Azimuthal distribution

Azimuthal distribution Horizontal direction of a celestial point from Random Lemeur, 1973 ... 

X-ray diffraction (in crystalline polymers) Unoriented crystalline polymers show X-ray diffraction patterns, which resemble powder diagrams of low-molecular crystals, characterised by diffraction rings rather than by spots. As a result of orientation the rings contract into arcs and spots. From the azimuthal distribution of the intensity in the arcs the degree of orientation of the crystalline regions can be calculated (Kratky, 1941). 

Figure 11 Cu(100)c(2 x 2)-0. Azimuthal distribution of 0 s photoelectron intensity at a polar emission angle 6 = 7°. In the upper half of the figure the dotted curve represents the raw data and the full curve the data with the minimum value subtracted. The lower half of the figure is a theoretical curve for O atoms in fourfold holes at a z-position co-planar with the surface Cu atoms...

When two CH2 molecules combine, it seems reasonable to assume the superposition of two identical charge distributions of this kind, with a common Z-axis. An alternative is to assume that the two carbon atoms line up with respective electronic configurations of C(p p+i) and C p p i) as in 7(b)E. The azimuthal distribution of the angular momentum vectors of such a pair is in opposite sense as shown in Figure 20-7(e). The actual distribution of electron density in the ethylene molecule is conjectural. The obvious guess that anti-parallel 2p solutions quench the angular momentum, is favoured, but not confirmed, by chemical intuition. 

There are only two papers describing the physical properties of discotic LCEs [35, 36]. X-ray diffraction showed that a narrower azimuthal intensity distribution can be obtained when a second crosslinking step is done under an external load reflecting a macroscopic alignment parallel to the direction of the external stress [35]. X-ray diffraction experiments on monodomains (LSCE) show the occurrence of sharp maxima in the azimuthal distribution. Thermal expansion has also been studied for polydomain LCEs and for LSCEs in [36]. It is found that well inside the discotic phase, that is sufficiently far below the discotic-isotrop-ic transiton, the length parallel to the stress axis of both, poly- and monodomain samples, increases approximately linearly with increasing temperature as expected from the isobaric expansion of the network. 

They as well revealed that the LC monolayer adjacent to the rubbed polymer substrate remains aligned also in the isotropic phase [20], consistent with NMR and AFM experiments (see Chaps. 2 and 3). The opposite is also true on several rubbed polymer films that induce a strong bulk alignment, SHG experiments show the first LC layer has an isotropic azimuthal distribution. An example of that will be shown in Sect. 5.3 below. 

The meridional width is used to calculate the apparent length parallel to the fiber axis, Lafiber axis, L,j., by the Ruland and Wamen formulas, respectively [1] [4] [32]. The interlayer spacing door, is derived from the 002 reflection along the equatorial direction. It characterizes the turbostratic feature of the carbon. A parameter characterizing the misorientation of the carbon layers with respect to the fiber axis, Z, is derived from the half maximum width of the azimuthal distribution intensity curve along the 002 arc (arc opening). 

Fig. 17. Azimuthal distribution of intensity measured from the data of Fig. 16 at i = l-4A , the scattering position of the main interchain peak. The broken line represents the background level used in the analysis. ...

Figure 14.21. Azimuthal distribution of the scattered intensity in the 2D SAXS images of two HDPE/PA6/YP UDP MFCs with compositions of (a) 80/20/0 and (b) 70/20/10 (1) the initial MFC at 30 C, (2) the MFC with in-beam heating at 160°C, and (3) the MFC at 30°C after heating to 160°C. The dashed line indicates the fiber direction [63]...

At the transition to the SmC phase, the layers usually shrink just as we would see even from the simplest cartoon. However, in a few examples the layers do not shrink at all upon the transition to the SmC phase. Such transitions can be explained by assuming that the director is already tilted in the individual layers, but the tilt direction is not correlated in the subsequent layers. In these models, it is assumed that only the azimuthal distribution of the molecules becomes asymmetric, and the layer spacing may not be affected by the transition. ... 

Fig. 11 Evolution of the lamellar morphology (top row), ED pattern (middle row), and azimuthal distribution of the (1 0 0) reflection (bottom row) obtained for epitaxied P3HT films grown by slow-DEC in TCB for three P3HT samples with different molecular weights. The insets in the top sequence correspond to the fast Fourier transform...

Typical methods obtain the MFA by measuring the azimuthal distribution of intensity of the [200] equatorial reflection. The sample is oriented so that the axis of the fiber was in a vertical direction and then the azimuthal position is changed until the intensity of the peak reaches minimal value. The sample can be located under angle 45° or 90° to a beam of X-ray, while azimuthal distribution of intensity of the [200] reflection is photographed on a photo-film. If the sample is under 45° to beam, the MFA can be calculated by the equation of Eichhorn et al. [2001] ... 

FIGURE 3 (a) Cylindrical distributicm function f(x atactic PS obtained from l(Q,a) using EQN (10). (b) Sharpened CDF. Orientational distribution D(a) was approximated by a function of the form cos a which closely describes the azimuthal distribution of the halo with the narrowest profile. Dashed contours represent negative, i.e. lower than average, values. The alignment axis is vortical. From [3] with permission. 

Alternatively, Leadbetter and Norris [12] have derived a direct relationship between the orientational distribution function D(a) and the azimuthal distribution of the equatorial intensity I(y) (FIGURE 5) ... 

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