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Use of the sampling theorem for collimation corrections in small‐angle X‐ray scattering
Author(s) -
Gerber T.,
Walter G.,
Schmidt P. W.
Publication year - 1991
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/s0021889891001322
Subject(s) - collimated light , optics , function (biology) , scattering , slit , physics , series (stratigraphy) , fourier series , noise (video) , mathematical analysis , mathematics , paleontology , laser , evolutionary biology , artificial intelligence , computer science , image (mathematics) , biology
A technique has been developed for correcting small‐angle X‐ray scattering curves for the distortions produced by the width and the length of the collimating slits. The method is based on applying the frequency function, which can be calculated from discrete measured points by a series expression. When this function is divided by the Fourier transform of the slit‐width weighting function, the slit‐width corrected frequency function is obtained. The slit‐length correction makes use of slit functions calculated by Deutsch & Luban [ J. Appl. Cryst. (1978), 11 , 87–97, 98–101] and Schmidt & Fedorov [ J. Appl. Cryst. (1978), 11 , 411–416]. Then the slit‐length‐corrected scattering curve is evaluated from the analytically differentiated slit‐width‐corrected intensity function by numerical integration. The latter intensity, which is partially smoothed, is calculated from the slit‐width‐corrected frequency function by a series expression. By use of the same technique, an adequate expression for calculating the collimation corrected scattering curve for the case of infinitely long slit collimation was derived. A possibility is shown of how this method can be applied with a small computer and very short computing time. The procedure was tested by applying it to simulated experimental data. The influence of noise and the sensitivity of the corrected data to distortions produced by the slit width are discussed in detail.