Tilt boundaries intersecting

For tilt boundaries, the value of E can also be calculated if the plane of the boundary is specified in the coordinate systems for both adjoining grains. This method is called the interface-plane scheme (Wolfe and Lutsko, 1989). In a crystal, lattice planes are imaginary sets of planes that intersect the unit cell edges. The tilt and twist boundaries can be defined in terms of the Miller indices for each of the adjoining lattices and the twist angle, , of both plane stacks normal to the boundary plane, as follows ... 

Facetting of the twist boundary must also be considered. In this case, BPF and symmetrical tilt facets do not intersect the macroscopic current path. Since we already have indication from the (103) films that the twist boundary does not degrade current, we do not expect that indirect paths crossing these boundaries will be preferred. However, when the nominal twist boundary deviates from its macroscopic direction it may form (110)(103) facets which do... 

Real crystals differ from ideal crystals in that they are composed of mosaic blocks, i.e. small crystallites. Typical values of their sizes lie in the range of some 100 up to some 1000 lattice constants. They are tilted relative to one another by a few minutes of arc (Fig. 4.6). The intersection of two of these blocks forms a so called small-angle grain boundary. This is known from the maxima in X-ray diffraction spectra their widths for an ideal crystal should theoretically be only a few seconds of arc. In reality, one measures instead values of at least one minute of arc, due to the distribution of orientations of the mosaic blocks. 

Twin boundaries and low-angle grain boundaries can often be observed directly on intersection with the surface. Frequently, the small tilts associated with internal twinning (as in antiferroelectrics) may be observed by reflection microscopy or multiple-beam interferometry. The etch technique used to identify optical and electrical twins in quartz is of considerable practical importance in fabricating piezoelectric oscillator plates. ... 

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