Sixfold axes

Rotational Symmetry of 2D Lattices. Each of the five lattices has rotational symmetry about axes perpendicular to the plane of the lattice. For the oblique lattice and both the primitive and centered rectangular lattices these are twofold axes, but there are several types in each case. The standard symbol for a twofold rotation axis perpendicular to the plane of projection is . In the case of the square lattice there are fourfold as well as twofold axes. The symbol for a fourfold axis seen end-on is For the hexagonal lattice there are two-, three-, and sixfold axes the latter two are represented by a and , respectively. In Figure 11.4 are shown all of the rotation axes possessed by each lattice. 

Finally, if reflection lines passing through all the sixfold axes of p6 are added, we get the symmetry p6m, which, of course, has other automatically generated symmetry elements. 

Lattices with 3 or 6 Axes. The occurrence of six- and/or threefold axes in actual crystals occasions certain difficulties in classification. So long as we are concerned only with lattices, however, there need to be no ambiguity, if we proceed carefully and rigorously. For that reason we shall take a different (and more fundamental) approach in demonstrating that there are only two distinct lattice types consistent with the presence of three- or sixfold axes that are oriented in one direction only. The latter qualification is necessary to distinguish the present case from that of the isometric lattices where there... 

If, as in Figure 11.14, we make the sixfold axes of all 2D nets coincide, we obtain a primitive lattice that retains all the symmetry present in the 2D lattice p6. We call this the primitive, hexagonal lattice. However, we can also choose the stacking pattern shown in Figure 11.15a, where we place the origin of the cell in the nth layer over the point 1, i in the (n - l)th layer. The result of this stacking scheme is seen in elevation in Figure 11.156. It has several important properties. 

Hexagonal symmetry is lost Since the sixfold axes of the 2D net have been made coincident with threefold axes, only the threefold symmetry survives. The reason that this is nevertheless still called a hexagonal lattice will be given in Section 11.5. 

The 2D space group p6 arises by explicitly introducing one set of sixfold axes. Show with a diagram the other symmetry elements that arise automatically. 

Trigonal, tetragonal, and hexagonal crystal systems have three, four and sixfold axes of symmetry, respectively, while the cubic crystal contains four threefold axes along with diagonals of the cube as well as two-fold axes passing through the faces (see Fig. 15-14). 

When a crystal is rotated about an axis and inverted about the central point and at that point repeats itself, it is said to have an rnrit of rotary inversion. It is a twofold axis of rotary inversion if the geometrical figure is rolaled 180° and then inverted. Additionally, there are threefold, fourfold, and sixfold axes of rotary inversion possible. 

Figure 9-19. The eleven screw axes. The simple twofold, threefold, fourfold and sixfold axes are also shown for completeness. After Azaroff [24] copyright (1960) McGraw-Hill, Inc. used with permission.

Lastly, molecular motion affects solid-state spectra just as chemical exchange does in solution (Chapter 10 and Section 14.2). Nuclear magnetic resonance provided the proof that benzene molecules, structure 15-2, rotate in place about their sixfold axes (Example 4.3) in the crystal above 90 K (Kelvins absolute temperature 0°C = 273 K) ... 

The experimental result, Acr p = -5.3 ppm, which the theoretical community (see, e.g., Jameson, 1993) likes to cite for comparison with and confirmation of its calculations, is based on a single m.p. powder spectrum (Ryan et al., 1977). The asymmetry of that spectrum is desparately small and the value for Ao-g p that can be inferred is hardly more than an estimate of the upper limit of Act. Moreover, the experiment was done at a temperature of 77 K, which is too high to freeze out the well-known reorientational jumps of the benzene molecules about their sixfold axes. As Ryan et al. (1977) state explicitly, these jumps lead to a motional averaging of the in-plane shielding components. We feel, therefore, that the comparison of experimental and (converging) theoretical results for Aproton shielding in benzene is by no means fully specified by the... 

Similar kinds of relationships would arise among equivalent reflections for threefold or sixfold axes in the crystal as well. This tells us that symmetry elements in real space, the crystal, may be identified by searching the appropriate zones, or planes of reciprocal space for symmetrical patterns of diffraction intensities. If we see fourfold or threefold or sixfold distributions of reflections in the diffraction patterns, then they must imply corresponding symmetry relationships in the crystal. 

The hexagonal and trigonal systems yield special complications owing to the relationship between crystal axes and angles. One finds that face-centering or body-centering the primitive lattice types with simultaneous preservation of the threefold or sixfold axes is not possible. It suffices to say that only the two Bravais lattices of Fig. 1 are... 

Voids are frequently located at or along symmetry elements. Solvent molecules on those sites are generally highly disordered or fill one-dimensional channels along three-, four- or sixfold axes. 

Figure 2.10 shows that the presence of rotation or rotoinversion axes implies a characteristic metric for the lattice. Twofold axes do not impose any special metric reflection lines are only compatible with a rectangular or diamond lattice fourfold axes are only compatible with a square (tetragonal) lattice threefold and sixfold axes are only compatible with a triangular (hexagonal) lattice. 

Fig. 2.10. Tiling of two-dinnensional Euclidean planes (a) arbitrary lattice, twofold axes 2 (b) rectangles, reflection lines nn (c) diannonds, rectangular centered cell, reflection lines m and glide lines g (d) squares, fourfold axes 4 (e) triangles, threefold axes 3 (f) hexagons, sixfold axes 6, same type of cell as (e)...

Two-dimensional illustration of why only certain rotational axes can fill all space without any gaps or overlaps. Threefold, fourfold, and sixfold axes are allowed, for instance, but fivefold and eightfold axes are not. 

Since, however, R should transform the lattice vector a into the lattice vector a = R a it follows that all the elements of D (R) and hence its trace, must be integers. It follows that cos 2, it will also contain the subgroup C . The above three limitations ensure that the point group of the lattice can only be one of the seven point groups S2>C2/t> This is why there... 

For screw axes, the nomenclature is of the form n, indicating a 360°/n rotation, followed by a xjn translation along one of the unit cell vectors, a, b, or c. For example, the 6i and 63 screw axes would imply sixfold axes of rotation followed by... 

In principle molecules may belong to any of the possible point groups permitted by mathematical group theory. In reality, however, the actual chemical examples of specific point groups are somewhat restricted. In dealing with crystals only one-, two-, three-, four-, and sixfold axes are permissible. With such a restriction, there are only thirty-two possible combinations of symmetry elements yielding the thirty-two crystal point groups. 

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