Misfit structures

In the book by Hyde and Andersson (1989), the Nowotny phases are presented as a special case of a group of ID, columnar misfit structures which also include compounds such as Bam(Fe2S4) and other complex sulphides. Layer misfit structures, such as those of some oxide-fluorides, arseno-sulphides, etc., are also presented and classified with reference to a concept of structure commensurability based on the recognition that (along one or more axes) the ratios between the different repeat units of various interpenetrating substructures can (or cannot) be represented as ratios between integer numbers. 

Makovicky Hyde (1981) have reviewed incommensurate misfit structures in graphite intercalation compounds, brucite-type compounds, sulphides and related layered systems. A simple two-dimensional incommensurate system is provided by graphite with adsorbed rare gas monolayers. At low densities and high temperatures. 

A general interface is not energy-minimized with respect to any of its degrees of freedom, and is far from any singular-interface values of the parameters that set its degrees of freedom. Such an interface cannot reduce its energy by adopting a fit-misfit structure (as in the vicinal case) and therefore cannot support localized dislocations or steps. Two examples serve to clarify these distinctions ... 

After an introductory discussion of such misfit structures, various terms that have previously been applied are reviewed, and degrees of incommensurability are used as the basis for a systematic nomenclature. The known structures of specific examples are then discussed graphite intercalates minerals with brudte-like layers as one component (koenenite, valleriite, tochilinites) silicates heavy metal sulphides (cylindrite, incaite, franckeite, cannizzarite, lengenbachite, lanthanum-chromium sulphide) anion-excess, fluorite-related yttrium oxy-fluorides and related compounds. 

A limited number of organic substances with two lattices is also known all columnar misfit structures. The crystal structure of hepta(tetrathiofulvalene)-pentaiodide, (TTF)7l5, determined by Johnson and Watson is one of the best described. [The substructures of the two components (TTF and iodine) were also determined - but in projection only - by Daly and Sanz ] In it, columns of TTF molecules form a matrix, the channels within which are filled by rows of iodine atoms. The 7 5 ratio of subcell multiplicities is only approximate within about 1% of the true ceD value. Similar structures have been determined for (TTF)Br , with a variable ratio of component subcells... 

But our concern here is with the two-dimensional cases layered misfit structures, in which the lack of commensurability is between the intralayer periodicities of layers of two types, which alternate regularly through the structure. The layers may be simple or complex (i.e. composite groups of several, physically distinct layers). In most cases the two layer types compensate each other s valency and consequently alternate with strict regularity, forming double-layer or two-component layered structures. Both intralayer identity vectors of one layer set A) may differ fi-om those of the other layer set (B), so that each layer set has its own periodicities, and the vectors defining the net common to both (if it exists) are more or less complicated resultants (e.g. lowest common multiples) of these basic, intralayer vectors. In some cases the basic vectors are identical in one... 

Known examples of layered misfit structures are listed in Tables 1 to 5. It is interesting that they have been recognised, and independently described as such, in at least four instances Cowley and Ibers Allmann, Lohse and Hellner Makovicky " and Hyde et al.. (The general absence of rigid, two-dimensional layers in organic structures suggests that layered misfit structures will be improbable for organic compounds. We know of no examples.)... 

In the layered misfit structures each layer set A and B can be described in terms of three basic translations, i.e. by its own component lattice. [The existence of the third vector is contingent upon a strict sequence in the layer stacking. If this is absent, the two three-dimensional subcells/lattices will, in the following discussion, be replaced by two two-dimensional subcells, i.e. by submeshes (nets) built only on the intralayer vectors.] In normal layered structures the unit cells of A and B are commensurate, i.e. their unit vectors are commensurable and the periodicity of the entire structure may be described in terms of a single unit cell. In contrast, we deal with those less-frequent cases in which this is not so at least one of the basic periodicities of A and of B are incommensurate. Then the component unit cell of set A has at least one intralayer unit cell parameter which is not commensurable with the corresponding parameter of set B. Such structures have two incommensurate, interpenetrating, component lattices and can be defined as composite) layered structures with two incommensurate component unit cells. Intermediate cases, in which the nodes of the two component lattices coincide at relatively large... 

The terms incommensurate and semi-commensurate are analogous to incoherent and semi-coherent for interfaces - in grain boundaries, heterophase interfaces and epitaxial layers (cf. also Nabarro - with which layered misfit structures have much in common. In extreme cases noncommensurability may arise by mutual rotation (to varying degrees) of component layers with identical component lattices... 

A considerable number of layer-misfit structures have brucite-like hydroxide layers M(0H)2 (see Fig. 7 a and A in Fig. 5) as one component, the cation sites being occupied by Mg and Fe and/or AP. The presence of the trivalent cations ensures a positively-charged layer, i.e. unsaturated valences. 

Among the complex natural and synthetic sulphides of lead, tin, antimony, bismuth and arsenic, often with minor amounts of iron, copper and/or silver, there is another group of layer-misfit structures. And recently a synthetic lanthanum-chromium sulphide has been described which is closely related to them structurally. 

In layered misfit structures of the type we are discussing, bonds at the layer surfaces (within and between the layers) will be strained periodically along a non-commensurate lattice direction parallel to the layers after a certain number of subcells there is a near match of the layers. Clapp has pointed out that, for a simple case, layer mismatch will cause tension in one layer type and compression in the other. The resulting strain energy may be relieved by the introduction of periodic antiphase boundary (apb) planes so that alternate contraction and extension occurs in all layers (Fig. 22) and hence cancels out (at the price of a small deformation of coordination polyhedra). 

It would seem that such a strain-relieving mechanism could be an alternative to, or additional to, non-commensurability in layer misfit structures (with only one non-com-mensurate lattice vector). It is observed as an addition, converting non-commensurable to commensurable layer misfit structmes. ... 

Table 7. Selected compounds transitional between the layer-misfit structures and the CC structures. (Intralayer vectors are underlined.)... 

Misfit structures consist of two or more different, often mutually non-commensurate, units which are held together by electrostatic or other forces no basic stmcture can be defined. The composition of compounds with misfit structures is determined by the ratio of the periodicities of their structural units and by electroneutrality. 

Figure 17. A non-commensurate, layer misfit structure (o), composed of alternating layers of two types, and the recombination structures (i-iii) derived from (o) by means of (i) composition-conservative and of (ii-iii) two kinds of composition non-conservative antiphase boundaries (or glide planes), (iv) A layered non-commensurate structure with corrugated layers that are composed of alternating strips of two kinds that exhibit layer misfit on interfaces.

Layered cobaltite ( 3300409) presents a misfit structure builtup with alternating single Cdl2-type [Co02]oo layers and triple [CaO-CoO-CaO] rock salt t3qie layers. The proposed structural model for involving a supercell with b 8hi 13 2 is shown in Scheme 5.4. This layered structure enables electrons behave in an anisotropic... 

Figure 4.17 The misfit structure of Ca3Co409 . The plane of edge-shared octahedral corresponds to a Cdli-type C0O2 layer. The second sublattice (in the centre) is made of three layers, 2 CaO layers surrounding a central layer made of...

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