A novel and holistic approach for experimental x-ray fundamental parameter determination—the Ru L-shell

The present work aims to improve both the experimental and the data evaluation procedures necessary for a determination of L-subshell atomic fundamental parameters (FPs). The main motivation lies in the fact that the established approach, consisting of a sequential evaluation of recorded transmission and fluorescence spectra in the vicinity of the three L-absorption edges, provides relatively large uncertainties for the Coster–Kronig (CK) factors. Up to now, reliable uncertainty budgets were in the order of 15%–100% and more. The novel holistic evaluation procedure addresses these large uncertainties, by employing a much larger experimental dataset extending far above the L1 absorption edge and a combined evaluation approach for a more reliable determination of the CK factors. Here, using the Ru-L shells as an example, their uncertainties are reduced to under 15%. In addition, this enables also the determination of other relevant FP data, such as the L-subshell Auger yields, the L-subshell photoionization cross sections as well as the L-subshell fluorescence production cross sections with significantly lower uncertainties as compared to the established approach.


Introduction
The knowledge of x-ray atomic fundamental parameters (FPs) such as the fluorescence yields or the photoionization cross section is of great relevance for most quantitative analysis involving x-ray fluorescence (XRF). However, most of the available FP data in the literature for the different chemical elements and the different FP values are either relatively old, only interpolated from adjacent elements or theoretically calculated without an experimental verification. In addition, the uncertainties of most of these tabulated FP data are either not available or only estimated. To improve this situation, the International initiative on x-ray FPs [1] and others are working on revisiting and updating FP databases with new experiments and calculations employing state-of-the-art techniques for a number of years now. At the Physikalisch-Technische Bundesanstalt (PTB), well-characterized physically calibrated instrumentation is employed for dedicated experiments in order to update existing FP data for K-shell fluorescence yields [2][3][4][5], L-shell fluorescence yields and Coster-Kronig (CK) factors [4,[6][7][8][9][10], transition probabilities [11], line energies [12] and even subshell photoionization cross sections [4,13].
Even though this experimental approach can provide fluorescence yield values with drastically lower uncertainties as compared to the Krause estimates [14], especially for low-Z elements, it only provides limited access to CK transition probabilities, where achievable experimental uncertainties are relatively large. For the L-subshells, the derived CK uncertainties are usually in the order of 15%-100%. This is due to the fact, that they are derived from subtractions, which depend on the different subshell photoionization cross sections. Thus, an error propagation usually results in large uncertainties especially for the factor f 12 . Even though, comparison to other existing or reliably experimental determinations suggested that these derived uncertainties may be too conservative, an alternative approach resulting in smaller uncertainties is needed. In consequence, a calculation of dependent FPs such as the Auger yields ω Aug Li did not make sense for L-shells despite their simple relation to the fluorescence yield ω Fluo Li and CK factors f ij . As ω Aug Li = 1 − ω Fluo Li − f ij , an error propagation would yield impractical large results due to the large uncertainties of the CK factors.
Even worse is the consequence, that a transfer of this approach towards the M-shells, where a much higher number of CK transitions exist in between the shells, is not feasible at all. But especially for high-Z elements, the M-shell transitions and thus the corresponding FP data is very relevant as the inner shells are often not accessible with most available or commonly used excitation sources. Thus, it is a very disappointing situation that existing M-shell FP data cannot be validated or experimentally determined and should be addressed.
In this work, a new experimental FP determination scheme, which takes into account all experimental data in a combined rather than in a sequential approach, is presented. It uses a much larger set of probed photon energies for sample transmission and fluorescence emission and can be combined with modern mathematical methods to derive the FPs of interest and in order to learn about the uncertainties of the derived FPs. The FP data determined in this work can be found on Zenodo [15].

Experimental
The experiments were performed in a very similar way as for previous FP determinations scheme. Thus, both transmission and XRF emission are recorded for photon energies ranging from just below the L 3 edge to above the L 1 absorption edge. But in contrast to the previous approach, the photon energy range covered above the L 1 edge is extended much further in order to gain a more reliable access to the CK factors. In the present case, where the L-edges of Ru were targeted, transmission measurements from about 2.1 keV to 8 keV and XRF measurements from 2.84 keV up to 8 keV were performed. The tabulated absorption edge energies for Ru are 2.838 keV (L 3 ), 2.967 keV (L 2 ) and 3.224 keV (L 1 ) [16].
As a sample, a Ru coated silicon nitride membrane was used. The nominal coating thickness was 150 nm and the nominal membrane thickness 500 nm. All experiments were carried out at the four crystal monochromator beamline [17] of PTB at the BESSY II electron storage ring. This beamline provides monochromatic radiation between 1.75 keV and 10.5 keV by means of either four InSb(111) or Si(111) crystals. The contribution of higher harmonics in the spectral range applied is well below 10 −5 [17].
The experiments were carried out using an in-house developed vacuum chamber [18] equipped with calibrated photodiodes and an energy-dispersive silicon drift detector (SDD) with experimentally determined response functions and radiometrically calibrated detection efficiency [19]. The sample as well as a blank silicon nitride foil (for subtraction of the carrier contribution) was placed into the center of the chamber by means of an x-y scanning stage and the incident angle θ in between the surface of the sample and the incoming beam was set to 45 • .

Holistic evaluation approach
In general, an experimental determination of FPs for the L-subshells of Ru should provide access to all relevant measurands. These include the L-subshell fluorescence yields, the CK transition probabilities, the L-subshell Auger yields as well as the photon energy dependent FPs including the subshell fluorescence production cross sections (FPCSs), the subshell photoionization cross sections and the mass attenuation coefficients. By performing transmission and fluorescence experiments in the vicinity of the L-absorption edges with a sequential excitation of the three L-subshells all of these parameters can be determined also with the conventional approach for such FP determinations. However, the achievable uncertainties, especially for the CK factors are so large that a meaningful calculation of e.g. Auger yields is not feasible. By significantly extending the studied photon energy range and a novel combined data evaluation scheme this is addressed here.
The starting point for the novel holistic approach is similar to the conventional procedure for L-subshell fluorescence yields and CK factors determination. For the definition of the CK factors the description of Jitschin et al is used [20]. Via the selective excitation approach, where the incident photon energy is subsequently tuned across the L absorption edges, a possible transfer of vacancies from the higher binding energy shell to the lower bound ones can be turned on. As an example, the CK transition from L 2 to L 3 can only occur if E 0 is above the L 2 ionization threshold. The fluorescence production factor σ L i (E 0 ) at an incident photon energy of E 0 for each L-subshell is then written as follows: where E 0 is the incident photon energy, ω L i is the fluorescence yield of the subshell L i , τ L i (E 0 ) the photoionization cross section of the subshell L i and f i,j the CK factors (with i = 1, 2 or 3 and depending on the value of i, j = 2 or 3). So, if E L3 ⩽ E 0 ⩽ E L2 , the fluorescence production factor σ L3 reduces to with where θ in and θ out are incident and exit angles, respectively. The fluorescence photon flux Φ d i (E 0 ), the solid angle of detection Ω 4π , the sample specific attenuation correction factor M i,E0 and the recorded fluorescence photon flux Φ d i (E 0 ) need to be derived from the experimental data. The areal mass of the sample, which is the product of density ρ and thickness d of the Ru coating, does not need to be known for determination of the subshell fluorescence yields and CK factors. Here, only the so-called sample specific data or the product of τ L2 (E 0 ) or µ S (E 0 ) respectively and ρd is required and can be derived from the transmission data.
With the help of PTB's physically calibrated instrumentation for reference-free x-ray spectrometry [21], all of the relevant measurands can be accessed. The fluorescence photon flux Φ d i (E 0 ) can be derived from the recorded fluorescence spectra by means of a spectral deconvolution procedure using the detector response functions for all relevant fluorescence lines as well as relevant background contributions. Employing fixed line sets for each of the three L-shells the deconvolution can be numerically stabilized so that a reliable separation of the different fluorescence lines and thus a reliable assignment to the respective L i shell of origin can be performed [6]. An exemplary fluorescence spectrum including the deconvolution is shown in figure 1. Here, it is crucial to accurately normalize the deconvolved events for each line set to the spectrum's integration time and the SDD detector's detection efficiency for the corresponding photon energies in order to derive an accurate Φ d i (E 0 ). Any errors will directly affect the derived results. The incident photon flux Φ 0 (E 0 ) and the solid angle of detection Ω 4π are known due to the use of calibrated instrumentation [22]. The sample specific attenuation correction factor M i,E0 for the incident (E 0 ) as well as the fluorescence radiation (E i ) is calculated according to equation (5) using the sample specific attenuation coefficients µ S (E 0 )ρd and µ S (E i )ρd as derived from transmission experiments. For the transmission experiments it is crucial to choose the coating thickness correctly to stay within the 5% and 95% transmission range in order to limit possible artifacts by stray light contributions. A good knowledge on the spectral purity of the employed beamline is helpful in this manner. The sample specific photoionization cross section τ L3 (E 0 )ρd can also be derived from these transmission experiments by removing the scattering contributions according to equation (6) (employing tabulated mass attenuation coefficients mu Ru (E 0 ), coherent sigma C,Ru (E 0 ) and incoherent scattering cross sections sigma I,Ru (E 0 ) from X-raylib) and a seperation of the total photoionization cross section τ Tot (E 0 )ρd into the different subshell contributions as shown in figure 1.
This separation of τ Tot (E 0 )ρd into the three subshell contributions is realized by scaling the Ebel polynomials [23] for the lower bound shells as well as the three L i shells into the transmission dataset. These polynomials are exponential functions of the form: Figure 1. Examplary fluorescence spectrum recorded with the SDD at an incident photon energy of 3.3 keV with the deconvolved fixed line sets for the three Ru-L shells (left side). On the right side, the sample specific photoionization cross section, which is derived from the transmission data, as well as its separation into the subshell contributions is shown (see text for further details). The black dotted lines indicate the cut off positions at which the datasets where truncated to study the benefit of the dataset extension on the achievable uncertainties.
Here, the A i are variable parameters which are modified for the scaling into the experimental τ Tot (E 0 )ρd dataset.
In contrast to the former FP determination procedure, a much larger transmission and fluorescence dataset with excitation energies far above the L 1 shell is used and it is not performed in a step-by-step manner. For the holistic approach, the separation of the sample specific photoionization cross section into the subshell contributions is performed within an optimization procedure together with the determination of the CK-factors and the fluorescence yield values. The separation result is depicted in figure 1, where each L-shells contribution is marked by a different color. From this, the sample-specific τ T Li (E 0 )ρd for each incident photon energy (note the superscripted 'T' as a label for data derived from transmission experiments) are derived.
From the set of equations for the production cross sections 1-3, one can derive the following criteria for an automated parameter optimization: (1) The fluorescence yields ω Li are independent of the excitation photon energy and should thus be the same for all excitation photon energies above the respective threshold. So for data points with E 0 being below the subsequent absorption edge, the respective fluorescence yield can be directly calculated from equation (4) and averaged over all available points for this shell in order to derive ω M Li (the superscripted 'M' labels the averaged fluoresence yields). Employing the equations (1)-(3), one can calculate ω F Li (E 0 ) (superscripted 'F' as a label for data derived from fluorescence experiments) for each data point up to an incident photon energy of 8 keV from the experimentally determined fluorescence photon fluxes and the τ T Li (E 0 )ρd. (2) If the equations (1)-(3) are solved for the respective τ L i (E 0 )ρd, one can calculate the sample specific subshell ionization cross section τ F L i (E 0 )ρd (superscripted 'F' as a label for data derived from fluorescence experiments) from the derived fluorescence photon fluxes. For τ F L3 (E 0 )ρd, one ends up with a similar set of equations, depending on the excitation photon energy: for τ L2 (E 0 )ρd: and for τ L1 (E 0 )ρd: As we now have two different approaches to determine the subshell fluorescence yields ω Li as well as the sample specific τ Li (E 0 )ρd. Of course, their ratios ( Li (E0) ) should be unity for each photon energy above the respective ionization threshold if the correct CK factors and an accurate extrapolation of the transmission derived cross sections is performed.
Based on these equations, an objective function for an optimization algorithm can be defined in order to determine the CK factors and the optimal scaling functions for the Ebel-Polynomial based subshell photoionization cross section extrapolation. The objective function is calculating a total χ 2 Tot , which is the sum of the partial χ 2 PCS for the subshell photoionization cross section extrapolation (see figure 1), the three partial χ 2 Ratio-PCS for the subshell photoionization cross section ratios and the partial χ 2 Ratio-Yields for the subshell fluorescence yield ratios. For a correct determination of the uncertainties for the free parameters, it is essential that correct uncertainties for the experimental input (∆τ ρd(E 0 ) and ∆σ Li (E 0 )) data are used for the χ 2 calculations. with and and The parameter set, which the optimizer must probe for their optimal value consists of up to 18 parameters: the three CK factors, up to six factors for the L 3 -shell and L 1 -shell PCS polynomials, two for the combined polynomial of the PCS of lower bound shells as well as a factor for the ratio of the L 2 to L 3 PCS. As they have identical energetic behaviour, the L 2 cross section is only a scaled version of L 3 in order to reduce the degrees of freedom.
For parameter optimization and determination of their respective uncertainties, a Markov-Chain-Monte-Carlo (MCMC) [24] algorithm was used. This algorithm performs a calculation of the previously shown evaluations as a function of the respective model parameters and evaluates the goodness of these parameter values using equation (14). A minimization of χ 2 Tot through adjustment of the model parameters, allows a probability density function for each model parameter to be estimated. From these, the mean value and the one-sigma uncertainties can easily be derived if all relevant instrumental and experimental uncertainty contributions are taken into account.
To study the stability of the present approach, several separate parameter minimizations have been performed. Here, both the number of free parameters as well as the extent of the used experimental dataset was varied. The model parameter vector always contains, the three CK factors, the L 2 to L 3 PCS ratio and the two parameters for the combined polynomial of the PCS of lower bound shells. Varied was the number of free polynomial factors for both the L 3 and L 1 PCS: in the minimal case only the A 0 parameter was varied, whereas A 1−5 were fixed to the values from Ebel et al [23]. In the maximal case all parameters were varied. With respect to the extension of the dataset, the high energy part of the experimental data was cut off at different energies above the L 1 absorption edge (these energy cutoff positions are indicated on the right side of figure 1. The results of this evaluation procedure are shown in figure 2 for the three CK factors. The left part shows the derived CK value as a function of free model parameters (E max = 8 keV), as well as a combined value employing all of these results. For each CK factor, the respective results are slighty scattered without a clear trend with respect to the number of free parameters. For this reason, the combined result is an average of all 5 partial results. All CK results are relatively constant for N p ⩾ 12, thus the results in the right part of figure 2, which shows the derived CK values as a function of the upper dataset energy E max , were obtained with N p = 12 as this seems to be the best compromise between a minimal amount of free parameters and reliable results. When comparing the results as a function of the upper dataset energy E max , some trends a clearly visible: for f 23 and f 12 , the result increases with larger extent of the dataset. The reason for this is the growing contribution of the L 1 subshell PCS due to its weaker decay versus photon energy. In addition, more evaluation points at higher photon energies reduce the influence of fine-structure artifacts, which can influence the data evaluation for energies close to the absorption edges. More importantly, the derived uncertainties decline with larger dataset extension due to both of these effects.
In the case of the L-subshell fluorescence yields, similar behavior is observed. In contrast to the CK factors, the derived L 2 and L 1 fluorescence yields are slightly increased in the case of 14 ⩽ N p ⩽ 18 as compared to the values obtained using less modeling parameters. With respect to the upper dataset energy E max similar trends are observed and stable results are obtained for E max ⩾ 5.3 keV. As a consequence, the final yield results are the ones obtained with N p = 12 and E max = 8 keV.

Determination of mass attenuation coefficients
The mass attenuation coefficients of Ru can be derived from the transmission experiments using the Beer-Lambert law. Here, the contribution of the Si 3 N 4 backing foil must be removed using the transmission data recorded on a blank substrate. The resulting sample specific mass attenuation coefficients µ S (E 0 )ρd still contain the areal mass of Ru present on the sample. In order to calculate the Ru mass attenuation coefficients µ S (E 0 ), one needs to determine the areal mass of Ru ρd. In this work we employed a reference-free quantification employing the determined FPs and the recorded Ru-L 3 fluorescence intensity at an incident photon energy of 8 keV. Of course this quantification contributes to the overall uncertainty budget of the determined mass attenuation coefficients. Here, a contribution of ρd determination of about 5% is achieved. This leads to an overall uncertainty of the derived MACs in the order of about 5.2%. Employing more sophisticated methods, e.g. precise weighing and sample area measurement [25] the uncertainty contribution of the ρd determination and thus also the final uncertainty could be drastically reduced.
In figure 3 the obtained results are shown in comparison to selected data from the literature. As can be seen in the plots, the agreement with respect to the data taken from X-raylib [16], Ebel et al [23] and Chantler [26] is relatively good for photon energies above the Ru-L 1 absorption edge. Here, all datasets match our experimental data within its uncertainty regime. However, both the near-edge fine structure as well as the overall absorption edge heights are not adequately represented by any of the three literature datasets. Similar differences for photon energies below the L 3 absorption edge were observed already in other works on mass attenuation coefficients [25,27] and demonstrate the need for such experimental work. As it is not straightforward to take into account the near edge fine structure in tabulated datasets as it is dependent  [16,23,26]. The data from this work (t.w.) is derived from the transmission experiments and the samples areal mass was determined by reference-free quantification at 8 keV. In the bottom of the plot, the respective ratios are plotted.

Determination of subshell fluorescence production yields
Employing equation (4) and similar relations for the L 2 and L 1 shells, one can directly derive the sample specific FPCSs. By taking into account the sample's areal mass, also the absolute FPCS for Ru-L i fluorescence generation can be determined at the probed photon energies. Due to the fact that here also the reference-free quantification result is used, the same statements are valid for the uncertainty of these derived subshell FPCS.
Here, overall untertainties in the order of 5.5% where achieved.
The resulting FPCS for the Ru-L shells are shown in figure 4. For comparison, the X-raylib data without cascade effects is also plotted. Similar to the determined MACs also here the tabulated data does not contain any near-edge fine structure for the very same reasons. Apart from this, the agreement for the L 3 PCS is very good throughout the studied photon energy regime. For the L 2 -and L 1 shells, the X-raylib data is lower than our experimental values. In the case of L 2 the observed discrepancy is larger than the uncertainty of our results. Figure 5. Comparison of the derived subshell fluorescence yields for the Ru-L shells in comparison to recently published data employing the classical evaluation approach [28], selected data in common fundamental parameter compilations by Schoonjans et al [16] and Krause [14] as well as theoretically calculated factors by McGuire [29], Puri et al [30] and Xu and Rosato [31].

Determination of fluorescence yields
The Ru L-subshell fluorescence yields derived with the presented new holistic evaluation approach are shown in figure 5. As stated earlier, they were obtained using N p = 12 and E max = 8 keV. For comparison, also the results of recently published data [28], which was evaluated according to the classical approach based on subsequent determination of sample specific subshell ionization cross sections, subshell fluorescence yields and CK-factors as presented in detail in [6]. Another big difference of these classical results to the present approach is the extent of the used experimental dataset, which was limited to an E max of 3.4 keV. In addition, also the available data from common compilations and theoretical calculations are shown.
In general, the agreement with respect to our classically determined results is very good. Only for the L 3 -shell yield, minor differences were found. Here the reason is that the derived fluorescence yield using the holistic approach is less affected by the fine structure modulating the subshell photoionization cross section in the vicinity of the L 3 absorption edge as the fluorescence yields are derived from a much broader incident energy range than just the regime between the L 3 and L 2 absorption edges. The determined uncertainty budget for the L 3 fluorescence yield is very similar to the classical approach as it is dominated by the uncertainties of the experimental sample-specific τ T Li (E 0 )ρd and σ L i (E 0 )ρd. For the L 2 and L 1 yields, a slightly reduced uncertainty is obtained. Here, the reason is that they depend on a correct separation of the subshell photoionization cross sections which in turn must use the extrapolated L 3 and L 2 shell data. This is expected to be much more reliable in the present approach.
With respect to the available data from compilations and calculations the agreement is within expectancy: X-raylib and Krause usually provide quite realiable data but minor deviation are found for L 1 . But the uncertainty of the data from this work is significantly lower as compared to Krause's estimates. Similar trends are found for the different theoretical flourescence yields.

Determination of CK factors
For the CK factors, the comparison to the classical approach reveals large differences, especially with respect to the uncertainties. The derived CK factors are shown in figure 6 in comparison to results obtained using the classical approach and available data from the literature. Even though, the determined CK values are in good agreement, except for f 23 , the achieved uncertainties are significantly lower in the holistic approach. Instead of absolute uncertainty budgets in the order of 0.05 for f 23 , between 0.1 and 0.15 for f 13 and 0.1 for f 12 , the uncertainties are reduced to the 0.03 regime and lower. Even though this needs to be further validated in future work, the presented holistic approach seems more reliable. As already shown in figure 2, the achieved uncertainty gets reduced with increased size of the employed dataset. In addition, the new approach provides a more reliable determination of the subshell photoionization cross sections as their extrapolated energy dependence is validated in contrast to the classical approach. As already stated above, the growing L 1 contribution with increasing photon energy as well as the reduced influence of fine structure artifacts also contribute to the higher reliability and lower achievable uncertainty here.
With respect to the available CK factors from databases or theoretical works, there is basically no agreement within the determined uncertainty regime. Even though this is not overwhelmingly surprising, it demonstrates the need for an additional verification.

Determination of subshell-photoionization cross sections
Employing the recorded fluorescence signals from the Ru-L shells and the determined fluorescence and CK yields, one can also calculate each subshell's sample specific photoionization cross section at the given excitation photon energy [13]. If the areal mass (ρd) of the sample is known, also the absolute cross sections Figure 6. Comparison of the derived Coster-Kronig factors for the Ru-L shells in comparison to recently published data employing the classical evaluation approach [28], selected data in common fundamental parameter compilations by Schoonjans et al [16] and Krause [14] as well as theoretically calculated factors by McGuire [29] and Puri et al [30].

Figure 7.
Comparison of the derived subshell photoionization cross sections for the Ru-L shells in comparison to the most common theoretical calculations by Scofield [32] and two datasets by Trzhaskovskaya et al [33,34] as well as constant jump ratio data from Elam et al [35]. The Ru-L3 and Ru-L2 shell are shown in the top left plot, the Ru-L1 shell next to it. In the lower row, ratios between the experimentally determined cross section and the three theoretical datasets are shown. The grey shaded area depicts the uncertainty regime of the experimental data and the dashed line depicts a ratio of unity.
can be calculated. In the present work, we have derived the areal mass using reference-free quantification of the experiments performed at 8 keV incident energy. This adds a significant contribution to the achievable uncertainty budget, which may be reduced using other approaches such as precise weighing and area measurement.
The derived experimental subshell photoionization cross sections for the incident energy range observed here are shown in figure 7 in comparison to common theoretical calculations by Scofield [32] and by Trzhaskovskaya [33,34]. As pointed out in our earlier work [13,36], subshell cross sections calculated using the constant jump ratio approach lead to wrong results as they do not consider the different photon energy dependence of the L 1 -shell. This can also be observed here when comparing the jump-ratio based Elam data [35] to the experimental dataset.
With respect to the theoretical calculations, a good agreement with the three theoretical datasets is observed for the L 3 and L 2 subshells. The theoretically predicted values agree well with the data both with respect to the absolute values of the subshell cross section and with respect to the photon energy dependence in the observed photon energy regime. In the case of the L 1 subshell, all theoretical datasets underestimate the subshell's photoionization cross section but predict a correct photon energy dependence. The more recent dataset by Trzhaskovskaya and Yarzhemsky [34] shows slightly less good agreement.  [16] and Zschornack [37]. For easier comparison, the uncertainty of the derived data is marked in grey.

Determination of Auger yields
Employing the relation the Auger yield a Li as the probability for an Auger decay following an ionization of the L i shell can be derived by subtracting the determined fluorescence yield and relevant CK factors from 1. As the uncertainties of the experimental fluorescence yield and the CK-factors add up for a combined uncertainty of the Auger yield, such a calculation only provides a meaningful result if both uncertainty contributions are reasonably small. With the new holistic approach, this can be realized and the resulting Auger yields for the Ru L-shells can be derived. The resulting Auger yields for the Ru-L shells are shown in figure 8. A comparison to tabulated values from X-raylib [16] and Zschornack [37] is also presented. For a L3 , which is only dependent on the fluorescence yield the agreement is good. For the other two a Li factors, which are dependent also on the CK transitions, the agreement is less good.

Conclusions
In the present work, a novel approach for the experimental determination of x-ray FPs is demonstrated. It is based on transmission and fluorescence measurements over a wide range of incident photon energies far above the absorption shells of interest. Both of these datasets are evaluated by a unified parameter optimization rather than in a sequential manner in order to ensure a more reliable extrapolation of the subshell photoionization cross sections. In addition, the experimental uncertainties are evaluated employing MCMC based analysis. Both of these developments allow for a significant reduction of the overall uncertainties, especially for the CK factors and thus for a determination of nearly all relevant x-ray FPs of the Ru-L subshells. The FP data determined in this work can be found on Zenodo [15].
This novel approach can be applied also for the L-subshell FPs of other elements and is expected to also be applicable towards M-subshell FP determination, where the complexity is significantly larger due to the higher number of subshells and possible CK factors. In fact, in the case of M-subshells the previously performed methodology could not be succesfully applied due to the large number of relevant FPs. In addition, a further reduction of the achievable uncertainties is foreseen, especially if the areal mass of the employed samples is determined by precision weighing and area measurements.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors. The atomic fundamental parameter data determined in this work can be found at https://zenodo.org/record/7768251. programme and the Netherlands, Belgium, Germany, France, Austria, Hungary, United Kingdom, Romania and Israel.