High Resolution Photoexcitation Measurements Exacerbate the Long-standing Fe XVII Oscillator Strength Problem

For more than 40 years

For more than 40 years, most astrophysical observations and laboratory studies of two key soft X-ray diagnostic 2p − 3d transitions, 3C and 3D, in Fe XVII ions found oscillator strength ratios f (3C)/ f (3D) disagreeing with theory, but uncertainties had precluded definitive statements on this much studied conundrum.Here, we resonantly excite these lines using synchrotron radiation at PETRA III, and reach, at a millionfold lower photon intensities, a 10 times higher spectral resolution, and 3 times smaller uncertainty than earlier work.Our final result of f (3C)/ f (3D) = 3.09 (8) (6) supports many of the earlier clean astrophysical and laboratory observations, while departing by five sigmas from our own newest large-scale ab initio calculations, and excluding all proposed explanations, including those invoking nonlinear effects and population transfers.
A direct probe of these lines using an EBIT at the Linac Coherent Light Source (LCLS) X-ray free-electron laser (XFEL) found again their oscillator strength ratio f (3C)/ f (3D) to be lower than predicted, but close to astrophysical observations [25].This highlighted difficulties with oscillator strengths calculations in many-electron systems [24][25][26][27][28][29][30][31].FIG. 1. (Top) Experimental setup: an electron beam (orange) aimed at the trap center produces Fe XVII ions, which are then resonantly excited by a monochromatic photon beam (red).Subsequent X-ray fluorescence is registered by a silicon drift detector.(Bottom) Fluorescence yield and photon energy vs. time with three different methods: (1) line scans with open photon shutter; (2,3) by closing it at each step (purple areas), we subtract the electron-beam-induced background at (2) each line center and (3) scanning over a one-full-width-at-half-maximum range.
In this Letter, we report on new measurements of resonantly excited Fe XVI and Fe XVII with a synchrotron source at tenfold improved spectral resolution and millionfold lower peak photon flux than in [25], suppressing nonlinear dynamical effects [32,34,38] and undesired ion population transfers [35].We also carry out improved large-scale calculations using three different advanced approaches [39][40][41], all showing a five-sigma departure from our experimental results.
We used the compact PolarX-EBIT [42], in which a monoenergetic electron beam emitted by an off-axis cathode (see Fig. 1) is compressed by a magnetic field.At the trap center, it collides with a beam of iron-pentacarbonyl molecules, dissociating them, and producing highly charged Fe ions with a relative abundance of Fe XVI to Fe XVII close to unity.These ions stay radially confined by the negative space charge of the ∼2-mA, 1610-eV (≈ 3 times the Fe XVI ionization potential) electron beam, and axially by potentials applied to surrounding electrodes.
Monochromatic, circularly polarized photons from the P04 beamline [43] at the PETRA III synchrotron photon source enter through the electron gun, irradiate the trapped ions, and exit through the collector aperture.They can resonantly excite X-ray transitions on top of the strong electron-induced background due to ionization, recombination, and excitation processes.A side-on-mounted energy-resolving windowless silicon drift photon detector (SDD) equipped with a 500-nm thin aluminum filter registers these emissions.
By scanning the P04 monochromator between 810 and 830 eV, we excite the Fe XVII lines 3C and 3D, as well as the Fe XVI lines B ( (2p 5 ) 1/2 (3s3d 3/2 ) 1 J=1/2 → 2p 6 3s J=1/2 ) and C.They are also nonresonantly excited by electron-impact as the electron beam energy is well above threshold [21,23].This leads to a strong, nearly constant X-ray background at the same energies as the photoexcited transitions but independent of the exciting photon-beam energy.In our earlier work [25], we rejected this background by detecting the fluorescence in time coincidence with the sub-picosecond-long LCLS pulses of ≈ 10 11 photons each at 120/s.Due to much longer and weaker 50-ps-long pulses (≈ 10 3 photons, 6 × 10 6 /s repetition rate) at P04 and the limited time resolution of SDD, we could not use the coincidence method, and reached a signalto-background ratio of only ∼5%.To improve this, we used a shutter to cyclically turn on and off the P04 photon beam.
Using a 50-µm slit width, we reached a resolving power of E/∆E ≈ 10000, ten to fifteen times higher than that of Chandra and XMM-Newton grating spectrometers [44,45], and tenfold that of our previous experiment [25] (See Supplemental Material (SM)).We find a separation of 3C from 3D of ∆E 3C−3D = 13.398(1)eV, and resolve for the first time the Fe XVII 3D line from the Fe XVI C one at ∆E 3D−C = 154.3(1.3) meV.This gives us the 3C/3D intensity ratio without having to infer a contribution of Fe XVI line C (in [25] still unresolved) from the intensity of the well-resolved Fe XVI A line.Thereby, we largely reduce systematic uncertainties and exclude the resonance-induced population transfer mechanism [35] that may have affected the LCLS result [25].
We apply three different methods (Fig. 1) to systematically measure the 3C/3D oscillator strength ratio.In method 1, we did not operate the photon shutter; instead, we repeatedly scanned the lines C and 3D (812.0 -812.5 eV), as well as 3C (825.5 -826.0 eV), in both cases using scans of 100 steps with 20-s exposure each (see Fig. 2).The SDD fluorescence signal was integrated over a 50-eV wide photon-energy region of interest (ROI) comprising 3C, 3D, and C, and recorded while scanning the incident photon energy.By fitting Gaussians to the scan result, we obtain line positions, widths, and yields, modeling the electron-impact background as a smooth linear function [23].The ratio of 3C and 3D areas is then proportional to the oscillator strength ratio [33].However, given the low 5% signal-to-background ratio and long measurement times, changes in the background cause systematic uncertainties.In method 2, we fixed the monochromator energy to the respective centroids of C, 3D, and 3C found with method 1, and cyclically opened and closed the shutter for equal periods of 20 s to determine the background.The backgroundcorrected fluorescence yields at the line peaks were multiplied with the respective linewidths from method 1, to obtain the 3C/3D ratio.Still, slow monochromator shifts from the selected positions could affect the results.To address this, in method 3, we scanned across the FWHM of C, 3D, and 3C in 33 steps with on-off exposures of 20 s, which reduced the effect of possible monochromator shifts.After background subtraction, we fit Gaussians to the lines of interest fixing their widths to values from method 1.
All three methods share systematic uncertainties caused by energy-dependent filter transmission and detector efficiency (∼1%) and by the incident photon beam flux variation (∼2%).Additionally, for method 1, we estimate systematic uncertainties from background (∼1.2%) and ROI selection (∼2.7%).In method 2, possible monochromator shifts from (set) line centroids and widths taken from method 1 cause a systematic uncertainty of ∼3.5%.Analogously, for method 3, we estimate a ∼3% uncertainty due to the use of linewidth constraints from method 1.The weighted average of all three methods is f (3C)/ f (3D) = 3.09(8) sys (6) stat , see Fig. 3 (see SM for individual ratio and uncertainties).Note that the circular polarization of the photon beam does not affect these results, since 3C and 3D (both ∆J = 1) share the same angular emission characteristics [46][47][48][49].
Calculations using a density-matrix approach by Oreshkina et al. [32,33], and the time-dependent collisional-radiative model of Loch et al. [34], pointed to a possible nonlinear response of the excited upper state populations in [25], reducing the observed oscillator strength ratio [38], which would depend on photon pulse parameters, like intensity, duration, and spectral distribution.It has been estimated that peak intensities above 10 12 W/cm 2 would give rise to nonlinear effects [32,38].Fluctuations of the self-amplified spontaneous emission process at LCLS can conceivably generate some pulses above that threshold.At P04, we estimate a peak intensity of ≈10 5 W/cm 2 , more than 6 orders of magnitude below that threshold (see SM).This could explain why the 3C/3D ratio in Bernitt et al. [25] is in slight disagreement with the present result.Nonetheless, our experiment validates the main conclusion of that work with reduced uncertainty.Our work implies that future experiments at ultra-brilliant light sources should take possible nonlinear effects into account.
ble excitations from the 2s 2 2p 6 , 2s 2 2p 5 3p even and 2s 2 2p 5 3s, 2s 2 2p 5 3d, 2s2p 6 3p, 2s 2 2p 5 4d, 2s 2 2p 5 5d odd configurations, correlating eight electrons.We separately calculate triple excitations and fully correlate the 1s 2 shell, and also included dominant quadruple ones, finding them negligible.The line strengths S and 3C/3D oscillator strength ratio after several computation stages are summarized in the SM to illustrate the small effect of all corrections.Theoretical uncertainties are estimated based on the variance of results from the smallest to largest runs, size of the various effects, and small variances in the basis set construction.We verified that the energies of all 18 states considered, counted from the ground state, agree with the National Institute for Standards and Technology [50] database well within the experimental uncertainty of 0.05%.The theoretical 3C-3D energy difference of 13.44 eV is in agreement with the experiment to 0.3%.
We also carried out entirely independent large-scale calculations using the multireference multiconfiguration Dirac-Hartree-Fock (MR-MCDHF) approach [40] with up to 1.2 million configurations.First, the 2s 2 2p 5 3s, 2s 2 2p 5 3d and 2s2p 6 3p J = 1 levels were used as reference states to generate the list of configuration state functions with single and double exchanges from all occupied orbitals up to 12spd f ghi.Virtual orbitals were added in a layer-by-layer manner.Subsequently, the role of triple excitations was studied by the CI method.In a second step, the multireference list was extended to include all J = 1 odd parity states, generated from the Ne-like ground state by single and double electron exchanges.Monitoring the convergence of the results for the addition of layers of virtual orbits, we arrive at an oscillator strength ratio of 3.55 (5), and to a 3C-3D energy splitting of 13.44 (5) eV.Another full-scale CI calculation with more than a million configurations was carried out in the particle-hole formalism using AMBiT [41], agreeing well with the other theoretical results.Full details of all calculations can be found in SM.
We emphasize that there are no other known quantum mechanical effects or numerical uncertainties to consider within the CI and MCDHF approaches.With modern computational facilities and MPI codes, we have shown that all other contributions are negligible at the level of the quoted theoretical uncertainties.The significant improvements in experimental and theoretical precision reported here have only further deepened this long-standing problem.This work on the possibly so far most intensively studied many-electron ion in experiment and theory, finally demonstrates convergence of the dedicated atomic calculations on all possible parameters, excluding an incomplete inclusion of the correlation effects as potential explanation of this puzzle.
Our result is the presently most accurate on the 3C/3D oscillator strength ratio.Its excellent resolution suggests promising direct determinations of the natural linewidth.They depend on the Einstein A coefficients, hence, on the oscillator strengths [62].Thus, future accurate measurements of individual natural linewidths of 3C and 3D not only would test theory more stringently than their oscillator strength ratio does, but also deliver accurate oscillator strengths.
Moreover, 3C and 3D with their, among many transitions, strong absorption and emission rates can also dominate the Planck and Rosseland mean opacity of hot plasmas [63][64][65].Therefore, an accurate determination of their oscillator strengths may help elucidating the iron opacity issue [12,14,66], if, e. g., Rosseland mean opacity models [67,68] were found to use predicted oscillator strengths also in departure from experiments.Our result exposes in simplest dipole-allowed transitions of Fe XVII a far greater issue, namely, the persistent problems in the best approximations in use, and calls for renewed efforts in further developing the theory of many-electron systems.
Shortcomings of low-precision atomic theory for L-shell ions had already emerged in the analysis of high resolution Chandra and XMM-Newton data [13,19,20,24,69].Similar inconsistencies were recently found in the high resolution K-shell X-ray spectra of the Perseus cluster recorded with the Hitomi microcalorimeter [70,71].Moreover, recent opacity measurements [12,14] have highlighted serious inconsistencies in the opacity models used to describe the interiors of stars, which have to rely on calculated oscillator strengths.
All this shows that the actual accuracy and reliability of the opacity and turbulence velocity diagnostics are still uncertain, and with them, the modeling of hot astrophysical and highenergy density plasmas.The upcoming X-ray observatory missions XRISM [72] and Athena [73] will require improved and quantitatively validated modeling tools for maximizing their scientific harvest.Thus, benchmarking atomic theory in the laboratory is vital.As for the long-standing Fe XVII oscil-lator strength problem, our results may be immediately used to semiempirically correct spectral models of astrophysical observations.3C/3D oscillator strength ratio EBIT [18] EBIT [21] LCLS [25] Present experiment

C. Comparison between experimental data, theoretical predictions and astrophysical observations D. Estimation of photon peak intensity on sample at P04 PETRA III
For an estimation of the photon beam intensity on the plasma sample, we refer to the technical parameters of beamline P04 [S13] and generously assume a total time-averaged photon flux after monochromatization of the beam of Φ Beam = 4 × 10 12 photons/s, a focal spot size of 1 × 10 −6 cm 2 , and a photon energy of 825 eV.The 16 ns pulse separation ensures that the upper state population has sufficient time to completely relax into the ground state in between the pulses.We obtain a total energy per bunch of Combined with the typical pulse duration of 44 ps given by the official PETRA III datasheet (see [S14] and [S15]), we deduce an average intensity per pulse of For typical pulse shapes, the peak intensity can be expected to be well below 1 × 10 6 W cm −2 .

II. CALCULATIONS OF 3C AND 3D OSCILLATOR STRENGTHS A. Very large-scale CI calculations
We start from the solution of the Dirac-Hartree-Fock equations in the central field approximation to construct the one-particle orbitals.The calculations are carried out using a configuration interaction (CI) method, correlating all 10 electrons.The Breit interaction is included in all calculations.The QED effects are included following Ref.[S16].The basis sets of increasing sizes are used to check for convergence of the values.The basis set is designated by the highest principal quantum number for each partial wave included.For example, [5spd f 6g] means that all orbitals up to n = 5 are included for the spd f partial waves and n = 5, 6 orbitals are included for the g partial waves.We find that the inclusion of the 6, 7h orbitals does not modify the results of the calculations and omit higher partial waves.The CI many-electron wave function is obtained as a linear combination of all distinct states of a given angular momentum J and parity [S17]: The energies and wave functions are determined from the time-independent multiparticle Schrödinger equation We start with all possible single and double excitations to any orbital up to 5spd f 6g from the 2s 2 2p 6 , 2s 2 2p 5 3p even and 2s 2 2p 5 3s, 2s 2 2p 5 3d, 2s2p 6 3p, 2s 2 2p 5 4d, 2s 2 2p 5 5d odd configurations, correlating 8 electrons.We verified that inclusion of the 2s2p 6 3s, 2s 2 2p 5 4 f , 2s 2 2p 5 5 f even and 2s2p 6 4p, 2s 2 2p 5 4s, and 2s 2 2p 5 5s odd configurations as basic configurations have negligible effect on either energies of relevant matrix elements.
The only unusually significant change in the ratio, by 0.07, is due to the inclusion of the 2s 2 2p 3 3d 3 and 2p 5 3d 3 configurations.These are obtained as double excitations from the 2s 2 2p 5 3d odd configuration, prompting the inclusion of the 2s 2 2p 5 4d, 2s 2 2p 5 5d to the list of the basic configurations.
Contributions to the energies of Fe 16+ calculated with different size basis sets and a number of configurations are listed in Table S3.The results are compared with experimental data from the NIST database [S18] and from a revised analysis of the experimental data [S19].We use LS coupling and NIST data term designations for comparison purposes, but note that j j coupling would be more appropriate for this ion.Contributions to the E1 reduced matrix elements D(3D) = D(2p 6 1 S 0 − TABLE S3.Contributions to the energies of Fe 16+ calculated with increased size basis sets and a number of configurations.The results are compared with experiment.All energies are given in cm −1 with exception of the last line that shows the difference of the 3C and 3D energies in eV.The basis set is designated by the highest quantum number for each partial wave included.For example, 12spd f g means that all orbitals up to n = 12 are included for spd f g partial waves.Contributions from triple excitations, excitations from the 1s 2 shells, and QED contributions are given separately.S4.Contributions to the E1 reduced matrix elements D(3D) = D(2p 6 1 S0 − 2p 5 3d 3 D1) and D(3C) = D(2p 6 1 S0 − 2p 5 3d 1 P1) (in a.u.) and the ratio of the respective oscillator strengths R. See caption of Table S3 for designations.L and V rows compared results obtained in length and velocity gauges for the [12spd f g] basis.All other results are calculated using the length gauge.Transition rates are listed in the last row in s −1 .2p 5 3d 3 D 1 ) and D(3C) = D(2p 6 1 S 0 − 2p 5 3d 1 P 1 ) and the ratio of the respective oscillator strengths

D(3C) D(3D)
are listed in Table S4.The energy ratio is 1.01655.We include a very wide range of configurations obtained by triple excitations from the basic configurations as well as excitations from the 1s 2 shell and find negligible corrections to both energies and matrix elements as illustrated by Tables S3 and S4.These contributions are listed as "Triples" and "1s 2 " in both tables.A significant increase of the basis set from [5spd f 6g] to [12spd f g] improves the agreement of energies with experiment but gives a very small, -0.009, contribution to the ratio.We find that the weights of the configurations containing 12 f g orbitals are several times higher than those containing 12spd orbitals, so we expand the basis to include more d f g orbitals.We also include 2s 2 2p 3 nd 3 and 2p 5 nd 3 configurations up to n = 14.The contributions to the energies of the orbitals with n = 13 − 17 are 3 − 5 times smaller than those with n = 6 − 12, clearly showing the convergence of the values with the increase of the basis set.The effect on the ratio is negligible.The uncertainty of the NIST database energies, 3000 cm −1 is larger than our differences with the experiment.The energies from the revised analysis of Fe 16+ spectra [S19] are estimated to be accurate to about 90 cm −1 and the scatter of the differences of different levels with the experiment is reduced.The last line of Table S3 shows the difference of the 3C and 3D energies in eV, with the final value 13.44 (5)eV.We explored several different ways to construct the basis set orbitals.While the final results with an infinitely large basis set and complete configurations set should be identical, the convergence properties of the different basis sets vary, giving about 0.04 difference in the ratio and 0.04 eV in the 3C − 3D energy difference at the 12spd f g level.Therefore, we set an uncertainty of the final value of the ratio to be 0.05.As an independent test of the quality and completeness of the current basis set, we compare the results for D(3C) and D(3D) obtained in length and velocity gauges for the [12spd f g] basis, see rows L and V in Table S4.The difference in the results is only 0.001.The final results for the line strengths S and the 3C/3D oscillator strength ratio after several stages of computations are summarised in Table S5, which clearly illustrates a very small effect of all corrections.This work was supported in part by U.S. NSF Grant No. PHY-1620687 and RFBR grants No. 17-02-00216 and No. 18-03-01220.

B. Multiconfiguration Dirac-Hartree-Fock calculations
In the multiconfiguration Dirac-Hartree-Fock (MCDHF) method, similarly to the CI approach outlined in the previous section, the many-electron state is given as an expansion in terms of a large set of j j-coupled configuration state functions [see Eq. ( S3)].In contrast to the CI calculations, in the case of MCDHF, the single-electron wave functions (orbitals) are self-consistently optimized.We use the method in one of its most recent implementations, namely, applying the GRASP2018 code package [S20].For the virtual orbitals, the optimization of the orbitals was done in a layer-by-layer approach, i.e. when adding a new layer of orbitals (in our case, orbitals in the same shell) in the configuration expansions, the lower-lying single-electron functions are kept frozen.
In a first set of calculations, we use the 2s 2 2p 6 configuration for the ground state and the 2s 2 2p 5 3s, 2s 2 2p 5 3d, and 2s the n = 2, 3 spectroscopic (occupied) orbitals were taken into account up to virtual orbitals n max spd f g, where the maximal principal quantum number n max is varied in the computations to study the convergence of the results.Such an approach is helpful in estimating the final theoretical uncertainty.Test calculations also using virtual orbitals with h and i symmetry have shown that these high angular momenta do not play a noticeable role.The ground and excited states were treated separately, i.e. two independently optimized sets of orbitals were used.After these multireference MCDHF calculations, the possible effects of further higher-order electron exchanges were included in a subsequent step, when a CI calculation was performed with the extended configuration lists (triple excitations from the multireference states up to n = 4 orbitals, yielding approx.800 thousand configurations for n max = 13), employing the radial wave functions obtained from the previous MCDHF calculations.Furthermore, the effects of the frequency-independent Breit relativistic electron interaction operator, the normal and specific mass shift, and approximate radiative corrections are accounted for (see [S20] and references therein).The QED effects have been included in the calculation of transition energies (which also enter the oscillator strengths), however, not in the electric dipole matrix elements, as such corrections are anticipated to be on the 1% level and thus can be neglected.The oscillator strengths were evaluated with the biorthogonal basis sets, each optimized separately for the ground-and excited states, to include orbital relaxation effects.The results of these calculations are presented in the first column of Fig. S4.The bottom panel shows that the oscillator strength ratio is converged from n max = 9 on.In a subsequent set of calculations, the multireference set describing the ground and excited levels were expanded to include all J = 0 even and J = 1 odd states with 1 or 2 electrons in the M shell.The maximal principal quantum number of the virtual orbitals was set to 9 to limit the computational expense of calculations.Results are shown in the 2nd column of Fig. S4.In a third setting, calculations were performed with the smaller list as described in the previous paragraph, however, with all spectroscopic orbitals (those with n = 1, 2, 3) included in the active set of orbitals when generating the configuration list.With the triple electron exchanges also included, this procedure yielded approx.1.2 million configurations in the description of the excited states.The energies and strengths are shown in the last column of the figure.The converged 3C/3D oscillator strength ratios agree well for all 3 calculations.Comparing the different results, the final value for the ratio is 3.55 (5), which agrees well with earlier large-scale MCDHF results [S1, S21], and also with the results of the other theoretical methods described in this Supplement.For the difference of the energies of the 3C and 3D lines -which can be more accurately determined in the experiment than the absolute X-ray transition energies -we obtain 13.44(5) eV.

C. AMBiT: particle-hole CI method calculations
A separate CI calculation of the 3C and 3D lines in Fe 16+ has been performed with the AMBiT code [S22].Our calculation begins with a Dirac-Hartree-Fock calculation of Ne-like Fe to construct the core 1s, 2s, and 2p orbitals in the V N potential.The Breit interaction is included throughout the calculation.We diagonalize a set of B-splines in the Dirac-Fock potential to obtain valence orbitals.Configuration interaction is performed using the particle-hole CI method [S23], however, this can be mapped exactly to the electron-only approach described in Section II A.
Our basic calculation is presented on the first line of Table S6.The CI space consists of all possible single and double excitations up to 10spd f from the same set of leading configurations presented previously: 2s 2 2p 6 , 2p −1 3p, 2p −1 3s, 2p −1 3d, 2p −1 4d, 2p −1 5d, and 2s −1 3p.At this stage, we do not include excitations from the frozen 1s 2 core.Even for this calculation, the matrix size for the odd-parity J = 1 levels is N = 479 075.To reduce the number of stored matrix elements we use emu CI [S24], where interactions between high-lying configuration state functions are ignored.We limit the smaller side of the matrix to only including double excitations up to 5spd f and limit the number of 2s and 2p holes to single removals from an expanded set of leading configurations which include, in addition to those listed above, 2p −2 3d 2 , 2p −1 2s −1 3d 2 , and 2s −2 3d 2 .This results in a reduced small side N small = 80497.We have checked that expanding the configuration state functions included in N small makes no difference to our results at the displayed accuracy.
All of our calculations include the Breit interaction at all stages, and the dipole matrix elements are calculated in the relativistic formulation with ω = 30 a.u.In the second and third lines of Table S6, we show the effects of removing the Breit interaction and using the static dipole matrix element (ω = 0), respectively.
We then expand our calculation to include g-wave excitations, up to basis 10spd f g.The difference from 10spd f is shown in the fourth line of Table S6.We see in the AMBiT calculation very little effect from the inclusion of these waves.In the fifth line, we show the effect of allowing excitations from 1s 2 , and in the sixth line we see the effect of including the Uehling potential [S25] and self-energy [S26] using the radiative-potential method [S27].This broadly agrees with the model-operator QED presented in Table S4.
The final row of Table S6 gives the results including excitations to g-waves, excitations from 1s 2 , and QED effects.The total number of configuration state functions accounted for is over 1.25 million for the odd-parity J = 1 symmetry.Nevertheless, the CI is not quite converged with respect to including orbitals with n > 10.We estimate based on calculations for 8spd f and 12spd f that the uncertainty in level energies is conservatively of order 3000 cm −1 and for the ratio g f 3C /g f 3D is of order 0.05.These results are consistent with the CI calculation presented in Sec.II A and the MCDHF results presented in Sec.II B.

FIG. 2 .
FIG. 2. Fluorescence photon yield and energy vs excitation-photon energy for the Fe XVI C, Fe XVII 3C, and 3D transitions recorded by a silicon-drift detector.Black dots: total fluorescence within a 50-eV region of interest.Red solid lines: Fits to 3C and 3D.Red dashed line: fit to C.

TABLE S1 .
Measurements Exacerbate the Long-standing Fe XVII Oscillator Strength Problem: Supplemental Material I. EXPERIMENT AND DATA ANALYSIS Comparison between experimental values and theoretical predictions of the 3C/3D oscillator strength ratio and relative line energy positions achieved within this work.
FIG. S1.Experimental 3C/3D ratio for the three different methods.The individual measurements are shown as solid green circles.The weighted mean experimental values and associated 1-σ statistical and systematic uncertainties are indicated as red band.

TABLE S2 .
3C/3D oscillator strength ratios obtained from three diffferent measurement methods and their statistical and systematic uncertainties. .Resolving Fe XVII 3D and Fe XVI C lines FIG.S2.Spectral resolution comparison between the present work at PETRA III/P04, our previous work at LCLS/XFEL [S1], and the high resolution Chandra High Energy Transmission Grating (HETG) [S2] spectrum of Capella [ObsId: 1103].In this work, for the first time, we have resolved previously-blended Fe XVI C from Fe XVII 3D line, which are 154.3(1.3)meV apart from each other.This enables us to obtain the line intensity ratio of Fe XVII 3C and 3D without having to subtract the contribution of the Fe BXVI C line, in contrast to all other previous works.Moreover, this has largely reduced the systematic uncertainties and eliminated the need for taking resonance-induced population transfer[S3]into account, which may have affected the accuracy of our LCLS work[S1].

TABLE S5 .
Contributions to the 3C and 3D line strengths S and the 3C/3D oscillator strength ratios (energy ratio 1.01655 is used).Energies in eV, transition rates A in s −1 and natural linewidths Γ in meV are listed in the last three rows of the tables.
1 2p 6 3p J = 1 odd configurations for the excited states to generate the configuration lists.Single and double electron exchanges from Convergence of the MCDHF calculations: The X-ray transition energies of the 3C and 3D lines, their difference, the weighted oscillator strengths g f and their ratio vs. the maximal quantum number nmax used.Oscillator strengths are given both in the relativistic length (Babushkin) and velocity (Coulomb) gauges to numerically control gauge invariance.The different columns display results from different sets of calculations, as described in the text.

TABLE S6 .
Particle-hole CI calculations using AMBiT of the level energies E3C and E3D (cm −1 ), reduced matrix elements D(3C) and D(3D) (a.u.), and ratio of oscillator strengths of the 3C and 3D transitions in Fe 16+ .