Projections with fiber symmetry

Figure 8.27. Steps preceding the computation of a CDF with fiber symmetry from recorded raw data The image is projected on the fiber plane, the equivalent of the Laplacian in real space is applied, the background is determined by low-pass filtering. After background subtraction the interference function is received... 

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... 

This chapter presents an overview of work that has been performed by the author aiming to develop adapted (i. e. a structural model is chosen as late as possible) evaluation methods for SAXS diagrams with fiber symmetry and the results obtained so far [I ]. The basic principle of the methods to be presented is the extraction of curves from the 2D data in the scattering patterns by catain kinds of integ tions ( projections ). The importance of such projections has early been recognized (5]. Novel is the analysis of the extracted curves in terms of ID and 2D structural models. Applicability is assessed by comparison of the results with the obvious features of the scattering images. 

It has been demonstrated, how the utilization of the concept of projections can help to analyze two-dimensional SAXS patterns with fiber symmetry quantitatively. Because the computation of long tanging integrals is involved in this method, a prerequisite for successful analysis is the careful choice of a small beam stop, a wide vacuum tube, a rather short distance between sample and detector and a fast and linear detector. By doing so one can hope to register both all the reflections and also the important background scattering with required accuraQ and spatial resolution. 

Equation (1.4) gives the full expression, in which intensity varies with both polar angle 0 and azimuthal angle yt. K fiber symmetry is assumed, the intensity will be the same with given 0 and s and the integration of yields a constant In. Usually, a flat 2D detector is used to record the scattering pattern, which is a stereo-projection of 3D scattering intensity distribution. In this case, Fraser correction is needed to correct the distortion due to the stereo-projection [94]. The Hermans orientation function varies between -0.5 and 1. When /= -0.5, the normal and the reference axes are perpendicular to each other when /= 1, the normal is in parallel to the reference axis when / = 0, the system has random orientation. 

On a stereographic projection parallel to the surface of a rolled sheet, show (a) the positions of the (110) poles, represented by small ellipses, for the ideal orientation 111 (110), including the positions due to reflection symmetry, and (b) the lines showing the positions of the (110) poles for a (111) fiber texture, with the fiber axis normal to the plane of the sheet. 

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