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This study extends a mathematical concept for the description of heterogeneity and polydispersity in the structure of materials to multiple dimensions. In one dimension, the description of heterogeneity by means of Mellin convolution is well known. In several papers by 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 is particularly attractive considering the advantageous mathematical properties enjoyed by the Mellin convolution. Thus, average particle size, and width and skewness of the particle size distribution 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 extensible 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 generalized 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 in the field of soft condensed matter when anisotropic nanostructured materials are to be characterized by means of small-angle scattering (SAXS, USAXS, SANS).

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