Incommensurate modulated structure

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

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. 

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. 

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. 

The essential properties of incommensurate modulated structures can be studied within a simple one-dimensional model, the well-known Frenkel-Kontorova model . The competing interactions between the substrate potential and the lateral adatom interactions are modeled by a chain of adatoms, coupled with harmonic springs of force constant K, placed in a cosine substrate potential of amplitude V and periodicity b (see Fig. 27). The microscopic energy of this model is ... 

Once Te-III was identified as incommensurate, subsequent analysis was conducted on the previously-collected powder-diffraction data using the formalism of 4D superspace [234], and the JANA2000 software for structure refinement [235]. The Rietveld refinement of the incommensurate Te-III diffraction profile is shown in Fig. 9, and the modulated structure is shown in Fig. 10. Tellurium was only the second element found to have a modulated crystal structure at high-pressure, the... 

Figure 6.5. ED in (001) of (a) O2-annealed 2212 and (c) the HRTEM modulated image—the atom columns are shown with an incommensurate (modulated) structure, (c) N2-annealed with 2212. (Temperature of anneal 400 °C.) (d) its HRTEM image (/ , d) show the commensurate structure, (e) Changes in magnetic flux inclusion annealed in (a) N2 (b) in oxygen. The resulting changes in the electronic structure due, e.g., to oxygen interstitials, influence the catalytic process, (f) (001) CBED of sample annealed in oxygen. HOLZ is arrowed. (After Gai J. Solid State Chem. 104 119.)...

Dzyaloshinskii (1964) belong to this category. The term modulated structures has been used to describe incommensurate perturbation (i.e. one in which the ratio of the imposed periodicity to that of the unit cell is irrational). A broader definition of modulated structure can be used to describe any periodic or partly periodic perturbation of a cystal structure with a repetition distance appreciably greater than the basic unit cell dimensions such a definition would include a variety of superstructures as well (Cowley et al, 1979). 

Bagautdinov, B., Hagiya, K., Kusaka, K., Ohmasa, M., and lishi, K. (2000). Two dimensional incommensurately modulated structure of (Sro.i3Cao.87)2-CoSi207 crystals. Acta Cryst. B56, 811-21. 

Dertinger (2001) also found that the a-axis modulated structure a of Figure 39 is much more sensitive to pressure, compared to the other two magnetic structures of Figure 39, and it even disappears at relatively low values of P. Interestingly he observed near-reentrant behavior also at temperatures and pressures where the a structure had disappeared. Therefore he concluded that the near-reentrant behavior in H0M2B2C cannot mainly be caused by the presence of the a incommensurate magnetic structure. This problem will be further discussed in the next Section 4.9.4. 

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. 

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. ... 

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