Incommensurately modulated structures

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

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 ... 

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.)...

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. 

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. ... 

As noted above, when a is irrational, the incommensurate modulation occurs, and this is shown schematically in Figure 1.53. The exact description of incommensurately modulated structure is impossible using only conventional crystallographic symmetry in the unit cell of any size smaller than the crystal. The periodicity of the structure can only be restored by using two different periodic functions. The first function is the conventional crystallographic translation, and the second one is the sinusoidal modulation function with certain period, which is incommensurate with the corresponding translation, and amplitude. 

Figure 1.53. The ideally periodic one-dimensional structure, the corresponding modulation function with the period X= Mq, which is incommensurate with a, and the amplitude. 4 (top), and the resulting incommensurately modulated structure (bottom).

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). 

Fig. 24. Variation of the phase

The three intermediate phases II, III, and IV of thiourea are believed to be incommensurate, and the incommensurately modulated structures in phase II and rv have been analyzed [50]. Another commensurate phase which is stable between phase I and phase II in a narrow temperature region of about 2 K has been recognized many years ago [27c,/], and was refined at 170 K by Tanisaki et al. [27b]. This ninefold superstructure of thiourea is characterized by a rotation of the (NH2)2CS molecule along the c axis coupled with a displacement of the center of mass in a plane perpendicular to the axis. Around mirror planes (y = 1/4 and 3/4) the local structure is isostructural with phase I. Therefore the superstructure is constructed of alternately polarized layers which are sandwiched by domain walls (or discom-mensurations) whose local structure is that of the paraelectric room temperature phase (V) around y = 0 and 1/2. 

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