# Periodic Multilayer for X-ray Spectroscopy in the Li K Range
## 4 Discussion  
With respect to the requirements presented in the Introduction regarding the suitability of a periodic multilayer for X-ray spectroscopy in the Li K range, the proposed trilayer stack:  
* has a quite high reflectance (0.32 at 51.5 eV, for example); let us note that the measured reflectance is lower than the one expected for a perfect multilayer, i.e., without roughness or interdiffusion (0.404 @ 51.5 eV), so there is still some room to improve the reflectance; an optimized reflectance is mandatory when working in this spectral range, as the K fluorescence yield of lithium is very low, 9 10\({}^{-5}\)[25], and as the windows used in front of the detector or between the sample and spectrometer chambers of EPMA absorb many useful photons; if no care is taken to choose these windows, their transmission can be as low as 10\({}^{-10}\) (a polymer having a density of 1.11 g.cm\({}^{-3}\), such as polyimide, having a total thickness of 2 \(\mu\)m), jeopardizing the measurement of the Li K emission; this transmission can be as high as 5 10\({}^{-3}\) if the setup is optimized (suppression of the window between the sample and spectrometer chambers, use of a low density polymer of 0.9 g.cm\({}^{-3}\), such as polypropylene, having a thickness of 0.5 \(\mu\)m and supported by a grid having a 50% transmission);
* presents a quite narrow bandwidth (mean of 3.5 eV, leading to a resolving power E/\(\Delta\)E = 0.07), see Table 1; this value is smaller than the width of the Li K emission band, which can be 10 eV wide [5; 6; 7; 8; 9]; thus, it should be possible to obtain an idea of the chemical state of the lithium atoms from the examination of the shape of the Li K spectrum and its comparison to those of the reference samples, whereas with a grating spectrometer having a resolving power of 0.01 or better, it is possible to determine the chemical state of a lithium atom directly from the examination of the shape of its Li K emission band;
* rejects the radiation diffracted at high orders (see Table 1 and Figure 3) moderately for the second order and efficiently for the third order; the rejection of the second order diffracted radiation with a perfect multilayer is expected from simulations to be between 150 and 200, i.e., two or four times better than the measured one;
* works around a glancing angle around 30\({}^{\circ}\), which is in the middle of the angular range scanned by standard spectrometers equipping EPMA. The Li K emission corresponds to the electronic transition between the Li 2\(p\) valence states to Li 1\(s\) states. Thus, it is an emission band whose shape (its energy distribution) describes the density of occupied valence states. As a consequence, the natural width of \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Photon Energy** & **Angle** & **Bandwidth** & & & \\
**(eV)** & **(\({}^{\circ}\))** & **(eV)** & **R1** & **R1/R2** & **R1/R3** \\ \hline
47.6 & 33.7 & 3.1 & 0.282 & 82 & 737 \\
49.5 & 32.1 & 3.3 & 0.300 & 83 & 667 \\
51.5 & 30.6 & 3.5 & 0.318 & 64 & 432 \\
53.4 & 29.2 & 3.8 & 0.336 & 58 & 618 \\
55.3 & 27.9 & 4.1 & 0.357 & 50 & 643 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the reflectivity curves shown in Figure 3. Ri is the reflectance measured at the i-th diffraction order. The uncertainty is estimated to be around 0.1 eV for the bandwidth, 1% for R1 and 5% for the reflectance ratios. the emission is closely related to the width of the valence band, i.e., up to 10 eV wide (for example 4 eV in the case of Li metal [26]). The shape of the emission band is sensitive to the chemical state of the lithium atom, as the Li \(2p\) valence electrons are the ones participating in the chemical bond. This sensitivity is very good when observing the Li band with a grating spectrometer, introducing a spectral broadening smaller than 1 eV. The bandwidth of the designed multilayer is around 3.5 eV. This will lead to a broadening of the emission band.