TIT# Analysis of Scattering from Polydisperse Structure Using Mellin Convolution
AUT# Stribeck, Norbert;
SOU# J. Appl. Cryst. (2005), revised
LOC# xv096
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MAT#
ABS# This study extends a mathematical concept for the description of
     heterogeneity and polydispersity of materials structure to multiple
     dimensions. In one dimension the description of heterogeneity by means of
     Mellin convolution is well-known. In several papers of the author the
     method has been applied to the analysis of data from materials with
     one-dimensional structure (layer stacks or fibrils along their principal
     axis). According to this concept heterogeneous structures built from
     polydisperse ensembles of structural units are advantageously described
     by the Mellin convolution of a representative template structure with the
     size distribution of the templates. Hence the polydisperse ensemble of
     similar structural units is generated by superposition of dilated
     templates. This approach gains particular attractivity from advantageous
     mathematical properties enjoyed by the Mellin convolution. Thus average
     particle size, width of the particle size distribution and their skewness
     can be determined from scattering data without the need to model the
     size distributions themselves. The present theoretical treatment
     demonstrates that the concept is generally extended to dilation in
     multiple dimensions. Moreover, in an analogous manner a representative
     cluster of correlated particles (e.g. layer stacks or microfibrils) can
     be considered as a template on a higher level. Polydispersity of such
     clusters is, again, described by subjecting the template structure to the
     generalised Mellin convolution. The proposed theory leads to a simple
     pathway for the quantitative determination of polydispersity and
     heterogeneity parameters. Consistency with the established theoretical
     approach of polydispersity in scattering theory is demonstrated. The
     method is applied to the best advantage, if in the field of soft
     condensed matter anisotropic nanostructured materials shall be
     characterised by means of small-angle scattering (SAXS, USAXS, SANS).