Fourfold axes

Th = symmetry of an octahedron without fourfold axes [2/m3, short m3 ]. 

The occurrence of twinned crystals is a widespread phenomenon. They may consist of individuals that can be depicted macroscopically as in the case of the dovetail twins of gypsum, where the two components are mirror-inverted (Fig. 18.8). There may also be numerous alternating components which sometimes cause a streaky appearance of the crystals (polysynthetic twin). One of the twin components is converted to the other by some symmetry operation (twinning operation), for example by a reflection in the case of the dovetail twins. Another example is the Dauphine twins of quartz which are intercon-verted by a twofold rotation axis (Fig. 18.8). Threefold or fourfold axes can also occur as symmetry elements between the components the domains then have three or four orientations. The twinning operation is not a symmetry operation of the space group of the structure, but it must be compatible with the given structural facts. 

There is a second long interface stretching between the threefold and fourfold axes, involving both hydrophobic and hydrophilic interactions. Close to the threefold axis is an intersubunit salt bridge between Asp-139 of subunit I and His-128 in III, which links the N-terminal end of helix D (III) to a position near the kink in helix D (I). Further along the interface, N-terminal residues 6-12 of subunit III make several interactions with the C-helix of subunit I, including several which are mediated... 

Figure 6.6 E helices around the fourfold axes in four ferritins and EcoBFR. (a) Superposition of -carbon traces for three mammalian ferritins, HuHF, HoLF and RaLF (b) BfLF (c) EcBFR. (a) is viewed perpendicular to and (b) and (c) along the fourfold axis. Reprinted from Harrison et al., 1998, by courtesy of Marcel Dekker, Inc.

Figure 1. The protein coat of ferritin from horse spleen. N refers to the N-terminus and E refers to the location of the E-helix, a short helix with an axis perpendicular to the long axis of the subunit and which lines the channels formed at the fourfold axes. (Reproduced with permission from Ref. 5. Copyright 1983 Elsevier.)...

Fig. 16. a. The atomic arrangement in sodium chloride, and some of its axes of symmetry, b and c. Fourfold axes of cube and octahedron, d and e. Twofold axes of cube and octahedron. 

There are also two ways to add reflection lines to symmetry p4. In one case they pass through all fourfold axes, thus giving the symmetry designated p4m. In this symmetry (as we shall see in more detail presently) there are... 

The diagrams for the remaining symmetries show clearly all of the symmetry elements in these more elaborate cases. For instance, in pAm we see the twofold axes that were required by the presence of the fourfold axes of p4 as well as the glide lines that arise automatically when the reflection lines are introduced. In p4g we see there are actually two networks of glide lines. 

Example /. The net of vertical and horizontal reflection lines is rather obvious, as are the twofold axes. The fourfold axes may be slightly less obvious. Once they are found, however, it becomes clear that we need to turn the pattern 45° in order to put it into the standard orientation for one of the square symmetries, / 4, p4m, or p4g. Since we have seen the net of reflection lines we know it must be either p4m or / 4g, and when we note that the reflection lines pass between, not through, the fourfold axes we conclude that it is p4g. The presence of the two different nets of glide lines, only one net passing through the fourfold axes, is not obvious. The reader should convince himself that they are there. The second diagram in column C shows one example of each type of glide, g, takes brick AB to A B while g takes brick... 

If in addition to orthogonality of the translation vectors we also require two vectors to be of equal length, say a = b, we have a tetragonal lattice. This now has the same mirror planes and twofold axes as an orthorhombic lattice but has fourfold axes parallel to the c direction. In this case there is only one form of centering possible, namely, / centering. 

Start with blank paper and a closed book and develop, systematically, the seventeen 2D space groups, or selected subsets (e.g., those with only g and m lines, those with fourfold axes). 

A molecule with four threefold axes and no proper fourfold axes is tetrahedral. If there is a centre of inversion (unusual but possible, for example, for an M(H20) + cluster) the system has point group Th- If the molecule has three proper twofold axes and six planes of reflection the point group is T4, otherwise it is T (the latter is very unusual). 

A molecule with four threefold axes and three proper fourfold axes is cubic or octahedral. If it has a centre of inversion the point group is Oh, otherwise it is O (very unusual). 

Symmetry elements of a cube. There are three fourfold axes, four threefold axes, and six twofold axes of rotation. There are three 100 mirror planes and six 110 mirror planes (only two are shown). 

Axial symmetry according to Bleaney s approach is maintained in polymetallic lanthanide complexes when the metals lie on the molecular threefold or fourfold axes. For n magnetically non-coupled lanthanide ions packed along the symmetry axis, contact and pseudo-contact contributions can be considered as additive and the original model-free equation (eq. (47)) is transformed into eq. (61) in which the sum runs over the n paramagnetic centres, each being located at the origin of its own reference frame associated with a specific set of axial coordinates O 1 and r" (the z axis corresponds to the molecular symmetry axis, fig. 54),... 

Axial symmetry in trimetallic lanthanide complexes requires the location of the metal ions along the molecular threefold or fourfold axes. Since the terminal coordination sites are different from the central coordination site for symmetry reasons, two different crystal-field parameters 2terminal ancj /)2ccntral must be considered. Equation (61) holds for the general case of three (n = 3) magnetically non-coupled lanthanide metal ions packed along the symmetry axis and eqs. (62)-(64) can be used for homotrimetallic axial complexes. To the best of our knowledge, only one partial study of the NMR data for a >3-symmetrical axial trimetallic complex has been reported (Bocquet et al., 2002 Floquet et al., 2003 see sect. 5.1.2). The Dih-symmetrical complexes [R3(L16-3H)2(OH2)6]3+ do not fit the requirements for axial symmetry since the metal ions are located on mirror planes and not on the threefold axis, but... 

E six rotations C4, C4 1 about the fourfold axes three rotations Q2 = C2 about the fourfold axes eight rotations C3/ C3-1 about the threefold axes six rotations C 2 about the twofold-axes. This set of twenty-four rotations by itself forms the group O. 

Oh. Three mutually perpendicular fourfold rotation axes and four threefold rotation axes which are tilted with respect to the fourfold axes in a uniform manner, and a center of symmetry. Examples ... 

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