Cell rotation rules

By simply extending the methods used for the point group selection rules, one can obtain selection rules for molecules involving rotation-translation and reflection-translation. Two approaches are available. The older method is the Bhagavantum-Ventkatarayudu (BV) method (50), and necessitates the availability of the structure of the material being studied. The other method is that of Halford-Hornig (HH) (51-53) and considers only the local symmetry of a solid and the number of molecules in the unit cell and is simpler to work with. This method is also called the correlation method and depends on the proper selection of the site symmetry in the unit cell. 

The four unit cells shown in Figure 1.21 have the same symmetry (a twofold rotation axis, which is perpendicular to the plane of the projection and passes through the center of each unit cell), but they have different shapes and areas (volumes in three dimensions). Furthermore, the two unit cells located on top of Figure 1.21 do not contain lattice points inside the unit cell, while each of the remaining two has an additional lattice point in the middle. We note that all unit cells depicted in Figure 1.21 satisfy the rule for the monoclinic crystal system established in Table /.//. It is quite obvious, that more unit cells can be selected in Figure 1.21, and an infinite number of choices is possible in the infinite lattice, all in agreement with Table 1.11. 

Symmetry operations apply not only to the unit cells shown in Fig. 2-3, considered merely as geometric shapes, but also to the point lattices associated with them. The latter condition rules out the possibility that the cubic system, for example, could include a base-centered point lattice, since such an array of points would not have the minimum set of symmetry elements required by the cubic system, namely four 3-fold rotation axes. Such a lattice would be classified in the tetragonal system, which has no 3-fold axes and in which accidental equality of the a and c axes is allowed. 

Quasicrystals or quasiperiodic crystals are metallic alloys which yield sharp diffraction patterns that display 5-, 8-, 10- or 12-fold symmetry rotational axes - forbidden by the rules of classical crystallography. The first quasicrystals discovered, and most of those that have been investigated, have icosahedral symmetry. Two main models of quasicrystals have been suggested. In the first, a quasicrystal can be regarded as made up of icosahedral clusters of metal atoms, all oriented in the same way, and separated by variable amounts of disordered material. Alternatively, quasicrystals can be considered to be three-dimensional analogues of Penrose tilings. In either case, the material does not possess a crystallographic unit cell in the conventional sense. 

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