System Hermann-Mauguin

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

Heck, R. F., 711 Heme, and dioxygen. 895-897 Hemerythrin. 908 Hemoglobin physiology of, 900-902 structure of, 902-908 Hermann-Mauguin system. 

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. 

A rotoinversion l about an axis is a rotation by the angle (f> followed by an inversion through a point on the axis. This is also a combined operation of the second type which is neither a pure rotation nor a pure inversion. It is easily seen that each rotoinversion is equivalent to a rotoreflection ) = S n(j>), S() = n + ). Thus, operations of the second type may be represented by either rotoreflections or by rotoinversions. We could limit ourselves to one or other of these two representations. However, the two most commonly used systems of nomenclature applied to geometrical symmetry do not use the same convention. The Schoenfiies system is based on rotoreflections, whereas the Hermann-Mauguin (or international) system is based on rotoinversions. In crystallography we prefer to use the Hermann- Mauguin system. The correspondence between l and S is shown in Table 2.1. 

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) ... 

The final symmetry element is described differently by the two systems, although both descriptions use a combination of the symmetry elements described previously. The Hermann-Mauguin inversion axis is a combination of rotation and inversion and is given the symbol tl -The symmetry element consists of a rotation by l/n of a revolution about... 

Table 2.4. Space groups, crystal systems, point groups, and the Hermann-Mauguin Symbols.

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... 

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)... 

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. 

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