CONTENTS 1 Symmetry considerations

The minimum amount of information needed to specify a crystal structure is the unit cell type, that is, cubic, tetragonal, and so on, the unit cell parameters, and the positions of all of the atoms in the unit cell. The atomic contents of the unit cell are a simple multiple, Z, of the composition of the material. The value of Z is equal to the number of formula units of the solid in the unit cell. Atom positions are expressed in terms of three coordinates, x, y, and z. These are taken as fractions of a, b, and c, the unit cell sides, say and j. The x, y, and z coordinates are plotted with respect to the unit cell axes, not to a Cartesian set of axes. The space group describes the symmetry of the unit cell, and although it is not mandatory when specifying a structure, its use considerably shortens the list of atomic positions that must be specified in order to built the structure. 

The Ti4+ distribution in TS-1 has also been studied by computational methods (34,62,160-163). The actual location of the Ti atoms in the framework of titanosilicates is difficult to determine experimentally because of the low Ti content (Section II), and information obtained from theoretical methods is, therefore, of considerable interest. In the orthorhombic MFI structure, substitution can take place at 12 crystallographically different tetrahedral (T) sites (T1-T12) (Fig. 1 and Section II.A.l.b). In the monoclinic MFI framework, the mirror symmetry is lost and 24 crystallographically different T sites can be distinguished (Fig. 31) (160). 

Figure 3.12 shows typical Raman spectra of several doped ZnO thin films. Additional modes (AM), occurring at to 275, 510, 582, 643, and 856 cm-1 (the first four of them are shown and marked by vertical solid lines in Fig. 3.12), were first assigned to N-incorporation [49-51], because the intensity of these modes was reported to increase with increasing N-content [50], However, the AMs appear also in Raman spectra of ZnO samples doped with other elements (Fig. 3.12a), [48,52,53]). Therefore, it was suggested that the AMs are related to defect-induced modes [48]. Theoretical considerations confirmed this assignment [131]. It was discussed that the AMs could be related to modes of ZnO, which are Raman-inactive within a perfect crystal. Upon doping-induced defect formation, the translational crystal symmetry can be broken, and Raman-inactive modes may become Raman-active. The Raman spectra of the ZnO thin films with transition metals in Fig. 3.12b show a different behavior than those in Fig. 3.12a [43,48], Raman spectra of Fe0.08Zn0.92O contain the above described AMs, but with different intensity ratios. For MnZnO, CoZnO, and NiZnO a broad band between iv 500 cm-1...

Consideration of the octahedral model in accordance with symmetry knowledge also has been used to predict the presence of the mirror isomerism in complex compounds with definite content and structure. Its discovery was made by Werner in 1911, and is a confirmation of the coordination theory. ... 

So far, our discussion of symmetry of the lattice was limited to lattice points and symmetry of the unit cell. The next step is to think about symmetry of the lattice including the contents of the unit cell. This immediately brings translational symmetry into consideration to reflect the periodic nature of crystal lattices, which are continuous or infinite object. As... 

The permutational isomers under consideration correspond to the orbits of R = Dj in the set of distributions 5 e 7 of content c = (4,4). Since the skeleton is planar, P R, but the ligands are achiral, thus all isomers are achiral. Concerning the construction, we are faced with the problem of evaluating a transversal of a subset of the set of symmetry classes of distributions... 

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