Crystal symmetries characterization

A pecuhar sohd phase, which has been discovered not too long ago [172], is the quasi-crystalline phase. Quasi-crystals are characterized by a fivefold or icosahedral symmetry which is not of crystallographic type and therefore was assumed to be forbidden. In addition to dislocations which also exist in normal crystals, quasi-crystals show new types of defects called phasons. Computer simulations of the growth of quasicrystals [173] are still somewhat scarce, but an increasing number of quasi-crystalline details are studied by simulations, including dislocations and phasons, anomalous self-diffusion, and crack propagation [174,175]. 

The examples of polymer crystal structures shown in the previous sections are ideal structures, which can be described with the traditional concepts of the principles of equivalence and close packing or the new concepts of symmetry breaking146 and frustration.154 The models of perfect crystals are characterized by a long-range positional order for all the atoms (disregarding thermal motion). The X-ray diffraction patterns of such crystals, oriented with the chain axes along one direction (as in oriented fibers), present sharp reflections organized in layer lines. 

The advantage of being able to record diffraction intensities over a range of incident beam directions makes CBED readily accessible for comparison with simulations. Thus, CBED is a quantitative diffraction technique. In past 15 years, CBED has evolved from a tool primarily for crystal symmetry determination to the most accurate technique for strain and structure factor measurement [16]. For defects, large angle CBED technique can characterize individual dislocations, stacking faults and interfaces. For applications to defect structures and structure without three-dimensional periodicity, parallel-beam illumination with a very small beam convergence is required. 

These systems can be described in terms of their symmetry elements. A triclinic crystal has only a center of symmetry. Monoclinic crystals have a single axis of twofold rotational symmetry. Orthorhombic crystals have three mutually perpendicular axes of twofold symmetry. With tetragonal symmetry, there is a single axis of fourfold symmetry. Cubic crystals are characterized by four threefold axes of symmetry, the <111> axes. There is a single axis of threefold symmetry in the rhombohedral system. The hexagonal system involves a single axis of sixfold symmetry. 

In recent years, theoretical calculations of the molecular vibrations that characterize solid polymers have become sophisticated and complete enough to permit detailed comparison with measured spectra. These calculations include the internal frequencies of the basic repeat unit, skeletal modes involving cooperative or phased vibrations of an entire chain, and interchain vibrations. The skeletal modes are determined by intrachain forces and chain conformation interchain modes depend upon crystal symmetry and spacing and upon interchain forces (usually much weaker than intrachain forces). 

Such inversion must occur to a considerable degree in the glass-ceramics Ac as evident from the appearance of additional lines due to E A2 emissions resulting from the modified Cr(III) energy levels in the low symmetry characterizing cases of inverted spinel sites. This behaviour in crystals was thoroughly studied by Mikenda et al. ° . ... 

In principle, any modification of the intra- or intermolecular relationships that break the averaged crystal symmetry on a macroscopic scale corresponds to a structural phase transition. Diffraction techniques (X-ray or neutron) are thus of primary importance to characterize the different phases. From the Bragg peaks that measure the long-range order of the mean crystal structure, the space groups, atomic positions and thermal parameters can be determined in each phases (Fig. 1). Moreover, in the most favorable cases, these methods can directly measure information on the order parameter and pretransitional ordering by following the superstructure at 7< Tc,... 

In fact, with increasing temperature, these materials may not completely lose their translational order while retaining their orientational order. All liquid crystals are characterized by their orientational order, but liquid crystal phases show varying amounts of translational order with the only exception of nematics. Apart from the above basic symmetries, there is another important symmetry — bond orientation symmetry. This symmetry is important when dealing with hexatic phases. Liquid crystals are classified in terms of following criterion ... 

What is the relation between the space group which describes the symmetry of a crystal structure at the atomic level and the point group which describes the symmetry of the corresponding macroscopic crystal By analogy with the Bernhardi principle (Section 1.3.1), a crystal is characterized by its properties in different directions, for example rate of growth, thermal and electrical conductivity, elasticity and piezoelectricity. Let us consider the periodic structure shown in Fig. 2.12. Clearly, the macroscopic properties in the directions of the two large... 

We do not observe an image of the crystal in the focal plane of the objective of the microscope, but an interference pattern. Each point in this plane corresponds to the direction of a wave normal characterized by a birefringence, and hence, by an interference color or light intensity. The intensity goes to zero for all waves polarized according to the polarizer and the analyzer. The interference figures are characteristic of the crystal symmetry, the optical sign and the orientation of the indicatrix. 

A novel crystal form II of the benzodiazepine chlordiazepoxide was reported [49] the standard form I and the new one II are similar with respect to the cell volume or crystal symmetry. Both structures consist of a pair of hydrogen-bonded dimers in the asymmetric unit. The major difference is the crystal packing of the dimers in the form II the dimers are displaced with respect to each other. Both forms were characterized by solid-state C NMR and the obtained spectra were different. The splitting of the methyl group signal was observed in II. Notable differences were also found for aromatic and olefinic carbons. The differing chemical environments in the two crystal structures can explain these effects. 

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