Isotropization of Anisotropic Scattering Patterns

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 problem. For the field of polymer science the correct integration procedure has already been described in 1967 by DESPER and Stein [148]. 

In general, only a 2D scattering pattern will be available. In this case isotropization can only be performed if the pattern shows fiber symmetry and the fiber axis is contained in the scattering pattern. This symmetry axis must be known. Complete is the available information under these conditions only if SAXS data are evaluated. For WAXS data there are blind regions about the meridian (cf. Fig. 2.6 on p. 28), and missing information must be completed either by extrapolation or by extra experiments in which the sample is tilted with respect to the primary beam. 

Completeness means that from the recorded data we have to be able to reconstruct the scattering intensity for every point inside a sufficiently big volume of reciprocal space. 

Isotropization in the Case of Fiber Symmetry. If methods for the analysis of isotropic data shall be applied to scattering patterns with uniaxial orientation, the corresponding isotropic intensity must be computed. By carrying out this integration (the solid-angle average in reciprocal space) the information content of the fiber pattern is reduced. One should consider to apply an analysis of the longitudinal and the transversal structure (cf. Sect. 8.4.3). 

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