Incoherent boundaries

Fig. 4.10 Coethite twinning. Upper left Twins grown at pH 4 and 25 °C consist of two (a) or three armed (b) twin pieces (Schwertmann. Murad, 1983,with permission). Upperright Multidomainic star-like twin grown at [OH] = 0.3 ML" and 70°C with stirring (courtesy P. Weidler).Lower/ejt Schematic drawing of a twin-zone in a singly-branched goethite twin showing the lattice planes and the coherent and incoherent boundary (Maeda. Hirono, 1981, with permission).

In Figure 12.9 (a), (i) there are six 111 diffraction poles, rather than three, around the central (111) pole, and (ii) three diffraction poles, (111), (111), and (111) are more intense that the other poles, (111), (111), and (111). The result (i) indicates that there are two possible orientations of diamond crystals that are rotated by 60° in plane with respect to each other. A similar type of orientational structure has been observed for p-SiC(l 11) epitaxially grown on 6H-SiC(0001) [395]. In this case, incoherent boundaries, called double positioning boundaries, are formed in the film by 60°-rotated domains. On the other hand, the result (ii) indicates that the three orientations, (111), (111), and (111), are preferable to other three orientations. The existence of non-equivalent diamond orientations on the Pt(lll) surface was attributed to interactions with the second nearest neighbor Pt atoms. However, the incoherent boundaries due to the existence of 60°-rotated diamond crystals have not been identified yet. It is of interest that an azimuthally unidirectional surface was formed as the diamond CVD was continued for a longer time, as seen in Figure 12.7 (b). 

A step (fig. 19) has [OTO] for the component of the Burgers vector in the (20T) plane, determined as shown on fig. 19, nearly 30° off the twinning plane and nearly parallel to the local incoherent boundary (fig. 22a). The height of the step of about 20 A fits with 11 1 reticular distances 313. ... 

The different kinds of incoherent twin boundaries that have been observed can be related to the very different roles they play isolated wedge microtwins are equivalent to disclinations. They are probably stabilized by the A or B steps visible on the images, while repeated wedge microtwins which do not present any step are equivalent to superdislocations. They allow the junction between different regions and are very stable. The incoherent boundaries are as different as the role that the microtwins play. 

In our case neither the first nor the second condition are satisfied. The first condition can perhaps be left unsatisfied assuming, for example, the existence of some incoherent boundary. For the second, if it is not satisfied, at least the displacement must be the same on each side of the preexisting twin. If not, it would create a fracture inside the crystal. We shall see now how it can be realized. It occurs in such a sophisticated way that the observed phenomenon was first differentiated from a twin intersection (Boulesteix and Loier, 1973). However, this phenomenon fits exactly the definition of a twin intersection, i.e., the situation where one twin passes right through another, rather than tapering to a point and starting afresh on the other side (Cahn, 1953), and so must be really treated as a twin intersection. 

We have studied and defined four basic twin boundaries that form during the growth of diamond film from the gas phase. The basic E=3 boundary which occurs apparently due to strain build-up in the growing diamond crystal is usually coherent but can form as an incoherent boundary when two parts of the crystal which have a E=3 relationship grow toward each other. E=9 and E=27 boundaries form in a similar way. 

Twins are commonly found or formed in all types of crystals. Their boundaries are of two general types coherent and incoherent. The coherent boundaries are usually also symmetric, so they offer little resistance to dislocation motion. However, the incoherent ones are not symmetric and may resist dislocation motion considerably. 

Fig. 3.20 NSE data obtained from the incoherent scattering from a fully protonated PE melt at 509 K (M =190 kg/mol). The data close to the 0.1 ns boundary grey bar) are, due to technical difficulties at the range boundaries of the two spectrometer configurations, more uncertain than the bulk of the data points, as seen by the size of the error bars. Lines see text. (Reprinted with permission from [43]. Copyright 2003 The American Physical Society)...

Up to this point we have dealt with the thermodynamics of planar boundaries. Let us add several relations for curved interfaces. First, we have to establish an equivalent to the Gibbs-Thomson equation which holds for curved external surfaces in a multi-component system. For incoherent (fluid-like) interfaces, this can be done by considering Figure 10-5. From the equilibrium condition at constant P and T, one has... 

Finally, before carrying out the calculation, it is necessary to sketch the boundary between the crystalline peaks and the amorphous background. This line can be calculated if an amorphous sample has been used as a reference, such as for PET and cellulose fibers. If no amorphous standards are available, the background is drawn manually, following a line parallel to the theoretical curve (jt,5) (total scattering power summing up coherent and incoherent scattering). 

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