Incommensurate satellites

Figure 12. Simulation of the temperature dependence of the neutron intensities of the first and third harmonics of the incommensurate satellite reflection of copper metaborate at zero magnetic field.

A particular feature of the bismuth cuprates deals wih the existence of incommensurate satellites on their electron diffraction patterns as shown for instance for Bi2Sr2CaCu208+8 (Fig. 34). This modulation of the structure is also observed on the single crystal X-ray... 

Figure 34 ED patterns observed in Bi2Sr2CaCu20g+6. (a) [010]. (b) [001] with incommensurate satellites along [100], q = 4.6. (c) Superposition of two areas [001] with extra spots in two perpendicular directions.

Contrary to bismuth cuprates, the thallium cuprates exhibit rarely incommensurate satellites. Moreover, the intensity of the latters is weak. In fact the existence of... 

X-ray measurements also clarified that B7 materials have at least two different phase structures. Some of them (for example, the first B7 material [25]) have distinct sharp peaks in the low angle range [16] which is characteristic to columnar structure. Other B7-type materials (we denote them by Bl ) have strong commensurate reflections -which indicate a layer structure - and small incommensurate satellites, which are very close to each other and hard to resolve with normal X-ray technique. These peaks were attributed to a one-dimensional undulation of the smectic layers with... 

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. 

From LEED measurements of H monolayers adsorbed on Fe(110) Imbihl et al. proposed a phase diagram as shown in Fig. IS. In addition to lattice gas and lattice fluid phases, two commensurate ordered phases were identifled, denoted as (2 x 1) and (3 x 1) in the figure (cf. Fig. 16). The shaded regions are interpreted as incommensurate phases or as phases composed of antiphase domains their signature is that the LEED spot does not occur at the Bragg position but rather the peak is splitted and satellites appear (Fig. 17). 

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... 

Fig. 9 Rietveld refinement of incommensurate Te-III at 8.5 GPa. The upper and lower tick marks below the profile mark the peak positions of the main and first order satellite reflections, respectively. The reflections shown in Fig. 8 are identified by their (hklm) indices, where the main reflections are indexed (M/O)...

The first important point is that satellites in incommensurate positions are observed in all superconductive bismuth cuprates. They are directed along the [100] or [010] directions and can appear along two perpendicular directions due to the existence of domains at 90°, characterized by a perfect coherent interface (Figure 30a). 

Figure 4.21 Portion of the (010) plane in the reciprocal lattice of K2Se04. On the left is the lattice for the high-temperature phase and on the right that for the ferroelastic phase. In the high-temperature phase, a softening occurs at a point displaced by 5 from q = (j, 0, 0), shown by a cross. In the incommensurate phase a satellite reflection develops at X.

Previously, sudden changes of amplitudes of magnetic satellites observable below 2 K [7] were interpreted in terms of lock-in transitions into phases with commensurate propagation vectors. Within the framework of the thermodynamic potential (5) it is impossible to describe a low-temperature transition from an incommensurate phase into a commensurate phase at zero external magnetic field. 

In some crystalline materials a phase transition on lowering the temperature may produce a modulated structure. This is characterized by the appearance of satellite or superstructure reflections that are adjacent reflections (called fundamental reflections) already observed for the high temperature phase. The superstructure reflections, usually much weaker than fundamental reflections, can in some cases be indexed by a unit cell that is a multiple of the high temperature cell. In such a case the term commensurate modulated structure is commonly used. However, the most general case arises when the additional reflections appear in incommensurate positions in reciprocal space. This diffraction effect is due to a distortion of the high temperature phase normally due to cooperative displacements of atoms, ordering of mixed occupied sites, or both. Let us consider the case of a displacive distortion. 

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