Rotoinversion

Rotoinversion. The symmetry element is a rotoinversion axis or, for short, an inversion axis. This refers to a coupled symmetry operation which involves two motions take a rotation through an angle of 360/N degrees immediately followed by an inversion at a point located on the axis (Fig. 3.3) ... 

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. 

Associated with this change from rotoreflections (5,/s) to rotoinversions, there are other notational changes and it is most efficient to deal with them all at once. To describe point symmetry in a crystal we use the following symmetry elements and operations. 

Second, there are five rotoinversion operations, symbolized 1, 2, 3, 4, and 6. We need to examine these more closely to see how they relate to the Schonflies symbols 5, m, and / (where it is well to recall that m = Sx and / s S2). 

Figure 11.16. Diagrams comparing rotoinversion and rotoreflections operations.

Chirality in Crystals. When chiral molecules form crystals the space group symmetry must conform with the chirality of the molecules. In the case of racemic mixtures there are two possibilities. By far the commonest is that the racemic mixture persists in each crystal, where there are then pairs of opposite enantiomorphs related by inversion centers or mirror planes. In rare cases, spontaneous resolution occurs and each crystal contains only R or only S molecules. In that event or, obviously, when a resolved optically active compound crystallizes, the space group must be one that has no rotoinversion axis. According to our earlier discussion (page 34) the chiral molecule cannot itself reside on such an axis. Neither can it reside elsewhere in the unit cell unless its enantiomorph is also present. 

Since handedness (left-handed versus right-handed) is important in molecules the eight symmetry operations can be rethought as (C) 4 operations of the first kind (which preserve handedness) translation, identity, rotation, and screw rotation (D) 5 operations of the second kind (which reverse handedness, and produce enantiomorphs) inversion, reflection, rotoinversion, and glide planes. 

The body-centered position in the cubic Bravais lattice (a = fa = c) is an inversion center and for this reason is taken as the origin in space group Im 3 m. The position possesses three-fold rotoinversion symmetry with three perpendicular mirror planes. 

Apart from the symmetry elements described in Chapter 3 and above, an additional type of rotation axis occurs in a solid that is not found in planar shapes, the inversion axis, n, (pronounced n bar ). The operation of an inversion axis consists of a rotation combined with a centre of symmetry. These axes are also called improper rotation axes, to distinguish them from the ordinary proper rotation axes described above. The symmetry operation of an improper rotation axis is that of rotoinversion. Two solid objects... 

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,... 

Only the two-fold rotation about c of the twin lattice is a correct twin operation, in the sense that it restores the lattice, or a sublattice, of the individuals. If however twin operation. The rotations about c give simply the (approximate) relative rotations between pairs of twinned mica individuals, but are not true twin operations. Similar considerations apply also to the rotoinversion operations, s depends upon the obliquity of the twin but, at least in Li-poor trioctahedral micas, is sufficiently small to be neglected for practical purposes (Donnay et al. 1964 Nespolo et al. 1997a,b, 2000a). 

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

A rotoinversion l about an axis is a rotation by the angle (f> followed by an inversion through a point on the axis. This is also a combined operation of the second type which is neither a pure rotation nor a pure inversion. It is easily seen that each rotoinversion is equivalent to a rotoreflection ) = S n(j>), S() = n + ). Thus, operations of the second type may be represented by either rotoreflections or by rotoinversions. We could limit ourselves to one or other of these two representations. However, the two most commonly used systems of nomenclature applied to geometrical symmetry do not use the same convention. The Schoenfiies system is based on rotoreflections, whereas the Hermann-Mauguin (or international) system is based on rotoinversions. In crystallography we prefer to use the Hermann- Mauguin system. The correspondence between l and S is shown in Table 2.1. 

The groups formed by rotations and rotoinversions (or rotoreflections) are called point groups. 

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... 

Table 2.3. Equivalence between rotoinversion and rotoreflection axes...

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...

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