Angle-dependent X-ray absorption correction in small- and wide-angle X-ray scattering: accounting for constrained...

A GPU-accelerated numerical method has been developed to calculate angle-dependent transmission factors A(2θ) for small- and wide-angle X-ray scattering (SWAXS) data collected with a Kratky-type camera using a line-collimated beam and a horizontal cylindrical capillary with a vertically oriented detector. The procedure extends path-length-based absorption correction to the Kratky-type geometry by averaging Beer-Lambert attenuation over the constrained illuminated scattering volume and by including experimentally measured primary beam profiles and capillary-wall absorption. Sensitivity analysis shows that the vertical illumination geometry dominates the calculated angle-dependent transmission factor, whereas the horizontal beam-width effect is negligible. Using measured beam profiles makes the correction instrument specific and avoids introducing an effective uniform beam height as an approximation. For strongly absorbing nona-fluoro-tert-butanol, the conventional scalar primary beam transmission correction fails after subtraction of empty-capillary scattering, while the numerical correction enables physically meaningful absolute-scale data reduction. Comparisons between independently calculated theoretical SWAXS intensities and the two reduced experimental datasets obtained sequentially using two different detectors in the same experiment on the same Kratky-type SWAXS camera provide a direct experimental cross check for the data-reduction workflow based on calculated transmission factors. They indicate that the observed limitation is associated with transmission measurement of a spectrally modified primary beam, rather than with the Kratky-type geometry itself. Importantly, the results show that the limitations of conventional scalar transmission corrections are not confined to extreme cases but may also affect routine water-based absolute-scale SWAXS calibration. A simple Beer-Lambert-type numerical scalar correction can still provide a useful qualitative approximation, whereas reliable quantitative analysis requires the full angle-dependent A(2θ) correction.