Misfit ratio

The structural form of brookite (TiO ) is expected to be bounded by 210 and 111, both being F faces by PBC analysis, but the actual growth form observed is platy Habitus bounded by largely developed 100, which is an S face. The misfit ratio between the PBC on the (0110) face of quartz and that on (100) ofbrookite is the smallest among any misfit ratios between the two crystal species. From this, it was found that the platy Habitus ofbrookite arose because quartz adsorbed in an epitaxial relation on 100 ofbrookite, thus diminishing thegrowthratekof(lOO) [30]. 

Figure 7.13. Misfit ratio, (a) One-to-one corresponding relation. Open circles indicate the unit cell of the host phase, and the solid circles show that of the guest phase.

The corresponding relation between the host and guest crystals when evaluating the misfit ratio may be a one-to-one lattice relation in the same direction (a X b to a xb axes), or in different axial directions (aX b axes versus aX <110> axes), or on the basis of one unit cell versus a few unit cell sizes (see Fig. 7.13). Royer s misfit ratio is generally a two-dimensional correspondence, but Hartman [13] extended this relation to the misfit ratio in PBCs (see Section 4.2), which is a one-dimensional correspondence. Royer s epitaxial relations correspond to a relation between the F faces of the host and guest crystals containing more than two PBCs, and an epitaxial relation is not allowed between S faces or K faces. In Hartman s analysis, rela-... 

The freedom of the dangling bonds on the crystal surface increases with increasing temperature. As a result, there is a critical temperature below which an epitaxial relation cannot be realized. This temperature is called the epitaxial temperature, and it depends on interface orientation. If the misfit ratio is small, the epitaxial temperature is low if the misfit ratio is large, the epitaxial temperature is high, and an epitaxial relation will not be achieved unless the temperature is higher than the epitaxial temperature. 

Coaxial intergrowth is a paragenetic relation that describes crystals of two different species growing with a common axis the misfit ratios between the two crystals in the direction of the common axis are small, without exception. The formation of coaxial intergrowth can be understood to be one crystal conjunct to the other in an epitaxial relation, where both continue to grow. If a liquid of eutectic A-B component is solidified from one side (unidirectional solidification), crystals of the two phases A and B precipitate in dotted, columnar or lamellar (with common axis) form, and show unique textures for unidirectional solidification. This is a well known phenomenon in metallurgy. 

Fig. 8. 2D critical temperatures as a function of misfit. Ratio of 2D to 3D critical temperatures for Ar, Kr and Xe on various lamellar halide surfaces (from Ref 63). The dimensional incompatibihty is the excess above one of the ratio of the spacing between neighboring anions of the substrate and the interatomic spacing of the (111) plane of the bulk adsorbate. 

Figure 4.20 The transport properties in the = 4 Bi-based family, with the p(T) measured on single crystals (a), S(T) measured on polycrystals (b) and 5 as a function of the misfit ratio bllb (c)...

While the c/a ratio deviates only by about 2% from one, it is not ideal and this has significant consequences for the pseudotwin and 120° rotational fault. It results in a misfit at these interface which is compensated by a network of misfit dislocations (Kad and H2izzledine 1992). In contrast, the non-ideal c/a ratio does not invoke any misfit at ordered twins. However, the misfit dislocations present at interfaces are about fifty lattice spacings apart and thus there are large areas between them where the matching of the lamellae is coherent. The structures and... 

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. 

Here, G denotes the shear modulus, and f(c/r) is a function of the ratio c/r in which c and r are the spheroidal semiaxes of the precipitate. For spheres, f(c/r= 1) = 1 = /max. For discs as well as for rods, /< 1. In principle, shear stress energies and energies arising from misfit dislocation networks also have to be added. They influence AG by additional energy terms. 

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... 

Compound Misfit Component Component Sym- Coincidence Symmetry Coincidence cell Ratio of Ref. 

Compound Misfit modulation type Component layers Component sublattices Sym- metry Subcell Component intralayer lattices axial ratio bic Sym- metry Unit cells subcells in terms of Ratio of unit cell areas Ref. 

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. 

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