Rotational and Mirror Symmetry

Figure 10-13 The recognition of rotational and mirror symmetry in organic moiecuies aiiows the identification of chemicai-shift-equivaient hydrogens. The different colors distinguish among nuclei giving rise to separate absorptions with distinct chemical shifts.

It is the mentioned symmetry properties additional to the discrete translational symmetry that lead to a classification of the various possible point lattices by five Bravais lattices. Like the translations, these symmetry operations transform the lattice into itself They are rigid transforms, that is, the spacings between lattice points and the angles between lattice vectors are preserved. On the one hand, there are rotational axes normal to the lattice plane, whereby a twofold rotational axis is equivalent to inversion symmetry with respect to the lattice point through which the axis runs. On the other hand, there are the mirror lines (or reflection lines), which he within the lattice plane (for the three-dimensionaUy extended surface these hnes define mirror planes vertical to the surface). Both the rotational and mirror symmetry elements are point symmetry elements, as by their operation at... 

The Bravais lattices are classified according to the applying group of rotational and mirror symmetries. Figure 4.11 presents these classes with the translation vectors and the point symmetry elements indicated within one unit cell. 

There are two other phases that are similar to (3- and (3"-alumina but are built from spinel blocks that are six close-packed oxygen layers thick. The material (3" -alumina is the analog of (3-alumina, with the spinel blocks related by 180° mirror-plane (hexagonal) symmetry, while the phase (3""-alumina, the analog of (3"-alumina, has blocks related by 120° rotation and rhombohedral symmetry. 

The notation of the symmetry center or inversion center is 1 while the corresponding combined application of twofold rotation and mirror-reflection may also be considered to be just one symmetry transformation. The symmetry element is called a mirror-rotation symmetry axis of the second order, or twofold mirror-rotation symmetry axis and it is labeled 2. Thus, 1 = 2. 

An inversion may also be represented as the consecutive application of two simple symmetry elements, namely, a twofold rotation and mirror reflcc-... 

The notation of the symmetry center or inversion center is 1 while the corresponding combined application of twofold rotation and mirror reflection... 

It is always possible to choose a primitive cell. The discussion of the conditions which lead us, in certain cases to choose a multiple cell will be left until later (Section 2.6.1, Bravais lattices). For the moment, it is sufficient to indicate that, in the presence of rotational or mirror symmetry, we choose the vectors a, b and c to... 

The crystal class mirrors the internal symmetry of the crystal. The internal symmetry of any isolated object, including a crystal, can be described by a combination of axes of rotation and mirror planes. 

Schoenf lies system A system for categorizing symmetries of molecules. C groups contain oniy an n-foid rotation axis. C groups, in addition to the n-foid rotation axis, have a mirror piane that contains the axis of rotation (and mirror pianes associated with the existence of the n-foid axis). 

A semi-infinite crystal is a three-dimensional (3D) object. Its symmetry group contains, in addition to translations in the surface plane, only the rotations and mirror reflections that keep the atoms in the planes parallel to the surface. Only 17 such groups exist. Formally, these groups are isomorphic with diperiodic space groups in two dimensions. They are called plane groups. 

The second structure, Ni(100)-Hp4g(2x2)-2C, contains two (equivalent) C adatoms in each (2x2) unit cell, related by p4g symmetry (i.e. glide-plane symmetry, as well as 4-fold rotation and mirror planes). The no entries indicate that neither the clean Ni(100)-(lxl) nor the C-covered substrate are reconstructed (in the sense that no Ni-Ni bonds are made or broken relative to the ideally-terminated substrate) admittedly, this is debatable for the C-covered surface, since the Ni-Ni distances within the first Ni layer do change appreciably. 

The symmetry of the structure we are looking for is imposed on the field 0(r) by building up the field inside a unit cubic cell of a smaller polyhedron, replicating it by reflections, translations, and rotations. Such a procedure not only guarantees that the field has the required symmetry but also enables substantial reduction of independent variables 0/ the function F (f)ij k )- For example, structures having the symmetry of the simple cubic phase are built of quadrirectangular tetrahedron replicated by reflection. The faces of the tetrahedron lie in the planes of mirror symmetry. The volume of the tetrahedron is 1 /48 of the unit cell volume. 

In the Hermann-Mauguin Symbols, the same rotational axes are indicated, plus any inversion symmetry that may be present. The numbers indicate the number of rotations present, m shows that a mirror symmetry is present and the inversion symmetry is indicated by a bar over the number, i.e.- 0. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

Figure 8.4 Illustration showing layer normal (z), director (n), and other parts of the SmC structure. Twofold rotation axis of symmetry of SmC phase for singular point in center of layer is also illustrated. There is also mirror plane of symmetry parallel to plane of page, leading to C2h designation for the symmetry of phase. This phase is nonpolar and achiral.

Polar structures may have rotation symmetry and reflection symmetry. However, there can be no rotation or reflection normal to the principal rotation axis. Thus, the presence of the mirror plane normal to the C2 axis precludes any properties in the SmC requiring polar symmetry the SmC phase is nonpolar. 

Chirality (or a lack of mirror symmetry) plays an important role in the LC field. Molecular chirality, due to one or more chiral carbon site(s), can lead to a reduction in the phase symmetry, and yield a large variety of novel mesophases that possess unique structures and optical properties. One important consequence of chirality is polar order when molecules contain lateral electric dipoles. Electric polarization is obtained in tilted smectic phases. The reduced symmetry in the phase yields an in-layer polarization and the tilt sense of each layer can change synclinically (chiral SmC ) or anticlinically (SmC)) to form a helical superstructure perpendicular to the layer planes. Hence helical distributions of the molecules in the superstructure can result in a ferro- (SmC ), antiferro- (SmC)), and ferri-electric phases. Other chiral subphases (e.g., Q) can also exist. In the SmC) phase, the directions of the tilt alternate from one layer to the next, and the in-plane spontaneous polarization reverses by 180° between two neighbouring layers. The structures of the C a and C phases are less certain. The ferrielectric C shows two interdigitated helices as in the SmC) phase, but here the molecules are rotated by an angle different from 180° w.r.t. the helix axis between two neighbouring layers. 

Fig. 24. Stabilizing orbital interactions in cis and trans 1,2-difluoro-l,2-dihydroxy ethylene. Symmetry labels are with respect to a rotational axis (trans isomer) and mirror plane (cis isomer)...

To the extent that a crystal is a perfectly ordered structure, the specificity of a reaction therein is determined by the crystallographic symmetry. A crystal is built up by repeated translations, in three dimensions, of the contents of the unit cell. However, the space group usually contains elements additional to the pure translations, such as a center of inversion, rotation axis, and mirror plane. These elements can interrelate molecules within the unit cell. The smallest structural unit that can develop the whole crystal on repeated applications of all operations of the space group is called the asymmetric unit. This unit can consist of a fraction of a molecule, sometimes fractions of two or more molecules, a single whole molecule, or more than one molecule. If, for example, a molecule lies on a crystallographic center of inversion, the asymmetric unit will contain half... 

Finally, reference must be made to the important and interesting chiral crystal structures. There are two classes of symmetry elements those, such as inversion centers and mirror planes, that can interrelate. enantiomeric chiral molecules, and those, like rotation axes, that cannot. If the space group of the crystal is one that has only symmetry elements of the latter type, then the structure is a chiral one and all the constituent molecules are homochiral the dissymmetry of the molecules may be difficult to detect but, in principle, it is present. In general, if one enantiomer of a chiral compound is crystallized, it must form a chiral structure. A racemic mixture may crystallize as a racemic compound, or it may spontaneously resolve to give separate crystals of each enantiomer. The chemical consequences of an achiral substance crystallizing in a homochiral molecular assembly are perhaps the most intriguing of the stereochemical aspects of solid-state chemistry. 

Lord Kelvin lla> recognized that the term asymmetry does not reflect the essential features, and he introduced the concept of chiralty. He defined a geometrical object as chiral, if it is not superimposable onto its mirror image by rigid motions (rotation and translation). Chirality requires the absence of symmetry elements of the second kind (a- and Sn-operations) lu>>. In the gaseous or liquid state an optically active compound has always chiral molecules, but the reverse is not necessarily true. 

Fig. 8a-c. r rotation photographs of H. marismortui SOS crystals at 0 "C and at cryotemperature (obtained at XI1/EMBL/DESY and at SSRL/Stanford U.) a The hkO-orientation of a nearly perfectly aligned (although split) crystal reveals the mirror symmetry of the C-centred lattice plane. The severe overlap problem in this orientation caused by the large mosaic spread is obvious from this picture, b The Okl-orientation shows the extinctions of the twofold screw axis, c The best crystals have a Bragg resolution limit of about 6 A, which decreases to about 9 A in the course of a hundred exposures... 

S , S indicates a mirror symmetry (Spiegel in German) and n invariance by a -fold rotation around the perpendicular axis to the plane. An added superscript m number indicates sometimes that the symmetry operation is apphed m times. 

Actions such as rotating a molecule are called symmetry operations, and the rotational axes and mirror planes possessed by objects are examples of symmetry elements. 

Only certain symmetry operations are possible in crystals composed of identical unit cells. In three dimensions these are one-, two-, three-, four- and six-fold rotations and each of these axes combined with inversion through a centre to give I, 2 ( = m, mirror plane), 3, 4, and 6 operations. Five-fold rotations and rotations of order 7 and higher, while possible in a finite molecule, are not compatible with a three-dimensional lattice. 

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