Rotoinversion axes

The requirement for the existence of enantiomers is a chiral structure. Chirality is solely a symmetry property a rigid object is chiral if it is not superposable by pure rotation or translation on its image formed by inversion. Such an object contains no rotoinversion axis (or rotoreflection axis cf. Section 3.1). Since the reflection plane and the inversion center are special cases of rotoinversion axes (2 and 1), they are excluded. 

The equivalence of the important rotoreflection axes with rotoinversion axes or other symmetry operators is given in Table 4.1. In crystallography the rotoinversion operation is always preferred,... 

The ensemble of fixed points (points, lines or planed) of a symmetry operation are called symmetry elements. To the fixed point of l or S , we must add the fixed line corresponding to the operation A or the plane which is perpendicular to it. Rotation axes correspond to the operations A , centers and rotoinversion axes to the operations l , and mirror planes and rotoreflection axes to the... 

Fig. 2.5. Rotoinversion axes 4, 3 and 6. These axes represent cyclic groups. In contrast, non-cyclic groups are obtained by combining an even-order axis with a center of symmetry or with a perpendicular mirror plane...

In a periodic structure we find series of symmetry elements, i.e. series of rotation and rotoinversion axes as shown in Fig. 2.9. 

The only allowed values of n are thus w = 1,2,3,4,6. Periodic structures can only be invariant with respect to the rotation axes 1,2,3,4,6 and with respect to the rotoinversion axes T, 2 = m, 3,4,6 ("or to the corresponding rotoreflec-tion 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. 

The presence of symmetry elements without a translation component, i.e. rotation axes X and rotoinversion axes x, cannot be detected by systematic absences. 

For each crystal system, three axes (four in the hexagonal system) are assigned that coincide with the symmetry axes or are perpendicular to mirror planes, ha the isometric system, these axes coincide with the four-fold rotation or rotoinversion axes or the two-fold rotational axes. They are mutually perpendicular and are labeled ai, a.2, and a3, rather than the conventional labels of a, b, and c, because they are identical in every respect other than orientation. By convention, the positive end of the ai axis is toward the reader, the positive end of the aa axis is to the right in the plane of the paper, and the positive end of the 3 axis is up in the vertical direction. The axes are shown for the octahedron in Figure 27. 

There is also a center of symmetry. Although it is probably not obvious, there are also three-fold rotoinversion axes coaxial with the three-fold axes (a hand-held model is almost essential to see this). The highest order axes are the three four-fold axes that mn through the middle of each face. (The axis goes through the sodium ion at the center of the face in Figure 41 below.)... 

By combining the rotation with the reflection are obtained the axes of rotoreflexion or the gyroids, and combining the rotation with the inversion are obtained the axes of inversion or rotoinversion. These are complex axes (elements) of symmetry. Are noted by the barred order of rotation (ii) or for the gyroids, respectively with A for the inversion axes. It will be demonstrated that only one of those operations of symmetry is self-consistent, all others may be reduced to the association of simple operations is about the tetra-gyroid A or the fourth order axis of inversion Af. Let s follow the effect of the rotoinversion axes. 

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