Crystal orientation distribution

The crystal orientation distribution for a cubic material in sheet form can be calculated from any two experimental pole figures, for example, the (100) and (110). Once the distribution is known, any other desired pole figure can be cal-ctllated, for example, the (111) it need not be measured. It is even possible to calculate the orientation distribution from a set of partial pole figures, determined by a reflection method out to 60° from the center of the pole figure (a = 30°) [9.28]. The crystal orientation distribution itself is usually presented in the form of crystal density plots, in which the density is shown as a contour map on, for example,... 

Finally, we must not forget that the practical reason for investigating preferred orientation is to understand the properties of the aggregate. The old problem still remains how do we calculate the physical and mechanical properties of an aggregate from the corresponding properties of the single crystal and the measured texture The crystal orientation distribution affords the most rational basis so far available for this calculation, and considerable progress has been made in this direction [9.18, for example]. 

Sheet textures may also be represented by inverse pole figures. Here three separate projections are needed to show the distribution of the sheet normal, rolling direction, and transverse direction. Figure 9-24(b) is such a projection for the normal direction of the steel sheet whose (110) pole figure was given in Fig. 9-20 it was calculated from the crystal orientation distribution mentioned in Sec. 9-8. The distribution of the normal direction is also shown in (c), for the same material. This distribution was measured directly in the following way. A powder pattern is made of the sheet in a diffractometer by the usual method, with the sheet equally... 

The inverse pole figure is the best way to represent a fiber texture, but it offers no advantage over a direct pole figure in the description of a sheet texture. Inverse or direct, a pole figure is a two-dimensional plot that fixes, at a point, only a direction in space, be it crystal space or specimen space. Only the three-dimensional plot afforded by the crystal orientation distribution (Sec. 9-8) can completely describe the orientations present, and this approach, being quite general, is just as applicable to fiber textures as it is to sheet. 

Fig. 3.31). The length axis L of the tape is vertical, so the (110), (040) and (13 0) diffraction peaks lie on the equator of the figure. When semicrystalline polymers are stretched, the crystal c axes tend to align with the tensile direction, here the L axis. The assumed crystal orientation distribution is of c axes perfectly aligned with the L axis, with a and b axes randomly distributed in the plane perpendicular to L. Therefore, the hkO) poles, at 90° to the c axis, should be perpendicular to the tape L axis. The diffraction peaks in Fig. 3.31 are consistent with this assumption. If in Fig. 3.29 the L axis is normal to the paper, the diffracting planes in the crystal shown are of the hkQ) type. As the [hkO) poles in the PP tape are randomly oriented in the plane of the diagram, many crystals will be positioned to produce diffraction spots on either side of the equator of the pattern. To confirm the orientation distribution, further diffraction patterns should be taken as the sample is rotated around its L axis. 

A more advanced approach to analyzing crystal orientation distribution is to compute the entire 2D diffraction pattern and then compare simulated intensity with experimental data. In this case, it is possible to derive the complete orientation distribution of crystals in real space, and the Hermans orientation function can be analytically or numerically given, depending on integration kernel used for simulation. Details of 2D pattern computation and mathematical treatment of orientation distribution function was reviewed by Burger [92]. Examples can be found in References [95,96]. 

Typical shapes of the orientation distribution function are shown in figure C2.2.10. In a liquid crystal phase, the more highly oriented the phase, the moreyp tends to be sharjDly peaked near p=0. However, in the isotropic phase, a molecule has an equal probability of taking on any orientation and then/P is constant. 

Figure C2.2.10. Orientational distribution functions for (a) a highly oriented liquid crystal phase, (b) a less well...

Theoretical studies of diffusion aim to predict the distribution profile of an exposed substrate given the known process parameters of concentration, temperature, crystal orientation, dopant properties, etc. On an atomic level, diffusion of a dopant in a siUcon crystal is caused by the movement of the introduced element that is allowed by the available vacancies or defects in the crystal. Both host atoms and impurity atoms can enter vacancies. Movement of a host atom from one lattice site to a vacancy is called self-diffusion. The same movement by a dopant is called impurity diffusion. If an atom does not form a covalent bond with siUcon, the atom can occupy in interstitial site and then subsequently displace a lattice-site atom. This latter movement is beheved to be the dominant mechanism for diffusion of the common dopant atoms, P, B, As, and Sb (26). 

Fig. 2. Schematic representation of the orientational distribution function f 6) for three classes of condensed media that are composed of elongated molecules A, soHd phase, where /(0) is highly peaked about an angle (here, 0 = 0°) which is restricted by the lattice B, isotropic fluid, where aU. orientations are equally probable and C, Hquid crystal, where orientational order of the soHd has not melted completely.

A method that creates patterned micro-structures distributed on the bottom wall of the micro-channel was proposed by Yang et al. (2006). A roughened bottom wall was created using the crystal orientation characteristics of the wafers. 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

We begin, however, with the singlet orientational distribution function which is shown for the three liquid crystal phases in Fig. 6. In each phase the distribution is peaked at cos of 1 showing that the preferred molecular orientation is parallel to the director. The form of the distribution function is well represented by the relatively simple function... 

Aluminium on Silicon. Low Contact Resistance. Improved Corrosion Resistance c/f Evaporated A1. Grain Size and Crystal Size Distribution is Function of Acceleration Voltage. Crystal Orientation is strongly (111) under High Acceleration Voltage... 

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Now we compare the isotropic-liquid crystal phase boundary concentrations for various polymer solution systems with the scaled particle theory for the wormlike spherocylinder. If the equilibrium orientational distribution function f(a) in the coexisting liquid crystal phase is approximated by the Onsager trial... 

Fuhs et al.m investigated P p0 Aj in multilayers of Synechocystis PCC 6803 oriented on mylar sheets by transient W-band EPR. They could show an enhanced resolution of structural parameters of the RP in this model system. A problem is the uncertainty of the orientation distribution (width 30 10°). Limitations and possibilities of the method are discussed in this work. The technique is interesting for all systems for which no single crystals are available. 

There are pros and cons for each method of electrode preparation. The polycrystalline electrodes are cheap and also are nearest in character to those used in practical reactors inindustiy. However, a polycrystal consists ofinumerable grains (bits) of the electrode material, each having a different crystal orientation and hence a different catalytic property. One way of manufacturing an original metal may differ from another in the distribution of crystal faces of different kinds. Thus, irreproducibility of results in electrode kinetics is not only due to inadequate purification of solution,... 

Orientational Distribution Function and Order Parameter. In a liquid crystal a snapshot of the molecules at any one lime reveals that they arc not randomly oriented. There is a preferred direction for alignment of the long molecular axes. This preferred direction is called the director, and it cun be used to define- an orienlalional distribution function, f W). where flH win Vilb is proportional to the fraction of molecules with their long axes within the solid angle sinbdw. 

It has also proven possible to directly compare the structure of the monolayer film at the A/W interface with the structure of the monolayer film transferred onto a solid substrate using conventional L-B methods. For DPPC monolayer films transferred to Ge ATR crystals at low-to-intermediate pressures, the transferred monolayer films have a constant conformational order independent of the transfer pressure, and an orientational distribution that is more ordered than that of the in-situ monolayer. For those monolayer films transferred at high surface pressures, the hydrocarbon chains have a similar conformational order but are more oriented than the in-situ monolayer at the same surface pressure,... 

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