Commensurate modulated composite structures

In cases where the two c-parameters come into perfect register after small distances, the stmcture is described as a commensurate modulated composite structure. In such circumstances, the value of y is a rational fraction, p q where p and q are (small) integers and the c lattice parameter of the phase is given by... 

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. 

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. 

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... 

In the case of semi-commensurability in one or two directions it is likely that, structurally, the two layer sets are not quite independent. Semi-coherent structural and/or compositional modulation is then present i.e. a cooperative periodic variation in the size and/or content of the component subcells. Each modulation vector (one only in Fig. 2a, two in Fig. 2 b) will be equal to (or a multiple or sub-multiple of) that of the coincidence net. For each modulated layer set, A or B, a true-structure (component) lattice and unit cell can then be defined, based on its modulation period or periods plus the basic vectors or vector in the direction(s) in which there is no visible modulation of the basic structure. The longer-range modulation pattern of the two layer sets is imposed on the short range approximate periodicities which, in turn, describe sub-motifs manifested as a subnet (or subcell) of each layer set. If, as may be the case, the separate component unit cells of each of the two sets are identical, then they are also the coincidence cell of the two sets (Fig. 2 b). In the more general case, when this is not so, the vectors of the coincidence net will be multiples of the identity vectors of the unit nets of the two layer sets (or some simple summations of them). 

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. 

Cylindrite is a collective name for a homologous series of compounds with very closely related vernier structures. These have slightly different chemical compositions and consequently different numbers of matching pseudo-tetragonal (T) and pseudo-hexagonal (H) subcells in the modulated, semi-commensurate direction . Some are modulated in two directions and are therefore SS structures, but the majority have a... 

IR-11.6.1 Introduction Several special problems of nomenclature for non-stoichiometric phases have arisen with the improvements in the precision with which their structures can be determined. Thus, there are references to homologous series, non-commensurate and semi-commensurate structures, Vernier structures, crystallographic shear phases, Wadsley defects, chemical twinned phases, infinitely adaptive phases and modulated structures. Many of the phases that fall into these classes have no observable composition ranges although they have complex structures and formulae an example is Mo17047. These phases, despite their complex formulae, are essentially stoichiometric and possession of a complex formula must not be taken as an indication of a non-stoichiometric compound (cf. Section IR-11.1.2). 

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