Fine structure of the atomic scattering factors near the iridium L-edges

Abstract. We measured the reflectivity of an Athena silicon pore optics sample coated with 10-nm thick iridium near the iridium L-edges (L3  ,  L2, and L1) in a step of 1.5 eV. The derived atomic scattering factor f2 was similar to a shape of the absorption coefficient μ near L3 and L2 obtained by previous x-ray absorption spectroscopy (XAS) measurements. The fine structures of f2 of L3 and L2 can be represented by a strong sharp line referred to as a white line (WL) and two weak lines at center energies of ∼17 and ∼31  eV from each edge energy. The branching ratio (L3  /  L2) of the WL is >2, which reflects the initial core-electron states available for the L2 (2p1/2) and L3 (2p3/2) processes, and the ratio remains high to the energy of +7  .  5  eV from WL. The fine structure seen in L1 also has two weak lines, which were seen in XAS at L1-edge. Our measurements near L3, L2, and L1 edges demonstrated a different technique to provide atomic structural information as XAS. The ground calibration to measure fine structures near the edges may potentially be simplified using f2 estimated based on μ.


Introduction
The x-ray absorption fine structure (XAFS) is often measured with x-ray absorption spectroscopy (XAS) to find useful information on the environment (geometry) and electronic structure of the absorbing atoms. XAS is performed in a transmission setup, providing a direct measure of the linear x-ray attenuation coefficient μðEÞ. The x-ray absorption edge spectra of iridium, which is one of the elements having high electron density, such as Au, Pt, etc., have been obtained to reveal the fundamental characteristics of the 5d material 1-3 . Clancy et al. 1 (hereafter Clancy12) investigated spin-orbit coupling effects in a series of Ir-and Re-based 5d compounds using XAS techniques and observed anomalously large L 3 ∕L 2 branching ratios in all Ir-based compounds, indicating a very large expectation value for the spin-orbit operator in these systems.
On the other hand, small oscillations are seen in the reflectance setup, referred to as the diffraction anomalous fine structure (DAFS). Stragier et al. 4 studied the DAFS as a new x-ray *Address all correspondence to Hisamitsu Awaki, awaki@astro.phys.sci.ehime-u.ac.jp structural technique and demonstrated that the DAFS measurements provide the same local atomic structural information as XAFS. They used XAFS standards to analyze the DAFS data by shifting the phase. Pickering et al. 5 applied the Kramers-Kronig relation to the atomic scattering factors f 1 and f 2 and determined the scattering factor f 2 to fit the DAFS data. The scattering factor f 2 is suitable for research of XAFS because f 2 is related to the absorption coefficient μ: 6 where C ¼ ðπr 0 hcÞ −1 , h is the Planck's constant, r 0 is the classical electron radius, and c is the speed of light. However, the f 1 value in the wide energy band is required for calculating the f 2 value using the Kramers-Kronig relation.
The atomic scattering factors f 1 and f 2 of iridium can be derived from the reflectance obtained by angular scan and/or energy scan without using the Kramers-Kronig relation. This is demonstrated by measurements on the ground for building the response function of the x-ray mirror (e.g., ASTRO-H 7 and Chandra 8 ). We measured the reflectance of a sample fabricated by employing ESAs silicon pore optics (SPO) technology in the high energy region from 9000 to 15,000 eV at the synchrotron radiation facility SPring-8 9 and obtained the atomic scattering factors of iridium near L-edges with an energy pitch of 1.5 eV. We compare the f 2 fine structure with XAFS and show that the method using reflection is as effective as XAFS methods. In addition, it is becoming increasingly important to measure the fine structure of the atomic scattering factors because of the improvement in energy resolution of focal plane detectors in recent years. 10,11 Although fine energy scans for measuring f 1 and f 2 near the edges are possible, estimating f 2 from μ can reduce the time required for ground calibration tests. 7,9 In Sec. 2, we introduce our x-ray measurements and calculate the atomic scattering factors from the reflectance. In Sec. 3, we characterize the fine structure seen in the atomic scattering factor f 2 to compare the previous work on XAFS and then analyze the derived f 2 in a way similar to XAS.

X-Ray Measurement
Our SPO sample was prepared at the coating facility dedicated for the Athena mirror production. 12,13 The sample has a length and width of 110 and 49 mm, respectively. A 10-nm thick iridium thin film coating was deposited using DC magnetron sputtering. The details of the sample preparation are described in Awaki et al. 2021. 9 We measured the x-ray reflectivity of the SPO sample at BL20B2 14 in SPring-8. The crystal plane of 311 of the Si double crystal monochromator 15 was selected to monochromatize continuum x-rays in our experiment. The energy resolution (ΔE∕E) of the available x-ray beam is <10 −4 in the energy range from 9000 to 15,000 eV, and the energy scale was calibrated by measurements of the Cu K-edge structure. We measured the intensity of a reflected x-ray on the SPO sample as well as the intensity of an incident x-ray and derived the reflectance R, defined as the ratio of the intensity of a reflected x-ray to that of the incident x-rays. Since the reflectance R is described as a function of the energy E of the incident x-ray and grazing angle θ, we measured the reflectivity by performing angle and energy scans at fixed energies and fixed incident angles, respectively.
In the energy scans, three fine pitch scans with an incident angle of 0.2, 0.32, and 0.4 deg were performed around the energies of the iridium L-edges in steps of 1.5 eV to clarify the iridium L-shell structure of f 1 and f 2 . The scanning energy ranges are 11 to 11.5 keV for L 3 edge, 12.5 to 13 keV for L 2 edge, and 13.2 to 13.7 keV for L 1 edge. Figure 1 shows the reflectance obtained in the energy scans. We can clearly see small oscillations in the post-edge region. These small oscillations are referred to as DAFS. Angle scans were performed at x-ray energies of 11, 12, and 14 keV, spanning the range of the iridium L-edges, specifically. The incident angle was scanned up to 1.5 deg in steps of 0.05 deg to measure the thickness of the iridium layer.

Estimation of the Atomic Scattering Factors
The reflectance of a single thin layer can be calculated using the refractive indices of the thin layer and the substrate. By fitting the measured data with a single thin layer model, we can estimate the parameters describing the iridium layer, specifically, the thickness of the iridium layer (d), interfacial roughness between the iridium layer and SPO substrate (σ b ), and the surface roughness of the iridium layer (σ). However, the angle data points in the energy scan were not enough to estimate the parameters. Thus, the fit to the angle scan data was used to identify the parameters describing the iridium layer, specifically, d, σ b , and σ. In this calculation, the loss of reflectance owing to roughness was considered as the Nevot-Croce factor, 16 and the atomic weight and the corresponding mass density of iridium were taken as 192.22 and 22:421 g cm −3 , 17 respectively. We employed an overlayer model, assumed as a hydrocarbon chain of the form CH 2 with a density of 1 g∕cm 3 (Graessle et al. 8 ). The best-fit values were obtained from the angle scan. The details of the measurements are given in Awaki et al. 9 In the fitting procedure of the energy scan data, σ, σ b , and d were set as constant at the weighted mean values of the best-fit values of the angle scan data (σ ¼ 0:30 nm, σ b ¼ 0:64 nm, d ¼ 10:01 nm). The thicknesses of the overlayer were set to be 0.9 nm because we found that an overlayer thickness of 0.9 nm led to a better fit to the f 2 data reported by Henke et al. 6 Figure 2 shows the derived f 2 near the iridium L-edges to clarify their structures. (c) Fig. 2 The atomic scattering factor f 2 near the L-edges. The dash-dotted lines indicate the energies (11215.7 AE0:3 eV, 12824.3 AE0:9 eV, and 13426.5 AE0:6 eV) of L 3 , L 2 , and L 1 edges, respectively. 9 The gray line in the panels of L 2 and L 3 shows μðE Þ (arb. unit) obtained by Clancy12.
3 Fine Structures Derived from the Atomic Scattering Factor f 2

Comparison with the XANES Obtained from XAS
It is known that the absorption coefficient μðEÞ of iridium near L 2 and L 3 edges contains three main features: (i) a sharp line feature known as "white line" 18,19 (hereafter WL), (ii) a step-like edge feature, and (iii) a small wave-like feature. 1,20 The scattering factor f 2 near L 2 and L 3 edges also shows these features (see Fig. 2). To confirm the similarity between μðEÞ by Clancy12 and our result f 2 ðEÞ, the Clancy12 data (arb. unit) was scaled to fit our data and then overlaid on our data in Fig. 2. Both data were found to be similar, including the small wave feature in the energy ΔE > 50 eV. The similarity suggests that (1) the fine structures of f 2 obtained from reflectance are consistent with XAFS obtained from XAS and (2) there is no apparent difference of optical characteristics between a thin iridium film formed by a DC magnetron sputter and an iridium powder with a purity of 99.99%. The similarity to Clancy12 can also be seen in the best-fit parameters of L3. Comparing the best-fit parameters obtained by Awaki et al. 9 with previous works, 1,20 the best-fit parameters (center energy and line width) of L3 WL were consistent with previous works 1,20 (see Table 1). Clancy12 also measured L 2 WL and estimated the branching ratio to be 3.2, which is larger than that of our result. In the fitting procedure by Clancy12, the line width of L 2 WL was fixed to the same line width as L 3 WL. Thus, we fitted the L 2 WL with the same line width as L 3 WL. The branching ratio is changed to 2:9 AE 0:4, which is consistent with their result.

Fine Structures of the Scattering Factor f 2 Near L-Edges
There are two distinct peaks (p1 and p2 in Fig. 2) in the f 2 fine structures at L 2 and L 3 -edges, and these peaks are also seen in the x-ray absorption near edge structure (XANES) results obtained by previous works. 1 Each peak was fitted with a Lorentzian model. The best-fit parameters are shown in Table 2. The center energy of each peak is represented by the energy ΔE from the absorption edge. Since the center energies ΔE and line widths of p1 in the L 3 edge are consistent with those of p1 in L 2 edge in the 1σ error region, p1 detected in L 2 edge is considered to be a transition to the same state as detected at L 3 edge. It would be the same for p2 as for p1. Sham 21 studied the x-ray absorption fine structures at L-edges for 4d elements, Rh, Pd, and Ag, and there were prominent weak peaks at20 and ∼38 eV in their absorption spectrum at the L-edge for these materials. The two weak lines p1 and p2 in Fig. 2 would be assigned to peaks 2 to 4 (→ pd) and 5 (→ df) in Table 1 of Sham. 21 To find branching ratios of these lines,   Fig. 8 of Sham, 21 and these two peaks were associated with 2 s → p, pd and 2 s → f, respectively. This assignment of the two peaks will be helpful to consider the p1 and p2 assignments found in the iridium L 1 edge, although we cannot eliminate the possibility that p2 might be similar to extended XAFS (EXAFS) due to the large line width of 11 þ6 −6 eV.

Ratio of the Scattering Factor f 2 Between L 2 and L 3 -Edges
The measured step size, which presents the depth of an edge, of the L 3 edge was twice that of the L 2 edge. Since the step-like edge features are associated with the 2p → continuum transition, the ratio of the step size reflects the ratio, which is 1:2, of the initial core-electron states available for the L 2 (2p 1∕2 ) and L3 (2p 3∕2 ) processes. On the other hand, the WL, which is associated with 2p → 5d transition, has a branching ratio ðL 3 ∕L 2 Þ > 2. We investigated how the ratio of the scattering factor f 2 (L 3 ∕L 2 ) changes with energy offset from WL. At first, we estimated a floor value of f 2 in the energy region of ΔE from −100 to −50 eV, and then subtracted the floor value from the measured f 2 . Next, the measured f 2 was smoothed by applying savgol_filter in python 3.8.2 with a window length of 5 and an order of 2 to reduce scattering of the ratio due to poor statistics. Figure 3 shows the f 2 after subtraction of the floor value and the ratio of f 2;L3 ∕f 2;L2 , where f 2;L3 and f 2;L2 are the f 2 at L 3 edge and L 2 edge, respectively. The solid line indicates the smoothed curve with savgol_filter. The energy ΔE 0 is the energy from the center energy of WL, because the uncertainty of the edge energy in our fitting results makes a discrepancy of the energy ΔE between L 2 and L 3 regions, and then the discrepancy makes pseudo peaks in the ratio near WL. A typical 1σ error of the ratio at the energy ΔE 0 > 0 is 0.14. Table 2 Best fit parameters of two peaks in L-edge structure. The branching ratio around the energy of WL is about 2.4, which is significantly >2. The branching ratio remains large up to ΔE 0 ∼ 7:5 eV. This result is consistent with a larger line width of WL in L 3 than that in L 2 . In the energy region ΔE 0 ¼ 8 to 18 eV, the ratio is 2.1-2.2, which is slightly larger than 2. This result is also in agreement with the estimation of the results of the branching ratio of p1. Above 20 eV, the branching ratio becomes to be a mean value of 2.04 with a standard deviation of 0.05, which is consistent with the ratio of the density of states between 2p 1∕2 and 2p 3∕2 .

Fine Structures at ΔE >50 eV Seen in the Scattering Factor f 2
It is found that the f 2 structure of the L 3 data at ΔE > 50 eV is similar to that of the L 2 data (see Fig. 2). The similarity can also be seen in the constant ratio of f 2;L3 ∕f 2;L2 at ΔE > 20 eV (see Fig. 4). The origin of the fine structure in EXAFS is the interference between the incoming and scattered photoelectron waves. The kinetic energy of a photoelectron wave is given by where hν is the photon energy and E edge is the edge energy. Interference patterns should depend on E K because the wavelength of a photoelectron is related to 1∕E K . Thus, the fine structure in EXAFS of L 3 is expected to be similar to that of L 2 . We supposed that the fine structure of the L 1 data is also similar to that of the L 3 data, but we did not see a clear similarity. If we shift the energy of the L 1 f 2 data by þ20 eV, we may see a weak similarity between the f 2 fine structures of L 3 and L 1 .

Measurement of the Atomic Scattering Factors
We have demonstrated that the f 2 fine structure obtained from the reflectance is similar to the XANES results obtained using XAS. This confirms the relationship between f 2 and μ and shows that f 2 can be estimated from μ. Estimating f 2 from μ may lead to reducing the time for the ground calibration of an x-ray mirror because the detailed structure of f 2 near the absorption edge is measured without a fine energy scan of the reflectance. In addition, because μ is accurately determined via XAS, the error margin in f 1 value obtained using μ is small. We demonstrate this possibility by fixing f 2 in our data. Figure 4 shows the error of f 1 when f 2 was fixed. It is found that the error in the f 1 value decreased by 0.3 to 0.8 times. As the critical angle for iridium at 12 keV is ∼0:385 deg, the x-ray reflectivity at an incident angle of 0.4 deg at an energy of 12 keVor less is not sufficiently low (Fig. 1). Thus, the uncertainty in the f 1 value is large when f 2 is set to a free parameter. Using μ may be worthwhile when using data wherein the incident angles are close to or below the critical angle.

XAFS Measurement
Our result suggests that the 10-nm-thick iridium formed by the DC magnetron sputtering device has almost the same optical property as that of iridium powder. This result indicates that a study of XAFS is possible with a thin layer instead of powder.
The reflectance measurements will be useful for fine structure research on thin atomic layer samples that are difficult to do with XAS technique. For thin films, the absorption coefficient is estimated by measuring the intensity of a fluorescence line (fluorescence XAFS measurements 22,23 ). However, the self-absorption effect causes distortion in the measured spectra in fluorescence line, and a correction for this distortion is needed. 22 In addition, a sufficient intensity of fluorescence line is required to obtain μ with a high signal-to-noise ratio. Our reflectance measurement is an alternative method to estimate the absorption coefficient. The similarity between f 2 and μ suggests that the effects observed in the fluorescence XAFS measurements are small.
We have provided the fine structures of f 2 in the energy region ΔE < 50 eV from iridium L-edges (L 3 ; L 2 , and L 1 ). The properties of the two peaks are similar to those of 4d material obtained by Sham 1985. We have also calculated the ratio f 2;L3 ∕f 2;L2 in the energy range ΔE 0 ¼ 0 ∼ 150 eV. The ratio is about 2.4 in the energy range ΔE 0 < 7:5 eV, and as the photon energy increases, the ratio decreases; at the energy ΔE 0 > 20 eV, the ratio comes to the mean value of ∼2, which is consistent with the ratio of the density of states between 2p 1∕2 and 2p 3∕2 .
The origin of the fine structure in EXAFS is the interference between the incoming and scattered photoelectron waves. Thus, each fine structure in the EXAFS region can be compared because we obtained the f 2 fine structures of the L-edges (L 3 ; L 2 , and L 1 ). The f 2 fine structure of L 2 is, as expected, similar to that of L 3 , but the f 2 fine structure of L 1 is not similar to that of L 3 . If we shift the energy of the L 1 f 2 data by þ20 eV, we may see a weak similarity between the f 2 fine structures of L 3 and L 1 . This shift may be because of the difference between the L 1 transition (→ 6p) and the L 2 ∕L 3 transition (→ 5d). Detailed analysis using high quality data is required to investigate the fine structures of iridium L-edges.

Conclusion
We measured the reflectance of an SPO plate coated with an iridium layer having a thickness of 10 nm at Spring-8 in the energy range covering all iridium L-edges.
We have shown that the shape of the fine structures of f 2 near L 2 and L 3 edges are remarkably similar to those in previous works on XAFS. The similarity suggests that we could obtain f 2 for a reflector substrate with an Ir film with a thickness of 10 nm by measuring the reflectance. In measuring the fine structure of the metal on the reflective surface at the fine energy pitch, the method used in this study leads to a reduction in the time of the ground calibration. Moreover, as the value of μ is well determined via XAS, the error in f 1 is reduced using f 2 estimated based on μ.
By comparing f 2 of L 3 ; L 2 , and L 1 , the ratio of f 2 values corresponding to L 3 and L 2 was found close to 2 and exceeded 2 near WL. Regarding the structure of XAFS, L 3 and L 2 are similar, and the energy apparently shifts between L 3 and L 1 . This is probably because the final state of the transition is 5d for L 3 and L 2 , whereas it is 6p for L 1 . Further detailed analyses of the fine structures using an analysis tool such as Demeter 0.9.26 are therefore required. 24