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Figure 4
Optimization of the curvature of crystal sheets to realize Laue mode analysers; along its curvature, at every 1°, the diffraction planes of these thin curved crystals are symbolized by a large plane with the corresponding direction of its incident X-ray beam. (a) Examples of crystal plane orientations for positioning the thin Si single-crystal sheet in a transmission MAD system. (b) Beam paths from two sources at 7 keV and 9 keV (green and purple, respectively) arriving on the Si(111) planes of two single-crystal sheets which have ideal curvatures for 7 keV and 9 keV, respectively; the origins of these representations have been translated so that the midpoints of the two curved crystal sheets are coincident. (b′) Zoom-in on the directions and impacts of the beams at the upper ends of the single-crystal sheet which enlarge the most important curvature difference existing between 7 keV and 9 keV (green and purple), an energy range close to that of a Cu Kα1 X-ray source. (c) Beam paths coming from these two different energy sources 7 keV and 9 keV, positioned to be filtered by Si(111) planes of a single-crystal sheet having its curvature determined for 9 keV in both cases; the purple curve represents the ideal paths of the 9 keV beams filtered by a MAD system having an ideal curvature for 9 keV, the blue curve represents the paths of the 7 keV beams filtered by a MAD system having a modified curvature corresponding to 9 keV filtering and corrected due to a δD optimization of the sample-to-MAD-system distance. (c′) Enlargement showing this variation δD in the distance of the sample from the MAD system which can modify and reduce the deviations of the Bragg angle ΘA of the beams on this curved crystal of the MAD filter and can allow a wider angular range of filtering, as discussed in the text and in Fig. 5[link]. (c′′) Zoom-in on the impact of these 7 and 9 keV beams, at the upper ends of the single-crystal sheet with curvatures determined for 9 keV ideally and with the modification due to a small δD variation of the sample-to-MAD-system distance.

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