Selected Unit Cells

A number of selected crystal structures of polymers are shown in this section in projection along the helix axes. The poly(ethylsilylethylene) of Fig. 5.23 has its silicon atoms marked by solid circles. Neighboring helices are related by a center of symmetry, so that they must be enantiomorphous and also anticlined, i.e., the helix pairs have different handedness (d-RH and f-LH helices) and opposite inclinations of side-groups (up- and down-helices) as discussed in Sect. 5.1.8. The coordination number for the helices is three instead of the expected four because the 31 and 32 screw axes of the 2 3/1 helices match the trigonal lattice symmetry and permit a closer overall packing with CN = 3 rather than 4 (see Fig. 5.21). 

Unit Cell of Crystals of Poly (o -f luorosty rene) 

2) has a less extended chain because of the more compact cii-configuration of the double bond. To arrive at an extended chain, the translational repeat has to be doubled, as illustrated in the sketch in Fig. 5.28. Chains A and B have the methyl groups substituted on carbon atoms 3 and 5, but have their chain axes reversed. Chains C and D have the methyl groups substituted on carbon atoms 2 and 6, and again, the action of the screw axis in the ab-plane turns the chain in going from C to D. These four possible arrangements of the same chain lead to disorder in the crystals, as in the vinyl helices of Fig. 5.22. The most frequent disorder occurs when pairs A-B and C-D do not alternate regularly. 

A coordination number of four is demonstrated by isotactic poly(o-methylstyrene) crystals, as shown in Fig. 5.29. The neighboring 2 4/1 helixes are created by the indicated glide planes c and are of opposite handedness, but isoclined (drawn are the up chains). Many of the vinyl helices with larger side-groups have 2 4/1 helices in their crystal structures (see also Fig. 5.14). 

The diamond glide plane d has been left out for clarity. 

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Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Aluminum distribution in zeolites is also important to the catalytic activity. An inbalance in charge between the silicon atoms in the zeolite framework creates active sites, which determine the predominant reactivity and selectivity of FCC catalyst. Selectivity and octane performance are correlated with unit cell size, which in turn can be correlated with the number of aluminum atoms in the zeolite framework. ... 

In this zeolitic material a very low percentage of Ti(IV), dispersed in a pure siliceous microporous matrix (with the MFI framework, the same as that of the ZSM-5 zeolite), is able to oxidize in mild conditions many substrate with extremely high activity and selectivity (see Sect. 2). However, after more than three decades, a complete picture of reaction mechanisms is still missing. Major problems related to characterization are due to the extremely high dilution of Ti(IV) in the zeolitic matrix and the presence of high amounts of water in the reaction media. The first point requires characterization techniques very sensitive and selective towards Ti(IV). For instance, XRD measurements have been able to recognize the presence of Ti(IV) in the framework only indirectly, via the measured unit cell volume increase [21,22], but attempts to... 

The atomic positions were allowed to relax until all forces were smaller than 0.005 eV/A The local structure of the Eu2+ impurity in CsMgBr3 is tackled via geometry optimization based on the cluster approach.40 The optimized lattice structure of CsMgBr3 is used. An appropriate cluster is obtained as a selective cut along the 3-fold c axis of the unit cell of CsMgBr3 (Figure la). A moiety containing five units... 

For the origin of the unit cell a geometrically unique point is selected, with priority given to an inversion center. 

Figure 3.13. Crystal structure of N4H9CU7S4 viewed down [010], with the unit cell shown using a dashed line. For clarity, atoms are represented as spheres, with uniform sizes selected for each atom type. Within the Cu7Sl slabs, Cu—Cu bonding is not shown to improve clarity.

Unit Cell Parameters and Selected Interatomic Distances and Angles in m-Quartz PON and in the Corresponding Si02 Phase ... 

Figui 8. Comparison of the experimental and calculated Na + Ag selecti ities in zeolite Y (113) as a function of the number of Ag ions/unit cell. The set of selectivity... 

The parallelepiped built upon the three translations selected as unit translations is defined as the unit cell. 

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