Use of the sampling theorem for collimation corrections in small-angle X-ray scattering

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.