Commensurate structures

The nematic phase of all the compounds CBn is characterized by a coherence length of about 1.4 times the elongated structure of the molecule. Based on this behaviour local associations in form of dimers with cyano-phenyl interactions were postulated. For the smectic A phase a partial bilayer arrangement of the molecules (SAd) is most likely. But there are also example for the smectic A phase with a monolayer (Sai) or a bilayer (Sa2) arrangement of the molecules as well as a commensurate structure A large number of X-ray measurements were carried out in the liquid crystalline state to clear up the structural richness and variability (see Chap. 2, this Vol. [52]). 

The simulated LEED pattern is shown in Figure 7-B. Comparison of the simulated LEED pattern with the observed LEED pattern (lower camera shutter speed) is shown in Figure 7-C. As can be seen, there is good agreement. This PYR layer structure is incommensurate that is, the mesh vectors of the layer are not exact multiples of the substrate mesh. The nearest commensurate structure would have been (2/3x/21, 79°)R30°, Figure 7-D. Although this commensurate alternative is numerically quite similar... 

Figure 7. Continued. E. Diagram of calculated LEED pattern corresponding to nearest commensurate structure, (2v/3xv/21, 79.1 )R30. Continued on next page.

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.)...

Figure 4.14 One-dimensional model showing interaction between a row of atoms (circles) and a periodic potential (wavy lines) giving rise to (a) commensurate structure, (b) incommensurate structure and (c) chaotic structure. (Following Bak, 1982.)...

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. 

The vector F(002) is parallel to the vector 2g, and these vectors generally show the relation of F(002) / n x 2g, in contrast to the commensurate structure. This relation is called a spacing anomaly and a structure showing a spacing anomaly is a kind of incommensurate structure. 

Generally the vector g can change continuously. For example, the structure with F(002)/2g = 14.60 (p/q = 1.068), which has a composition of Ba78(Fe2S4)73, shows a rather longer c-axis of about 400 A. It is to be noted that the structural principle of the incommensurate structure type is the same as that of the commensurate structure type, although the former type shows incommensurate behaviour on EDPs. 

Fig. 2.44 EDPs from the commensurate structures Baj + i(Fe2S3) (i = 7, 8,...) with [100]B zone axis. ...

Elastic neutron diffraction was first performed (analyzer in Fig. 1 set to zero energy transfer) to establish the structure of the monolayer at low temperature. Three Bragg reflections were observed which could be indexed by a triangular lattice having a nearest-neighbor distance about 10% smaller than required for a 3 X /3 R30° commensurate structure (every third carbon hexagon in the graphite basal plane occupied). 

It is significant that the second calorimetric peak and the associated isotherm sub-step were detectable only if the graphitized thermal black had been heated at temperatures above 1700°C. These results suggest that the 2-D phase transformation is very sensitive to the perfection of the surface basal planes and this is a further indication that the phase change leads to the development of a commensurate structure. 

A discussion of types of inter-layer bonding, layer types and layer matching is followed by a consideration of disorder in such structures, as well as symmetry aspects and some growth mechanisms. Structures transitional between this type of structure and commensurate structures, i.e. structures which are really non-commensurate layers joined at anti-phase boundaries which restore commensurability, are then considered and classified with a detailed consideration of known examples from inorganic chemistry and mineralogy. Finally, some broad, general conclusions are enumerated. 

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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