Structure incommensurate

This has a number of interesting consequences. One is that the oxidation state of Nb is lowered from the ideal value of -1-3.5 to +3.425. A second consequence is that each Eu ion at the surface of the (EuS)3 layer has a different environment since each sees a different part of the NbS2 layer. Some Eu + ions lie directly over an ion and form a very short Eu S bond, while others lie between the ions and form two or more longer bonds. Not surprisingly, the bond valence sums around the Eu ions, as well as around the ions of the NbS2 layer, show considerable variation depending on the relative positions of the layers at any given point in the crystal. 

Not all incommensurate structures are composite. It is possible to have incommensurate modulations in a structure composed of a single infinite building block, particularly if a weak cation fits rather loosely into a hole in a flexible framework. The polyhedra that compose the framework tend to twist to give the cation a distorted environment. These twists can often be described by a wave with a wavelength that may or may not be commensurate with the lattice translation of the crystal. If it is commensurate, the twisting is described as 

These examples illustrate how fundamental building blocks that are only weakly bonded to eaeh other ean give rise to incommensurate structures, and how such structures relax by generating waves of small displacements that ensure conformanee to the valenee sum rule around all atoms. An excellent and full account of modulated struetures has recently been published by Withers et al. (1998) where many further details of these faseinating materials can be found. 

The examples discussed in this chapter show that there are many different ways in which lattice-induced strain can be relaxed or aeeommodated, the partieular mode depending on the properties of the elements and the struetures involved. Many of these compounds have unusual properties resulting from non-integral stoichiometry, the presence of non-integral oxidation states, or the spontaneous breaking of symmetry, all of which are the direct consequence of lattice-induced strain. 

Lattice-induced strains are characterized by large values of the GII because the environments around some atoms are stretched and around other atoms are compressed but, since the valence is still distributed as uniformly as possible among the bonds, the BSI remains small. This contrast with the electronically driven distortions discussed in Chapter 8 where the GII is small (the valence sum rule is obeyed) but the BSI is necessarily large. 

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Figure 10.11 A well-ordered 2D AuS phase develops during annealing to 450 K. (A) The structure exhibits a very complex LEED pattern, which can be explained by an incommensurate structure with a nearly quadratic unit cell. (B) STM reveals the formation of large vacancy islands by Oswald ripening which cover about 50% of the surface, thus indicating the incorporation of 0.5 ML of Au atoms into the 2D AuS phase. The 2D AuS phase exhibits a quasi-rectangular structure (inset) and uniformly covers both vacancy islands and terrace areas. (Reproduced from Ref. 37).

Fig. 11.11, p. 112). Incommensurate structures related to bismuth-III are also observed for strontium and barium. Magnesium, calcium and strontium are remarkable in that they transform from the normal closest-packing of spheres to a body-centered packing upon exertion of pressure. Even more remarkable is the following decrease of the coordination number to 6 for calcium and strontium (Ca-III, a-Po type Sr-III, /3-tin type). 

Ex situ LEED and XPS studies were conducted to demonstrate the effects of CUand Br" these anions form densely packed incommensurate structures on Cu underpotential deposition at the full monolayer. ... 

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.

AFM image of Cu adlayer on Au(lll) in copper perchlorate solution, showing close-packed structure with a rotation to the Au(lll) substrate, (b) Schematic of the incommensurate structure of the Cu adlayer. (c) ( J3 X y/3 )/ 30° structure of the Cu adlayer on Au(lll), in copper sulfate solution, (d) Schematic diagram of that structure. The open circle represents Au atom at the topmost layer, the hatched circle represents the Cu adatom. (Reproduced from Manne et al., 1991a, with permission.)... 

Fig. 12 The composite incommensurate structure of Sc-II, as viewed down the c axis. The eight-atom host framework is shown in grey, and the ID guest chains are shown in black. The insets show perspective views of (a) the body-centred guest structure of Fujihisa et al. and (b) the C-centred guest stmcture of McMahon et al. The crystallographic axes are labelled...

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.)...

Incommensurate structure This type can be grouped into the following two sub-classes. 

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. 2.45 EDPs from the incommensurate structures Ba2j+j (Fe2S4)2j. 1 ( = 8,9,...) with [100] zone axis These are a special case of incommensurate structure and satisfy the equation F(002) = ( - i) X 2g.

Fig. 2.46 EDPs from the incommensurate structures with [100] zone axis. (a) EDP shows spacing anomaly, (b) EDP shows orientation anomaly. (See text.)...

Obviously, in systems with more than one order parameter, when the different ordering modes are coupled in one way or the other, the ordering kinetics are appreciably more complicated. In order to produce mismatched periodic patterns in a crystal (incommensurate structures), Landau and Lifshitz [L.D. Landau, E. M. Lifshitz (1980)] proposed a G expansion of the form... 

Crystals, however, are not always so perfectly ordered. Atomic mobility exists within the crystal lattice however, it is greatly reduced relative to the amorphous state. Partial loss of solvent from the lattice can result in static disorder within the lattice where the atomic positions of a given atom can deviate slightly within one asymmetric unit of the unit cell relative to another. Lattice strain and defects occur for many reasons. Solvents can be present within channels of the lattice in sites not described by the symmetry of the crystal structure itself, resulting in disordered solvent molecules or incommensurate structures and potentially nonstoichiometric solvates or hydrates. 

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