Modulated structures commensurate

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. 

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.)...

Fig. 11. Orthorhombic distortion of tetragonal HoNi2B2C (a) upon cooling from 15 K to 1.5 K the neutron diffraction reflection (332) splits into two peaks, (b) Schematic presentation of the distortion a,b — original tetragonal axes dashed line tetragonal basal plane —f shift of the Ho atoms leading to the orthorhombic cell with the axes onh- Thick arrows Ho magnetic moments in the commensurate c-axis modulated structure...

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. 

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... 

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).

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). 

IR-11.6.2 Modulated structures Modulated structures possess two or more periodicities in the same direction of space. If the ratio of these periodicities is a rational number, the structures are called commensurate ... 

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. 

The diffraction pattern from a normal crystal is characterised by an array of spots separated by a distance /a = a that arise from the parent structure, together with a set of commensurate superlattice reflections that arise as a consequence of the additional ordering. In this case the spot spacing is 1 /na = a /n, where n is an integer, (Figure 8.21a, b). In modulated structures, the modulation might be in the position of the atoms, called a displacive modulation, (Figure 8.21c). Displacive modulations... 

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. 

The three intermediate phases II, III, and IV of thiourea are believed to be incommensurate, and the incommensurately modulated structures in phase II and rv have been analyzed [50]. Another commensurate phase which is stable between phase I and phase II in a narrow temperature region of about 2 K has been recognized many years ago [27c,/], and was refined at 170 K by Tanisaki et al. [27b]. This ninefold superstructure of thiourea is characterized by a rotation of the (NH2)2CS molecule along the c axis coupled with a displacement of the center of mass in a plane perpendicular to the axis. Around mirror planes (y = 1/4 and 3/4) the local structure is isostructural with phase I. Therefore the superstructure is constructed of alternately polarized layers which are sandwiched by domain walls (or discom-mensurations) whose local structure is that of the paraelectric room temperature phase (V) around y = 0 and 1/2. 

Incommensurate and Commensurate Structures, p. 712 Inelastic Neutron Scatter-ing, p. 727 Modulated Structures, p. 873... 

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. 

Modulated structures can be obtained from the structures having translational symmetry by say a displacement of atomic layers by an integral number of lattice translations. This results into a superlattice still retaining translational symmetry. But if the displacement is not an integral multiple of lattice vectors, the resulting lattice wiU lose its commensurability with the basic structure and also the translational symmetry. The diffraction of the modulated wave in addition to the determination of r of the atoms is within the domain of Incommensurate Crystallography. ... 

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