Incommensurate modulated composite

In cases where Cj and fail to come into register at all, that is, when piq is irrational, or at least when q is large, the stmctures are called incommensurate modulated composite structures. In order to analyse these incoimnensurate structures, a more complex superspace description needs to be adopted, which involves the introduction of a modulation vector q =yc, where c is a vector corresponding to the basic repeat along the c-axis, corresponding to Cj. 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

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. ... 

Ideally, incommensurately modulated structures have two fairly distinct parts. One part of the crystal structure is conventional and behaves like a normal crystal. An additional, more or less independent part, exists that is modulated in one, two, or three dimensions. For example, the fixed part of the structure might be the metal atom array, while the modulated part might be the anion array. The modulation might be in the position of the atoms, called a displacive modulation or the occupancy of a site, for example, the gradual replacement of O by F in a compound M(0, F)2, to give a compositional modulation. In some more complex crystals modulation in one part of the structure induces a corresponding modulation in the fixed part. 

Not all incommensurate structures are composite. It is possible to have incommensurate modulations in a structure composed of a single infinite building block, particularly if a weak cation fits rather loosely into a hole in a flexible framework. The polyhedra that compose the framework tend to twist to give the cation a distorted environment. These twists can often be described by a wave with a wavelength that may or may not be commensurate with the lattice translation of the crystal. If it is commensurate, the twisting is described as... 

Incommensurate structures have been known for a long time in minerals, whereas TTF-TCNQ is one of the very first organic material in which a incommensurate phase has been observed. There are two main types of incommensurate crystal structures. The first class is that of intergrowth or composite structures, where two (or more) mutually incommensurate substructures coexist, each with a different three-dimensional translational periodicity. As a result, the composite crystal consists of several modulated substructures, which penetrate each other and we cannot say which is the host substructure. The second class is that of a basic triperiodic structure which exhibits a periodic distortion either of the atomic positions (displa-cive modulation) and/or of the occupation probability of atoms (density modulation). When the distortion is commensurate with the translation period of the underlying lattice, the result is a superstructure otherwise, it is an incommensurately modulated structure (IMS) that has no three-dimensional lattice periodicity. 

The symmetry treatment of incommensurate structures is beyond the scope of this chapter. From Equation (33) it is readily seen that for indexing, whatever the reflection of the diffraction pattern of an incommensurately modulated structure, we need to specify 3 + d integers (h, k, I, m, m2... m fl. It can be demonstrated that the observed 3D structure can be considered as a projection of a periodic structure m3 + d dimensions over the real 3D space, which is a hyper-plane not cutting the points of the 3 + d lattice except the origin. The superspace approach of de Wolff, Janssen and Janner is now well established and has become the routine way of treating the symmetry of the displacive incommensurate structures. The same approach has been extended to study general quasiperiodic structures (composite structures and quasicrystals). 

XND XND code from X ray laboratory data to incommensurately modulated phases. Rietveld modelling of complex materials, J. F. Berar and G. Baldinozzi, Int. Union Crystallogr., Commission Powder Diffr. Newsletter, 1998, No. 20, 3 5 Incommensurate and composite structures, refinement of multiple powder data sets... 

Let the period of the basic structure be a and the modulation wavelength be the ratio a/X may be (1) a rational or (2) an irrational number (Fig. 1.3-7). In case (1), the structure is commensurately modulated we observe a qa superstructure, where q= /X. This superstructure is periodic. In case (2), the structure is incommensurately modulated. Of course, the experimental distinction between the two cases is limited by the finite experimental resolution, q may be a function of external variables such as temperature, pressure, or chemical composition, i. e. = f T, p, X), and may adopt a rational value to result in a commensurate lock-in stmcture. On the other hand, an incommensurate charge-density wave may exist this can be moved through a basic crystal without changing the internal energy U of the crystal. 

Composite crystals are crystalline structures that consist of two or more periodic substructures, each one having its own 3-D periodicity to a first approximation. The symmetry of each of these subsystems is characterized by one of the 230 space groups. However, owing to their mutual interaction, the fine structure consists of a collection of incommensurately modulated subsystems. All known composite stractures to date have at least one lattice direction in common and consist of a maximum of three substructures. There are three main classes ... 

Those I and S structures without modulation may also, within certain limits, be infinitesimally adaptive. Atomic substitutions within the two layer sets (which includes the introduction of vacancies ) will change the dimensions of the component unit cells and hence also the (approximate) coincidence cell, as well as the chemical composition and the layer valence balance. And so, once again, the division between semi- and incommensurability - and sometimes even commensurability - may, in these compounds, be rather arbitrary. 

Compositionally induced incommensurate-commensurate transitions also were documented by Seifert et al. (1987), who examined synthetic melilites in the akermanite system (Ca2(Mgi c,Fejc)Si207) by transmission electron microscopy. They found that superstructures with modulations of 19 A are present in these minerals at room temperature for 0 < < 0.7. However, the incommensurate phases disappear on heating,... 

Quasicrystals may thus be regarded as a special type of incommensurate system, which may be described by space groups of dimension larger than three in a similar way to modulated crystal phases and incommensurate composite structures. In the case of icosahedral quasicrystals the above model based on the icosahedral quaternion group //, fits well into the idea of 6D space groups. Each of the three standard coordinates, namely x, y, and z, corresponds to two coordinates in 6D space, namely a rational and an irrational coordinate corresponding to the rational and irrational portions of variables, of the form a -H o V5, where a and d are integers. Projection of the lattice points of this 6D space of icosahedral symmetry into conventional 3D space leads to the icosahedral quasicrystal lattice. 

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