Roto-reflections

Figure 7.2 The different symmetry elements of the center ABg. (a) A trigonal axis, C3 (b) A binary axis, C2. (c) A symmetry axis belonging to both the 6C4 and SCj classes, (d) A symmetry plane, au- (e, f) Two of the six aj, reflection planes, (g) A view down the C3 axis in (a) to show a roto-reflection operation, S. ...

It can be seen that there are seven other symmetry operations (roto-reflections) of this class, which is then denoted by SSe, the subscript 6 indicating a rotation through lit 16. In a similar way, we can analyze other symmetry operations of classes 65 4,15 2 (commonly called an inversion symmetry operation and denoted by /), and IE (the identity operation that leaves the octahedron unchanged). 

Roto-reflection, that is, a rotation followed by a reflection through a plane perpendicular to the axis of rotation... 

Roto-inversion operators (combinations of rotation plus inversion 1,2,3,4, or 6) these are sometimes replaced by roto-reflection operators. Some scientists use roto-reflection operators, while crystal-lographers have chosen roto-inversion. 

One n-Fold Roto-Reflection Axis S , (Rotation by C Plus One Mirror Perpendicular to Cj)... 

Alternatively, roto-reflection axes eombining simultaneous rotation and reflection may be used, however, each of them is identical in its action to one of the roto-inversion axis. 

Figure 1. Symmetry of the. single wall CNT (8,6), (6,0) and (6.6). The horizontal rotational axes U and U are symmetrie.s of all the tubes, while the mirror planes (o , o),), the glide plane and the roto-reflectional plane are. symmetrie.s of the achiral tubes only (from [3])...

Table 2.1. Correspondence between roto-reflections S and rotoinversions l ...

For instance, through the combinations (are noted with x ) between the rotations and the reflections are obtained new symmetry operations, the roto-reflections, which generate spatial changes, so that the final results overlapped to the initial structure. 

Symmetry operations, therefore, can be visualized by means of certain symmetry elements represented by various graphical objects. There are four so-called simple symmetry elements a point to visualize inversion, a line for rotation, a plane for reflection and the already mentioned translation is also a simple symmetry element, which can be visualized as a vector. Simple symmetry elements may be combined with one another producing complex symmetry elements that include roto-inversion axes, screw axes and glide planes. 

The mirror plane (two-fold inversion axis) reflects a clear pyramid in a plane to yield the shaded pyramid and vice versa, as shown in Figure 1.12 on the right. The equivalent symmetry element, i.e. the two-fold inversion axis, rotates an object by 180" as shown by the dotted image of a pyramid with its apex down in Figure 1.12, right, but the simultaneous inversion through the point from this intermediate position results in the shaded pyramid. The mirror plane is used to describe this operation rather than the two-fold inversion axis because of its simplicity and a better graphical representation of the reflection operation versus the roto-inversion. The mirror plane also results in two symmetrically equivalent objects. 

Further on, from the nine mirroring planes three are perpendicular to axis of 4 -order (and if only one is unique results that there is associated an unique reflection plane) other six planes being perpendicular on axis of 2 -order (and, of these only one axes in unique, the uniqueness is reflected on the correspondent perpendicular plane), and there is no reflection plane perpendicular on the roto-inversion axes of 3 -order finally the symbol will be... 

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