Diffraction incommensurate

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. 

The resulting (A + T) sequence is perfectly ordered (its diffraction pattern would be built by peaks ) but it is not periodic (the two periods a and ar/2 are incommensurate) (See Fig. 3.45). Approximations to this composite structure may be built considering the representation of r through a continuous infinite fraction ... 

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. 

Striped (SI) and from a hexagonal incommensurate (HI) phase. The structure and the corresponding schematic diffraction patterns are shown in Fig. 31 the diffraction patterns have been calculated for fully relaxed walls, i.e. the SI and the HI phase are in fact uniaxially and uniformly compressed phases, respectively. By inspection of Fig. 31 it is obvious that the various incommensurate structures can easily be identified by their characteristic diffraction patterns. 

Fig. 31. Schematics of (a) real lattice and (b), (c) the (n,n) and (n,2n) diffraction features of incommensurate layers. SI - striped incommensurate, HI - hexagonal incommensurate, HIR -hexagonal incommensurate rotated. All phases are assumed to be fully relaxed. O denotes the (93 X 93)R30° commensurate and the incommensurate structures.

The analysis in the last paragraph has shown that the incommensurate Xe layer on Pt(lll) at misfits of about 6% is a striped phase with fully relaxed domain walls, i.e. a uniaxially compressed layer. For only partially relaxed domain walls and depending on the extent of the wall relaxation and on the nature of the walls (light, heavy or superheavy) additional statellites in the (n, n) diffraction patterns should appear. Indeed, closer to the beginning of the C-I transition, i.e. in the case of a weakly incommensurate layer (misfits below 4%) we observe an additional on-axis peak at Qcimm + e/2 in the (2,2) diffraction pattern. In order to determine the nature of the domain walls we have calculated the structure factor for the different domain wall types as a function of the domain wall relaxation following the analysis of Stephens et al. The observed additional on-axis satellite is consistent with the occurrence of superheavy striped domain wails the observed peak intensities indicate a domain wall width of A=i3-5Xe inter-row distances. With... 

The basic modem data describing the atomic stmcture of matter have been obtained by the using of diffraction methods - X-ray, neutron and electron diffraction. All three radiations are used not only for the stmcture analysis of various natural and synthetic crystals - inorganic, metallic, organic, biological crystals but also for the analysis of other condensed states of matter - quasicrystals, incommensurate phases, and partly disordered system, namely, for high-molecular polymers, liquid crystals, amorphous substances and liquids, and isolated molecules in vapours or gases. This tremendous... 

The comparison of PTCDA with its parent perylene molecule is extremely interesting. For perylene MLs on Ag(l 11), electron diffraction suggests an orientational liquid, in which the molecules are positionally ordered in an incommensurate close-packed superlattice but orientationally disordered and mobile. The same activated Raman peaks as for PTCDA are observed but they are, however, orders of magnitude weaker, indicating that, while a molecular reaction centre may still exist in the perylene backbone, its residual activity would be too small for the molecule to recognize a preferred site. 

Once Te-III was identified as incommensurate, subsequent analysis was conducted on the previously-collected powder-diffraction data using the formalism of 4D superspace [234], and the JANA2000 software for structure refinement [235]. The Rietveld refinement of the incommensurate Te-III diffraction profile is shown in Fig. 9, and the modulated structure is shown in Fig. 10. Tellurium was only the second element found to have a modulated crystal structure at high-pressure, the... 

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