Tilt systems

More recently, Glazer developed a very influential notation for describing the 23 different tilt systems that have been observed (Glazer, 1972). The description is based on the rotations of the octahedra about each of the three Cartesian axes. The direction of the rotation about any one axis, relative to the rotations about the other two axes, is specified by a letter with a superscript that indicates whether the rotations in adjacent layers, which are coupled to the other layers via the vertex-sharing octahedra, are in the same (-I-) or opposite ( —) directions. For example, signifies an equivalent... 

In order to relate these to crystallographic structures, it is necessary to take into account the relative magnitudes of the tilts. In this case, three unequal tilts produce three unequal pseudocubic axes, two equal tilts and one different tilt produce two equal pseudocubic axes and one different and three equal tilts produce three equal pseudocubic axes. Moreover, whenever a superscript is 0 or +, a mirror plane must exist perpendicular to the tilt axis. By examination of the 10 tilt combinations described in terms of these and related constraints, 23 different tilt systems can be derived together with the pseudocubic unit cell dimensions and the ideal space group of the four octahedra unit cell (Table 1.2). It is, though, important to... 

Perovskites Structure-Property Relationships Table 1,3 Lattice parameter equations for some tilt systems... 

Tilt system Space group Lattice parameters... 

The 15 allowed transformations are imagined to operate alone. In reality many perovskite phases show additional octahedral cation displacements or distortions which will change things. For example, the tilt system 14, a"a"a" has an ideal trigonal space... 

Figure 1.16 The space groups allowed by the possible tilt systems of octahedra in pseudocubic perovskites...

The nitride antiperovskites show other structures, analogous to those displayed by oxide perovskites, but the number of known examples of these phases is minimal to date. The phases PNCaj and AsNCaj have tilted octahedra and belong to the GdFeOj structure type, tilt system a b"b". AsNCr is tetragonal with tilt system aVc . 

The symmetry of ordered rock-salt A2(BB X) double perovskites, taking into account octahedral tilting, has been systematically derived via group theoretical analysis. If the stmcture is cubic, with a tilt system aVa , the space group is Ftriitn (225), the new axes run parallel to the simple cubic perovskite axes, and the lattice parameter is double the simple cubic lattice parameter a. Thus, in the case of Ba FeMoO, a=0.81865 nm Relatively small A cations usually lead to octahedral tilting, as... 

Tungsten trioxide, WO3, is similar to ReOj, but the octahedra now exhibit cation displacement and octahedral tilting so that a variety of temperature-sensitive lower symmetry forms occur. Below -43°C the e-structure is monocUnic, with an a b c tilt system coupled with displacement of the W cations away from the centres of the octahedra. Between -43 and -tl7°C the 5-form has a triclinic structure with the... 

To contribute to a voxel (r, s, z) for z 0 in the cone beam geometry, the fan beams must be tilted out of the r, s plane to intersect the particular voxel (r, s, z) from various x-ray source orientations. As a result, the location of the reconstruction point in the tilted system is now determined by a new coordinate system (r, s) (Fig. 26.25). Consequently, the fan beam geometry in these new coordinates will change. Specifically, the new source distance is defined by... 

Another type of deformation of the perovskite structure is known to occur among the LT modifications of cerium and praseodymium aluminates and solid solutions based on them. In the orthorhombic Imma structures, 12 nearest distances in the RO12 polyhedra split into a set of five imique R-O distances, typically grouped as 1-4-2-4-1. The six Al-O bonds within the octahedra split into two shorter and four longer distances, but this deformation is very weak and octahedra are distorted mainly due to considerable deviations of the O-Al-O angles from 90° (see Table 28). Moreover, the octahedra tilt either about the twofold [110]p axis or about both [010]p and [001]p pseudo-axes of the perovskite cell, which corresponds to the tilt system a% b (Figure 39C). 

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