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

Synthetic multilayers sml have been utilized since

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Synthetic multilayers (SML) have been utilized since the 1980s as dis- persion devices for soft X-rays or EUV [91]. Typically, SML is formed as a multiple periodic stack of refraction layers, which consist of a heavy element in most cases, and transmission, or spacer layers which consist of a light element, as illustrated in Fig. 4.65. There are factors which influence the characteristics of an SML, such as the refraction coefficients of layers, the thickness ratio of the refraction layer and transmission layer (called Γ value, see Fig. 4.65 for the definition), total layer number, sharpness of the layer boundaries, etc. The optimal substance pair should have low absorption and large Fresnel coefficient, i.e., a large difference of refraction indices between the spacer and refraction layer. The complex refraction index n is described using atomic scattering factor f 1 + i f 2 as (4.65) n = 1 δ + i β = 1 r 0 λ 2 2 π N A ρ A ( f 1 i f 2 ) . (4.65) Here, δ and β are the real and imaginary part of refractive index, r 0 is the clas- sical electron radius, λ is the wavelength of incident X-rays, N A is Avogadro number, ρ is the density, and A is the atomic weight. Yamamoto and Namioka [92] have presented a way to find such a proper pair using a plot of complex refraction indices. Figure 4.66 is an example of the plot for C-K α . The scattering factors used are that of Henke et al. [93]. Yamamoto and Namioka also proposed designing methods of each layer thickness and total layer number using complex amplitude reflectance. It should be mentioned that realizing a sharp interface between layers is also an important property for good SML and it limits the choice of layer materials. SMLs have a number of advantages over natural crystals; much higher reflectivity, physical stability, small thermal expansion coefficient, and suppression of higher order reflection. Regarding higher order reflection, dras- tic suppression is achieved by adjusting Γ value. For example, a second (third) order reflection is virtually eliminated with an SML of which Γ is 1/2 (1/3). d t d r d G = d r / d Fig. 4.65. Schematics of an synthetic multilayer (SML)
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288 N. Kawahara and T. Shoji Ru Ag Rh Cd Al Sn Real part of refractive index Ca Pr Lu La Nd Tm Ho Si Pb Tl Ti Sb BN Sm Tb Yb In B Bi Cs C Na P K Sc Ge Se Rb Sr Y Ba V Cr Mn Fe Co Ni Cu Zn Ga As Zr Nb Mo Pd Hf Ta W Re Os Ir Pt Au 0.000 0.005 0.010 0.015 0.000 0.005 0.010 0.015 Imaginary part of refractive index 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.000 0.001 0.002 0.003 0.004 0.005 0.006 Bi Be B C Na Mg Al Si P S K Ca Sc Ti Ge Se Rb Sr Y Zr Ru Rh Ag Cd In Sn Sb Te Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Tl Pb B 4 C BN Fig. 4.66. Material map in complex refractive index plane at 277 eV (C-K α ). Bulk densities at room temperature are used A noteworthy matter on the use of SML is that the dispersion may not obey Bragg’s equation. Considering the refraction index (4.62) has to be modified as = 2 d sin θ 1 δ i β sin 2 θ . (4.66) As refraction index depends on the wavelength of incident X-rays, the apparent 2 d value calculated by using “simple” Bragg’s equation varies de- pending on wavelength.
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