Space-Group Symmetries

The symmetry operators of the space-group G of a crystal are of the form 

v is not necessarily a lattice translation t, since w may be either the null vector 0 or the particular non-lattice translation associated with some screw axis or glide plane. If v C a VR, then there are no screw axes or glide planes among the symmetry elements 

Note that in eq. (3), as in Chapter 16, no special symbol is used to signify when (R w +1) is a space-group function operator since this will always be clear from the context. It will often be convenient (following Venkataraman et al. (1975)) to shorten the notation for a space-group operator to 

In addition, to minimize the need for multiple subscripts, anK will now be denoted by the alternative (and completely equivalent) notation a(nn) and similarly rnK will be denoted by r (nn). 

because of the space-group symmetry, the displacement at (NK) is equal to Ru(nK), the rotated displacement from the equivalent site (nk). 

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Energy minimisation and normal mode analysis have an important role to play in the study of the solid state. Algorithms similar to those discussed above are employed but an extra feature of such systems, at least when they form a perfect lattice, is that it is can be possible to exploit the space group symmetry of the lattice to speed up the calculations. It is also important to properly take the interactions with atoms in neighbouring cells into account. 

Any perturbation from ideal space-group symmetry in a crystal will give rise to diffuse scattering. The X-ray diffuse scattering intensity at some point (hkl) in reciprocal space can be written as... 

Inspired by experimental observations on bundles of carbon nanotubes, calculations of the electronic structure have also been carried out on arrays of (6,6) armchair nanotubes to determine the crystalline structure of the arrays, the relative orientation of adjacent nanotubes, and the optimal spacing between them. Figure 5 shows one tetragonal and two hexagonal arrays that were considered, with space group symmetries P42/mmc P6/mmni Dh,), and P6/mcc... 

International Tables for Crystallography, Vol. A, Space-Group Symmetry, edited by Theo Hahn, 2nd ed. (Kluwer Academic, Dordrecht, 1989). 

The first was not the structure of brookite. The second, however, had the same space-group symmetry as brookite (Ft,6), and the predicted dimensions of the unit of structure agreed within 0.5% with those observed. Structure factors calculated for over fifty forms with the use of the predicted values of the nine parameters determining the atomic arrangement accounted satisfactorily for the observed intensities of reflections on rotation photographs. This extensive agreement is so striking as to permit the structure proposed for brookite (shown in Fig. 3) to be accepted with confidence. 

During the investigation of the structure of brookite, the orthorhombic form of titanium dioxide, another method of predicting a possible structure for ionic compounds was developed. This method, which is described in detail in Section III of this paper, depends on the assumption of a coordination structure. It leads to a number of possible simple structures, for each of which the size of the unit of structure, the space-group symmetry, and the positions of all ions are fixed. In some cases, but not all, these structures correspond to closepacking of the large ions when they do, the method further indicates... 

We believe that our conclusions can be accepted with considerable certainty, for the agreement between the predicted structure and the experimental results in regard to space-group symmetry, size of the unit of structure, and intensities of reflections on rotation photographs is so striking as to remove nearly completely from consideration the possibility of its being accidental. 

The Unit of Structure and Space-group Symmetry of Bixbyite. 

Our investigation of zunyite has shown the cubic unit of structure with a0 = 13.82 A to contain four molecules of composition Al Sifi OH, F)18Cl and to have the space-group symmetry T%, and has led to the formulation of a detailed atomic arrangement. 

A further complication is that an icosahedron sharing four faces loses its symmetry operations of the second sort and becomes chiral, D or L. The eclipsed configuration described above requires that D and l alternate. Hence, in a ring of five icosahedra one DD or LL bond must occur, with a strain in the ring of 40°, which is a rotation of 4 per shared face in each icosahedron. The space-group symmetry is thus changed from that of diamond to a subgroup. 

Two antiparallel helices, related by space group symmetry, are packed in an orthorhombic unit cell (Fig. 39b). There is substantial interdigitation between the helices so that side chains and main chains are linked by hydrogen bonds, such as 0-4E-0-4D (2,73 A) and 0-4D-0-3F (2.84 A) involving parallel and antiparallel strands, respectively. Plausible sites for sodium ions are near the... 

Fig. 14.3 Polyhedral packing plots for the two-dimensional layers of [RE(P2S6),/2(PS4)P in the series of solids A2RE(P2S6)i/2(PS4), where A=K, Cs RE = Y, La. Rare-earth polyhedra are striped PS4 polyhedra are black phosphorous atoms in P2S6 are shown as black circles. Alkali atoms are not shown for clarity. Although these phases have distinctly different structures based on space group symmetry and atomic positions, the compounds are clearly related upon close inspection of the building blocks.

In the following problems the positions of symmetrically equivalent atoms (due to space group symmetry) may have to be considered they are given as coordinate triplets to be calculated from the generating position x,y,z. To obtain positions of adjacent (bonded) atoms, some atomic positions may have to be shifted to a neighboring unit cell. 

Table 17.1 Crystallographic data of the hexagonal and cubic closest-packings of spheres. +F means +(j,0), +(j,0, j), +(0, j, j) (face centering). Values given as 0 or fractional numbers are fixed by space-group symmetry (special positions)...

The crystal structures of (EDT-TTFBr2)2MX4 and (EDO-TTFBr2)2MX4 are quite similar, although the space group symmetry is different in these two systems. However, this difference comes only from the conformation of terminal six-membered rings of the donor molecules, which plays no important role in the physical properties of the present salts. The donor molecules are stacked in a head-to-tail manner to form quasi-one-dimensional columns as shown in Fig. 6a. 

Adoption of numerical techniques that maximally exploit space group symmetry, the periodic boundary condition, and the nature of the basis set ... 

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