Two-fold axes

Fig. 7. Cross-sectional representation of the cavities in 8 when crystallised with benzene. The cavities, which are linked along two-fold axes parallel to c, are shown as the van der Waals surfaces due to the hydrogen atoms of the host molecules. Benzene guest molecules are shown in their major orientation 13>...

If c is an anticlockwise rotation through 7r/2, satisfying c4 = e, about an axis through the centre O perpendicular to the plane of the figure, and b = b, is the two-fold rotation about the axis OX, other two-fold axes can be identified as b2 about AC, b3 about OY and b4 about BD, i.e. 

In a famous paper by Shechtman et al. (1984) electron diffraction patterns were shown of rapidly quenched and solidified aluminium-manganese alloys. Sharp diffraction peaks, suggesting long-range translational order, were observed with the presence however of five-fold symmetry (that is of a non-crystallographic symmetry see 3.6.1.1). By different orientation of the specimen five-fold axes (in 6 directions), three-fold axes (in 10 directions) and two-fold axes (in 15 directions) were identified with the subsequent observation of the existence also of an inver-sion centre the assignment of this phase to the icosahedral point group, m36, was defined. 

A simple calculation for urea by Spackman is instructive. Urea crystallizes in an acentric space group (it is a well-known nonlinear optical material), in which the symmetry axes of the molecules coincide with the two-fold axes of the space group. All molecules are lined up parallel to the tetragonal c axis. If the electric field is given by E, and the principal element of the diagonalized molecular polarizability tensor along the c axis by oc , the induced moment along the polar c axis is... 

Figure B.2 Flow chart for point-group assignment. A symmetry plane that is perpendicular to a proper axis of rotation is a <7 , plane, one that includes the unique proper axis of rotation is a cr plane, and one that includes the highest order proper axis of rotation and bisects the remaining two-fold axes of rotation is a a, plane...

Groups with one main axis of symmetry the z axis points along the main axis of symmetry and, whore applicable, the x axis lies in one of the rrT planes or coincides with one of the C2 axes. For 2 and 2h the x, y, z axes coincide with the three equivalent two-fold axes. 

Trigonal, tetragonal, and hexagonal crystal systems have three, four and sixfold axes of symmetry, respectively, while the cubic crystal contains four threefold axes along with diagonals of the cube as well as two-fold axes passing through the faces (see Fig. 15-14). 

Particular care should be taken to check for two-fold axes perpendicular to the principal axis. Overlooking these is probably the commonest error made in identifying point groups. Note also that the group that results from the product of a centre of inversion with Dn is Dnh when n is even but Dnd when n is odd. 

Equations (21) and (23) showthat M is invariant under C3Z and C2x- Therefore the compatible point groups are those that contain a proper or improper three-fold axis, with or without proper or improper two-fold axes normal to the principal axis. These point groups are 3,3,32, 3m, 3m (or C3, S6, D3, C3v, D3d). To generate a 3-D lattice with three-fold rotational symmetry, the second and third layers of the hexagonal planar net in Figure 16.5 must be translated so that Pj lies over P2 and P3, respectively, i.e. at (i/3 2/3 y3) and (2/3 /3 2/3). 

Figure 17.4. Labeling of the symmetry elements of a cube as used in the derivation of Ym and of the symmetrized bases. The points 1, 2, 3, and 4 label the three-fold axes. The poles of the two-fold axes are marked a, b, c, d, e, and f. The points 1,2,3,4,5,6,7, and 8 provide a graphical representation of the permutations of [1 1 1] (cf. Table 17.10).

The room-temperature crystal structure of (NH3)K3C60 was first determined by Rosseinsky et al. [17]. Ammoniation of K3C60 leads to an orthorhombic distortion of the fee structure with one K+ and one NH3 per octahedral site. The C60 units are orientationally ordered with their two-fold axes aligned with the unit cell vectors (space group Fmmm), while K+ and NH3 are oppositely displaced from the site centre along the [110] direction and the K+-NH3 pairs are randomly oriented along one of four K+-NH3 directions. 

In the triclinic, monoclinic and orthorhombic space groups, the types of symmetry operations that alter a molecule s orientation are planes (mirror and glide), two-fold axes (rotation and screw) and centers of symmetry. These operations involve changing the signs of one, two or all three of the coordinates of the atomic positions. The unique direction of a plane or axis must be parallel... 

As a second example, suppose that the crystal has three mutually perpendicular, two-fold axes of symmetry. In this case, if we take these axes as the axes of coordinates, each term in the Fourier series may contain the product of three cosines, but, unless the coordinate planes are also planes of symmetry, the series must contain terms with trigonometric sines. These sines, however, must occur in pairs. A term cannot contain the product of one sine and two cosines, nor can it be the product of three sines for the term must have the same value when we change the algebraic signs... 

FIG. 3.1. (a) Molecular structure of Buckminsterfullerene Cgg- The structure on the backside is also visible. The figure is taken from Ref. [1, p. 110], (b) A view of Buckminsterfullerene Cgo in one of the two standard forms. In each of these two forms the two fold axes are aligned with cubic (100) axes. The thick lines show the double bonds, see Ref. [1, p. 154]. 

However, in the case of N03 , there are also two-fold axes that lie along each N-0 bond ... 

The capsid proteins of the Leviviridae family are all similar. The main difference is that the dimers of QJ3 (Golmohammadi et al, 1996) and PP7 (Tars et al, 2000) are both linked by disulfides between the loops close to the 5-fold and quasi-two-fold axes. In these viruses, there are fewer dimer-dimer contacts, and the capsids are less stable when the disulfides are not formed (Cielens et al, 2000). 

Figure 2.8 The regular orbit of the Oj point group constructed as the first 48-membered shell for Cubium. Two distinct bond lengths are evident in the projected structure, especially about the two-fold axes through the equator, but the structure remains of Oj point symmetry.

There are 15 two-fold axes in an object of Ih point symmetry. Sets of four vertices about the 30 poles of these axes are shown in Figure 2.19c. The 30-vertex cage of the icosidodecahedron to result on contraction of these local sets onto the pole positions on the unit sphere. 

Figure 2.23 Pairwise coalescence contractions of the regular orbit cage of I[, on the four-membered local sets about the poles of the two-fold axes, to return in the first column a copy of the 3-valent Ogo orbit isomer of the fuUerene cage of Figure 2.21, the truncated dodecahedron of geometry, while the small rhombicosidodecahedron results, in the second column, for the alternative sequence of pairwise contractions, which maintain point symmetry.

Orthorhombic Three mutually perpendicular two-fold axes, either rotation or inversion Trigonal Unique three-fold axis, either rotation or inversion... 

Figure 2.47 Structural relationship between (a) SSZ-53 and (b) SSZ-59. The layers in SSZ-53 are related by rotations about two-fold axes within the plane of the Figure. The layers in SSZ-59 are related by rotations about two-fold axes perpendicular to the plane of the Figure. Reproduced with permission from [117], Copyright (2003) Wiley-VCH...

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