# New Measurement Resolves Key Astrophysical Fe xvii Oscillator Strength
Problem

Steffen Kühn<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Heidelberg Graduate School of
Fundamental Physics, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer
Feld 226, 69120 Heidelberg, Germany  Charles Cheung  Department of Physics and
Astronomy, University of Delaware, Newark, Delaware 19716, USA  Natalia S.
Oreshkina  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany  René Steinbrügge  Deutsches Elektronen-Sychrotron DESY,
Notkestraße 85, 22607 Hamburg, Germany  Moto Togawa  Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Sonja Bernitt  Max-
Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität Jena,
Max-Wien-Platz 1, 07743 Jena, Germany  Helmholtz-Institut Jena, Fröbelstieg 3,
07743 Jena, Germany  GSI Helmholtzzentrum für Schwerionenforschung,
Planckstraße 1, 64291 Darmstadt, Germany  Lukas Berger  Max-Planck-Institut
für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Jens Buck
Institut für Experimentelle und Angewandte Physik (IEAP), Christian-Albrechts-
Universität zu Kiel, Leibnizstr. 11-19, 24098 Kiel, Germany  Moritz Hoesch
Deutsches Elektronen-Sychrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
Jörn Seltmann  Deutsches Elektronen-Sychrotron DESY, Notkestraße 85, 22607
Hamburg, Germany  Florian Trinter  Institut für Kernphysik, Goethe-Universität
Frankfurt am Main, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany
Molecular Physics, Fritz-Haber-Institut der Max-Planck-Gesellschaft,
Faradayweg 4-6, 14195 Berlin, Germany  Christoph H. Keitel  Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Mikhail
G. Kozlov  St. Petersburg Electrotechnical University “LETI”, Prof. Popov Str.
5, St. Petersburg, 197376, Russia  Sergey G. Porsev  Department of Physics and
Astronomy, University of Delaware, Newark, Delaware 19716, USA  Ming Feng Gu
Space Science Laboratory, University of California, Berkeley, CA 94720, USA
F. Scott Porter  NASA/Goddard Space Flight Center, 8800 Greenbelt Rd,
Greenbelt, MD 20771, USA  Thomas Pfeifer  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Maurice A. Leutenegger
NASA/Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA
Zoltán Harman  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany  Marianna S. Safronova  Department of Physics and
Astronomy, University of Delaware, Newark, Delaware 19716, USA  José R. Crespo
López-Urrutia  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany  Chintan Shah<EMAIL_ADDRESS>NASA/Goddard
Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA  Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

(Received 22 January 2022; revised 27 July 2022; accepted 23 August 2022;
published 5 December 2022)

###### Abstract

One of the most enduring and intensively studied problems of X-ray astronomy
is the disagreement of state-of-the art theory and observations for the
intensity ratio of two Fe xvii transitions of crucial value for plasma
diagnostics, dubbed 3C and 3D. We unravel this conundrum at the PETRA III
synchrotron facility by increasing the resolving power 2.5 times and the
signal-to-noise ratio thousandfold compared with our previous work. The
Lorentzian wings had hitherto been indistinguishable from the background and
were thus not modeled, resulting in a biased line-strength estimation. The
present experimental oscillator-strength ratio
$R_{\mathrm{exp}}=f_{\mathrm{3C}}/f_{\mathrm{3D}}=3.51(2)_{\mathrm{stat}}(7)_{\mathrm{sys}}$
agrees with our state-of-the-art calculation of $R_{\mathrm{th}}=3.55(2)$, as
well as with some previous theoretical predictions. To further rule out any
uncertainties associated with the measured ratio, we also determined the
individual natural linewidths and oscillator strengths of 3C and 3D
transitions, which also agree well with the theory. This finally resolves the
decades-old mystery of Fe xvii oscillator strengths.

Soft X-ray spectra from present space observatories such as Chandra Weisskopf
_et al._ (2000) and XMM-Newton Jansen  _et al._ (2001); Den Herder  _et al._
(2001) offer deep insight into massively energetic astrophysical sources, and
together with next-generation future missions such as XRISM Tashiro  _et al._
(2018) and Athena Barret  _et al._ (2016); Pajot  _et al._ (2018) will
continue improving our understanding of phenomena driving the growth of
galaxies, star formation, and the reionization phase of the universe. Since
ubiquitous iron ions are among the strongest emitters in most X-ray sources,
decades of efforts have been aimed at understanding their spectra, and
delivering valuable diagnostics of electron temperature, density, chemical
abundances, and opacities of various astrophysical objects Parkinson (1973);
Smith  _et al._ (1985); Schmelz  _et al._ (1992); Waljeski  _et al._ (1994);
Phillips  _et al._ (1996); Behar  _et al._ (2001); Gu (2003); Paerels and Kahn
(2003); Beiersdorfer  _et al._ (2018); Gu, L.  _et al._ (2019); Gu  _et al._
(2020); Grell  _et al._ (2021). One of the key diagnostics uses the ratio of
two strong emission lines from the same ion which have different oscillator
strengths to probe for resonance scattering in an otherwise optically thin
plasma. This provides constraints on gas turbulence and column density in
stellar coronae, as well as in diffuse hot gas in galaxy clusters and groups,
and their lower mass analogs, giant elliptical galaxies Xu  _et al._ (2002);
Werner  _et al._ (2009); de Plaa  _et al._ (2012); Hitomi Collaboration  _et
al._ (2016); Ogorzalek  _et al._ (2017).

The L-shell emission of Ne-like Fe xvii (Fe16+), namely the resonance line 3C
($[2p^{6}]_{J=0}\rightarrow[(2p^{5})_{1/2}\,3d_{3/2}]_{J=1}$) and the
intercombination line 3D
($[2p^{6}]_{J=0}\rightarrow[(2p^{5})_{3/2}\,3d_{5/2}]_{J=1}$), are often the
best candidates for such diagnostics in plasmas with temperatures of 1–10 MK,
given their brightness. However, their intensity ratio has long been the
subject of controversy Bernitt  _et al._ (2012). Early solar observations gave
a ratio of 3C/3D below 2.7, whereas atomic theory and astrophysical models
predicted a ratio close to 4.0 Blake  _et al._ (1965a); Parkinson (1973);
McKenzie  _et al._ (1980); Smith  _et al._ (1985); Loulergue and Nussbaumer
(1975); Waljeski  _et al._ (1994); Bhatia and Doschek (1992); Cornille  _et
al._ (1994); Saba  _et al._ (1999); Brown  _et al._ (1998). This difference
was initially attributed to resonance scattering in the Sun Schmelz  _et al._
(1992); Phillips  _et al._ (1996). However, systematic measurements using an
electron beam ion trap (EBIT) Levine  _et al._ (1988); Brown  _et al._ (2001a)
showed significant blending of inner-shell Fe XVI line C
($[2p^{6}3s]_{J={1/2}}\rightarrow[(2p^{5})_{1/2}(3s3d)_{5/2}]_{J={3/2}}$) with
line 3D, which could potentially reduce the apparent 3C/3D ratio and, if not
accounted for, lead to an overestimate of the strength of resonance scattering
Brickhouse and Schmelz (2005). Theoretical predictions also improved over time
with the gradual inclusion of more configurations and relativistic effects,
converging close to a ratio of $\approx 3.5$ Zhang and Sampson (1989);
Cornille  _et al._ (1994); Kaastra and Paerels (2011); Dong  _et al._ (2003);
Loch  _et al._ (2005); Chen (2007); Gu (2009); Jönsson  _et al._ (2014); Wu
_et al._ (2021), but still exceeding measurements by $\sim$10–15% Brown  _et
al._ (1998); Laming  _et al._ (2000); Beiersdorfer  _et al._ (2001, 2002,
2004); Brown  _et al._ (2006); Gillaspy  _et al._ (2011); Shah  _et al._
(2019). This has compromised the diagnostic utility of Fe xvii lines, and
repeatedly raised questions regarding the accuracy of calculations and models.

Figure 1:  Fluorescence yield of the soft X-ray transitions 3C and 3D of Fe
xvii as well as B and C of Fe XVI as a function of the exciting-photon energy.
Fitted Voigt profiles (orange) show residuals (bottom panel) due to a
nonperfectly Gaussian distribution of the monochromator spectral bandwidth.
Inset (right): 3C comparison between present and previous Kühn  _et al._
(2020) works. Inset (left): resolved C and 3D from this work compared to
unresolved in LCLS measurements (green Bernitt  _et al._ (2012)) and Capella
observations (red Huenemoerder  _et al._ (2011)).

The line strengths of 3C and 3D in a collisionally excited plasma are expected
to scale with their oscillator strengths, with only small contributions from
resonant excitation. Further complications in astrophysical plasmas and
experiments can arise from neglect of dielectronic recombination, charge
exchange, inner-shell ionization, collisional quenching, polarization, and
non-Maxwellian electron-energy distribution. Thus, an experiment using only
photons to populate the upper levels of 3C and 3D has the advantage of testing
and validating only atomic structure and avoiding more complex collision
physics or other plasma processes. A resonant photoexcitation experiment at
the free-electron laser Linac Coherent Light Source (LCLS) measured an
oscillator-strength ratio $R=f_{\mathrm{3C}}/f_{\mathrm{3D}}=2.61\pm 0.21$,
which unexpectedly departed from theory even more than earlier works measuring
collision strengths Bernitt  _et al._ (2012). Nonlinear excitation dynamics
Oreshkina  _et al._ (2014, 2016); Loch  _et al._ (2015); Li  _et al._ (2017)
and charge-state population transfer Wu and Gao (2019) were suggested as
sources of this surprising discrepancy. Renewed experimental efforts with a
synchrotron-based technique brought tenfold resolution improvement compared
with the LCLS experiment and furthermore excluded population transfer and
nonequilibrium effects because of their much lower instantaneous photon flux.
The result $R=3.09\pm 0.1$ Kühn  _et al._ (2020) still disagreed with the
greatly improved, large-scale converged calculations from the same work, which
took into account all known quantum mechanical effects, and predicted
$R=3.55\pm 0.05$ Kühn  _et al._ (2020).

In this Letter, we present the most precise measurements of the oscillator-
strength ratio of resonantly photoexcited 3C and 3D lines in Fe xvii ions,
with an increase in both the resolving power by a factor of 2.5 and the
signal-to-noise ratio (SNR) by 3 orders of magnitude compared with our
previous work Kühn  _et al._ (2020). Both these improvements allowed us to
determine the individual natural linewidths of 3C and 3D. Since natural line
broadening is directly proportional to the oscillator strengths rad (1997),
the complexities arising from the various upper-level population mechanisms,
perhaps the most intractable difficulty in previous measurements, can now be
eliminated. Our improved large-scale atomic calculations also have reduced
uncertainty by a factor of 2.5 compared to Kühn  _et al._ (2020). Finally, the
present experiment agrees with our present and previous predictions and
resolves the decades-old Fe xvii oscillator-strength paradox.

Figure 2: Left: Synthetic Voigt profiles for C (purple) and 3D (red).
Predicted natural linewidths are convolved with a Gaussian component. At
similar line intensities, the overlapping product is only 1% of the total 3D
intensity, which suppresses feeding of the 3D lower state in Fe xvii from the
upper level of C in Fe xvi through an autoionization decay branch of $\approx
60$%. Right: Synthetic 3C Voigt profile without (blue) and with a strong
background (green) as in Kühn  _et al._ (2020) for comparison. A Gaussian
model (orange) fitted to this curve can hardly separate the Lorentzian wings
from the background and recovers only 94% of the true intensity. There, a
Voigt fit to the data from Kühn  _et al._ (2020) also does not work well at
the wings due to the low SNR.

The experiment was performed using the PolarX-EBIT Micke  _et al._ (2018) at
the PETRA III beamline P04 Viefhaus  _et al._ (2013). PolarX-EBIT generates a
nearly monoenergetic electron beam—using a novel off axis gun designed for
photonic studies of highly charged ions (HCI) Epp  _et al._ (2007); Simon  _et
al._ (2010); Rudolph  _et al._ (2013); Steinbrügge  _et al._ (2015);
Leutenegger  _et al._ (2020); Togawa  _et al._ (2020)—which is compressed to a
radius of 50 $\mu$m by a 0.87 T magnetic field. The HCIs are produced by
successive electron impact ionization, and are simultaneously trapped in the
radial direction by the negative space-charge potential of the electron beam
and in the axial direction by applied potentials on the drift tubes. The
electron impact ionization cross sections peak at energies 2 to 3 times higher
than the threshold. Thus, we set the beam energy
$E_{\mathrm{B}}\approx$1200\text{\,}\mathrm{eV}$$, which is approximately 2.5
times the production threshold of Fe xvii, 490 eV. However, this beam energy
exceeds the excitation thresholds of 3C and 3D, causing a very strong and
constant X-ray background induced by direct electron-impact excitation Brown
_et al._ (2006); Shah  _et al._ (2019). At free-electron lasers (FELs), such
high background can be suppressed by using time coincidence between the
photon-excited fluorescence signal and the pulsed photon train Epp  _et al._
(2007, 2010); Bernitt  _et al._ (2012). However, this was not possible here
because synchrotrons exhibit a much shorter photon pulse interval compared
with FELs and due to the limited time resolution of silicon drift detectors
(SDDs). Indeed, this was the main reason behind a poor SNR of only $\approx
0.05$ in Kühn  _et al._ (2020).

To overcome this difficulty, we reduced the strong electron-impact induced
background by cyclically varying $E_{\mathrm{B}}$ within tens of nanoseconds
between a breeding and probing phase, using fast switching power supplies (see
Fig. 2 in the Supplemental Material Sup  ). First, we produced a sufficient
amount of Fe xvii at breeding energy of
$E_{\mathrm{B}}=$1200\text{\,}\mathrm{eV}$$ for 200 ms. Then, we reduced
$E_{\mathrm{B}}$ to the probing energy of $250\text{\,}\mathrm{eV}$ within a
few tens of ns. The advantage of such low probing energy is that we can
eliminate the background from dielectronic recombination, resonant excitation,
and direct electron-impact excitation channels (cf. Fig. 1 of Shah  _et al._
(2019)). Moreover, the ion population remained mostly unchanged within the
switching time because the collision rates are relatively low at electron
densities below $10^{10}~{}\mathrm{cm}^{-3}$, typically found in PolarX-EBIT
Grilo  _et al._ (2021). On the other hand, the probing energy cannot further
produce Fe xvii, leading to a continuous depletion of Fe xvii due to radiative
recombination and charge exchange Grilo  _et al._ (2021), as well as due to
loss of ions from the trap Penetrante  _et al._ (1991). We observed a
confinement half-life of $\sim 50$ ms for Fe xvii. Once the Fe xvii population
is completely depleted, the cycle starts again to produce ions at the breeding
energy. By optimizing both the duty cycle and the slew rate for a maximum
resonant photoexcitation signal, we achieved a SNR $\approx 45$, approximately
a thousandfold improvement compared with our previous work Kühn  _et al._
(2020).

The monochromator spectral resolution increases linearly with decreasing
aperture size up to a limit set by the source size. Thus, with a significant
improvement in SNR, we were also able to decrease the monochromator exit-slit
width from $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (as in Kühn  _et al._
(2020)) to $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to improve the spectral
resolving power while maintaining the strong resonant photoexcitation signal
for lines 3C and 3D. This in turn allowed us to fine tune other X-ray optical
elements of the beamline, resulting in a further increase in spectral
resolution Viefhaus  _et al._ (2013); Follath (1998). All these optimizations
resulted in an excellent resolving power of $E/\Delta E=20\,000$ at 830 eV
photon energy.

Both these unprecedented advances in SNR and spectral resolution are essential
for accurately determining the line intensities and their profiles required to
extract the natural linewidth of a given transition. By scanning the
monochromator energy, we measured 3C and 3D of Fe xvii along with two other
lines of Fe xvi, named B
($[2p^{6}3s]_{J={1/2}}\rightarrow[(2p^{5})_{1/2}(3s3d)_{3/2}]_{J={1/2}}$) and
C. Fluorescence X-rays from the decay of photoexcited ions were detected using
two identical SDDs mounted side-on to the photon beam axis. As shown in Fig.
1, the statistical data quality allowed us to reliably fit each transition
with a Voigt function, which is the convolution of Lorentz and Gaussian
functions. The Gaussian part of the line shape stems from the thermal motion
of the trapped ions, so-called Doppler broadening, and from the limited
monochromator instrument resolution, while, in principle, the Lorentzian part
of the line shape should only result from the natural linewidth of an
electronic transition due to its finite intrinsic lifetime. We derived the
area under the peak by fitting the Voigt function to lines 3C, 3D, C, and B,
each scanned over 20 times. The weighted ratio of 3C flux to that of 3D, which
is proportional to the oscillator-strength ratio, is $R=3.51(2)$.

Our own recent work Kühn  _et al._ (2020) clearly resolved C from 3D. However,
fits considering only a Gaussian model underestimated the Lorentzian line
wings, which were buried in the background Laming  _et al._ (2000), and
resulted in a biased line flux determination, as shown in Fig. 2. Now, with
blends resolved, with perfectly exposed Lorentzian wings, and with
continuously monitoring of background and photon-flux variations using side-on
SDDs and a downstream calibrated photon diode, respectively, we could fit
Voigt profiles with relative intensity uncertainties below $1\%$, achieving a
statistical accuracy in $R$ that supersedes all previously reported
experimental results.

Armed with this high resolution, we found in a later campaign at beamline P04
on a different ion an additional source of uncertainty. While linearly
scanning the monochromator for fluorescence, we let the photon beam passing
through the EBIT reach an electron spectrometer with very high resolution
(asphere Rossnagel  _et al._ (2001)) for simultaneously monitoring the excess
energy of copious photoelectrons emitted from a clean gold surface. Periodic
excursions of their energy from the expected values revealed that the angular
encoder for the grating was inaccurately interpolating readings between fixed
0.1 degree marks Krempaský  _et al._ (2011). This causes an apparent cyclic
compression and stretching of the photon-energy increments by $\approx\pm
0.04$ eV with $\approx 1$ eV period (see Supplemental Material Sup ). We
modeled the effects of this distortion on the Voigt profiles, resulting in a
systematic uncertainty of $\approx$2% that now dominates the final error
budget of the 3C/3D ratio. The final experimental result is
$R=3.51(2)_{\mathrm{stat}}(7)_{\mathrm{sys}}$.

Given the observed line shape at an excellent resolution, the individual
natural linewidths of 3C and 3D can also be determined. In general, the
oscillator-strength ratio of two given electronic transitions, as in the case
of 3C and 3D, is correlated with the corresponding ratio of natural linewidths
through the following relation,

$\frac{f_{\mathrm{3C}}}{f_{\mathrm{3D}}}=\left(\frac{\Gamma_{\mathrm{3C}}}{\Gamma_{\mathrm{3D}}}\right)\Big{/}\left(\frac{E_{\mathrm{{3C}}}}{E_{\mathrm{{3D}}}}\right)^{2}.$
(1)

Hence, by combining the observed $R=3.51\pm 0.09$ with the energy quotient
$\left(E_{\mathrm{3C}}/E_{\mathrm{3D}}\right)^{2}=1.032$ determined in Brown
_et al._ (1998), a linewidth ratio $\Gamma_{\mathrm{3C}}/\Gamma_{\mathrm{3D}}$
of $\approx$3.6 is expected. In contrast, the present ratio of the inferred
Lorentzian widths of 3C and 3D lies well below 2.0. We have found the reason
behind this by measuring the Lorentzian widths of well-known He-like F viii
$K_{\alpha},K_{\beta}$, and $K_{\gamma}$ transitions, for which the theory is
deemed accurate. This revealed a pseudo-Lorentzian instrumental contribution
of $7.0\pm 0.3$ meV, which artificially increases the apparent natural
linewidth of all observed transitions. It results from X-ray diffraction from
elements of the beamline and mimics a Lorentzian profile at finite resolution
Follath (1998). This effect was found to be nearly constant within the photon-
energy range of $E_{\mathrm{3C}}-E_{\mathrm{3D}}\approx 13.4$ eV. Furthermore,
the convolution of two Lorentzians results in another Lorentzian whose width
is simply the sum of the two widths. Thus, we can accurately determine the
difference $\Delta\Gamma_{\mathrm{3C}-\mathrm{3D}}=$10.92(175)\text{\,}$$ meV.
Now by combining this difference with Eq. 1, we can extract the individual
natural linewidths of 3C and 3D as,

$\Gamma_{\mathrm{3C}}=\frac{\Delta\Gamma_{\mathrm{{3C-3D}}}}{1-{\frac{f(\mathrm{{3D}})}{f(\mathrm{{3C}})}\,\,\left(\frac{E_{\mathrm{{3D}}}}{E_{\mathrm{{3C}}}}\right)^{2}}},\mathrm{and\,\,\,}\Gamma_{\mathrm{3D}}=\frac{\Delta\Gamma_{\mathrm{{3C-3D}}}}{\frac{f(\mathrm{{3C}})}{f(\mathrm{{3D}})}\,\,\left(\frac{E_{\mathrm{{3C}}}}{E_{\mathrm{{3D}}}}\right)^{2}-1}.$
(2)

Analogously, the natural linewidths of lines B and C are determined by adding
their respective difference from 3D Lorentzian width to the inferred 3D
natural linewidth. In a second approach, we used the extracted pseudo-
Lorentzian beamline component from measured narrow transitions of F viii to
validate the values determined by Eq. 2. Both independent analysis methods
give consistent values, and they agree very well with the predictions (see
Supplemental Material Sup ). The influence of the periodic energy shifts of
the monochromator on linewidths was also modeled, giving a relative systematic
uncertainty of $5\%$ in the final error budget, as shown in Tab. 1.

Figure 3: Present experimental 3C/3D oscillator-strength ratio compared with predictions, including our own, shown in an inset. Theoretical approaches Zhang and Sampson (1989); Bhatia and Doschek (1992); Cornille  _et al._ (1994); Kaastra  _et al._ (1996); Safronova  _et al._ (2001); Dong  _et al._ (2003); Loch  _et al._ (2005); Chen (2007); Gu (2009); Jönsson  _et al._ (2014); Santana  _et al._ (2015); Oreshkina  _et al._ (2016); Mendoza and Bautista (2017); Wu and Gao (2019); Kühn  _et al._ (2020); Gu (2021); Cheung  _et al._ (2021) are marked as follows: Distorted wave (magenta hexagons), multiconfiguration Dirac-Fock (teal diamonds), R-matrix (blue triangles), many-body perturbation theory (orange crosses), and configuration interaction (black circles). Blue band: observed line ratios in astrophysical sources Behar  _et al._ (2001); Mewe  _et al._ (2001); Xu  _et al._ (2002); Blake  _et al._ (1965b); McKenzie  _et al._ (1980); Ness  _et al._ (2003), with color shades coding the distribution of values weighted by their reported accuracies. Purple band: spread of tokamak results Beiersdorfer  _et al._ (2004). Black diamonds: previous EBIT results Brown  _et al._ (2001a, 2006); Bernitt  _et al._ (2012); Kühn  _et al._ (2020). Navy blue squares: spectral plasma models and databases Kramida  _et al._ (2022); Foster  _et al._ (2012); Kaastra  _et al._ (1996); Del Zanna  _et al._ (2021). The data behind the figure are given in the Supplemental Material Sup . Table 1:  Average of two methods used to infer the natural linewidths ($\Gamma$ in meV) and oscillator strengths ($f$) compared with their predictions. Line | $\Gamma_{\mathrm{exp}}$ | $\Gamma_{\mathrm{th}}$ | $f_{\mathrm{exp}}$ | $f_{\mathrm{th}}$
---|---|---|---|---
3C | $15.07(107)\text{\,}$ | $14.74(1)\text{\,}$ | $2.32(17)\text{\,}$ | $2.271(2)\text{\,}$
3D | $3.79(52)\text{\,}$ | $4.03(2)\text{\,}$ | $0.60(8)\text{\,}$ | $0.641(3)\text{\,}$
B | $15.87(116)\text{\,}$ | 14.43 Oreshkina  _et al._ (2016) | $0.84(6)\text{\,}$ | 0.760 Oreshkina  _et al._ (2016)
C | $20.07(138)\text{\,}$ | 23.10 Oreshkina  _et al._ (2016) | $2.13(15)\text{\,}$ | 2.454 Oreshkina  _et al._ (2016)

We also performed a large-scale configuration-interaction (CI) calculation
using a new version of our highly scalable parallel CI code, where we
substantially enlarged the basis compared with our previous calculations Kühn
_et al._ (2020). Details are given in the Supplemental Material Sup . In
brief, the new basis includes all orbitals up to $24spdfg12h$ compared with
$12sp17dfg$ in Kühn  _et al._ (2020). With 1.2 million configurations and
almost 100 million Slater determinants, including triple excitations and
opening of the $1s^{2}$ shell, we have fully saturated the CI expansion. The
Breit interaction Breit (1929) and QED effects are included in these
computations following Refs. Tupitsyn  _et al._ (2016); Kühn  _et al._ (2020).
QED contributes $-0.016$ to the ratio, mostly through its effect on the dipole
matrix elements, whereas its influence on the 3C and 3D energies is
negligible. Conservatively, we consider the total QED correction as the
uncertainty of our final prediction of $R=3.55\pm 0.02$. Similarly, the
natural linewidths for 3C and 3D transitions are calculated to be
$14.74(1)\text{\,}\mathrm{meV}$ and $4.03(1)\text{\,}\mathrm{meV}$,
respectively. Theoretical $\Delta\Gamma_{\mathrm{3C}-\mathrm{3D}}$ is 10.71(2)
meV, with the QED contribution of 0.02 meV that is larger than the electronic
correlation uncertainty. Overall, the QED calculation uncertainties, resulting
from subtle differences in configuration mixing for the upper and lower states
of 3C and 3D, dominate our predictions.

Figure 3 shows that the experimental oscillator-strength ratio and natural
linewidths agree very well with our previous and present large-scale
calculations (see Tab. 1 in the Supplemental Material Sup  for details). It is
also noteworthy that certain earlier theoretical works agree with the present
experimental results, while others disagree for various reasons, including
differences in terms of number of configurations, incomplete description of
correlation and relativistic effects, as well as insufficient numerical
convergence. On the contrary, previous EBIT and tokamak measurements Brown
_et al._ (1998); Laming  _et al._ (2000); Beiersdorfer  _et al._ (2001, 2002,
2004); Brown  _et al._ (2006); Gillaspy  _et al._ (2011); Shah  _et al._
(2019), and astrophysical observations show deviations from our measurement
and from each other. At least some of the scatter in these results is
attributable to known effects such as neglected blending of satellite lines
Brown  _et al._ (2001b); Brickhouse and Schmelz (2005). Nevertheless, a
significant discrepancy remains between even the most careful measurements and
calculations of collision strength ratios, which remains to be solved in
future work Brown  _et al._ (2006); Shah  _et al._ (2019).

Our work exposes how critical a good understanding of non-Gaussian line shapes
in both theory and experiment is for accurately determining transition
amplitudes and line positions, especially when the resolving power is limited
such that line wings are indistinguishable from background Gomez  _et al._
(2021). Our experimental technique achieved hitherto-unfeasible spectral
resolution and SNR, both of which were crucial for allowing measurement of the
Lorentzian component of line shapes and thus for accurately determining the
natural linewidths of key astrophysical X-ray transitions.

Accurate oscillator strengths and natural linewidths are essential for
modeling and computation of the mean opacity of hot-dense matter Fontes  _et
al._ (2015); Pain  _et al._ (2017); Nagayama  _et al._ (2019); Gomez  _et al._
(2021). The present experimental confirmation of state-of-the-art theories
indicates that the oscillator strengths of Fe xvii cannot fully explain the
discrepancy between measured and modeled iron opacity Bailey  _et al._ (2007,
2015), and thus the radiative transport in the interior of the Sun Iglesias
and Hansen (2017). In addition, the results shown in Fig. 3 point to the
urgent need for updating oscillator strengths in spectral databases and plasma
modeling codes such as NIST Kramida  _et al._ (2022), AtomDB Smith  _et al._
(2001); Foster  _et al._ (2012), SPEX Kaastra  _et al._ (1996), and Chianti
Del Zanna  _et al._ (2021), given the upcoming launch of XRISM Tashiro  _et
al._ (2018) in early 2023, and of Athena Barret  _et al._ (2016) in the 2030s.
Scientific breakthroughs expected from these missions will only be realized
with an improved understanding of atomic theory through targeted laboratory
benchmarks Hitomi Collaboration  _et al._ (2018); Heuer  _et al._ (2021); Gu,
L.  _et al._ (2022).

###### Acknowledgements.

Financial support was provided by the Max-Planck-Gesellschaft (MPG) and
Bundesministerium für Bildung und Forschung (BMBF) through Project No.
05K13SJ2. C.S. acknowledges support by an appointment to the NASA Postdoctoral
Program at the NASA Goddard Space Flight Center, administered by Oak Ridge
Associated Universities under contract with NASA and by Max-Planck-
Gesellschaft (MPG). F.S.P. and M.A.L. acknowledge support from NASA’s
Astrophysics Program. The views and conclusions contained in this document are
those of the authors and should not be interpreted as representing the
official policies, either expressed or implied, of the National Aeronautics
and Space Administration (NASA) or the U.S.Government. The U.S.Government is
authorized to reproduce and distribute reprints for government purposes
notwithstanding any copyright notation herein. The theoretical research was
supported in part by US NSF Grant No. PHY-2012068, European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation program
(Grant Agreement No. 856415), the Russian Science Foundation under Grant No.
19-12-00157, and through the use of Caviness and DARWIN HPC systems at the
University of Delaware Dar . We acknowledge DESY (Hamburg, Germany), a member
of the Helmholtz Association HGF, for the provision of experimental
facilities. Parts of this research were carried out at PETRA III. We thank
Jens Viefhaus and Rolf Follath for valuable discussions on X-ray monochromator
resolution and performance, and the synchrotron-operation team and P04 team at
PETRA III for their skillful and reliable work. We also thank anonymous
referees whose comments and suggestions helped improve and clarify this
manuscript.

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