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

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0.03 0.04 0.05 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 Tungsten Carbon Wavelength (nm) Absorption edge for carbon Absorption index ß Fig. 3.31. 1- δ and β in the X-UV wavelength range index δ has a value on the order of 10 4 at the energy of 10 keV. Figure 3.31 gives δ and β values for the wavelength range from 0.1 to 11 nm, tabulated in [49]. The limitations of the total external reflectance phenomena at grazing incidence angles were discussed in detail in Sect. 3.2. Here is shown one simple equation which relates a complex refractive in- dex ˜ n and reflectance at grazing incidence angles R . The reflectance can be deduced from the well-known Fresnel formula R = ˜ n 1 ˜ n + 1 . (3.59) The reflectivity R decreases proportional to λ 4 for the shorter wavelengths. This reflectivity is too low for practical use at a large incidence angle (see (3.1) and (3.2)) for X-ray optics. A reflection takes place only at the angles below the critical angle θ c 2 δ . Enhancement of a reflectance at large incidence angles can be obtained by adding coherently and in phases all the reflections from several thin layers: multilayer structure. A combination of layers with different optical properties can lead to a structure similar to that of crystal. Effective Bragg diffraction can be obtained even at normal incidence. These multilayers can be considered as an extension of natural crystals to larger lattice spacing. Such “artificial” crystals became, in the last decade, the most important optical devices for a soft and middle photon energy range. Having a high reflectance which reaches 70–80% at normal incidence angles multilayer structures can provide also a beam monochromatization on the order of λ/ λ 25. The dif- fraction properties of a multilayer mirror can be described by the theory of Bragg diffraction on a crystal. The basic elements of this theory are given above in Chaps. 3.3, 3.4.3 and 3.4.4. Here one can find a description of the special properties of multilayer optics connected with the choice of materials and technological aspects.
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X-Ray Optics 131 The microscopic model used in the kinematic theory of X-ray diffraction on a crystal describes a medium atom by atom. This theory can be applied only to the ultrathin layers with a few atoms stacked in a layer. An application of the kinematic theory for the calculations of X-ray diffraction on a “thick” multilayer mirror cannot be done without considerable approximations due to the complexity of this model. In comparison with crystals the X-ray diffraction on a multilayer mirror needs “averaged” characteristics of the media, such as refractive and absorption indexes. The optical (also called dynamical) theory operates with average optical properties of the complete layer. In a real case one has to consider properly which theory should be applied to a particular structure. For example one can apply the Fresnel Equation (3.59) even to mono-atomic films for calculation of reflectance.
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