System Schoenflies

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. 

Hermann-Mauguin system followed by the Schoenflies system of notation. 

In the Schoenflies system the improper axis is an axis of rotation-relleclion (see page 52). In the International system the axis of rotatory inversion (7/) is one of n-fold rotation followed by inversion (see Fig. 3.29). 

For a flow chart or the 32 crystallographic point groups in the International system that is analogous to Fig. 3.16 for the Schoenflies system, see Breneman, G. L. J. Chem. Ethic. 1987,64, 216. 

The Schoenflies system denotes with the symmetry class with a single A axis. So, Cj is the S5mibol for class with A, is for the class willi A as the only element of S5mimetry, and analogous C3, and C the present discussion follows (Chiriac-Putz-Chiriac, 2005)... 

For completeness we must also define the identity operation ( leave alone ) as a symmetry operation, which applies to all molecules, so that strictly speaking we should never say that a molecule has no symmetry. This is given the symbol E (in some texts as /). Note that this set of five symmetry operations contains some hidden duplications a can also be described as 5i, and i as 2, while the identity operation is equivalent to Ci so all five symmetry operations can actually be described by either C or S . The various symmetry operations and elements, along with their conventional symbols (known as the Schoenflies system) are summarized in Table 2.1. 

At this point, you may find that the subject of symmetry in a crysted structure to be confusing. However, by studying the terminology carefully in Table 2-2, one can begin to sort out the various lattice structures and the symbols used to delineate them. All of the crystal systems can be described by use of either Schoenflies or Hermaim-Mauguin S5mbols, coupled with the use of the proper geometrical symbols. 

The coordinate system of reference is taken with the vertical principal axis (z axis). Schoenflies symbols are rather compact—they designate only a minimum of the symmetry elements present in the following way (the corresponding Hermann-Mauguin symbols are given in brackets) ... 

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). 

Note 3 The smectic C structure corresponds to monoclinic symmetry characterised by the symbol C2h, in the Schoenflies notation and the space group t Hm in the International System. 

Note 4 The point-group symmetry is C2h (2/m) in the Schoenflies notation, and the space group, 121m in the International System. 

Note 3 The relevant space group of a Colh mesophase is P 6lmmm (equivalent to P 6/m 2 m in the International System and point group Dhh in the Schoenflies notation). 

The equivalent symmetry element in the Schoenflies notation is the improper axis of symmetry, S which is a combination of rotation and reflection. The symmetry element consists of a rotation by n of a revolution about the axis, followed by reflection through a plane at right angles to the axis. Figure 1.14 thus presents an S4 axis, where the Fi rotates to the dotted position and then reflects to F2. The equivalent inversion axes and improper symmetry axes for the two systems are shown in Table 1.1. 

TABLE 1.1 Equivalent symmetry elements in the Schoenflies and Hermann-Mauguin Systems... 

Table 2.4 shows the crystal systems, point groups, and the corresponding space groups. The numbers for space groups are those as derived and numbered by Schoenflies. The space groups isomorphous to each point group are indicated by a superscript (e.g., Number 194, D6/,4). 

Crystal system Schoenflies symbol International symbol Crystal system Schoenflies symbol International symbol... 

Crystal system International symbol Schoenflies symbol ... 

To make matters even more complicated, it is much more convenient in crystallography to use an entirely different system of symmetry symbols, termed the Herman-Maugin (as opposed to Schoenflies) notation (Table 8.2). 

Table 8.2 Symmetry notation in the Schoenflies and Herman-Maugin systems commonly used in...

Crystal system Schoenflies symbol Hermann-Mauguin symbol Examples in minerals ... 

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. 

Crystal System Point Groups (Hermann-Mauguin) Point Groups (Schoenflies)... 

Schoenflies, A. Krystallsysteme und Krystallstruktur. [Crystal systems and crystal structures.] B. G. Teubner Leipzig (1891). (2nd edn. 1923.) Reprinted. Springer Berlin (1984). 

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