Modulation commensurate

Consider the simplest case, when the periodicity of the crystal lattice is perturbed in one dimension by periodic deviations of atoms from their ideal 

Dehlinger, fiber die Verbreiterung der Debyelinien bei kaltbearbeiteten Metallen, Z. Kristallogr. 65,615 (1927). 

Shechtman, I. Blech, D, Gratias and J.W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53,1951 (1984). 

de Wolff, The pseudo-symmetry of modulated crystal structures, Acta Cryst. A30, 777(1974). 

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Modulations are normally described as waves. The modulation wave can fit exactly with the underlying unmodulated component, or more precisely with the unit cell of the underlying component, in which case the structure is described as a commensurately modulated structure. In cases where the dimensions of the modulation are incommensurate (i.e., do not fit) with the unit cell of the underlying structure, the phase is an incommensurately modulated phase. Modulation changes are normally continuous and reversible. 

Bi2Sr2CaCu2Og but containing a commensurate modulated superstructure has been prepared and characterized (19) and gives further insight into the structures of the Bi-O layers. This is discussed in further detail below. A material containing three copper-oxygen sheets, with ideal formula Bi2Sr2Ca2Cu3O10, can also be made. The Tc s of the one, two, and three copper-sheet compounds are 10, 85, and 110 K, respectively. 

To conform with the ample literature on these compounds the following crystallographic orientation will be used a represents the lattice parameter in the layer-stacking direction, b the (long) intralayer parameter in the semi-commensurate, modulated direction, and c the intralayer parameter perpendicular, or nearly perpendicular, to both the other directions... 

Figure 1.52. The ideally periodic one-dimensional stmcture, the corresponding modulation function with the period X= Mq, which is commensurate with a, and the amplitude A (top), and the resulting commensurately modulated structure (bottom).

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. 

When the modulation wavelength exactly fits a number of parent unit cells, (that is, is commensurate with it), it will be possible to index the reflections in terms of a normal superlattice. Recently a number of structures that were once described in this way have been restudied and found to be better described as modulated structures with commensurate modulation waves - commensurately modulated structures. 

A qualitative change occurs only at about 1.056 monolayers, where it is no more possible to unambiguously identify commensurate stripes [182]. There, the situation can be described as commensurate modulations in an otherwise uniform uniaxial incommensurate structure. Following an earlier suggestion [362], test calculations where the lateral substrate energy is increased by a factor of two compared to the original potential [326] were performed at several coverages. The results are qualitatively similar to what is shown in Fig. 40, only the boundaries between the domain walls seem to... 

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... 

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. 

Fig. 4.5-80 SC(NH2)2- Sbias versus T phase diagram. The value of S means that the phase is commensurately modulated with a vector of wavenumber 5c. Inc., incommensurate phase...

Commensurately modulated structures strongly depend on the periodicity of the modulation wave where the periodicity is an integral nnmber of lattice translations of the unit cell. ... 

The PrMneSng, NdMneSn and SmMneSne are isotypic with HoFegSne. A new superstructure formation has been found in an X-ray diffraction study of a DyFeeSn single crystal by Oleksyn et al. (1995). The crystal structure is characterized by SG Cmmm with a=37.343, =21.560 and c=8.912 A. This structure may be described as a three-dimensionally commensurate modulated structure as well. 

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