[B._Beckhoff,_et_al.]_Handbook_of_Practical_X-Ray_(b-ok.org).pdf

The best results can be obtained with doubly bent

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The best results can be obtained with doubly bent crystals, when focusing takes place in both meridional and sagittal planes. The relationship for the meridional R 1 and sagittal R 2 curvature radii can be established geometrically (see Fig. 3.40) as R 1 = D/ 2 sin θ B , R 2 = D sin θ B , where D is the distance between the point source and the crystal centre ( distance between the focus and the crystal centre); θ B is the Bragg angle. The condition R 1 = R 2 (spherical shape) is valid only for θ B = 45 , which corresponds to E = 2 . 61 keV. In a general situation R 1 = R 2 , and the ellipsoidal shape could be offered as the most suitable one. Being a mosaic crystal, HOPG has some peculiarities that should be taken into account while designing X-ray optics. First, there is a spatial beam smear- ing induced by the deep penetration of the reflected beam into the bulk of the mosaic crystal [125] aggravated by the fact that HOPG consists of light carbon. Two main effects determine the penetration depth of the X-ray ra- diation into a crystal: true absorption and interference extinction [125, 126]. In ideal crystals extinction due to interference is very strong because of a perfect arrangement of reflecting crystal lattices. As a result, one can assume that Bragg reflection takes place practically on a surface of an ideal crystal. This is not the case for mosaic crystals. Although extinction in HOPG dom- inates over true absorption, it is not so strong as in ideal crystals due to the spread of mosaic blocks [126]. Typical values of the total penetration depth lie in the interval 20–100 µ m for the energies 6–30 keV and increase further S F R 1 R 1 S F R 2 R 2 (a) (b) Fig. 3.40. Principles of meridional ( a ) and sagittal ( b ) focusing
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150 A. Antonov et al. Beam profile HOPG Fig. 3.41. Spatial broadening of the beam after reflection with the raise of the energy. This fact has two simple consequences. First, a well-collimated beam becomes wider after the Bragg reflection (Fig. 3.41). This effect limits the possibility of exact focusing with HOPG crystals and should be always taken into account when estimating the spot size on a sample. Second, the thickness of standard HOPG crystals (usually 1–2 mm) is superfluous for forming a diffracted beam of the highest intensity. In practice the thickness of 2–3 times larger than the penetration depth for a given energy is already sufficient for a strong diffraction. Therefore flexible HOPG foils with a thickness optimized for a high effective reflectivity or for a spatially narrow reflected beam can be successfully employed offering the respective advantages in comparison to standard thick crystals. The second effect, which is characteristic for HOPG crystals and can be named as mosaic focusing, is illustrated in Fig. 3.42. A beam can be reflected even when its incidence angle has a certain deviation from the exact value determined by the Bragg condition. Such a beam penetrates into a crystal and somewhere in the depth meets a mosaic block with the necessary orien- tation. As a result, a flat mosaic crystal focuses incident radiation with the
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