Lattice planes extents

In electron micrographs, the native cellulose microfibrils are usually seen as bundles of lamellae containing an indefinite number of fibrillar units. A schematic representation of the cross section of a small lamellae of microfibrils is shown in Figure 3 (14). Two structural features should be pointed out. The cellulose lattice extends through the whole cross section of the microfibrils but the surface layers are supposed to be disordered to some extent because they represent a discontinuity. The microfibrils show a preferred orientation with their lattice planes 101 well aligned parallel with the flat surface of the lamellae—i.e., the tan-... 

Dislocations, and the associated internal strains, are thermodynamically unstable as we have mentioned, and to a certain extent can be eliminated by annealing. They have a strong influence on the mobility of charge carriers and of excitons in the crystal. In any case, they disturb the periodicity of the lattice and act as scattering centres. They also modify the distribution and the concentration of impurity molecules in the crystal. It thus becomes clear with the example of step dislocations (Fig. 4.3) that the lattice above the slippage plane is compressed by the insertion of an additional lattice plane, while it is expanded below the slippage plane. Smaller molecules than those of the host can then occupy lattice sites in the upper region. 

As the draw ratio increases, arcs first develop from the circular reflections at right angles to the draw direction and then point-shaped reflections in wide-angle X-ray pictures (Figure 5-35). The reciprocal length of an arc is a measure, therefore, of the extent of orientation of the crystallites, or, more precisely, the specific lattice planes. Arcs at various positions in the X-ray diagram correspond to the different lattice planes. Thus, an orientation factor / exists for each of the three spatial coordinates, and this is related to the angle of orientation P via... 

The potential between the molecule and the lattice plane of infinite extent is then simply the integral with respect to x from 0 to infinity ... 

In Fig. 4.3, an additional set of planes, and thus an additional source of diffraction, is indicated. The lattice (dark lines) is shown in section parallel to the ab faces or the xy plane. The dashed lines represent the intersection of a set of equivalent, parallel planes that are perpendicular to the xy plane of the paper. Note that the planes cut each a edge into two parts and each b edge into one part, so these planes have indices 210. Because all (210) planes are parallel to the z axis (which is perpendicular to the plane of the paper), the / index is zero. [Or equivalently, because the planes are infinite in extent, and are coincident with c edges, and thus do not cut edges parallel to the z axis, there are zero (210) planes per unit cell in the z direction.] As another example, for any plane in the set shown in Fig. 4.4, the first plane encountered from any lattice point cuts that unit cell at a/2 and b 3, so the indices are 230. 

Obviously, the critical thickness of the AB layer at which all the B atoms, capable of reaching interface 1 by a given moment of time, will be combined into the AB compound is six atomic planes corresponding to AB molecules (see Fig. 1.4). Indeed, in this case the reactivity of the A surface towards the B atoms is equal to one-sixth of B atom per second (one B atom per six seconds). The flux of the B atoms across the bulk of the AB layer is also equal to the same value (six consecutive displacements of the B atoms to adjacent sites within the AB lattice plus the transition of one of them through interface 2 last 6 seconds, so that one B atom crosses interface 1 as a result of these movements). At a greater thickness of the AB layer the rate of diffusion of the B atoms across its bulk is already insufficient to satisfy to the full extent the reactivity (combining ability) of the surface of phase A towards these atoms. 

Only quite small electric fields can be applied in a metal, because of the high conductivity in the time an electron moves before colliding with a defect or with the surface it can only be accelerated very slightly. Unless it happens to lie very close to a Bragg plane it will not be affected by the diffraction. Since most electrons are thus unaware of the lattice, conductivity can to a large extent be treated in the free-electron approximation, as we indicated earlier. 

The X-ray diffraction pattern shown in Figure 4 indicates that nearly phase-pure Hg-1212 films can be obtained after the annealing step. The films are epitaxially aligned with the c axis normal to the substrate plane. The c-axis lattice parameter varies to some extent with the annealing treatment and has been obtained in the range of 12.48-12.6 A, which is somewhat smaller than the value observed in the bulk (12.71 A). 

The needle-like crystallites, when packed into a flat sample, will also tend to align parallel to the surface. However, the preferred orientation axis, which in this case coincides with the elongated axes of the needles, will be parallel to the sample surface. In addition to the nearly unrestricted distribution of needles axes in the plane parallel to the sample surface (which becomes nearly ideally random when the sample spins around an axis perpendicular to its surface), each needle may be freely rotated around its longest direction. Hence, if the axis of the needle coincides, for example, with the vector d. then reflections from reciprocal lattice points with vectors parallel to will be suppressed to a greater extent and reflections from reciprocal lattice points with vectors perpendicular to d / will be strongly increased. This example describes the so-called in-plane preferred orientation. 

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