Distribution pole figure

The function Fh(y) defined by Equation (11) is called the reduced pole distribution (pole figure). Hereafter we will call it, simply, the pole distribution (or pole density), because />i,(y) will be used very rarely. The pole distribution is centrosymmetric and for crystal and sample symmetry higher than triclinic it... 

Lightning to distribution pole Figure 6.19 shows a measured result when 1q is 2,416 A. In this case, 8.1% of Iq flows into the grounding resistance of the neutral transformer, 8.5% flows into the grounding resistance of the air conditioner, and 2.8% flows into the grounding resistance of the telephone line SPD. 

The Miocene land distribution is compared with the present land distribution in Figure 7-24. There has been an increase in land fraction at the South Pole since the Miocene, a decrease in middle latitudes in the South-... 

In another set of experiments, tensile specimens initially 0.015 in. thick were elongated to various extents, released, and subsequently X-rayed. The resulting data can be analyzed by pole figure techniques to give the complete orientation distribution of unit cells in the specimen. For the case of axial specimen... 

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,... 

A pole figure shows the distribution of a selected crystallographic direction relative to certain directions in the specimen. Texture data may also be presented in the form of an inverse pole figure, which shows the distribution of a selected direction in the specimen relative to the crystal axes. The projection plane for an inverse pole figure is therefore a standard projection of the crystal, of which only the unit stereographic triangle need be shown. Both wire and sheet textures may be represented. 

Figure 9-24(a) is an inverse pole figure for the inside texture of an extruded aluminum rod, showing the density distribution of the rod axis on a times-random basis. It was derived by a trial-and-error method [9.36] from pole density curves, as in Fig. 9-22, for the ((X)l), (111), and (113) poles. We note concentrations of the rod axis at [001] and [111], indicating a double fiber texture the volume fractions of the [001] and [ill] components were estimated as 0.53 and 0.47, respectively. Note that an inverse pole figure shows immediately the crystallographic direction of the.scatter. In this double texture, there is a larger scatter of each component toward one another than toward [011]. 

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... 

Fig. 9-24 Inverse pole figures, (a) Distribution of axis of aluminum rod, extruded at 450°F to a reduction in area of 92 percent and a final diameter of 23 mm. letter, McHargue, and Williams [9.36]. (b) and (c) show the distribution of the sheet normal for the steel sheet of Fig. 9-20. Bunge and Roberts [9.18].

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. 

In eonneetion with the implementation in the Rietveld codes, the Dollase March model and the spherical harmonics approach, for pole distributions determination, is developed in the next two parts. The problem of pole figure inversion is outside the scope of this chapter. 

Consider a polycrystalline sample placed at a point O. A crystal within this sample diffracts for a family of planes (hkl) if and only if the reciprocal lattice point associated with this family is on the Ewald sphere with radius 1/X. By definition, the tip of the vector kg =(l/A,)So is the origin of the reciprocal lattice and the scattering vector S, which hnks point C with lattice point hkl, has a norm equal to 1/dhki. For all the crystals contained in the sample which diffract for a given family of planes (hkl), we observe a set of lattice points hkl located on a sphere with radius 1/dhn, which is centered in C. By definition, the directions [hkl] are normal to the (hkl) planes and therefore the distribution map of the hkl lattice points on the surface of this sphere is the hkl pole figure. This configuration is shown in Figure... 

Fig. 4a and b. Fig. 4a and b represent hypothetical pole figure plots of polyethylene as discussed in the text Numbers refer to relative diffracted intensities, (a) This shows uniform orientation distribution of a axes in the N-TD plane and a preference for perpendicular alignment to MD. (b) This shows a preference for alignment of the a axes along MD with a uniform (uniaxial) distribution about this same axis... 

In the case of PET the (100) and (—105) net planes are of special interest. The normals of the (100) planes are perpendicular to the planes of the benzene rings, while the normals of the (—105) planes are nearly parallel to the chain direction. The corrected pole figures of the two net planes are presented in Fig. 29. From these figures it can be seen immediately, that in the case of the investigated biaxially oriented sample the benzene rings are mostly aligned parallel to the film surface (see Fig. 29 a), while the chains exhibit a relatively broad distribution around the first drawing direction, MD (see Fig. 29 b). 

Similar to the WAXS pole figures, with SAXS one can obtain the orientation distribution of the normals onto the lamella surfaces. The measurements are based on the same instrumental technique, but with conventional X-ray sources the acquisition of one single SAXS pattern needs as much as several hours or days. Therefore the acquisition of a complete SAXS pole figure would require a tremendous amount of time and has never been carried out. The availability of S.R. reduces the required time to a few hours. 

For biaxial orientation the pole distribution t(, 4>) may be visualized as a density distribution defined on the surface of a sphere. The method of stereographic projection is then used to transcribe the density distribution from the spherical surface onto a sheet of paper. The contour map thus obtained is called a pole figure. 

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