Incommensurate composite crystal

This bismuth-III structure is also observed for antimony from 10 to 28 GPa and for bismuth from 2.8 to 8 GPa. At even higher pressures antimony and bismuth adopt the body-centered cubic packing of spheres which is typical for metals. Bi-III has a peculiar incommensurate composite crystal structure. It can be described by two intergrown partial structures that are not compatible metrically with one another (Fig. 11.11). The partial structure 1 consists of square antiprisms which share faces along c and which are connected by tetrahedral building blocks. The partial structure 2 forms linear chains of atoms that run along c in the midst of the square antiprisms. In addition, to compensate for the... 

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. 

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 ... 

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. 

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. 

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... 

The incommensurate, hexagonal monolayers are compressed compared to the bulk metal and they are rotated from the substrate by several degrees. From the results, the monolayer compressibility could be calculated. The restructuring (i.e. surface reconstruction) of top layers of single crystal metal surfaces as a function of solution composition and electrode potential has been studied [73]. The induced charge density was found to be the critical parameter [74]. Structural changes during... 

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. 

Neutron diffraction patterns of annealed NbCx (0.81 S x S 0.88) (4,8) and TaCx (0.79 S X S 0.90) specimens display weak superstructure peaks along with intense structural lines, suggesting an incommensurate ordered phase with a composition close to M Cs (4). The parameter of the fee sublattice of metals for the ordered phase is larger than that for the disordered carbides, indicating that the volume of the crystal varies discontinuously during ordering (4). 

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