Hermann-Mauguin Notation Point Groups

Figure 6. Complete subgroup lattice of continuous point groups. Solid circles represent point goups that can be represented by geometrical figures Ki, (sphere), (cylinder), Cw (cone). Open circles represent point goups that cannot be represented by geometrical figures. Schonflies notations are accompanied by Hermann-Mauguin (international) notations in brackets.

The Hermann-Mauguin notation for the description of point group symmetry (in contrast to the Schonflies system used in Chapter 6) is widely adopted in crystallography. An n-fold rotation axis is simply designated as n. An object is said to possess an n-fold inversion axis h if it can be brought into an equivalent configuration by a rotation of 360°/n in combination with inversion through a... 

Table 1-4 lists the point symmetry elements and the corresponding symmetry operations. The notation used by spectroscopists and chemists, and used here, is the so-called Schoenflies system, which deals only with point groups. Crystallographers generally use the Hermann-Mauguin system, which applies to both point and space groups. 

Table 7.3 Point Groups of Interest to Chemistry (in Schonflies and also Hermann-Mauguin Notation), with Examples of Molecules that Belong to Them3...

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...

The periodicity of a lattice limits the number of compatible rotation operations to onefold, twofold, threefold, fourfold, and sixfold. This, in turn, limits the number of point groups to thirty-two. Point groups are used to describe individual molecules. Table 14.1 shows the thirty-two point groups in both the Hermann-Mauguin notation and the Schoenflies notation divided into seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

The correspondence between the Schoenflies and Hermann-Mauguin notation for the 32 crystallographic point groups is given in Table A3.4. 

The complete designation of the symmetry of a crystal requires the correct assignment of axes and identification of (he symmetry elements. There are a total of 32 different combinations of symmetry elements. Each of these has a unique Hermann-Mauguin notation or point group and is called a crystal class. The 32 crystal classes can be divided into six crystal systems. We will (ry to give you an appreciation of point groups and crystal classes, but our main emphasis will be on the more general crystal systems. 

Molecular point group Hermann-Mauguin notation (Schoenflies notation) Percentage of crystal structures in noncentrosymmetric space groups Number of structures considered... 

I 1.9. Determine the crystallographic point group for each of the following crystals, where the rotational axes and mirror planes are indicated. Use both the Schoenflies and Hermann-Mauguin notations. 

Thirty-Two Point Groups of Symmetries in Hermann—Mauguin Notations... 

Table1.3-3 The 32 crystallographic point groups translation list from the Hermann-Mauguin to the Schoenflies notation...

Since the components of property tensors are quantities that can be measured experimentally, they have to be real. The results obtained from Equations [64] and [65] are real, if real rotation matrices and real irreducible representations are used. The latter can always be achieved for the multidimensional representations. However, there are 10 crystallographic point groups which have pairs of one-dimensional irreducible representations complex conjugate to each other. In a case like this, one first computes pairs of conjugate complex tensors and then forms two real linear combinations for each conjugate pair as illustrated below for C3 (or 3 in Hermann-Mauguin notation). 

The comparison of the Hermann-Mauguin notation for crystallographic point groups with the Schoenflies notation for symmetry point groups (Section 3) is apparent fi-om Table 5. 

Table 4.1 Survey of the lattice systems, point groups and plane groups together with the applying symmetry elements. The short notation for the Hermann-Mauguin symbols given in brackets is only occasionally used in the literature.

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