Six-fold inversion axis

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. 

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

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. 

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. 

Except for the center of inversion, which results in two objects, and three-fold inversion axis, which produces six symmetrically equivalent objects. See section 1.20.4 for an algebraic definition of the order of a symmetry element. 

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. 

The numbers 2, 3, 4 and 6 are used as symbols of the corresponding axes of symmetry while the symbols 3, 4 and 6 (3 bar, 4 bar, etc.) are used for the three-four- and six-fold (roto) inversion axes, corresponding to a counter-clockwise rotation of 360% around an axis followed by an inversion through a point on the axis. 

Hexagonal Unique six-fold axis, either rotation or inversion... 

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). 

The unit cell has the shape of a rhomb with angles of 60° and 120° (Figure 7). The point symmetry elements (six-fold axis normal to the plane, six two-fold rotation axes in the plane, six mirrors normal to the plane, and one in the plane, inversion) pass through the unit cell origin, at the center of the hexagons. There are two symmetry-related carbon atoms in the unit cell, labeled as A and B in Figures 7 and 9, with fractional coordinates (1/3,2/3) and... 

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