Commensurate composite crystals

A group of crystals show diffraction patterns in which two or more 3D lattices having periods commensurate or incommensurate to each other may be recognized. In other words, the crystal consists of two or more interpenetrating substructures (two or more different atom sets) with different periods at least along one direction (see Fig. 3.42). Names such as composite crystals, vernier structures, misfit-layer structures, and chimney-ladder structures have been used for this group of structures. 

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. 

CAI s that were once molten (type B and compact type A) apparently crystallized under conditions where both partial pressures and total pressures were low because they exhibit marked fractionation of Mg isotopes relative to chondritic isotope ratios. But much remains to be learned from the distribution of this fractionation. Models and laboratory experiments indicate that Mg, O, and Si should fractionate to different degrees in a CAI (Davis et al. 1990 Richter et al. 2002) commensurate with the different equilibrium vapor pressures of Mg, SiO and other O-bearing species. Only now, with the advent of more precise mass spectrometry and sampling techniques, is it possible to search for these differences. Also, models prediet that there should be variations in isotope ratios with growth direction and Mg/Al content in minerals like melilite. Identification of such trends would verify the validity of the theory. Conversely, if no correlations between position, mineral composition, and Mg, Si, and O isotopic composition are found in once molten CAIs, it implies that the objects acquired their isotopic signals prior to final crystallization. Evidence of this nature could be used to determine which objects were melted more than once. 

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... 

Commensurate structure In this case, the vector F(002) is parallel to the vector 2g and moreover these vectors are related as F(002) = n x 2g (n = positive integer) this equation leads to (cf — Cg) = Cb or ncp = ( + 1)cb. The supercell with this relation is generally called a commensurate structure. It has been concluded from the structural principle mentioned above that these have composition Bai+i(Fe2S4)j ( = integer). Figure 2.44 shows examples of EDPs for these compounds. In these, only the structures with i — 8 and 9 have been determined by X-ray single crystal analyses. 

For example, as one can see in Fig. 10, the specific surface area passes through a maximum, whose position coincides with that of the maximum in the Vs -composition curve. Before considering this problem, it is necessary to answer the question what determines the value of the specific surface area It is generally known that Ssp depends on the method and conditions under which adsorbents are obtained and is mainly determined by the nature of the hydroxide itself. Under the same conditions of production, hydroxides have different values of specific surface areas. This question is answered, to some extent by Berestneva and Kargin [27]. It appears that each metal hydroxide has a certain crystallization period determined by the rate of formation of ordered areas inside the amorphous phase and by the number of these areas. It cannot be excluded that the size and shape of various hydroxides differ from one another. Moreover, the specific surface area is a relative value that does not reflect the reality. The thing is that real densities (d) of adsorbents sometimes differ substantially and therefore, commensurable values of the surface areas can be obtained only if they are based not on gramm but on cm in this way ... 

Classification of composites by the phase inclusion size bears a philosophical aspect how small should a component in the matrix be not to make the term composite material so universal as to include in fact all materials Interatomic distances in molecules and crystals are of 1.5 10 m dimensionality, distances between iterative elements of the crystalline structure are 10 —10 m, while the size of the smallest intermolecular voids in polymers is 10 m. Note that mean nanoparticle size (plastic pigments are 10-8-10 m in size, the diameter of monocrystalline fibers or whiskers is 10 —10 m, glass microspheres are 10 —10 m) is commensurate with parameters of monolithic simple materials. This means that in the totality of engineering materials, nanocomposites occupy a place at the boundary between composite and simple materials. 

Another aspect relating to the structural studies of these phases is the close relationship between the polysulfide structures composed of the three mutually stacked R +, 8 - and (82) - layers. When the imit cell parameters of the phases of one or several related compositions are commensurate, block structures or microtwins are formed, as pointed out by all authors who studied these compounds (Tamazyan et al. 1994, 2000a,b, Podberezskaya et al. 1996, 1998). Twinning, in turn, may lead to incorrect determination of the crystal symmetry, especially if the photo method was used in the study. 8ince the majority of structural studies for the reduced polysulfides utilized this method, only investigations where the composition was refined on the basis of more exact recordings of the intensities and adequate corrections for absorption may be taken into account. 

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. 

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