Breadth azimuthal

Figure 15 Morphological map of linear polyethylene fractions. Plot of molecular weight against crystallization temperature. The types of supermolecular structures are represented by symbols. Patterns a, b and c represent spherulitic structures with deteriorating order from a to c. Patterns g and d represent rods or sheet-like structures whose breadth is comparable to their length g or display a different aspect ratio d. Pattern h represents randomly oriented lamellae. Neither h nor g patterns have azimuthal dependence of the scattering. Reproduced with permission from Ref. [223]. Copyright 1981 American Chemical Society. (See Ref. [223] for full details.) Note the pattern a is actually located as o in the figure this was an error on the original.

Motivation and Principle. Broadened reflections are characteristic for soft matter. The reason for such broadening is predominantly both the short range of order among the particles in the structural entities, and imperfect orientation of the entities themselves. A powerful method for the separation of these two contributions is Ruland s streak method [30-34], Short range of order makes that the reflection is considerably extended in the radial direction of reciprocal space - often it develops the shape of a streak. This makes it practically possible to measure reflection breadths separately on several11 nested shells in reciprocal space. As a function of shell diameter one of the contributions is constant, whereas the other is changing12. If the measurement is performed on spheres (azimuthal), the orientation component is constant. 

Separation of the two components is accomplished by means of data, in which the apparent azimuthal integral breadth... 

Figure 9.7. Separation of misorientation (Bg) and extension of the structural entities (1/ (L)) for known breadth of the primary beam (Bp) according to Ruland s streak method. The perfect linearization of the observed azimuthal integral breadth measured as a function of arc radius, s, shows that the orientation distribution is approximated by a Lorentzian with an azimuthal breadth Bs... 

Figure 7. Flux rope observations and models by Elphic and Russell [21]. Schematic representation of flux rope magnetic structure. The breadth of each arrow denotes the field strength the central axial field is strong, while the outer, more nearly azimuthal field is weaker.

The degrees of orientation for the crystallites in the injection-molded specimens of PECs were determined from the azimuthal breadth at half-maximum intensity in the strongest equatorial reflection by X-ray diffraction. Figure 6 showed the degrees of crystallite orientation of the as-molded and the annealed specimens. 

When a crystalline polymer is oriented, the random circular film pattern (random orientation) transforms to a collection of defined reflection arcs that are correlated with particular (hkl) planes that can be identified based on the crystal structure and Bragg relationship (see Fig. 12a-b). It follows that the magnitude of the azimuthal spread (x/2) of these reflections is indicative of the degree of orientation. (The breadth, k, of the reflection is related to crystal size and imperfection—see Ref. 32.) Also, the location of the reflection with respect to the sample axes indicates the orientation of the crystallographic planes. For example Fig. 5(a) and (b) show two X-ray photographs of polyethylene that had been cold rolled. From the (200) reflection in sample (a) one sees that the a-axis is aligned preferentially normal to Z whereas in (b) there are two distinct orientations of the a-axis—one along Z and one normal to this. 

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