Projections fiber patterns

Particularly useful in materials science is a special ID projection the projection of a fiber pattern on the fiber axis, S3... 

In order to demonstrate completeness of a SAXS fiber pattern in the 3D reciprocal space, it is visualized in Fig. 8.16. The sketch shows a recorded 2D SAXS fiber pattern and how it, in fact, fills the reciprocal space by rotation about the fiber axis. V3. Let us demonstrate the projection of Eq. (8.56) in the sketch. It is equivalent to, first, integrating horizontal planes in Fig. 8.16 and, second, plotting the computed number at the point where each plane intersects the S3-axis. 

Figure 16.5. (a) Fiber cross-section irradiated by an X-ray beam at an offset x from its center. The structure p pf) shows fiber symmetry. From all structures along the beam path a superposition is probed. y is the variable of the integration, (b) One-dimensional tomographic reconstruction turns the measured series of projected scattering patterns that carry the accumulated structure information passed by the beam (vertical bars) into the image patterns from voxels (quadratic boxes) residing on the fiber radius... 

Stribeck N (2000) Analysis of SAXS fiber patterns by means of projections, ACS Symp Ser 739 41-56. 

Analysis of SAXS Fiber Patterns by Means of Projections... 

Step 1 Projection on the Representative Plane For the analysis of the stmcture we are interested in a multidimensional image in real space but not in a projection of the structural image [46, 60]. In reciprocal space, however, we have to perform a projection on the representative plane (si, ss) of the fiber pattern (cf. Fig. 1.8), and compute... 

Figure 18.14 The diffraction pattern of helices in fiber crystallites can be simulated by the diffraction pattern of a single slit with the shape of a sine curve (representing the projection of a helix). Two such simulations are given in (a) and (b), with the helix shown to the left of its diffraction pattern. The spacing between the layer lines is inversely related to the helix pitch, P and the angle of the cross arms in the diffraction pattern is related to the angle of climb of the helix, 6. The helix in (b) has a smaller pitch and angle of climb than the helix in (a). (Courtesy of W. Fuller.)...

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the Lorentz correction yields the scattering curve of the isotropic material I (5) = I (s) / (2ns2). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity /, (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and <7C. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

In practice, either a pole figure has been measured in a texture-goniometer setup, or a 2D SAXS pattern with fiber symmetry has been recorded. In the first case we take the measured intensity g (pole figure. In the second case we can choose a reflection that is smeared on spherical arcs and project in radial direction over the range of the reflection. From the measured or extracted intensities I (orientation parameter by numerical integration and normalization... 

Senti and Witnauer206 have reported studies on the fiber diagrams from various alkali-amyloses. Specimens were obtained by deacetylating clamped specimens of amylose acetate with the appropriate alkali. The positions of the alkali ions and the lateral packing of the amylose chains were determined with the aid of Patterson projections. In the A - and B -modifica-tions, the fiber period was 22.6 A. (extension of 6 D-glucose units), whilst in the V -modification it was 8.0 A. These authors have also studied in detail the addition compounds of amylose and inorganic salts with special reference to the structure of the potassium bromide-amylose compound.206 Oriented alkali fibers were treated with the appropriate salt solution. Stoichiometric compounds were formed. The x-ray patterns from these showed that the addition compounds with potassium salts crystallized in... 

This is possible because the projection lens system, which for clarity was not shown in Figure 4.7, is normally included behind the objective lens and below the source image plane. This lens system allows the projection of both the diffraction pattern and the specimen image on the observation screen. In Figure 4.8, [50] the electron diffraction pattern of a Fe thin film is shown. In Figure 4.9, the transmission electron micrograph of the mordenite included in the sample CMT-C (see Table 4.1), where fiber-like crystals of mordenite are seen, is shown [51],... 

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