Defects modulated structure

Field effects on chiral nematics can be interpreted by adding a pitch term to the free energy, so it might be expected that the Fr6edericksz transitions observed for chiral nematics will be similar to those described above for achiral nematics. In reality this is not the case because the helical structure in chiral phases prevents the formation of uniformly aligned films, and so defects and defect-modulated structures are unavoidable in many field-induced orientational changes. The effects of external fields on chiral ne-... 

A very different model of tubules with tilt variations was developed by Selinger et al.132,186 Instead of thermal fluctuations, these authors consider the possibility of systematic modulations in the molecular tilt direction. The concept of systematic modulations in tubules is motivated by modulated structures in chiral liquid crystals. Bulk chiral liquid crystals form cholesteric phases, with a helical twist in the molecular director, and thin films of chiral smectic-C liquid crystals form striped phases, with periodic arrays of defect lines.176 To determine whether tubules can form analogous structures, these authors generalize the free-energy of Eq. (5) to consider the expression... 

In general, these defect-free modulated structures can, to a first approximation, be divided into two parts. One part is a conventional structure that behaves like a normal crystal, but a second part exists that is modulated5 in one, two, or three dimensions. The fixed part of the structure might be, for example, the metal atoms, while the anions might be modulated in some fashion. The primary modulation might be in the position of the atoms, called a displacive modulation (Fig. 4.35a). Displacive modulations sometimes occur when a crystal structure is transforming from one... 

The real structures of these phases are more complex. The coordination of the Ti atoms is always six, but the coordination polyhedron of sulfur atoms around the metal atoms is in turn modulated by the modulations of the Sr chains. The result of this is that some of the TiS, polyhedra vary between octahedra and a form some way between an octahedron and a trigonal prism. The vast majority of compositions give incommensurately modulated structures with enormous unit cells. As in the case of the other modulated phases, and the many more not mentioned, composition variation is accommodated without recourse to defects. ... 

There is little doubt that many materials that at present are described as containing ordered arrays of point or extended defects will be successfully described as notionally defect-free modulated structures. For example, the intergrowth Aurivillius phases, described as containing extended planar defects, have recently been described compactly as modulated structures. " The same formalism has been applied to hexagonal perovskite structures and superconducting copper oxides. Others will certainly follow. 

Since our surroundings are three-dimensional, we tend to assume that crystals are formed by periodic arrangements of atoms or molecules in three dimensions. However, many crystals are periodic only in two, or even in one dimension, and some do not have periodic structure at all, e.g. solids with incommensurately modulated structures, certain polymers, and quasicrystals. Materials may assume states that are intermediate between those of a crystalline solid and a liquid, and they are called liquid crystals. Hence, in real crystals, periodicity and/or order extends over a shorter or longer range, which is a function of the nature of the material and conditions under which it was crystallized. Structures of real crystals, e.g. imperfections, distortions, defects and impurities, are subjects of separate disciplines, and symmetry concepts considered below assume an ideal crystal with perfect periodicity. ... 

IR-11.4.5 Defect clusters and use of quasi-chemical equations IR-11.5 Phase nomenclature IR-11.5.1 Introduction IR-11.5.2 Recommended notation IR-11.6 Non-stoichiometric phases IR-11.6.1 Introduction IR-11.6.2 Modulated structures IR-11.6.3 Crystallographic shear structures IR-11.6.4 Unit cell twinning or chemical twinning IR-11.6.5 Infinitely adaptive structures IR-11.6.6 Intercalation compounds IR-11.7 Polymorphism IR-11.7.1 Introduction IR-11.7.2 Use of crystal systems IR-11.8 Final remarks IR-11.9 References... 

IR-11.6.1 Introduction Several special problems of nomenclature for non-stoichiometric phases have arisen with the improvements in the precision with which their structures can be determined. Thus, there are references to homologous series, non-commensurate and semi-commensurate structures, Vernier structures, crystallographic shear phases, Wadsley defects, chemical twinned phases, infinitely adaptive phases and modulated structures. Many of the phases that fall into these classes have no observable composition ranges although they have complex structures and formulae an example is Mo17047. These phases, despite their complex formulae, are essentially stoichiometric and possession of a complex formula must not be taken as an indication of a non-stoichiometric compound (cf. Section IR-11.1.2). 

The weak interchain coupling orders the modulated structure three-dimen-sionally at the Peierls transition temperature, Tp. The force due to an applied electric field, within regions where the CDW is coherent, is counterbalanced by the pinning force of impurities or other defects. For high-purity crystals the application of small electric fields may depin the CDW from the lattice, and the modulated structure slides as a whole. 

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