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

The crystal curvature radius can be as small as 5 mm

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The crystal curvature radius can be as small as 5 mm. The mosaic spread depends on the film thickness. It varies from 0 . 1 for 10 µ m thick films up to 0 . 4 ± 0 . 1 for the 200 to 400 µ m thick HOPG layers mostly used in focusing optics [124]. Another advantage of the new technology refers to the relatively low pro- duction costs of these HOPG crystals. Usually, the costs of a flexible HOPG in an X-ray device does not exceed those of a similar standard flat HOPG cry- stal of a high quality and is notably smaller than the cost of an analogous standard bent HOPG crystal. Figure 3.38 shows a set of cylindrically bent HOPG crystals with different curvature radii from 5 to 20 mm [120]. Fig. 3.38. Cylindrically bent HOPG crystals (radii from 5 to 20 mm) on aluminum substrates
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148 A. Antonov et al. Main Physical and Geometrical Relations The diffraction condition for an X-ray beam impinging on a crystal under the incidence angle θ is described by the well-known Bragg equation 2 d sin θ = nλ, where d is a crystal interplanar spacing for a given lattice plane, λ is the radiation wavelength and n is the order of the reflection. The maximum value of the wavelength, at which this relation can still remain valid, is determined by the condition λ max = 2 d . HOPG crystals have a hexagonal structure with the lattice constant d = 3 . 354 ˚ A for the 002 lattice planes. Therefore, the maximum wavelength, which can meet the Bragg equation, amounts to 6.708 ˚ A and the diffraction is possible only for photon energies above approximately 1.85 keV. HOPG crystals can be operated in a broad energy range (Fig. 3.39). Al- though the peak reflectivity reaches its maximum at about 17 to 20 keV, it preserves relatively large values up to very high energies where other optical elements (e.g., multilayers, capillaries, etc.) lose their efficiency drastically. Therefore, HOPG crystals can be also used for hard x-radiation in applica- tions such as medicine and X-ray astronomy. Focusing with bent crystals should meet normally two conflicting require- ments: the Bragg condition and the condition of geometrical focusing which imply different meridional curvature radii of the lattice planes involved and of 0.5 0.4 0.3 0.2 Reflectivity 0.1 0 0 5 10 15 20 25 30 35 40 45 50 55 60 E (keV) 65 70 75 80 85 90 95 100105 110115120 Fig. 3.39. Calculated reflectivity of a thick HOPG crystal (mosaic spread 0 . 4 )
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X-Ray Optics 149 the crystal surface. It is well known that the curvature radius R F for optimum focusing should be twice the Bragg radius R B . Application of the Johansson scheme, which is the best solution for single crystals, is not directly possi- ble for the mosaic case. However, one can use the crystal with the curvature radius R F for optimum focusing. Small angular deviations from the exact Bragg condition are mostly compensated by the mosaic spread of individual blocks.
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