comes about. In 1934 Neumann s principle was complemented by a powerful theorem proven by Hermann (1898-1961), an outstanding theoretical physicist with a passionate interest for symmetry, whose name is today mostly connected with the Hermann-Mau-guin crystallographic notation, internationally adopted since 1930. In the special issue on liquid crystals by ZeitschriftfUr Kristal-lographie in 1931 he also derived the 18 symmetrically different possible states for liquid crystals, which could exist between three-dimensional crystals and isotropic liquids [100]. His theorem from 1934 states [101] that if there is a rotation axis C (of order n), then every tensor of rank rcubic crystals, this means that second rank tensors like the thermal expansion coefficient a, the electrical conductivity Gjj, or the dielectric constant e,y, will be isotropic perpendicular to all four space diagonals that have threefold symme-... 

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