Point Hermann-Mauguin symbols

Examples of rotation axes. In each case the Hermann-Mauguin symbol is given on the left side, and the Schoenflies symbol on the right side. tni means point, pronounced dyan in Chinese, hoshi in Japanese... 

Cubic point groups have four threefold axes (3 or 3) that mutually intersect at angles of 109.47°. They correspond to the four body diagonals of a cube (directions x+y+z, -x+y-z, -x-y+z and x-y-z, added vectorially). In the directions x, y, and z there are axes 4, 4 or 2, and there can be reflection planes perpendicular to them. In the six directions x+y, x-y, x+z,. .. twofold axes and reflection planes may be present. The sequence of the reference directions in the Hermann-Mauguin symbols is z, x+y+z, x+y. The occurrence of a 3 in the second position of the symbol (direction x+y+z) gives evidence of a cubic point group. See Fig. 3.8. 

Symmetrical geometric figures and their point group symbols in each case, the short Hermann-Mauguin symbol is given to the left, and the Schoenflies symbol to the right... 

Figures 3.8 and 3.9 list point group symbols and illustrate them by geometric figures. In addition to the short Hermann-Mauguin symbols the Schoenflies symbols are also listed. Full Hermann-Mauguin symbols for some point groups are ...

Please refer to Table A.5.1. In each row a general face is shown on the left, and the symmetry elements appear on the right Hermann-Mauguin symbols are shown beneath. Points on the general face are distinguished by for the northern hemisphere and O for the southern hemisphere. For symmetry element symbols, refer to Appendix A.4. 

The symbol 3/m is used here because it is descriptive of the two operations that are being discussed. Conventionally this point group is designated by the equivalent Hermann-Mauguin symbol 6. 

Point-group Hermann-Mauguin symbols symbols... 

Point-group Hermann—Mauguin symbols Sehoenjlies symbols... 

Point-group Hermann-Mauguin symbol Schaenflies symbol... 

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

Space group number Point group Hermann- Mauguin symbol... 

Schoenflies symbols are given for all point groups. Hermann/Mauguin symbols are given for the 32 crystallographic point groups. 

Any solid object can be classified in terms of the collection of symmetry elements that can be attributed to the shape. The combinations of allowed symmetry elements form the general three-dimensional point groups or non-crystal-lographic three-dimensional point groups. The symmetry operators are described here by the International or Hermann-Mauguin symbols. 

The symmetry operators that characterise a point group are collected together into the point group symbol. Crystallographers use the International or Hermann-Mauguin symbols. [The alternative Schoenflies symbols are given in Appendix 3.] As in two-dimensions, the... 

Table 4.2 The order of the Hermann-Mauguin symbols in point groups...

The operation of the allowed symmetry elements on the 14 Bravais lattices must leave each lattice point unchanged. The symmetry operators are thus representative of the point symmetry of the lattices. The most important lattice symmetry elements are given in Table 4.3. In all except the simplest case, two point group symbols are listed. The first is called the full Hermann-Mauguin symbol, and contains the most complete description. The second is called the short Hermann-Mauguin symbol, and is a condensed version of the full symbol. The order in which the operators within the symbol are written is given in Table 4.2. 

The places in the Hermann-Mauguin symbol for a point group refer to ... 

Point group number Hermann-Mauguin symbol Crystal system Centrosymmetric Polar Sohncke group... 

A mirror plane (Hermann-Mauguin symbol m) is any plane that divides the lattice into two halves so that one half is the mirror image of the other half across the plane. As a lattice is made up of regular points (sites of motifs), the number of mirror planes in a plane pattern may be less than the number in a plane lattice. This depends on the shape and symmetry of the motifs which make that pattern. 

Crystallographic point groups are conventionally represented by symlwls called Hermann-Mauguin symbols, which have the following features ... 

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.

System and lattice symbol Point group Hermann-Mauguin symbol Short Full Plane group number Symmetry elements additional to discrete tranlations... 

A Hermann-Mauguin point-group symbol consists of a listing of the symmetry elements that are present according to certain rules in such a way that their relative orientations can... 

When two symmetry operations are combined, a third symmetry operation can result automatically. For example, the combination of a twofold rotation with a reflection at a plane perpendicular to the rotation axis automatically results in an inversion center at the site where the axis crosses the plane. It makes no difference which two of the three symmetry operations are combined (2, m or T), the third one always results (Fig. 3.6). Hermann-Mauguin Point-group Symbols... 

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