# Sc/SiC/Al Multilayer Optimization for Li K Spectroscopy
## 3 Results and Discussion
Figure 2: Reflectivity of the Sc/SiC/Al multilayer measured as a function of the incident photon energy at near-normal incidence (solid line) and calculated for the ideal stack (dashed line). of periods up to a saturation value of about 40. As can be seen in Figure 3, displaying reflectivity curves in the 50-58 eV photon energy range and for an incidence angle of \(7^{\circ}\), the peak reflectivity increases and the bandwidth decreases as the number of periods in the stack increases. This allows us to obtain a peak reflectance of about 49% and narrower bandwidth (about 2.0 eV), as shown in Figure 3.  
The oscillating features observed on both sides of the main peak (50-53 eV and 56-58 eV) are known as Kiessig fringes. They were first reported in 1931 [27] and result from interference between X-rays reflected at the different interfaces (from the vacuum/top layer to the bottom layer/substrate interfaces) of the multilayer stack. The position and spacing of the fringes can be used to determine the thickness of the various layers. To work in the Li K energy range at a grazing angle of \(30^{\circ}\), we increased the multilayer period to d = 25.2 nm. This configuration was extracted from a homemade Python script, able to compute X-ray reflectivity for a given set of parameters and geometry and to find the optimal thicknesses of the layers (in our case Sc and Al layers). Three constraints were imposed to reduce the number of combinations:  
1. The SiC intermediate layer thickness was restricted to two values: 1 and 1.5 nm;
2. The number of periods was limited to a range between 15 and 40;
3. The thicknesses of Al layers could vary between 1.5 nm and 23 nm, while Sc layers were allowed to vary between 22.8 nm and 1.5 nm, both with an increment of 0.25 nm. The simulations were conducted considering an ideal multilayer without roughness and interdiffusion. As observed in Figure 2, we expected that the real deposited multilayer would lose a few percent of reflectance with respect to the ideal multilayer. The code gave, as a result, a number of periods of 40, a period equal to 25.2 nm with \(\mathrm{d_{Sc}=1.8}\) nm, \(\mathrm{d_{SiC}=1.5}\) nm, and \(\mathrm{d_{Al}=21.9}\) nm. Figure 4 presents the calculated reflectivity at a fixed glancing angle of \(30^{\circ}\) as a function of the energy of the incident radiation: the structure maintains a peak reflectance of 35.6% at 55.2 eV, with a bandwidth of 3.9 eV. The latter is wider by 1.3 eV compared to the near-normal incidence design. Additional simulations with an interface roughness of 0.5 nm were conducted, but were not incorporated into the figure due to their negligible impact on the multilayer reflectance (decrease of less than 0.2%). Figure 3: Effect of period number on the Sc/SiC/Al multilayer reflectivity (calculated) in near-normal incidence condition. Figure 5 shows the reflectivity curve of the Sc/SiC/Al multilayer simulated at the fixed incident photon energy of 55.2 eV, which is the position of the maximum observed in Figure 4, as a function of the glancing angle. The same structural parameters of the multilayer as those of the simulation presented in Figure 4 were used. The reflectivity maximum is 35.1% and the bandwidth 2.7\({}^{\circ}\). This reflectivity curve, complementary to the previous curve obtained as a function of the photon energy, confirms the good performance of the multilayer in terms of its reflectance in the spectral range of the Li K emission band. Figure 4: Simulated reflectivity at a 30\({}^{\circ}\) grazing angle of the Sc/SiC/Al multilayer optimized for operation in the Li K emission spectral range (for an ideal multilayer) as a function of the incident photon energy. Figure 5: Simulated reflectivity at a 55.2 eV incident photon energy of the Sc/SiC/Al multilayer optimized for operation in the Li K emission spectral range (for an ideal multilayer) as a function of the glancing angle. Table 2 presents a comparison of the reflectance of various multilayer systems designed to work in a spectral range close to the one of the Li K emission band.