Structure modulated

The smectic X phase in the pure compound I shows additionally to the smectic C layering the two-dimensional (modulated) structure in which the smectic C layers are periodically shifted with a respect to one another by half a layer spacing. 

A -THC, the main psychoactive component of cannabis, is a moderately potent partial agonist of the CBi and CB2 receptors, while cannabidiol has little affinity for either receptor (Table 6.7). The term classical cannabinoids is used to describe cannabinoid receptor modulators structurally related to (67), which have a tricyclic dibenzopyran core. While several other structural types of cannabinoid receptor modulators have been discovered in recent years, the classical cannabinoids are still by far the most extensively studied group in terms of SAR and pharmacology. 

In X-ray diffraction, modulated structures reveal themselves by the appearance of satellite reflections. In between the intense main reflections which correspond to the structure of the approximant, weaker reflections appear they do not fit into the regular pattern of the main reflections. 

Four body-centered unit cells of the incommensurately modulated structure of tellurium-III. 

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... 

Take care to note that a modulated structure is not the same as a modular structure. 

Modulations are normally described as waves. The modulation wave can fit exactly with the underlying unmodulated component, or more precisely with the unit cell of the underlying component, in which case the structure is described as a commensurately modulated structure. In cases where the dimensions of the modulation are incommensurate (i.e., do not fit) with the unit cell of the underlying structure, the phase is an incommensurately modulated phase. Modulation changes are normally continuous and reversible. 

Some of the earliest examples of modulated structures to be unraveled were the fluorite-related vernier structures. These structures occur in a number of anion-excess fluorite-related phases and use a modulation to accommodate composition variation. They can be illustrated by the orthorhombic phases formed when the oxyfluoride YOF reacts with small amounts of YF3 to give composition YOxF3 with x in the range 0.78-0.87, but similar phases occur in the Zr(N, O, F) system with x taking values of 2.12-2.25 and other systems in which the Zr is replaced by a variety of lanthanides. 

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. ... 

Modulated structures, corresponding to substances in which there is an average structure with a three-dimensional periodicity showing modulated perturbations. 

In concluding this section in which some properties of modulated structures and of quasicrystals have been considered, we underline that the characteristics of these two types of structures do not coincide. Incommensurately modulated structures show main and satellite diffractions, an average structure and crystallographic point symmetry. The quasicrystals have no average structure, non-crystallographic point symmetry, and give one kind of diffraction only. 

Notes on the crystallography of quasi-periodic structures. A general way to face the problems related to the interpretation of quasi-periodic structures (modulated structures, quasicrystals) is based on the introduction and application of higher-dimensional crystallography (de Wolff 1974, 1977, Janner and Janssen 1980, Yamamoto 1982, 1996, Steurer 1995). 

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