Azimuthal averaging

Frequent malpractice is the application of the azimuthal averaging algorithm to anisotropic scattering data for the purpose of isotropization. The result appears isotropic, but the chosen integration is incorrect. Only in the case of low anisotropy this procedure is permitted, because then the introduced error is kept small. 

If azimuthal averaging is used for the purpose of isotropization, a geometric problem from the 3D world is taken for a 2D problem57. For the field of polymer science the correct integration procedure has already been described in 1967 by Desper and Stein [148],... 

Fig. 4. CT validation scan using (a) jar filled with water (b) cross-sectional attenuation profile of the liquid in the jar (c) azimuthally averaged attenuation in the radial direction ( - actual attenuation,- average attenuation).

Fig. 6. Determination of solid distribution at three axial positions (a) cross-sectional distribution (b) azimuthally averaged solids holdup and (c) overall solids holdup. The horizontal line indicates experimentally determined solids fraction.

Fig. 8 shows the time and azimuthally averaged radial liquid saturation profiles at varying superficial gas and liquid velocities at the middle axial position (2.5D). The figure shows that liquid saturation is nearly flat, which suggests a fair uniformity of liquid distribution. Moreover, with increasing liquid velocities, liquid saturation increases. Similar trends were obtained at all scan heights. 

Both primary and secondary electron models (Atoyan Voelk 2000, Brunetti et al. 2001, Blasi Colafrancesco 1999, Miniati et al. 2001) have been analyzed to reproduce the spectral and spatial features of the EUV excess in Coma without a definite solution. Additional experimental information has been recently added to the complexity of the problem in particular, the EUV intensity distribution seems to be highly correlated with the thermal X-ray intensity and produce a constant ratio between the azimuthally averaged EUV and X-ray intensifies (Bowyer et al. 2004). Specific secondary models seem, at present, one of the few viable possibilities to reproduce the EUV emission features of Coma. 

In order to calculate the RF PAD for a set of partial wave amplitudes and phases we use Eq. (54) to first calculate the MF PAD. The MF is defined with the z axis along the Ciy symmetry axis, the y-axis along the N—N bond, and the x axis perpendicular to the molecular plane. The RF plane is defined with the z axis along the N—N bond direction. In order to calculate the RF PAD from the MF PAD, a rotation is applied to bring the MF z axis to the RF z axis. The resulting PAD is then azimuthally averaged about the z axis (the N—N direction),... 

Figure 11 C Is XPD pattern, using a Mg source, from monolayer films of Ceo adsorbed on (a) Cu(lll), (b) Al(lll), (c) Cu(llO), and (d) Al(OOl). The patterns have been azimuthally averaged by considering the appropriate rotational symmetry. (From Ref. 34.)...

The first approach is called the multiple reference frame (MRF) or inner-outer approach (inner-outer approach in fact defines inner and outer zones with a finite overlap whereas in the MRF approach there is no overlap between inner and outer regions). In this approach, flow characteristics of the inner region are solved using a rotating framework. These results are used to provide boundary conditions for the outer region (after azimuthal averaging), flow in which is... 

Here, a denotes the single-scattering albedo, while the 2x2-matrix W(m,v) is the 7,g-component of the azimuthally averaged phase matrix, which obeys the relations of mirror symmetry and reciprocity, respectively ... 

Next we consider the application of quadratic integrals for polarized radiation. We analyze an optically semi-infinite or finite homogeneous atmosphere that scatters radiation according to the azimuthally-averaged Rayleigh-Cabannes law for molecular scattering. 

The results for the H2 phase are displayed in Figure 8.5 and in the first part of Table 8.4. The anisotropy of the T -order maximum shows high-order parameters (S = 0.44 resp. 0.56, not shown). Higher-order maxima were visible in the direction of the nematic director but less clear in the azimuthally averaged data. 

Figure 8.4 SAXS patterns of oriented lamellar morphologies in the system SI52C3E5I.6/D4 at 58 C (a) Azimuthally averaged scattering intensity, after background correction from bottom to top (poA = 0, 0.05, 0.05, 0.10, 0.16, 0.15, 0.15, 0.15, and 0.20, scaling factor -5 with respect...

Fig. 3 One-dimensional x-ray intensity profiles of a PET film at different deformation stages strain 0%, 20%, 58% and 100% (Calculated from WAXS on the meridian by azimuthal averaging from 85 °C to 95 °C)(after Ran et al. [96])...

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