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Application of spline functions to the correction of resolution errors in small‐angle scattering
Author(s) -
Schelten J.,
Hossfeld F.
Publication year - 1971
Publication title -
journal of applied crystallography
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.429
H-Index - 162
ISSN - 1600-5767
DOI - 10.1107/s0021889871006733
Subject(s) - spline (mechanical) , weighting , scattering , thin plate spline , small angle scattering , basis function , series (stratigraphy) , gaussian , resolution (logic) , mathematical analysis , mathematics , optics , physics , computer science , spline interpolation , statistics , paleontology , quantum mechanics , artificial intelligence , biology , acoustics , bilinear interpolation , thermodynamics
The concept of spline functions is introduced as a new tool for resolution error correction in small‐angle scattering. After discussing the approximating properties with respect to the estimation of experimental distributions and their derivatives from statistical data natural cubic splines are applied to the closed solution of the slit‐height effect with Gaussian weighting functions including infinite slit height and to the correction of the slit‐width effect based on Sauder's unfolding series. On the basis of an extended variational problem with one constraint taking adequately into account the statistical errors of the data a general procedure using cubic splines is developed for the solution of the integral equations associated with the three distinct types of resolution errors and arbitrary weighting functions. Computer simulations of small‐angle scattering experiments have been performed to demonstrate the efficiency of the spline function algorithms in the different applications using scattering functions and slit‐height and slit‐width distributions of various kinds.