on $n$ generators. Clearly each $\Gamma_{i}$ is the Milnor group on $n-1$
generators

######  Definition 7.2.

Let $f=f_{1}\sqcup\ldots\sqcup f_{n}:S^{2}_{1}\coprod\ldots\coprod
S^{2}_{n}\rightarrow S^{4}$ be a based link map then we define

$\sigma^{n}(f)=\left(M\left(\lambda\left(f_{1},f_{1}\right)\right),\ldots,M\left(\lambda\left(f_{i},f_{i}\right)\right),\ldots,M\left(\lambda\left(f_{n}f_{n}\right)\right)\right)\in\prod_{i=1}^{n}\mathbb{Z}\Gamma_{i}\text{.}$

######  Proposition 7.3.

Let $f$ and $g$ be link homotopic based link maps. Then

$\sigma^{n}(f)=\sigma^{n}(g)\text{.}$

###### Proof.

The proof of this result is analogous to the proof of Proposition 5.3. ∎

We now find the correct quotient to consider which removes the dependence on
basing and gives a unbased link map invariant.

Choose another set of basing paths for $f$, similarly to the three component
case, this pair of basing curves specifies elements of an $n$-tuple of
$\left(g_{1},g_{2},\ldots,g_{n}\right)\in\prod_{i=1}^{n}\Gamma_{i}$. Let
$\tilde{g}_{i}\in\Gamma_{j}$ where $i\neq j$ where $\tilde{g}_{i}$ is defined
to be the image of $g_{i}$ under the sequence of maps

$\Gamma_{i}\twoheadrightarrow\Gamma_{i}/\left<\left<x_{j}\right>\right>\hookrightarrow\Gamma_{j}\text{,}$

where the left most arrow is the quotient map and the right most map is the
natural inclusion map. The notation omits which generator we have omitted,
however it will be clear from the context. We define the following map
$\psi_{g_{i}}:\mathbb{Z}\Gamma_{i}\rightarrow\mathbb{Z}\Gamma_{i}$ where
$\psi_{g_{i}}(r)=g_{i}rg_{i}^{-1}$. Furthermore, let
$\phi^{i}_{g_{j}}:\mathbb{Z}\Gamma_{i}\rightarrow\mathbb{Z}\Gamma_{i}$ with

$\phi^{i}_{g_{j}}(x_{k})=\begin{cases}\tilde{g}_{j}^{-1}x_{k}\tilde{g}_{j}&k=i\\\
x_{k}&\text{ otherwise }\end{cases}$

extended linearly. Define
$w_{i}:\mathbb{Z}\Gamma_{i}\rightarrow\mathbb{Z}\Gamma_{i}$ where

$w_{i}:=\psi_{g_{\tau^{i-1}\left(1\right)}}\circ\phi^{i}_{g_{\tau^{i-1}\left(2\right)}}\circ\phi^{i}_{g_{\tau^{i-1}\left(3\right)}}\circ\ldots\circ\phi^{i}_{g_{\tau^{i-1}\left(n\right)}}\text{.}$

Define an action by

$\left(g_{1},\ldots
g_{n}\right)\cdot\left(r_{1},\ldots,r_{n}\right)=\left(w_{1}\left(r_{1}\right),w_{2}\left(r_{2}\right),\ldots,w_{n}\left(r_{n}\right)\right)\text{.}$

This action corresponds to the changing basing paths similarly to the action
defined in Lemma 5.11 in the three-component case. We define our $n$-component
invariant.

######  Definition 7.4.

Let $f$ be an $n$-component link map. Choose a collection of basing paths for
each sphere. Define

$\sigma_{i}\left(f\right):=M\left(\lambda\left(f_{i},f_{i}\right)\right)\text{.}$

Define

$\widetilde{\sigma}^{n}\left(f\right)=\left(\sigma_{1}\left(f\right),\ldots\sigma_{n}\left(f\right)\right)\in\prod_{i=1}^{n}\mathbb{Z}\Gamma^{i}/\sim\text{.}$

Let $1\leq i_{1}<\cdots<i_{k}\leq n$ let us define the map

$\left(i_{1},\ldots,i_{k}\right):\operatorname{LM}_{\underbrace{2,\ldots,2}_{n}}^{4}\rightarrow\operatorname{LM}_{\underbrace{2,\ldots,2}_{n-k}}^{4}$

where $\left(i_{1},\ldots,i_{k}\right)(f)$ is the link map $f$ where we forget
about each $i_{j}$ sphere. Define the
$p_{\left(i_{1},\ldots,i_{k}\right)}:\prod_{i=1}^{n}\mathbb{Z}\Gamma^{i}/\sim\rightarrow\prod_{i=1}^{n-k}\mathbb{Z}\Gamma^{i}/\sim$
to be the projection onto the factors which are not one of the $i_{j}$ and set
all $x_{i_{j}}=1$ and relabel.

######  Proposition 7.5.

The following diagram commutes

${\operatorname{LM}_{2,\ldots,2}^{4}}$${\operatorname{LM}_{2,\ldots,2}^{4}}$${\prod_{i=1}^{n}\mathbb{Z}\Gamma^{i}/\sim}$${\prod_{i=1}^{n-k}\mathbb{Z}\Gamma^{i}/\sim\text{,}}$$\scriptstyle{\widetilde{\sigma}^{n}}$$\scriptstyle{\underline{i}}$$\scriptstyle{\widetilde{\sigma}^{n-k}}$

where $\underline{i}:=\left(i_{1},\ldots,i_{k}\right)$.

The proof is omitted.

## References


# Comprehensive Laboratory Measurements Resolving the LMM Dielectronic
Recombination Satellite Lines in Ne-like Fe XVII Ions

Filipe Grilo Laboratory of Instrumentation, Biomedical Engineering and
Radiation Physics (LIBPhys-UNL), Department of Physics, NOVA School of Science
and Technology, NOVA University Lisbon, 2829-516 Caparica, Portugal;
<EMAIL_ADDRESS>Chintan Shah NASA Goddard Space Flight Center, 8800
Greenbelt Rd, Greenbelt, MD 20771, USA<EMAIL_ADDRESS>Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Steffen
Kühn 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
René Steinbrügge Deutsches Elektronen-Sychrotron DESY, Notkestraße 85, 22607
Hamburg, Germany  Keisuke Fujii Department of Mechanical Engineering and
Science, Graduate School of Engineering, Kyoto University, Kyoto 615-8540,
Japan  José Marques BioISI – Biosystems & Integrative Sciences Institute,
Faculdade de Ciências da Universidade de Lisboa, Campo Grande, C8, 1749-016,
Portugal  Laboratory of Instrumentation, Biomedical Engineering and Radiation
Physics (LIBPhys-UNL), Department of Physics, NOVA School of Science and
Technology, NOVA University Lisbon, 2829-516 Caparica, Portugal  Ming Feng Gu
Space Science Laboratory, University of California, Berkeley, CA 94720, USA
José Paulo Santos Laboratory of Instrumentation, Biomedical Engineering and
Radiation Physics (LIBPhys-UNL), Department of Physics, NOVA School of Science
and Technology, NOVA University Lisbon, 2829-516 Caparica, Portugal  José R.
Crespo López-Urrutia Max-Planck-Institut für Kernphysik, Saupfercheckweg 1,
69117 Heidelberg, Germany  Pedro Amaro Laboratory of Instrumentation,
Biomedical Engineering and Radiation Physics (LIBPhys-UNL), Department of
Physics, NOVA School of Science and Technology, NOVA University Lisbon,
2829-516 Caparica, Portugal

(Received 2020 August 8; revised 2021 March 27; accepted 2021 April 7;
published 2021 June 4 )

###### Abstract

We investigated experimentally and theoretically dielectronic recombination
(DR) populating doubly excited configurations $3l3l^{\prime}$ (LMM) in Fe
XVII, the strongest channel for soft X-ray line formation in this ubiquitous
species. We used two different electron beam ion traps and two complementary
measurement schemes for preparing the Fe XVII samples and evaluating their
purity, observing negligible contamination effects. This allowed us to
diagnose the electron density in both EBITs. We compared our experimental
resonant energies and strengths with those of previous independent work at a
storage ring as well as those of configuration interaction, multiconfiguration
Dirac-Fock calculations, and many-body perturbation theory. This last approach
showed outstanding predictive power in the comparison with the combined
independent experimental results. From these we also inferred DR rate
coefficients, unveiling discrepancies from those compiled in the OPEN-ADAS and
AtomDB databases.

atomic data — atomic processes — line: formation — methods: laboratory: atomic
— plasmas — X-rays: general

##  1 Introduction

Iron, the heaviest among the abundant chemical elements, has strong L-shell
transitions that dominate the X-ray spectra of astrophysical hot (megakelvin
temperature regime) plasmas in the range of 15$-$18 Å. Due to its closed-shell
configuration with a high ionization potential of $1260~{}\mbox{ eV}$, Fe XVII
(Ne-like Fe+16) is a very stable and abundant species under those conditions.
Collisional excitation of the $3d\rightarrow 2p$ and $3s\rightarrow 2p$
transitions in this ion generates the strongest observed lines in the X-ray
spectra (for an overview, see Brown (2008) and references therein). These,
together with less intense L-shell transitions from Fe in other charge states,
e.g., Fe XVI (Na-like Fe+15), provide means for diagnosing the physical
conditions of those plasmas (Paerels & Kahn, 2003). Therefore, over many years
numerous laboratory measurements have aimed at providing accurate values of
the wavelengths and relative intensities of L-shell transitions in Fe XV-XIX
(May et al., 2005), Fe XVI (Graf et al., 2009), Fe XVII (Laming et al., 2000;
Beiersdorfer et al., 2002, 2004; Brown et al., 2006; Gillaspy et al., 2011;
Shah et al., 2019), Fe XVIII-XXIV (Brown et al., 2002; Chen et al., 2006), Fe
XXI $-$ Fe XXIV (Chen et al., 2005), Fe XXIV (Gu et al., 1999; Chen et al.,
2002), and Fe xxi-xxiv (Gu et al., 2001). These works have revealed
significant discrepancies with theory; well-known problems include the 3C/3D
line ratio in Fe XVII (Bernitt et al., 2012; Kühn et al., 2020) and the Fe
solar opacity issue (Nagayama et al., 2019). Moreover, it is expected that
updated atomic data on Fe L-shells could resolve disparities among collision
models used for predicting the Fe abundance (Mao et al., 2019) in low-
temperature (and low-mass) elliptical galaxies (Yates et al., 2017; Mernier et
al., 2018). A recent review of astrophysical diagnostics of Fe L lines can be
found in Gu et al. (2019, 2020).

Dielectronic recombination (DR) is the dominant photorecombination channel for
Fe XVII in such plasmas. In the case of DR LMM, this means the capture of an
electron into a vacancy of the M shell with a simultaneous, energetically
resonant electron L$-$M excitation. The resulting doubly excited state can
either autoionize, resulting in resonant excitation ((Shah et al., 2019) and
(Tsuda et al., 2017) for Fe XVII and Fe XV-XVI), or radiatively decay,
completing the recombination:

$\displaystyle\text{Fe}^{16+}(1s^{2}2s^{2}2p^{6})+e^{-}$
$\displaystyle\downarrow\hskip 42.67912pt$
$\displaystyle\text{Fe}^{15+**}(1s^{2}2l^{7}3l^{\prime}3l^{\prime\prime})$ (1)
$\displaystyle\downarrow\hskip 42.67912pt$
$\displaystyle\text{Fe}^{15+*}(1s^{2}2s^{2}2p^{6}3l^{\prime\prime\prime})+h\nu~{}.$

Among the various processes exciting L$-$M emission, DR produces strong
”satellite” transitions very close to the main lines due to the perturbation
caused by the added spectator electron (Dubau & Volonte, 1980; Clementson &
Beiersdorfer, 2013). Such lines were seen with the Chandra X-Ray Observatory
in spectra from stellar coronae, like those of Capella and Procyon, and are
used for plasma temperature determination (Beiersdorfer et al., 2018; Gu et
al., 2020). DR also strongly influences plasma ionization equilibrium (Dupree,
1968). It is thus crucial to accurately know these dielectronic satellites
when diagnosing temperatures using collisional-radiative models (Savin &
Laming, 2002; Dudík et al., 2019), such as AtomDB (Foster et al., 2012) and
SPEX (Kaastra et al., 1996), or with the help of atomic databases like CHIANTI
(Dere et al., 2019) and OPEN-ADAS. 111https://open.adas.ac.uk

Figure 1: Time pattern of the electron beam energy sweeping at (a) FLASH-EBIT
(Epp et al., 2010) and (b) PolarX-EBIT (Micke et al., 2018). The dashed blue
lines indicate slow energy scans between 300 and 500 eV with a period of 40
minutes.

Except for direct observation of DR $3l5l^{\prime}$ and $3l6l^{\prime}$
satellites in Fe XXII-XXIV (Gu et al., 2001), no laboratory wavelengths or
intensities of Fe DR L-shell satellites are available, as mentioned in
Beiersdorfer et al. (2014). Only recently have DR cross sections for the
$3lnl^{\prime}$ series for Fe XVII been published, with the purpose of
investigating the $3d\rightarrow 2p$ and $3s\rightarrow 2p$ line ratios above
the collision excitation threshold (Shah et al., 2019). These data benchmark
the SPEX model and provide constraints on the global fit of Capella spectra
(Gu et al., 2020). Continuing those works, we focus on the DR $3l3l^{\prime}$
(LMM) satellites of Fe XVII, and provide experimental resonant strengths and
rate coefficients. Similar measurements were previously done for Au (Schneider
et al., 1992), Xe (DeWitt et al., 1992), and more recently for Si (Lindroth et
al., 2020).

In this work, we remeasured previously studied Fe XVII LMM region by our group
(see Shah et al. (2019)) with another electron beam ion trap (EBIT), PolarX-
EBIT (described in Micke et al. (2018)). By using a modified measurement
scheme, we obtained higher electron collision energy resolution compared to
that in previous works. We also simulated the dynamical charge-state
distribution for the present experimental conditions in order to exclude a
possible large depletion of Fe XVII ions due to DR. Furthermore, we inferred
the electron beam density for both devices, obtained experimental integrated
resonant strengths from the two different measurement schemes, and compared
them with those of earlier photorecombination studies performed at the
Heidelberg Test Storage Ring (TSR) (Schmidt et al., 2009).

In addition, our new calculations based on multiconfiguration Dirac$-$Fock
(MCDF) equations and our previous ones based on the Flexible Atomic Code (FAC)
were compared with configuration interaction predictions by Nilsen (1989).
Finally, our experimental and theoretical resonant strengths were converted to
DR rate coefficients for a few electron temperatures and compared with those
available in the OPEN-ADAS database, as well as with those in Zatsarinny et
al. (2004), which are included in AtomDB (Foster et al., 2012), a spectral
modeling code widely used in the X-ray astrophysics community.

##  2 Experiment

For accurate values of DR intensities, we relied on two complementary
measurements made on two different EBITs. Previous work with FLASH-EBIT at
Max-Planck-Institut für Kernphysik in Heidelberg is reported in detail in Shah
et al. (2019). We summarize here the method and emphasize the differences with
the new measurements performed with PolarX-EBIT. In both devices, Fe atoms
were injected into the trap and ionized by a magnetically compressed
monoenergetic electron beam with a radius of tens of micrometers. Its negative
space-charge potential confined the ions, allowing stages of high ionization
to be reached.

The DR LMM resonances appear at electron energies below the Na-like ionization
threshold into Ne-like. This requires one to first produce (breed) the Fe XVII
of interest before quickly changing the interaction energy to the values to be
probed, as described below. Since DR depletes the Ne-like population into Na-
like if the probe time is too long, we quantify this small effect in Sec. 4.
Our recorded signal, X-ray emission including the contribution from DR, was
observed at $90^{\circ}$ relative to the beam axis with a silicon drift
detector at both EBITs. Its photon energy resolution was around $120$ eV FWHM
at $6$ keV.

Figure 2: Top plot: Theoretical resonant strengths obtained from FAC.
Abscissa: electron beam energy; ordinate: emitted wavelength. The
experimentally resolved spectral lines (in electron beam energy) are labelled
as $s_{1}$, $s_{2}$, $p_{1}$, $p_{2}$, $d_{1}$, $d_{2}$, $d_{3}$ and $d_{4}$.
The first letter of the label is the $l$ of the spectator electron in the
$2p^{5}3l3d\rightarrow 2p^{6}3l$ transition. These lines correspond either to
a single strong isolated resonance, like $s_{1}$, or to multiple resonances
that are experimentally blended, such as $p_{1}$. The circle diameter shows
the relative resonant strength of the channels contributing to these spectral
lines. Projections onto both axes are displayed in the bottom left (b) and
right (c) plots and compared with our MCDF and Nilsen (1989) calculations. The
FAC and MCDF data shown in panels (b) and (c) are available as the data behind
the figure in the .tar.gz package. The package also includes FAC-MBPT data.
All three data sets can also be used to generate the synthetic spectra shown
in Figure 6.
(The data used to create this figure are available.)

###  2.1 FLASH-EBIT Measurements

The electron beam energy was at 1.15 keV for a 0.5 s breeding time, followed
by a linear ramp-down from 1.15 to 0.3 keV within 40 ms and a symmetric ramp-
up (as shown in Figure 1 (a); Shah et al. (2019)), a procedure introduced by
Knapp et al. (1989, 1993) and used in many other experiments (e.g. Yao et al.
(2010); Xiong et al. (2013); Hu et al. (2013)). For such scans, simulations of
the ion population predict a negligible DR depletion of the Fe XVII population
(see Section 4).

Every 5 s the ion inventory of the trap was dumped and regenerated to avoid
contamination by W and Ba ions, which take typically a few minutes to
accumulate. FLASH-EBIT uses superconductive coils inducing a magnetic field up
to 6 T (Epp et al., 2010) that strongly compresses the electron beam. This
beam efficiently produces ions up to the highest charge state allowed by the
ionization threshold, in this case Ne-like Fe XVII. The residual pressure at
the trap center stays below $10^{-12}$ mbar, making charge exchange (CX) with
residual gas negligible. Therefore, a high-purity sample of Fe XVII ions was
prepared (Shah et al., 2019). The beam current was adjusted according to
$n_{\mbox{e}}\propto I_{\mbox{e}}/\sqrt{E}$ to keep the electronic density
$n_{\mbox{e}}$ constant, having a value of 20 mA at the breeding energy. The
measured electron energy spread was $\sim$5 eV.

###  2.2 PolarX-EBIT Measurements

The new measurements used PolarX-EBIT, operating at PETRA III, Deutsches
Elektronen-Synchrotron (DESY), Hamburg. It uses an off-axis electron gun and a
magnetic field at the trap up to 0.86 T (Micke et al., 2018) produced by an
assembly of permanent magnets. It operates at room temperature and had at the
time of these measurements a rather poor vacuum ($10^{-8}$ mbar), which could
increase the recombination from Ne-like to Na-like ions through the CX
process. As observed in Section 4, the presence of Na-like ions is not
significant. PolarX-EBIT was also run at low beam current, in the present case
2 mA, and thus reached a lower (3.5 eV FWHM at 400 eV) electron energy spread
than FLASH-EBIT. The measurement scheme is different (see Figure 1 (b)): here,
the electron beam varies according to a square-wave scan, instead of a
sawtooth scan. A breeding time of 0.4 s at 1 keV is sufficient to reach Fe
XVII population equilibrium, according to both prospective measurements and
simulations. The probe energy was maintained for 1 s and varied slowly (in
minutes) between 300 and 500 eV. The electron energy was changed between
breeding (constant) and probing values (slowly scanned) by means of a fast
(tens of nanoseconds) high-voltage switch. The time evolution of the X-ray
spectrum can thus be observed, in direct dependence on the evolution of the
population of different charge states.

##  3 Calculations

DR is a resonant process involving two steps. At first, a dielectronic capture
of a free electron into an initial ionic state $i$ excites a bound electron
and forms a doubly excited (or intermediate) state $d$. Then, this state may
radiatively decay into a final state $f$, thereby completing the DR process.
Following our previous works (Amaro et al., 2017; Shah et al., 2018, 2019), we
calculated cross sections and resonant strengths in the isolated resonance
approximation, i.e., no quantum interference between DR resonances (Pindzola
et al., 1992), or with nonresonant recombination channels was considered
(Zatsarinny et al., 2006; González Martínez et al., 2005; Tu et al., 2015,
2016). This contribution only influences weak resonances as been previously
shown in Pindzola et al. (1992) and Zatsarinny et al. (2006). In this
approximation, the DR strength is given by

$\displaystyle S_{idf}^{DR}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\sigma^{DR}_{idf}(E_{e})dE_{e}$ (2)
$\displaystyle=$
$\displaystyle\frac{\pi^{2}\hbar^{3}}{m_{e}E_{id}}\frac{g_{d}}{2g_{i}}\frac{A^{a}_{di}A^{r}_{df}}{\sum_{i^{\prime}}A^{a}_{di^{\prime}}+\sum_{f^{\prime}}A^{r}_{df^{\prime}}},$

where $\sigma^{DR}_{idf}(E_{e})$ is the DR cross section as a function of the
free-lectron kinetic energy $E_{\mbox{e}}$. $E_{\mbox{id}}$ is the resonant
energy of the electron$-$ion recombination between states $i$ and $d$, with
respective statistical weights $g_{\mbox{i}}$ and $g_{\mbox{d}}$, and
$m_{\mbox{e}}$ is the electron mass. The autoionization rate
$A^{\mbox{a}}_{{\mbox{di}}}$ and radiative rate $A^{\mbox{r}}_{{\mbox{df}}}$
were calculated with both FAC and MCDF methods.

The details of the FAC calculation are given in Shah et al. (2019). FAC (Gu,
2008) provides atomic radial wave functions and respective eigenvalues
obtained in a configuration interaction method with orbitals from a modified
electron$-$electron central potential. This code uses the distorted$-$wave
Born approximation for calculating the autoionization rates. Beyond the
standard configuration interaction module of FAC, which we employed for the
initial calculations shown in Figure 4, we used the many-body perturbation
theory (MBPT) option of FAC (Gu et al., 2006) for the prediction of energies
and rates in the final detailed comparisons.

Calculations of the energies for the initial, intermediate, and final states
as well as their respective transition and autoionization rates were also
obtained with the Multiconfiguration Dirac$-$Fock General Matrix Elements
(MCDFGME) code (version 2008) of Desclaux and Indelicato (Desclaux, 1975;
Indelicato et al., 1987; Indelicato & Desclaux, 1990; Desclaux & Indelicato,
2008). Details of the method, including the Hamiltonian and the variational
processes employed for retrieving wave functions, can be found in Desclaux
(1993) and Indelicato (1995). In the present calculations, the electronic
correlation was restricted to mixing all states of a given $j$ within an
intermediate coupling (IC) scheme. Autoionization rates were evaluated using
Fano’s single-channel discrete$-$continuous expansion, which allows for non-
orthogonal basis sets between the initial and final states (see Howat et al.
(1978) for details). Figure 2 shows our calculations of DR resonant strengths
obtained using FAC and the MCDF method, as well as a comparison with Nilsen
(1989) values. Note that only DR resonant strengths and the corresponding
photon energies are given in Nilsen (1989). Thus, for comparing the respective
resonant strengths in Figure 2 with our data, we inferred the corresponding
resonant energies used in that work by subtracting from the photon energies
the binding energy of the recombined electron in the final state. Here, the
FAC-MBPT values of the binding energies were used.

Additionally, most LMM DR resonances decay through only one strong radiative
channel ($2p^{5}3l3d\rightarrow 2p^{6}3l$; see Figure 2 (a)). We predicted
with FAC the main spectral features that can be experimentally resolved in
both EBITs. We found eight spectral lines that have either a single resonant
contribution, like $s_{1}$, or a blend of resonances, such as $p_{1}$ or
$d_{1}$. Note that in our line nomenclature we use the $l$ of the $3l$
spectator electron for labeling. The observed line energies and strengths were
compared with all theoretical predictions and available literature. Details
are given in Appendix Appendix  DR Calculations for Fe XVII.

For spectral modeling, DR rate coefficients are convenient parameters. They
can be obtained by integrating the corresponding DR resonance strengths over a
Maxwellian velocity distribution of the electrons (Gu, 2003),

$\alpha^{DR}_{if}=\frac{m_{e}}{\sqrt{\pi}\hbar^{3}}\left(\frac{4E_{y}}{K_{B}T_{e}}\right)^{3/2}a^{3}_{0}\sum_{d}E_{id}S_{idf}^{DR}e^{-\frac{E_{id}}{K_{B}T_{e}}},$
(3)

where $E_{y}$ is the Rydberg constant in energy units, $a_{0}$ is the Bohr
radius, $K_{B}$ is the Boltzmann constant, and $T_{\mbox{e}}$ is the electron
temperature. A comparison between the present rate coefficients and those
available in the OPEN-ADAS database is shown in Section 5.1.

##  4 Simulations of the Charge-state Distribution

To measure DR resonant strengths for a given ionic species, it is necessary to
know the charge-state distribution of the ions trapped in the EBIT. This is
mostly determined by the following charge-changing processes: collisional
ionization (CI), radiative recombination (RR), DR and CX. Their competition,
depending on the measurement conditions and methods, determines the overall
charge-state distribution. Here, we simulate them following the work of
Penetrante et al. (1991) for computing the time evolution of the ion
population in the different charge states in an EBIT by using $Z+1$ steady-
state rate equations:

$\begin{split}\frac{dN_{q}}{dt}=&n_{e}v_{e}\left[N_{q-1}\sigma_{q-1}^{CI}+N_{q+1}\left(\sigma_{q+1}^{RR}+\sigma_{q+1}^{DR}\right)\right.\\\
&\left.-N_{q}\sigma_{q}^{CI}-N_{q}\left(\sigma_{q}^{RR}+\sigma_{q}^{DR}\right)\right]\\\
&-N_{0}N_{q}\sigma_{q}^{CX}\overline{v}_{q}+N_{0}N_{q+1}\sigma_{q+1}^{CX}\overline{v}_{q+1}.\end{split}$
(4)

Here, $N_{q}$ denotes the population of charge states $q$, $n_{\mbox{e}}$ the
electron density, $v_{\mbox{e}}$ the free-electron velocity, $\sigma$ the
cross section associated with a specific atomic process, and $\bar{v}_{q}$ the
mean (Maxwellian) velocity of an ion with charge $q$. The RR total cross
sections for Mg-like, Na-like, and Ne-like Fe ions under the present
experimental conditions were obtained using FAC, taking into account the
principal quantum numbers up to $n~{}=~{}15$. For other Fe charge states, we
used the analytical equation of Kim & Pratt (1983) to obtain RR cross
sections. CI cross sections from the measurements performed at the TSR
(Linkemann et al., 1995; Hahn et al., 2013; Bernhardt et al., 2014) for Mg-
like, Na-like, and Ne-like Fe ions were also used. These included, apart from
the usual direct CI channel, resonant ionization processes, such as excitation
and subsequent autoionization, which became strong starting from the collision
excitation threshold ($\sim$750 eV). For the remaining charge states, CI cross
sections were estimated using the Lotz formula (Lotz, 1968). As for the CX
cross sections used in our simulations, we applied the analytical expression
from Janev et al. (1983) to obtain them. However, for the more abundant charge
states of Ne-like, Na-like, and Mg-like Fe, we used CX cross sections obtained
using the multichannel Landau$-$Zener method as implemented in FAC. An ion
temperature of $\sim$60 eV for the CX cross sections was assumed, based on the
previous measurement by Schnorr et al. (2013) at FLASH-EBIT for Al-like Fe.
The CX rate is proportional to the residual gas density within the trap
region. Thus, by reducing the flow of the iron pentacarbonyl molecular beam in
our experiment, we could enhance the population of Fe in higher charge states.

Figure 3:  (a) Spectra observed at FLASH-EBIT by scanning upward and downward
the electron beam energy (see Figure 1(a) for the timing pattern). Simulations
of Fe charge-state distributions as a function of the electron beam energy for
effective electron densities of (b) $n_{\mbox{e}}$ = $2.8\times
10^{12}~{}\mbox{cm}^{-3}$ ($I_{e}=20~{}\mbox{mA}$) and (c) $n_{e}=2.8\times
10^{10}~{}\mbox{cm}^{-3}$ ($I_{e}=0.2~{}\mbox{mA}$). The simulations were
performed for an electron beam radius of $30~{}\mu\mbox{m}$. (d) Experimental
X-ray intensity ratio between downward and upward scans compared with
simulated X-ray intensity ratios for different effective electron densities.

For these experiments, it is very important to choose the ratio of ionization
time to recombination time appropriately to the simulation parameters. DR is a
very strong resonant process; within the resonant width it has cross sections
orders of magnitude higher than those of other collisional processes. This
means that for our measurement scheme the Ne-like population in the trap
should not be significantly depleted toward lower charge states by the
required electron beam energy across the DR resonances. Therefore, we
performed simulations for quantifying this depletion. We calculated the
corresponding DR rates for Mg-like, Na-like, and Ne-like Fe ions using FAC.
The principal quantum numbers of the recombined state up to $n~{}=~{}30$ for
Ne-like Fe and up to $n~{}=~{}10$ for all other relevant ions were taken into
account in our calculations, as well as radiative cascades for all relevant
atomic processes. With these calculations we then generated synthetic X-ray
emission spectra.

Furthermore, our charge balance simulations were restricted to electron
densities below $10^{13}$ cm-3, the relevant range for EBIT plasmas. At values
higher than those of our simulations, we note that the effect of DR
suppression has to be taken into account (see Nikolić et al. (2013, 2018) and
references therein).

###  4.1 FLASH-EBIT: Simulated Charge-state Distributions

Using Equation 4 and the scanning parameters presented in Figure 1, we
simulated the time evolution of the charge-state distribution during the
upward and downward electron beam energy scans in our FLASH-EBIT measurements
(Shah et al., 2019) (see Figure 3(a)). First, we investigated the effect of
the electron beam density on the Ne-like, Na-like, and Mg-like Fe trapped-ion
populations. Panel (b) of Figure 3 shows an extreme case of electron densities
of $n_{e}\sim 10^{12}$ cm-3, where the Ne-like population is drastically
depleted into the Na-like and Mg-like ones due to strong DR resonances. In
contrast, at $n_{e}\sim 10^{10}$ cm-3, this effect is found to be negligible
(panel (c)). In panel (d) of Figure 3, we display the experimental ratio
between the fast upward and downward energy scans, which shows a negligible
effect of the scanning direction on Fe XVII ion density in the trap. This
stands in contrast to the out-of-equilibrium slow DR scans, where the ion
populations have enough time for the decay during the scan. In this case, the
two scanning directions can show clear differences in the distribution of
charge states (Shah et al., 2016). We compared our present experimental ratio
with those of simulations for different electron densities and found only a
negligible charge-state depletion due to LMM DR resonances at electron
densities below $5\times 10^{10}$ $\text{cm}^{-3}$.

Figure 4: Theoretical resonant strengths (FAC) for Ne-like, Na-like, and Mg-
like Fe ions and simulated fluorescence yield under CX rates of
$6.1~{}\mbox{s}^{-1}$ and $1.1~{}\mbox{s}^{-1}$ in comparison with FLASH-EBIT
data.

Second, we investigated the influence of CX on the Fe XVII ion population
distribution. FLASH-EBIT has a four-stage differential pumping system for
injecting an atomic or molecular beam into the trap, where the ions are
generated. The first two stages operate at room temperature at pressures of
$\sim 8\times 10^{-9}$ mbar. Two additional stages operate cryogenically at 45
K and 4 K, and further constrain the gas flow into the trap region. This
brings the residual gas pressure well below $\leq 10^{-11}$ mbar at the trap
center and tremendously reduces the CX rates. For the study of a possible
influence of CX in our measurements, we simulated the ion population and
generated synthetic X-ray spectra for CX rates of 0.23 s-1 and 0.05 s-1 (see
Figure 4). When Fe XVI and Fe XV ions are produced by CX, distinct DR
resonances of these ions appear at beam energies of 380 and 440 eV. Since
those resonances were not observed, we conclude that under the present
conditions the dominant Ne-like Fe ion population was maintained during the
FLASH-EBIT measurements (Shah et al., 2019).

###  4.2 PolarX-EBIT: Simulated Charge-state Distributions

Figure 5: (a) Measured fluorescence yield as a function of electron beam
energy and probing time at PolarX-EBIT. (b) Simulation of the DR emission at
the experimental conditions.

We also simulated the time evolution of the charge-state distribution and its
effect on the observed LMM DR X-ray emission for PolarX-EBIT conditions,
according to the measurement technique shown in Figure 1(b). Here we compare
the measurement and simulation in Figure 5. For probing times shorter than 100
ms, we observed a dominant population of Ne-like ions with a small population
of Na-like and Mg-like Fe ions. However, the observed intensities of Na-like
and Mg-like LMM resonances are slightly higher than those in the FLASH-EBIT
measurements (see Figure 5 and Figure 4 for the resonant energy positions).
Simulations at the breeding electron beam energy of 1.2 keV used in the
PolarX-EBIT measurements yielded populations of 0.92$\pm$0.04, 0.07$\pm$0.03,
and 0.003$\pm$0.002 for Ne-like, Na-like, and Mg-like Fe ions, respectively.
These values are consistent with a negligible CX rate of
$0.47~{}\mbox{s}^{-1}$, as likewise observed in the FLASH-EBIT measurements.

The uncertainties given for the ion populations were estimated based on the
difference between the cross sections obtained with the analytical formula of
a given process (e.g., from Kim & Pratt (1983) for the RR case) and those of
FAC calculations. As for the CI rates, besides using the difference between
FAC and Lotz (1968) values, we also took into account the experimental
uncertainties from the CI measurements performed at the TSR (Linkemann et al.,
1995; Hahn et al., 2013; Bernhardt et al., 2014) for Mg-like, Na-like, and Ne-
like Fe ions. For the CX cross sections, we found relative deviations between
the expression of Janev et al. (1983) and FAC as high as 40%, which is in
accord with observations of Betancourt-Martinez et al. (2018). However, due to
the very low CX rate, uncertainties associated with the CX cross sections and
ion temperature do not significantly affect the predicted Ne-like populations.

The most intense Ne-like resonances exhibit decay times between 0.07 and 0.13
s in our experiment, while Na-like resonances (e.g., at 345, 380, and 435 eV
beam energies) reach their maxima between 0.2 and 0.4 s as the Ne-like
population starts to deplete. For probing times longer than 1 s, the spectra
are dominated by DR emission from the Mg-like and Al-like Fe populations. We
also observe a constant X-Ray emission background in our experiment, which
cannot be explained by simulations under any conditions of electron density
and CX rate. It might be attributed to the delayed photon emission from
metastable states. However, no exponential decay of the signal was observed in
the FLASH-EBIT data. Another possible source might be the electronic analog-
to-digital converter (ADC) noise caused by switching between power supplies.

Finally, when considering various recombination data either from FAC or MCDF
theories or from TSR measurements in the charge-state distribution
simulations, we did not see any change in the final synthetic spectrum. Also,
this does not change our conclusion that neither spurious charge states other
than Ne-like Fe nor charge-state depletion due to DR significantly affects any
of the two measurements.

##  5 Data Analysis and Results

Experimental spectra observed in the FLASH-EBIT, PolarX-EBIT, and TSR
measurements are shown in Figure 6. In the case of PolarX-EBIT, as explained
in Section 4, we have only selected the first 50 ms of data in order to avoid
possible charge-state depletion due to the DR and CX processes (see Figure 5).
Strong DR lines were identified in Section 3, which are clearly resolved with
an excellent collision energy resolution provided by both EBITs. The TSR data
have a significantly better resolution than the EBIT results. However, for our
comparison we synthetically broadened the TSR spectrum to match the PolarX-
EBIT spectral resolution (for a complete data set, see Schmidt et al. (2009);
Shah et al. (2019)).

We calibrated the electron beam energy axis using the TSR data of the $p_{1}$
and $d_{3}$ peaks. In order to compare the resonance strengths, we first
subtracted an assumed linear background from the RR continua (RR+ADC for
PolarX, see Section 4.2). The observed photon intensities in both EBIT spectra
were normalized to a DR resonance $d_{3}$ at 412 eV of electron beam energy.
Details of such cross-section normalization are explained in our previous work
(Shah et al., 2019). In EBITs, the unidirectional electron beam causes
anisotropic and polarized X-ray emission. We observed photons at $90^{\circ}$
with respect to the electron beam. We thus applied a correction factor defined
as $W(90^{\circ})=3/(3-P)$, where $P$ is the polarization for a specific
radiative transition (Shah et al., 2018). The calculated polarization values
were taken from Shah et al. (2019), which also agree with measurements
performed by Chen et al. (2004). With this correction, we obtained the total
cross sections using the
$S^{\mathrm{total}}=4\pi\,\,I^{90^{\circ}}/W(90^{\circ})$ relation, where
$I^{90^{\circ}}$ is the observed DR intensity. As observed in Figure 4 (Ne-
like case), there are more than $\sim$200 LMM DR resonances within the scanned
electron beam energy range. As most of the features observed in Figure 6
consist of a number of blended resonances, we only provide the integrated
resonance strengths. For this, we defined three energy regions: $s$, 300 eV to
335 eV; $p$, 335 eV to 380 eV; and $d$, 380 eV to 440 eV.

Figure 6:  Fe XVII LMM DR resonance strengths measured using FLASH-EBIT,
PolarX-EBIT and TSR, compared with the theoretical total DR cross sections
obtained using FAC, FAC-MBPT, MCDF, and literature values from Nilsen (1989).
The top panel compares the averaged integrated resonance strengths from three
measurements for the $s$, $p$, and $d$ regions with different theories. Note
that the calculations were convoluted with a Gaussian of 3.5 eV FWHM for the
comparison and FLASH-EBIT and PolarX-EBIT measurements were calibrated to each
of the theoretical cross sections.

###  5.1 DR Energies and Strengths

Figure 6 shows a comparison between the experimental data and predictions by
FAC, FAC-MBPT, MCDF, and Nilsen (1989). All theories except MCDF predict all
experimental features. FAC, in particular, shows a systematic shift in
resonance energies as compared with the observed data and other theories.
Also, disagreements of MCDF predictions are clearly visible for the $s_{1}$,
$s_{2}$, and $p_{2}$ features. DR resonances within these features show larger
energy splitting in the MCDF predictions than in the FAC and FAC-MBPT ones.
Moreover, a clear feature at 425 eV is not predicted by MCDF. The reason
behind this might be that the MCDFGME package (Desclaux & Indelicato, 2008)
used in the present work could not generate reliable Auger rates with
correlation to other configurations beyond minimal coupling, which would be
necessary to improve the energy centroid accuracy.

Table 1 presents experimentally inferred integrated resonant strengths for the
$s$, $p$, and $d$ regions. We estimated total uncertainties on the level of
$\sim$13% and $\sim$11%, respectively, for FLASH-EBIT and PolarX-EBIT
integrated resonance strengths, mainly arising from counting statistics and
cross-section normalization. For the TSR results, we used the quoted
uncertainty of 20% from the original work of Schmidt et al. (2009). By
combining these three independent measurements, we obtained final values for
the integrated resonant strengths $S_{\mathrm{EXP}}$ and their respective
uncertainties, found to be at the level of $\sim$10% (see Table 1). As shown
in the top panel of Figure 6, FAC-MBPT shows good agreement with the inferred
integrated strengths, except for region $s$. Interestingly, all theories
disagree with our inferred resonance strength for the $s$ region. We also note
that no calculations can effectively predict the resonant strength of the
$p_{2}$ feature. EBIT and TSR data also disagree here; since simulations seem
to exclude spurious features at this energy in EBITs, at present we do not
have an explanation for this.

Table 1: Experimental Integrated Resonant Strengths
($10^{-20}\mbox{cm}^{2}\mbox{eV}$) Compared to Values Obtained with FAC and
MCDF (This Work) and Those Reported by Nilsen (1989).

Label | $S_{\text{FLASH}}$ | $S_{\text{PolarX}}$ | $S_{\text{TSR}}$ | $S_{\text{EXP}}$ | $S_{\text{FAC}}$ | $S_{\text{FAC-MBPT}}$ | $S_{\text{MCDF}}$ | Nilsen (1989)
---|---|---|---|---|---|---|---|---
$s$ | $80\pm 10$ | $65\pm 9$ | $100\pm 10$ | $80\pm 6$ | 104.27 (-30%) | 97.77 (-22%) | 62.66 (22%) | 106.25 (-32%)
$p$ | $210\pm 30$ | $230\pm 20$ | $230\pm 50$ | $220\pm 20$ | 201.18 (15%) | 198.34 (11%) | 202.89 (25%) | 174.30 (20%)
$d$ | $300\pm 40$ | $280\pm 30$ | $330\pm 70$ | $290\pm 20$ | 271.76 (13%) | 283.43 (4%) | 270.29 (25%) | 306.67 (-5%)

Note. The listed experimental cross sections were calibrated with the FAC-MBPT
theoretical value of the d3 feature. Round brackets: Relative difference in
percent between measured and theoretical resonant strengths, where each
measurement was independently calibrated with the $d_{3}$ value from
respective theories. The labels $s$, $p$ and $d$ correspond to regions of
$300~{}\mbox{eV}$ to $340~{}\mbox{eV}$, $340~{}\mbox{eV}$ to
$380~{}\mbox{eV}$, and $380~{}\mbox{eV}$ to $420~{}\mbox{eV}$, respectively.

Table 2: Experimental DR Rate Coefficients ($\times
10^{-13}\mbox{cm}^{3}\mbox{s}^{-1}$) for a Few Electron Temperatures
$T_{\mbox{e}}$ (eV) Compared to Values Obtained with FAC-MBPT and MCDF and by
Nilsen (1989) and by Zatsarinny et al. (2004).

Label | $T_{\mbox{e}}$ | Exp. | OPEN-ADAS |  FAC-MBPT | MCDF | Nilsen (1989) |  Zatsarinny et al. (2004) 222 Only total DR LMM rates are provided.
---|---|---|---|---|---|---|---
|  |  | oizLS | oizIC | nrbLS | nrbIC |  |  |  |
$3s$ | 110.3 | $\cdots$ | 10.9 | 9.24 | 10.6 | 7.04 | 8.96 | 6.40 | 9.28 | $\cdots$
| 220.3 | $\cdots$ | 17.4 | 15.0 | 17.4 | 11.9 | 13.7 | 10.3 | 14.4 | $\cdots$
| 300 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 12.8 | 9.71 | 13.5 | $\cdots$
| 2000 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1.89 | 1.48 | 2.00 | $\cdots$
$3p$ | 110.3 | $\cdots$ | 20.1 | 20.4 | 16.8 | 17.2 | 18.8 | 17.1 | 16.5 | $\cdots$
| 220.3 | $\cdots$ | 40.3 | 42.2 | 36.4 | 38.9 | 33.0 | 30.9 | 29.5 | $\cdots$
| 300 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 31.9 | 30.0 | 28.6 | $\cdots$
| 2000 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 5.12 | 4.86 | 4.63 | $\cdots$
$3d$ | 110.3 | $\cdots$ | 18.9 | 21.2 | 15.9 | 16.9 | 19.2 | 16.8 | 20.6 | $\cdots$
| 220.3 | $\cdots$ | 46.2 | 52.6 | 41.4 | 45.4 | 40.7 | 36.6 | 43.8 | $\cdots$
| 300 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 41.3 | 37.5 | 44.5 | $\cdots$
| 2000 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 7.46 | 6.86 | 8.06 | $\cdots$
total | 110.3 | $47\pm 3$ | 49.9 | 50.9 | 43.3 | 41.2 | 46.9 | 40.3 | 46.4 | 46.6
|  |  | (-6.2%) | (-8.2%) | (7.9%) | (12%) | (0.2%) | (14%) | (1.3%) | (0.8%)
| 220.3 | $88\pm 4$ | 103.9 | 109.8 | 95.2 | 96.2 | 87.4 | 77.7 | 87.6 | 91.0
|  |  | (-18%) | (-25%) | (-8.2%) | (-9.3%) | (0.7%) | (12%) | (0.4%) | (-3.5%)
| 300 | $87\pm 4$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 86.0 | 77.2 | 86.6 | 90.3
|  |  |  |  |  |  | (0.8%) | (11%) | (0.1%) | (-4.2%)
| 2000 | $14.5\pm 0.7$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 14.5 | 13.2 | 14.7 | 15.3
|  |  |  |  |  |  | (0.3%) | (9.3%) | (-1.0%) | (-5.2%)

Notes. The first label indicates the orbital of the decoupled electron in the
final state after DR emission. Labels oizLS, oizIC, nrbLS and nrbIC refer to
data files available at adf09 of OPEN-ADAS.

###  5.2 DR Rate Coefficients

In addition to DR integrated resonant strengths, we also inferred DR rate
coefficients using Equation (3) at different plasma electron temperatures.
Such rate coefficients are important for collisional-radiative models of
single-temperature or multi-temperature plasmas. Table 2 lists our
experimentally inferred and theoretical DR rate coefficients from
$S_{\mathrm{EXP}}$ together with those available in the OPEN-ADAS and AtomDB
(Zatsarinny et al., 2004; Foster et al., 2012) databases.

To compare with OPEN-ADAS, we identified the DR rates by the respective final-
state configuration of the Na-like Fe ion. The combined total rates for the
3$s$, 3$p$, and 3$d$ final configurations are presented in Table 2. Since the
DR rates of OPEN-ADAS include all DR LM$n$ channels, we restricted our
comparison to the temperature range $T_{i}\leq 250$ eV, where the LMM channel
dominates ($\geq$ 95% of the total decay into a given final-state
configuration). Two tabulated DR rates in this temperature range
($T_{\mbox{e}}=110.3$ eV and 220.3 eV) were retrieved through adf09 files,
namely oiz00#ne_fe16ls23 (oizLS) and oiz00#ne_fe16ic23 (oizIC), provided by
the author O. Zatsarinny in LS coupling and IC, respectively, as well as
nrb00#ne_fe16ls23 (nrbLS) and nrb00#ne_fe16ic23 (nrbIC) given by the author N.
Badnell in respective atomic couplings. According to Foster et al. (2012), the
Fe XVII DR rates in the AtomDB database were taken from Zatsarinny et al.
(2004), calculated using the AUTOSTRUCTURE code, and summed over all final
states.

Since some minor final-state configurations are shared by the observed
experimental spectral features, only the total experimental rates are
presented in Table 2. Figure 7 depicts the comparison between our experiment
and all available DR rates over a large temperature range. Our experimental
rate coefficients agree well with the theoretical rates within $\sim$0.2–0.8%
for those calculated with FAC-MBPT and witin $\sim$7–14% for those calculated
with MCDF methods. The DR rates taken from Nilsen (1989) are in close
agreement with the FAC-MBPT calculations. The OPEN-ADAS data provided by
Badnell in both LS coupling and IC depart by $\sim-9$ to 12% from our
experimental results, while the data provided by Zatsarinny show differences
of $\sim$6–25%. The total DR rates obtained with AUTOSTRUCTURE (Zatsarinny et
al., 2004) (also used in the AtomDB database) display differences from our
results of $<$5% for the entire electron temperature range.

Figure 7: Experimental and theoretical DR rate coefficients obtained with FAC-
MBPT and MCDF and by Nilsen (1989) and Zatsarinny et al. (2004)
(AUTOSTRUCTURE), as well as tabulated values for two temperatures from adf09
files of OPEN-ADAS, namely oizLS, oizIC, nrbLS, and nrbIC.

##  6 Summary and Conclusions

The DR LMM resonances for Fe XVII ions have been measured using two different
EBITs and compared with results obtained at the TSR (Schmidt et al., 2009). We
simulated the time-dependent charge-state distribution to rule out systematic
effects due to the presence of spurious charge states and depletion of charge
states due to DR, which may affect our resonant strength measurements. The LMM
DR integrated strengths were extracted from all three experiments, and
compared with calculations performed using FAC, FAC-MBPT, and MCDF codes.
Among them, FAC-MBPT calculations show the best agreement with the
experiments. We also derived from our experimental data LMM DR rate
coefficients for several temperatures. Our experimental rates show good
agreement with rates obtained from Nilsen (1989) and FAC-MBPT, while rates
available in the OPEN-ADAS and AtomDB databases, which are frequently used for
the analysis of astrophysical spectra, show departures from our experimentally
inferred values. These differences highlight how crucial laboratory
measurements are for testing different spectral models. This is important not
only from the perspective of upcoming X-ray satellite missions XRISM (Tashiro
et al., 2018) and Athena (Barret et al., 2016), which will soon provide high-
resolution spectra using X-ray microcalorimeters (Durkin et al., 2019), but
also for interpreting available high-resolution spectra from the operating
Chandra and XMM-Newton observatories (Gu et al., 2020) needed for reliable
plasma diagnostics (Beiersdorfer et al., 2014, 2018).

## acknowledgments

F.G. and P.A. acknowledge support from Fundação para a Ciência e Tecnologia,
Portugal, under grant No. UID/FIS/04559/2020 (LIBPhys) and under contracts
UI/BD/151000/2021 and SFRH/BPD/92329/2013. C.S. acknowledges support from an
appointment to the NASA Postdoctoral Program at the NASA Goddard Space Flight
Center, administered by the Universities Space Research Association under a
contract with NASA, by Deutsche Forschungsgemeinschaft project No. 266229290,
and by Max-Planck-Gesellschaft. We also thank Dr. Stefan Schippers for
providing the raw data of Fe XVII DR rates measured at the TSR in Heidelberg,
Germany.

## Appendix
DR Calculations for Fe XVII

Table 3 contains the resonant energies and strengths for the main spectral
features that are benchmarked in this work. The emitted wavelengths of the
decay channels are listed in Table 4 with additional data provided by Nilsen
(1989) and Beiersdorfer et al. (2014).

The complete set of resonant energies, strengths, and emitted wavelengths
calculated within this work (FAC, FAC-MBPT, and MCDF methods) is available
online as machine-readable tables. These data can be used to make Figure 2 as
well as to produce the synthetic spectra shown in Figure 6.

Table 3: Theoretical Values of the Peak Resonant Energies $E$ (eV) and
Strengths $S$ ($10^{-20}$cm2eV) Obtained in This Work with FAC, FAC-MBPT, and
MCDF.

Label | Intermediate State |  | Final State | ${E}_{\text{FAC}}$ | ${E}_{\text{MBPT}}$ | ${E}_{\text{MCDF}}$ | ${S}_{\text{FAC}}$ | ${S}_{\text{MBPT}}$ | ${S}_{\text{MCDF}}$
---|---|---|---|---|---|---|---|---|---
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3s_{1/2})_{2}3d_{5/2}]_{1/2}$ | ${}^{4}D_{1/2}$ | $3s_{1/2}{}^{2}S_{1/2}$ | 313.06 | 310.23 | 313.08 | 6.04 | 6.42 | 2.74
$s_{1}$ | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3s_{1/2})_{1}3d_{5/2}]_{3/2}$ | ${}^{2}D_{3/2}$ | $3s_{1/2}{}^{2}S_{1/2}$ | 325.70 | 322.21 | 321.98 | 31.91 | 29.54 | 7.34
$s_{2}$ | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3s_{1/2})_{1}3d_{3/2}]_{3/2}$ | ${}^{2}P_{3/2}$ | $3s_{1/2}{}^{2}S_{1/2}$ | 334.62 | 330.72 | 328.60 | 25.89 | 23.22 | 20.61
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3p_{1/2})_{2}3d_{5/2}]_{3/2}$ | ${}^{4}P_{3/2}$ | $3p_{1/2}{}^{2}P_{1/2}$ | 341.50 | 338.94 | 339.95 | 1.04 | 1.11 | 0.24
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3p_{3/2})_{0}3d_{5/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 354.47 | 351.41 | 352.30 | 6.16 | 7.99 | 4.59
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{5/2}]_{5/2}$ | ${}^{4}F_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 355.8 | 353.04 | 354.05 | 5.14 | 4.95 | 0.04
$p_{1}$ | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{3/2}]_{3/2}$ | ${}^{4}F_{3/2}$ | $3p_{1/2}{}^{2}P_{1/2}$ | 353.65 | 355.05 | 352.28 | 1.06 | 9.70 | 0.16
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{3/2}]_{3/2}$ | ${}^{4}F_{3/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 353.65 | 355.05 | 352.28 | 0.008 | 0.22 | 0.32
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 358.36 | 355.09 | 356.22 | 0.03 | 0.61 | 0.48
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{3/2}]_{3/2}$ | ${}^{2}D_{3/2}$ | $3p_{1/2}{}^{2}P_{1/2}$ | 361.25 | 357.38 | 358.83 | 48.50 | 47.02 | 38.59
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3p_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 361.67 | 358.04 | 359.77 | 35.26 | 34.74 | 30.35
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{3/2}]_{1/2}$ | ${}^{2}S_{1/2}$ | $3p_{1/2}{}^{2}P_{1/2}$ | 361.71 | 358.42 | 367.70 | 12.90 | 12.36 | 12.68
$p_{2}$ | $[(2s_{1/2}2p^{2}_{1/2}2p^{4}_{3/2})_{1/2}3s^{2}_{1/2}]_{1/2}$ | $\cdots$ | $3p_{3/2}{}^{2}P_{3/2}$ | 364.85 | 361.80 | $\cdots$ | 5.56 | 4.45 | $\cdots$
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{3/2})_{2}3d_{3/2}]_{1/2}$ | ${}^{2}P_{1/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 367.42 | 363.36 | 364.06 | 10.77 | 11.87 | 19.61
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{3/2})_{1}3d_{3/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 368.99 | 364.74 | 371.33 | 8.69 | 4.43 | 29.69
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{3/2})_{1}3d_{5/2}]_{3/2}$ | ${}^{2}D_{3/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 369.50 | 365.35 | 366.12 | 12.17 | 11.39 | 16.64
$d_{1}$ | $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3d_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{4}G_{5/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 396.39 | 393.3 | 396.73 | 9.42 | 11.46 | 11.37
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3d^{2}_{5/2}]_{7/2}$ | ${}^{4}D_{7/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 397.93 | 394.65 | 396.22 | 11.89 | 13.96 | 12.55
$d_{2}$ | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d^{2}_{5/2}]_{7/2}$ | ${}^{2}G_{7/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 404.14 | 400.95 | 402.46 | 13.45 | 12.08 | 11.92
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{2}F_{5/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 404.45 | 401.05 | 402.21 | 10.19 | 24.19 | 1.33
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{2}F_{5/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 404.45 | 401.05 | 402.21 | 7.00 | 6.27 | 1.78
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d^{2}_{3/2}]_{5/2}$ | ${}^{2}F_{5/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 405.22 | 401.98 | 403.14 | 41.48 | 23.62 | 35.91
$d_{3}$ | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d^{2}_{5/2}]_{1/2}$ | ${}^{2}P_{1/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 411.16 | 407.57 | 408.61 | 6.25 | 7.53 | 9.59
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{1}3d_{5/2}]_{7/2}$ | ${}^{2}F_{7/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 411.90 | 407.96 | 410.07 | 60.63 | 58.13 | 52.73
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{1}3d_{5/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 412.85 | 408.75 | 410.29 | 14.74 | 13.79 | 16.84
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{1}3d_{5/2}]_{3/2}$ | ${}^{2}P_{3/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 415.20 | 411.12 | 412.39 | 12.26 | 12.66 | 17.84
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{1}3d_{5/2}]_{3/2}$ | ${}^{2}P_{3/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 415.20 | 411.12 | 412.39 | 15.55 | 16.92 | 18.24
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d^{2}_{3/2}]_{1/2}$ | ${}^{2}P_{1/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 420.27 | 416.73 | 415.67 | 6.01 | 5.39 | 7.49

Note. The resonant and final states are given in j-j and LSJ notations.

Table 4: Theoretical Values of Emitted Wavelengths (Å) Obtained in This Work
with FAC, FAC-MBPT, and MCDF.

Label | Intermediate State |  | Final State | $\lambda_{\text{FAC}}$ | $\lambda_{\text{FAC-MBPT}}$ | $\lambda_{\text{MCDF}}$ | Nilsen (1989) | Beiersdorfer et al. (2014)
---|---|---|---|---|---|---|---|---
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3s_{1/2})_{2}3d_{5/2}]_{1/2}$ | ${}^{4}D_{1/2}$ | $3s_{1/2}{}^{2}S_{1/2}$ | 15.47 | 15.52 | 15.46 | 15.52 | 15.49
s1 | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3s_{1/2})_{1}3d_{5/2}]_{3/2}$ | ${}^{2}D_{3/2}$ | $3s_{1/2}{}^{2}S_{1/2}$ | 15.23 | 15.29 | 15.29 | 15.28 | 15.27
s2 | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3s_{1/2})_{1}3d_{3/2}]_{3/2}$ | ${}^{2}P_{3/2}$ | $3s_{1/2}{}^{2}S_{1/2}$ | 15.06 | 15.13 | 15.17 | 15.12 | 15.11
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3p_{1/2})_{2}3d_{5/2}]_{3/2}$ | ${}^{4}P_{3/2}$ | $3p_{1/2}{}^{2}P_{1/2}$ | 15.59 | 15.63 | 15.61 | 15.58 | $\cdots$
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3p_{3/2})_{0}3d_{5/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 15.38 | 15.44 | 15.42 | 15.63 | $\cdots$
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{5/2}]_{5/2}$ | ${}^{4}F_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 15.36 | 15.41 | 15.39 | 15.41 | 15.41
p1 | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{3/2}]_{3/2}$ | ${}^{4}F_{3/2}$ | $3p_{1/2}{}^{2}P_{1/2}$ | 15.35 | 15.32 | 15.37 | 15.40 | 15.30
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{3/2}]_{3/2}$ | ${}^{4}F_{3/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 15.40 | 15.38 | 15.42 | 15.45 | $\cdots$
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 15.26 | 15.37 | 15.35 | 15.37 | $\cdots$
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{3/2}]_{3/2}$ | ${}^{2}D_{3/2}$ | $3p_{1/2}{}^{2}P_{1/2}$ | 15.21 | 15.27 | 15.25 | 15.26 | 15.26
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3p_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 15.25 | 15.31 | 15.28 | 15.55 | 15.29
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{1/2})_{1}3d_{3/2}]_{1/2}$ | ${}^{2}S_{1/2}$ | $3p_{1/2}{}^{2}P_{1/2}$ | 15.20 | 15.26 | 15.08 | 15.25 | 15.07
p2 | $[(2s_{1/2}2p^{2}_{1/2}2p^{4}_{3/2})_{1/2}3s^{2}_{1/2}]_{1/2}$ | $\cdots$ | $3p_{3/2}{}^{2}P_{3/2}$ | 15.19 | 15.24 | $\cdots$ | 15.23 | 15.24
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{3/2})_{2}3d_{3/2}]_{1/2}$ | ${}^{2}P_{1/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 15.14 | 15.21 | 15.20 | 15.19 | 15.19
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{3/2})_{1}3d_{3/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 15.11 | 15.18 | 15.07 | 15.17 | $\cdots$
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3p_{3/2})_{1}3d_{5/2}]_{3/2}$ | ${}^{2}D_{3/2}$ | $3p_{3/2}{}^{2}P_{3/2}$ | 15.10 | 15.17 | 15.16 | 15.20 | 15.16
d1 | $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3d_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{4}G_{5/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 15.48 | 15.53 | 15.47 | 15.54 | 15.50
| $[((2p^{2}_{1/2}2p^{3}_{3/2})_{3/2}3d^{2}_{5/2}]_{7/2}$ | ${}^{4}D_{7/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 15.45 | 15.51 | 15.49 | 15.52 | 15.49
d2 | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d^{2}_{5/2}]_{7/2}$ | ${}^{2}G_{7/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 15.34 | 15.39 | 15.37 | 15.40 | 15.37
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{2}F_{5/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 15.32 | 15.38 | 15.36 | 15.38 | $\cdots$
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{2}3d_{5/2}]_{5/2}$ | ${}^{2}F_{5/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 15.33 | 15.39 | 15.37 | 15.39 | $\cdots$
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d^{2}_{3/2}]_{5/2}$ | ${}^{2}F_{5/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 15.31 | 15.36 | 15.35 | 15.37 | 15.35
d3 | $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d^{2}_{5/2}]_{1/2}$ | ${}^{2}P_{1/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 15.20 | 15.26 | 15.24 | 15.26 | 15.24
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{1}3d_{5/2}]_{7/2}$ | ${}^{2}F_{7/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 15.19 | 15.26 | 15.22 | 15.25 | 15.23
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{1}3d_{5/2}]_{5/2}$ | ${}^{2}D_{5/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 15.17 | 15.24 | 15.22 | 15.23 | 15.22
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{1}3d_{5/2}]_{3/2}$ | ${}^{2}P_{3/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 15.12 | 15.19 | 15.17 | 15.35 | 15.18
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d_{3/2})_{1}3d_{5/2}]_{3/2}$ | ${}^{2}P_{3/2}$ | $3d_{5/2}{}^{2}D_{5/2}$ | 15.13 | 15.20 | 15.18 | 15.36 | 15.18
| $[((2p_{1/2}2p^{4}_{3/2})_{1/2}3d^{2}_{3/2}]_{1/2}$ | ${}^{2}P_{1/2}$ | $3d_{3/2}{}^{2}D_{3/2}$ | 15.03 | 15.09 | 15.11 | 15.09 | 15.08

Notes. The resonant and final states are given in $j$-$j$ and LSJ notations.
The respective data provided by Nilsen (1989) and by Beiersdorfer et al.
(2014) are also listed for comparison.

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