Six-fold rotation axis

The twofold mirror-rotation axis is the simplest among the mirror-rotation axes. There are also axes of fourfold mirror-rotation, sixfold mirror-rotation, and so on. Generally speaking, a 2 -fold mirror-rotation axis consists of the following operations a rotation by (360/2//) and a reflection through the plane perpendicular to the rotation axis. The symmetry of the snowflake involves this type of mirror-rotation axis. The snowflake obviously has a center of symmetry. The symmetry class m-6 m contains a center of symmetry at the intersection of the six-fold rotation axis and the perpendicular symmetry plane. In general, for all m n m symmetry classes with n even, the point of intersection of the //-fold rotation axis and the perpendicular symmetry plane is also a center of symmetry. When n is odd in an m-n m symmetry class, however, there is no center of symmetry present. 

Six-fold rotation axis and six-fold inversion axis... 

The six-fold rotation axis Figure 1.15, left) results in six symmetrically equivalent objects by rotating the original object around the axis by 60°, 120°, 180°, 240°, 300° and 360°. 

The six-fold rotation axis also contains one three-fold and one two-fold rotation axes, while the six-fold inversion axis contains a three-fold rotation and a two-fold inversion (mirror plane) axes as sub-elements. Thus, any N-fold symmetry axis with N > 1 always includes either rotation or inversion axes of lower order(s), which is(are) integer divisor(s) of N. 

Rotations around the six-fold rotation axis parallel to Z result in x-y, x, z ... 

Six-fold rotation axis with centre of symmetry... 

Six [(Me3Si)2N]3M units are orientated about a six-fold rotation axis such that a cylindrical channel, large enough to include a benzene ring, is formed. This phenomenon has been described in some detail (9a). The existence of the hollow cavity accounts for the low density of the solid (ca. lg cm 3). A single crystal, when viewed under a microscope, appears to contain a hole in the center, the macroscopic structure apparently mirrors the microscopic array (9b). The molecular packing shows that the molecules are three-dimensional molecular sieves. 

This set of unit cells shows that there is no six-fold rotational axis along the c-axis (remember that the top view looks down the c-axis). However, there is a three-fold rotational axis, and this is sufQcient to place wurtzite in the hexagonal system. 

We begin this section with an example of the X-ray diffraction on the nematic, smectic A and crystalline smectic B phases. In Fig. 5.16 there is a series of X-ray photos of the same mesogenic compound at different temperatures. In this experiment, the material flow induced by the electric current aligns molecular axes in the nematic phase parallel to the field direction, which is horizontal, but in the SmA phase parallel to the smectic layers. Correspondingly diffraction patterns of the nematic and smectic phase considerably differ from each other. In the crystalline SmBcr phase the picture shows the six-fold rotation axis perpendicular to the figure plane. Below we shall discuss such pictures in detail, but let us begin with solid crystals. 

We have recently demonstrated the ability of six resorcin[4]arenes and eight water molecules to assemble in apolar media to form a spherical molecular assembly which conforms to a snub cube (Fig. 9.3). [10] The shell consists of 24 asymmetric units - each resorcin[4]arene lies on a four-fold rotation axis and each H2O molecule on a three-fold axis - in which the vertices of the square faces of the polyhedron correspond to the corners of the resorcin[4]arenes and the centroids of the eight triangles that adjoin three squares correspond to the water molecules. The assembly, which exhibits an external diameter of 2.4 nm, possesses an internal volume of about 1.4 A3 and is held together by 60 O-H O hydrogen bonds. 

This trigonal distortion of the MFg ion is similar to that previously observed in the rhombohedral potassium hexafluoro-osmate(v) structured Each ion has twelve fluorine neighbours, six in a puckered ring and almost coplanar with the ion, the other six in sets of three, one above and one below this plane. The situation of the dioxygenyl ion with its long axis coincident with the three-fold rotation axis is more acceptable on the grounds of close packing than the alternative random orientation (equivalent to rotation) of the ion. Apart from the non-spherical nature of the cation, the structure is almost identical with that proposed for the potassium hexafluoroantimonate(v). ... 

Furthermore, as we will see in sections 1.5.3 and 1.5.5, below, transformations performed by the three-fold inversion and the six-fold inversion axes can be represented by two independent simple symmetry elements. In the case of the three-fold inversion axis, 3, these are the threefold rotation axis and the center of inversion acting independently, and in the case of the six-fold inversion axis, 6, the two independent symmetry elements are the mirror plane and the three-fold rotation axis perpendicular to the plane, as denoted in Table 1.4. The remaining four-fold inversion axis, 4, is a unique symmetry element (section 1.5.4), which cannot be represented by any pair of independently acting symmetry elements. 

It is easy to see that the six symmetrically equivalent objects are related to one another by both the simple three-fold rotation axis and the center of inversion. Hence, the three-fold inversion axis is not only the result of two simultaneous operations (3 and 1), Iwt it is also the result of two independent operations. In other words, 3 is identical to 3 then 1. 

Figure 1.15. Six-fold rotation (left) and six-fold inversion (right) axes. The six-fold inversion axis is tilted by a few degrees away from the vertical to visualize all six symmetrically equivalent pyramids. The numbers next to the pyramids represent the original object (1), and the first generated object (2), etc. The odd numbers are for the pyramids with their apexes up.

The six-fold inversion axis Figure 1.15, right) also produces six symmetrically equivalent objects. Similar to the three-fold inversion axis, this symmetry element can be represented by two independent simple symmetry elements the first one is the three-fold rotation axis, which connects pyramids 1-3-5 and 2-4-6, and the second one is the mirror plane perpendicular to the three-fold rotation axis, which connects pyramids 1-4, 2-5, and 3-6. As an exercise, try to obtain all six symmetrically equivalent pyramids starting from the pyramid 1 as the original object by applying 60° rotations followed by immediate inversions. Keep in mind that objects are not retained in the intermediate positions because the six-fold rotation and inversion act simultaneously. 

In the previous examples Figure 1.16 and Table 1.5), the two-fold rotation axis and the mirror plane are perpendicular to one another. However, in general, symmetry elements may intersect at various angles (( )). When crystallographic symmetry elements are of concern and since only one-, two-, three-, four- and six-fold rotation axes are allowed, only a few specific angles ( ) are possible. In most cases they are 0° (e.g. when an axis belongs to a plane), 30°, 45°, 60° and 90°. The latter means that symmetry elements are mutually perpendicular. Furthermore, all symmetry elements should intersect along the same line or in one point, otherwise a translation and, therefore, an infinite symmetry results. 

Screw axes perform a rotation simultaneously with a translation along the rotation axis. In other words, the rotation occurs around the axis, while the translation occurs parallel to the axis. Crystallographic screw axes include only two-, three-, four- and six-fold rotations due to the three-dimensional periodicity of the crystal lattice, which prohibits five-, seven- and higher-order rotations. Hence, the allowed rotation angles are the same as for both rotation and inversion axes (see Eq. 1.2). 

Figure 1.32. Characteristic distribution of the two-, three-, four-, and six-fold rotation axes parallel to the unique axis in the unit cell in tetragonal (left), trigonal (center), and hexagonal (right) crystal systems as the result of their interaction with lattice translations.

The structures of the two rhombohedral forms of elemental boron (Table 5) are of interest in illustrating what can happen when icosahedra are packed into an infinite three-dimensional lattice. In these rhombohedral structures the local symmetry of a Bj2 icosahedron is reduced from //, to Dsd because of the loss of the 5-fold rotation axis when packing icosahedra into a crystal lattice. The 12 vertices of an icosahedron, which are all equivalent under //, local symmetry, are split under iXd local symmetry into two nonequivalent sets of six vertices each (Figure 19a). The six rhombohedral vertices (labeled R in Figure 19a) define the directions of the rhombohedral axes. The six equatorial vertices (labeled E in Figure 19a) lie in a staggered belt around the equator of the... 

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