# High-Resolution X-Ray Spectroscopy of Interstellar Iron Toward Cygnus X-1
and GX 339-4

Lía Corrales Astronomy Department, University of Michigan, Ann Arbor, MI
48109, USA  Eric V. Gotthelf Columbia Astrophysics Laboratory, Columbia
University, 550 West 120th Street, New York, NY 10027-6601, USA  Efrain Gatuzz
Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrasse 1,
D-85748 Garching, Germany  Timothy R. Kallman NASA/GSFC, Code 662, Greenbelt
MD 20771, USA  Julia C. Lee Harvard University, John A. Paulson School of
Engineering & Applied Science, 29 Oxford Street, Cambridge, MA 02138, USA
Michael Martins Institut für Experimentalphysik, Universität Hamburg, Luruper
Chaussee 149, 22761 Hamburg, Germany  Center for Free-Electron Laser Science,
Notkestrasse 85, 22607 Hamburg, Germany  Frits Paerels Columbia Astrophysics
Laboratory and Department of Astronomy, Columbia University, 550 West 120th
St., New York, NY 10027, USA  Ioanna Psaradaki Astronomy Department,
University of Michigan, Ann Arbor, MI 48109, USA  Stefan Schippers I.
Physikalisches Institut, Justus-Liebig-Universität Giessen, Heinrich-Buff-Ring
16, 35392 Giessen, Germany  Daniel Wolf Savin Columbia Astrophysics
Laboratory, Columbia University, 550 West 120th Street, New York, NY
10027-6601, USA

(Received TBD; Revised TBD; Accepted TBD)

###### Abstract

We present a high-resolution spectral study of Fe $L$-shell extinction by the
diffuse interstellar medium (ISM) in the direction of the X-ray binaries
Cygnus X-1 and GX 339–4, using the XMM-Newton reflection grating spectrometer.
The majority of interstellar Fe is suspected to condense into dust grains in
the diffuse ISM, but the compounds formed from this process are unknown. Here,
we use the laboratory cross sections from Kortright & Kim (2000) and Lee et
al. (2009) to model the absorption and scattering profiles of metallic Fe, and
the crystalline compounds fayalite (Fe2SiO4), ferrous sulfate (FeSO4),
hematite ($\alpha$-Fe2O3), and lepidocrocite ($\gamma$-FeOOH), which have
oxidation states ranging from Fe0 to Fe3+. We find that the observed Fe
$L$-shell features are systematically offset in energy from the laboratory
measurements. An examination of over two dozen published measurements of Fe
$L$-shell absorption finds a $1-2$ eV scatter in energy positions of the
$L$-shell features. Motivated by this, we fit for the best energy-scale shift
simultaneously with the fine structure of the Fe $L$-shell extinction cross
sections. Hematite and lepidocrocite provide the best fits ($\approx+1.1$ eV
shift), followed by fayalite ($\approx+1.8$ eV shift). However, fayalite is
disfavored, based on the implied abundances and knowledge of ISM silicates
gained by infrared astronomical observations and meteoritic studies. We
conclude that iron oxides in the Fe3+ oxidation state are good candidates for
Fe-bearing dust. To verify this, new absolute photoabsorption measurements are
needed on an energy scale accurate to better than 0.2 eV.

††journal: ApJ††software:  SAS (Gabriel et al. 2004), XSPEC (Arnaud 1996),
ISMabs (Gatuzz et al. 2015), ISMdust (Corrales et al. 2016), newdust (Corrales
2023), Astropy (Astropy Collaboration et al. 2022; 2018; 2013)

##  1 Introduction

Determining the composition, size distribution, and spatial distribution of
iron (Fe)-bearing dust in the neutral interstellar medium (ISM) is important
for understanding the history of stellar nucleosynthesis, elemental abundances
in our galaxy, and stellar life cycles (Dwek & Arendt 1992, Jenkins 2009).
Furthermore, knowledge of the Fe-bearing dust is needed to calibrate
observations of the cosmic microwave background by constraining the sources of
the contaminating anomalous microwave emission attributed to dust (Hoang &
Lazarian 2016, Hensley & Draine 2017a). Fe-bearing dust also contributes to
the formation of rocky planets (Ishii et al. 2018, Westphal et al. 2019).
Ultraviolet (UV) and optical absorption studies find over $90-99\%$ of the Fe
in the ISM is locked up in the solid phase, namely in interstellar dust grains
(Savage & Sembach 1996, Jenkins 2009). However, those studies suffer from
uncertainties in the reference abundance tables (Jenkins 2009). More
importantly, they are unable to constrain the identity of the solid-phase Fe-
bearing compounds due to the lack of dust spectral features in UV or optical
(Savage & Sembach 1996, Jenkins 2009, Voshchinnikov & Henning 2010).

High resolution X-ray spectroscopy is a more powerful technique for studying
Fe in the neutral ISM. Gas-phase Fe and solid-phase Fe-bearing dust grains
with radii $\lesssim 1~{}\mu$m are semitransparent to X-rays (Wilms et al.
2000). As a consequence, high resolution X-ray spectroscopy is the sole
technique that can be used to study solid phase Fe in the neutral diffuse ISM,
for atomic hydrogen column densities of $N({\mathrm{H}})=10^{21}-10^{23}$
cm-2. Using high resolution X-ray spectra of bright Galactic X-ray binaries as
background light sources, Fe $L$-shell absorption and scattering features can
be used to identify the abundances, ionization states, and chemical
composition of the ISM. There have been numerous past studies of Fe in the
diffuse ISM (e.g., Paerels et al. 2001, Schulz et al. 2002, Takei et al. 2002,
de Vries et al. 2003, Juett et al. 2004, Ueda et al. 2005, Yao & Wang 2006,
Juett et al. 2006, de Vries & Costantini 2009, Lee et al. 2009, Hanke et al.
2009, Pinto et al. 2010, García et al. 2011, Costantini et al. 2012, Pinto et
al. 2013, Gatuzz et al. 2013; 2015, Schulz et al. 2016, Joachimi et al. 2016,
Zeegers et al. 2017; 2019, Rogantini et al. 2018; 2020, Psaradaki et al. 2020;
2023, Gatuzz et al. 2020; 2021). Many have primarily used the metallic Fe
$L$-shell absorption cross section of Kortright & Kim (2000) for all forms of
solid-phase Fe, due to a lack of data for other solid Fe-bearing compounds.
Most of these past studies have also approximated the extinction cross section
with only the absorption component, leaving out the important effects of dust
scattering (Corrales et al. 2016, Schulz et al. 2016, Hoffman & Draine 2016,
Zeegers et al. 2017).

Here, we present a new study of Fe $L$-shell extinction in the diffuse ISM.
This work improves on the current state of knowledge by taking advantage of
the laboratory data of Lee et al. (2009) for four different crystalline Fe-
bearing minerals that are candidates for forming ISM dust (fayalite [Fe2SiO4],
ferrous sulfate [FeSO4], hematite [$\alpha$-Fe2O3], and lepidocrocite
[$\gamma$-FeOOH]). We also use the metallic Fe data of Kortright & Kim (2000),
as was first used for astrophysical studies by Lee et al. (2001). We model
both the absorption and scattering components of the extinction cross section
for the Lee et al. (2009) data for the first time. We improve upon past
studies by adding the experimentally benchmarked theoretical photoabsorption
cross sections for Fe i-iv (Martins et al. 2006, Schippers et al. 2017,
Beerwerth et al. 2019, Schippers et al. 2021) to our extinction model. We also
determine the gas-phase neon (Ne) and oxygen (O) abundances in neutral and
ionic forms and use Ne and O as benchmarks for measuring the relative
abundances and depletion of Fe in the diffuse ISM. Chemically inert Ne is not
expected to be depleted in the ISM, while O appears to be depleted by only
$7-20\%$ in the diffuse ISM (Jenkins 2009, Psaradaki et al. 2020; 2023).

For this work, we have analyzed observations of the diffuse ISM along the
lines of sight to the black hole X-ray binaries (XRBs) Cygnus X-1 and GX
339-4. For our analysis, we choose to focus on high signal-to-noise archival
data sets from the Reflection Grating Spectrometer (RGS; den Herder et al.
2001) on the X-ray Multi-Mirror Mission Newton (XMM-Newton). In the energy
range for Fe $L$-shell extinction, RGS data provide $10-100$ times more
effective area111https://xmm-
tools.cosmos.esa.int/external/xmm_user_support/documentation/uhb/rgsaeff.html
than that of the Chandra X-ray Observatory
gratings222https://cxc.harvard.edu/proposer/POG/pdf/MPOG.pdf, mitigating the
slightly lower RGS spectral resolution. Consequently, at these energies, some
authors choose to rebin Chandra data to achieve a significant signal-to-noise
ratio (e.g., Gatuzz et al. 2015, Westphal et al. 2019), effectively reducing
the spectral resolution to less than that of RGS.

This paper is organized as follows. Section 2 reviews the properties and
current state of knowledge of Fe in the diffuse ISM. Section 3 describes the
Fe $L$-shell extinction models used here and the gas- and solid-phase data
incorporated into the models. Section 4 describes the sources, the X-ray
observations, and the RGS data reduction to generate spectra. Section 5
presents the results of our spectral analysis. Section 6 discusses the
astrophysical implications of our findings. A summary and conclusions are
given in Section 7.

##  2 Iron in the Diffuse ISM

The interstellar absorption signatures imprinted upon X-ray spectra of distant
Galactic XRBs represent the total attenuation across many kpc of interstellar
material, and thus samples all the known phases of the ISM (Draine 2011): the
cold neutral medium (CNM, $T\sim 100-1000$ K and $n_{\rm H}\sim 30$ cm-3), the
warm neutral medium (WNM, $T\sim 5000$ K and $n_{\rm H}\sim 0.6$ cm-3), and
the warm ionized medium (WIM, $T\sim 10^{4}$ K and $n_{\rm H}\sim 0.1$ cm-3).
Here, $T$ is the gas temperature and $n_{\rm H}$ is the hydrogen nuclei number
density. In all three phases, we expect the majority of gas-phase Fe to be
singly ionized (Fe ii). In the CNM, which makes up $\sim 1\%$ of the local
ISM, UV and optical observations find the gas-phase ratio of Fe ii : Fe i
$\sim 1.0:0.001$ (Snow et al. 2002, Welty et al. 2003, Miller et al. 2007,
Jensen & Snow 2007, Jenkins 2009). This is because the ionization potential
for Fe i at 7.8 eV is below the Lyman limit of 13.6 eV (912 Å), above which
the interstellar radiation field (ISRF) is heavily absorbed by interstellar
atomic H (Snow & McCall 2006). Thus Fe i is quickly ionized by the ISRF. Fe ii
has an ionization potential of 16.18 eV (738 Å), above the Lyman limit, making
the abundance of Fe iii generally negligible except near intense
photoionization sources. The WNM has a filling factor in the Galactic disk of
$\sim 40\%$ (Draine 2011). It is heated, in part, by photoelectrons from dust,
but the gas-phase temperature is still too low to form Fe iii, making Fe ii
the dominant ion (Jensen & Snow 2007, Bryans et al. 2009). The WIM contains
$\sim 16\%$ of the baryonic mass of the Galaxy, and about half of it lies
within 500 pc of the Galactic disk midplane (Draine 2011). In the WIM, UV and
optical observations find a gas-phase ratio of Fe iii : Fe ii : Fe i ratio of
$\sim 0.25:1.0:0.001$ (Snow 1976; 1977, Welty et al. 2003, Jenkins 2009).
Thus, in these three phases of the ISM, Fe ii is the only gas-phase
interstellar Fe species expected to produce an appreciable X-ray absorption
signature.

Studies comparing the gas-phase ISM Fe abundances to solar abundances
demonstrate that the majority of Fe is depleted relative to solar and hence
must have been incorporated into dust grains, which do not produce distinctive
solid-Fe spectroscopic features in the UV or optical. The amount of Fe
suspected to remain in the gas phase varies depending on the ISM temperature
and location in the Galactic disk, ranging from $\sim 0.5\%$ solar abundance
in cold clouds to $\sim 15-20\%$ solar abundance in the warm Milky Way halo
(Savage & Sembach 1996, Jones 2000). In a study of the atomic absorption from
over 200 sight-lines within 3 pc of the Milky Way disk, Jenkins (2009) finds a
maximum of 10% of interstellar Fe in the gas phase, depleting further as other
elements also disappear from the gas phase. Thus to achieve a full inventory
for the state of interstellar Fe, we must focus predominantly on identifying
the Fe-bearing compounds that make up interstellar dust, which account for the
majority of the X-ray attenuation in the Fe $L$-shell photoabsorption region.

The amount of interstellar Fe in dust grains is tied to their production by
stellar nucleosynthesis and the balance between their growth and destruction
rates (Dwek 1998, Nozawa et al. 2003, Ferrarotti & Gail 2006, Nozawa et al.
2007, Zhukovska et al. 2008; 2018). Fe is predominantly produced by Type-Ia
supernovae (SNe), with some contributions from core-collapse SNe and
asymptotic giant branch (AGB) stars (Dwek 2016). Type-Ia SNe have explosion
energies and expansion velocities similar to core-collapse SNe, but the
temperature and density of Type-Ia SNe ejecta decrease more rapidly, so there
is less time available in Type-Ia SNe ejecta for the formation of molecules
and grains. Dust growth in Type-Ia SNe ejecta is also impeded by radioactive
56Ni, which produces gamma rays and fast electrons that are effective at
dissociating molecular precursors of dust (Nozawa et al. 2011). Furthermore,
in the ejecta of Type-Ia SNe, there is no observational evidence to date for
significant dust formation with a mass comparable to that observed in core-
collapse SNe (Gomez et al. 2012). So it is puzzling that diffuse ISM
observations of gas-phase Fe find that $90-99$% of interstellar Fe is locked
up in dust. This implies that much of the Fe-bearing solids in the ISM must
form and grow in the diffuse ISM through a mechanism yet to be determined.

Recent studies of Fe-bearing ISM solids have made some progress in identifying
their composition. Kemper et al. (2004) studied $10~{}\mu{\rm m}$ absorption
towards the Galactic Center. Their findings suggests that interstellar
silicate dust consists by mass of $\sim 85$% olivine (Mg2xFe2(1-x)SiO4) and
$\sim 15$% pyroxene (MgxFe1-xSiO3). A broader analysis of the 10 and 20
$\mu{\rm m}$ features by Min et al. (2007) suggests equal olivine and pyroxene
abundances, by mass, and that Mg-rich silicates with Mg/(Fe+Mg)$\approx 0.9$
provide a better fit to the 10 and 20 $\mu{\rm m}$ features (Min et al. 2007).
This ratio aligns with the observed relative Mg and Fe abundances in
GEMS333While the relative Mg and Fe abundances in GEMS appear close to that
observed in the ISM, the relative O/Si and (Mg+Fe)/Si abundances are not (Min
et al. 2007). (Glass with Embedded Metals and Sulfides, thought to be remnants
of presolar dust grains that have not been destroyed or reworked by planetary
formation processes; Bradley 1994, Altobelli et al. 2016, Ishii et al. 2018).
Given that the solar and ISM abundances of Mg and Fe seem to be approximately
equal (Wilms et al. 2000), it is thought that the majority of interstellar Fe
is not incorporated in silicates but rather in compounds such as iron oxides,
ferrous sulfates, or metallic iron (Henning et al. 1995, Jones et al. 2013,
Köhler et al. 2014).

It has been hypothesized that much of the Fe in the ISM might be in the form
of Fe-rich nanoparticles (Hoang & Lazarian 2016, Gioannini et al. 2017).
Studies from Zhukovska et al. (2018) used hydrodynamical simulations for the
evolution of giant molecular clouds and analytical expressions for the
timescales of dust growth and destruction in order to match the depletion
rates of interstellar Fe observed across different phases of the ISM. They
found that a population of $1-10$ nm sized metallic iron nanoparticles could
reproduce the observed gas-phase Fe abundance profiles in the WNM, provided
that 70% of the nanoparticles were incorporated as inclusions in larger
silicate dust grains; otherwise, the destruction efficiency for metallic Fe
dust would have to be five times less than that for silicates. Choban et al.
(2022) came to similar conclusions when incorporating prescriptions for dust
growth and destruction into FIRE-2 simulations444See the FIRE (Feedback in
Realistic Environments) collaboration, http://fire.northwestern.edu – namely,
they require Fe nanoparticles in order to describe the observed ISM depletion
rates. Over the course of their 2 Gyr simulations, bare metallic Fe grains
accounted for $\leq 10\%$ of the total ISM dust mass, while it was assumed
that 70% of the total mass of Fe nanoparticles was incorporated into
silicates. The sources of these potential Fe nanoparticles remain uncertain,
as microgravity experiments of gas-phase Fe condensation in ISM-like
environments show no evidence that gas-phase Fe can spontaneously condense
into metallic particles (Kimura et al. 2017). Fe inclusions might instead form
spontaneously through thermal annealing or in the process of silicate
condensation (Davoisne et al. 2006, Matsuno et al. 2021).

Distinguishing among Fe bearing minerals in the ISM is also important for
other areas of astrophysics, as Fe in dust can have effects on the physical
conditions of the ISM. Whether free-flying or bound within a larger grain of
different composition, metallic Fe nanoparticles can affect photoelectic
heating in the ISM, can align dust grains along ambient interstellar magnetic
fields, and can emit at sub-millimeter or millimeter wavelengths (Draine &
Hensley 2012; 2013, Hoang & Lazarian 2016, Hensley & Draine 2017b). A
suspected population of free metallic Fe nanoparticles has been invoked to
explain the source of anomalous microwave emission (AME) observed at $\sim 30$
GHz (Hensley & Draine 2017b). However, Hensley & Draine (2017a) also showed
that Fe nanoparticles cannot fully account for the observed AME without
violating other observed infrared (IR) and extinction properties of
interstellar dust. Thus, the existence of metallic Fe nanoparticles in the ISM
is still a subject of debate.

Another question is the role of interstellar dust in contributing to planet
formation. Interstellar dust certainly delivers metals to protostellar
nebulae, but it is yet to be determined how much of that dust survives
processing in a protoplanetary disk and contributes to planetesimal formation.
Suspected interstellar and “presolar” grains, including GEMS, are identified
in meteorites via their isotopic anomalies relative to the solar system
(Zinner et al. 1987, Nittler 2003). Interstellar dust is more likely to
survive in the outer regions of planetary systems, leading researchers to
consider some cometary dust as a potential interstellar analog (Brownlee et
al. 2006). In situ collection of dust in the solar system has also yielded
suspected interstellar particles based on their isotopic ratios (Messenger
2000) and measured impact velocities in the collection medium (Westphal et al.
2014). In all of these cases, Fe appears in trace amounts. When it is
observed, Fe is found to be sequestered into Fe-Ni metals and Fe-rich sulfides
(Hanner & Zolensky 2010, Mann & Jessberger 2010, Bradley 2010). Confirming
whether or not these particles collected in our Solar System are truly
representative of unprocessed interstellar dust requires direct measurements
of solid-phase interstellar metals. High resolution absorption spectroscopy of
photoabsorption structure near the photoelectric absorption edge, in the X-ray
band, provides such an opportunity (Lee & Ravel 2005).

##  3 Modeling Iron L-Shell Spectroscopy in the Diffuse ISM

X-ray studies of Fe in the diffuse ISM focus primarily on the Fe $L$-shell
features in the $0.70-0.73$ keV ($16.9-17.7$ Å) bandpass. Past X-ray studies
of Fe in the diffuse ISM have been hindered by the lack of reliable publicly
available models for Fe $L$-shell extinction by gas- and solid-phase Fe. We
discuss this further in Sec. 6.3. The field has focused on the Fe $L$ shell,
in part, because the Fe $K$-shell features are more challenging to study in
the ISM. They have a much smaller absorption cross section and can only be
detected for very high hydrogen column densities of $N({\rm H})>10^{22}$ cm2
(Rogantini et al. 2018).

In this section, we briefly review the physics of the Fe $L$-shell features.
We then go on to describe the extinction models that we have updated with
state-of-the-art data for gas- and solid-phase Fe and that we use for the
present study. We begin with the simpler gas-phase Fe and then describe solid-
phase Fe.

###  3.1 Gas-Phase Fe

Near neutral Fe i, ii, iii, and iv [Fe(0,1,2,3)+, respectively] exhibit strong
resonant photoabsorption between $0.70-0.73$ keV (Martins et al. 2006,
Schippers et al. 2017, Beerwerth et al. 2019, Schippers et al. 2021). This is
due to their atomic structure, with a filled $L$-shell ($2s^{2}2p^{6}$) and a
partly filled $M$-shell (a filled $3s^{2}3p^{6}$ and partly filled $3d$
subshell). The strongest X-ray absorption features are due to $2p\to 3d$
dipole-allowed photoexcitations. These bound-bound transitions form $L$-shell-
hole states that have insignificant fluorescence yields and predominantly
autoionize (Krause 1979, Kaastra & Mewe 1993). Hence, the $L$-shell
photoionization and photoabsorption cross section are nearly identical for
these ions. The strongest spectral feature is due to $2p_{3/2}\to 3d$
photoabsorption ($L_{3}$). At slightly higher energies and less strong is the
$2p_{1/2}\to 3d$ photoabsorption feature ($L_{2}$). Going to higher photon
energies (shorter wavelengths), the $2p\to nd$ photoexcitations to higher
Rydberg levels $n$ blend together and remove any edge-like or step-like
structure in the photoabsorption cross section at the direct ionization
threshold, resulting in a smooth behavior above $\approx 0.73$ keV (below
$\approx 17$ Å).

In this work, we use the experimentally benchmarked theoretical
photoabsorption cross sections for Fe i (Martins et al. 2006), Fe ii
(Schippers et al. 2017), Fe iii (Schippers et al. 2021), and Fe iv (Beerwerth
et al. 2019). The uncertainty of the calculated resonance positions is $\sim
2-3$ eV. This is typical for calculations of transitions in the soft X-ray
regime for multi-electron systems using the Hartree-Fock method and is the
consequence mostly of neglecting electron correlation effects (Martins 2001,
Martins et al. 2006, Hirsch et al. 2012b). Smaller uncertainties can be
obtained using methods that include the electron-electron interaction, such as
the multiconfiguration Dirac-Fock approach, but these methods can rapidly
become computationally prohibitive for complex multi-electron systems, such as
the 3d metal atoms with many open shells. For these reasons, for this work we
have shifted the theoretical photoabsorption cross sections to best match the
experimental data by $-2.7$, $-2.8$, $-2.5$, and $-2.2$ eV, for Fe i, ii, iii,
and iv, respectively. The resulting energy scales are estimated to have an
accuracy of $\sim 0.2$ eV, based on calibration of the energy scale of the
experimental results that have been used to benchmark the theoretical
calculations (for details see Schippers et al. 2017; 2021). These specific
energy shifts were only applied to the gas-phase Fe i-iv and not to any of the
solid-phase Fe.

Our modeling work here builds on the model of Gatuzz et al. (2015), ISMabs, to
study gas-phase X-ray extinction in the diffuse ISM. ISMabs is designed for
high-resolution X-ray spectroscopy and meant to be used as a local model in
conjunction with the X-ray spectral fitting package XSPEC (Arnaud 1996). It
includes the gas-phase species H i [H0+], He i,ii [He(0,1)+], C i,ii,iii
[C(0,1,2)+], N i,ii,iii [N(0,1,2)+], O i,ii,iii [O(0,1,2)+], Ne i,ii,iii
[Ne(0,1,2)+], Mg i,ii,iii [Mg(0,1,2)+], Si i,ii,iii [Si(0,1,2)+], S i,ii,iii
[S(0,1,2)+], and Ar i,ii,iii [Ar(0,1,2)+]. The only non-gas-phase species
included is metallic Fe, for which ISMabs uses the photoabsorption data of
Kortright & Kim (2000). However, Gatuzz et al. (2015) did not include the
scattering portion of the metallic Fe extinction cross section and they also
blue shifted the photoabsorption cross section by 1.6 eV to best fit the
astronomical observations.

For this work, we did not use the metallic Fe component of ISMabs. Instead, we
incorporated the data of Kortright & Kim (2000) into our model for solid-phase
extinction, which includes the scattering component, as is described in more
detail below. In addition, we have modified ISMabs to include the energy-
shifted theoretical Fe i-iv gas photoabsorption data described above.

###  3.2 Solid-Phase Fe

When ionizing photons penetrate dust particles, the resulting absorption
spectrum differs from that of the gas phase for the same element. First, the
energy levels of the atom can be shifted by both the ionic bonds formed and
the coordination of the atom within the solid (e.g., van Aken & Liebscher
2002). The shape of the solid-phase photoabsorption cross section can be
further altered as the probability wave of the ejected photoelectron scatters
off nearby atoms, undergoing quantum interference that leads to oscillatory
variations in the absorption cross section (Woo 1995, Lee & Ravel 2005).
Together, the resulting shape of the absorption cross section is broadly
referred to as X-ray absorption fine structure (XAFS).555This may also be
referred to as near edge X-ray absorption fine structure (NEXAFS), X-ray
absorption near-edge structure (XANES), or extended X-ray absorption fine
structure (EXAFS). We use XAFS as a generic catch-all term for these effects.
In analyzing high-resolution spectra obtained from Galactic X-ray binaries,
XAFS provide a means to differentiate among gas and various solid phase-
species of interstellar elements.

####  3.2.1 Photoabsorption Data

Fe-bearing solids exhibit $L_{3}$ and $L_{2}$ photoabsorption features that
are similar to gas-phase Fe. In metallic Fe, the Fe $L_{3}$ and $L_{2}$
photoabsorption features arise, respectively, from $2p_{3/2}$ and $2p_{1/2}$
electrons undergoing dipole-allowed transitions to unoccupied $d$-like
orbitals near the Fermi level of the solid metal lattice (e.g., Mott 1949,
Miedema & de Groot 2013). Similar transitions occur in other Fe-bearing
compounds, though the energies and magnitudes of these $L_{3}$ and $L_{2}$
features can shift compared to that for metallic Fe and gas-phase near-neutral
Fe (Crocombette et al. 1995, van Aken & Liebscher 2002, Peak & Regier 2012).
In compounds, electrons are typically removed (i.e., oxidized) from the Fe
atoms to fill the outermost shell of other atoms within the solid. For
example, in fayalite (Fe2SiO4) and ferrous sulfate (FeSO4) the Fe is ferrous
(Fe2+), while in hematite ($\alpha$-Fe2O3) and lepidocrocite ($\gamma$-FeOOH)
it is ferric (Fe3+). The relevant properties of these compounds are summarized
in Table 1, which also gives the corresponding Fe nuclei column density
$N_{\rm Fe}$, determined from the stoichiometry of each compound. The chemical
shift of $\sim 1$ eV between the peaks in the photoabsorption cross section
for ferrous and ferric states (van Aken & Liebscher 2002) is readily
detectable with the spectral resolution of the XMM-Newton and Chandra grating
spectra (discuss in more detail below and in Section 4.2).

Table 1: Compound properties assumed for this work.

Compound | Formula | Effective | Density | $N_{\rm Fe}$
---|---|---|---|---
|  | charge |  |
|  | state | (g cm-3) | $\left(10^{17}~{}{\rm atoms}/10^{-4}~{}{\rm g}\right)$
Metallic Iron | Fe | Fe0+ | 7.6aaEncyclopedia of Minerals, 2nd edition (Roberts et al. 1990) | 10.78
Fayalite | Fe2SiO4 | Fe2+ | 4.39aaEncyclopedia of Minerals, 2nd edition (Roberts et al. 1990) | 5.91
Ferrous Sulfate | FeSO4 | Fe2+ | 3.65bbUnless otherwise noted, read off graphs with an estimated reading uncertainty of approximately $\pm 0.3$ eV. | 3.96
Hematite |  $\alpha$-Fe2O3 | Fe3+ | 5.3aaEncyclopedia of Minerals, 2nd edition (Roberts et al. 1990) | 7.54
Lepidocrocite |  $\gamma$-FeOOH | Fe3+ | 4.0aaEncyclopedia of Minerals, 2nd edition (Roberts et al. 1990) | 6.78
bbUnless otherwise noted, read off graphs with an estimated reading
uncertainty of approximately $\pm 0.3$ eV.

CRC Handbook of Chemistry and Physics (Lide 2000)

Spectroscopically, we require accurate photoabsorption data to distinguish the
various condensed matter forms of Fe ions from one another, and from their
gas-phase counterparts. Specifically, what are needed are absolute cross
sections on energy scales with an uncertainty much smaller than the accuracy
of the X-ray energy scale for the XMM-Newton and Chandra grating
spectrometers. Reliable data exist for gas-phase Fe, as described above. The
situation is not the same for solid-phase Fe. Theoretical cross-section data
for solid-phase Fe $L$-shell photoabsorption are beyond current computational
capabilities (e.g., van Aken & Liebscher 2002, Miedema & de Groot 2013). In
addition, the vast majority of laboratory measurements have reported only
relative cross section scales (Miedema & de Groot 2013) and are not suitable
for astrophysical modeling.

The purpose of this work is to utilize the absolute cross sections for Fe
$L$-shell photoabsorption of Lee et al. (2009) for the crystalline minerals
fayalite, ferrous sulfate, hematite, and lepidocrocite, which are predicted to
be possible candidates for dust in the ISM. Prior to this, the only similar
available cross sections were the metallic Fe results of Kortright & Kim
(2000). Their results have been used extensively for X-ray studies of the ISM
(e.g., Schulz et al. 2002, Juett et al. 2006, Hanke et al. 2009, Gatuzz et al.
2015). For our work, we use the Kortright & Kim (2000) and Lee et al. (2009)
data in our solid-phase Fe photoextinction models described below. Absolute
measurements for other Fe-bearing materials have also been reported by
Westphal et al. (2019) and Psaradaki (2021). The cross-sections from the Lee
et al. (2009) study are complementary to these data sets. Lee et al. (2009)
include iron-oxide and Fe-S compounds not used in these aforementioned
studies. In this work, we fit X-ray spectra with models that include the dust
scattering component derived from the Lee et al. (2009) measurements (Section
3.2.2), for the first time.

Table 2: Experimentally measured $L_{3}$ peak energies in eV.a,ba,bfootnotemark:  Reference | Metallic iron | Fayalite | Ferrous sulfate | Hematite | Lepidocrocite
---|---|---|---|---|---
| Fe | Fe2SiO4 | FeSO4 |  $\alpha$-Fe2O4 |  $\gamma$-FeOOH
Fuggle & Martensson (1980)ccExplicitly given in the reference and, if available, the uncertainty. | $706.7\pm 1.0$ |  |  |  |
Leapman et al. (1982) | 707.3 |  |  |  |
Fink et al. (1985)ccExplicitly given in the reference and, if available, the uncertainty. | 706.6 |  |  |  |
Krishnan (1990) |  | 710.5 |  |  |
Krivanek & Paterson (1990) |  |  |  | 708.4 |
Kuiper et al. (1993) |  |  |  | 710.0 |
Garvie et al. (1994)ccExplicitly given in the reference and, if available, the uncertainty. |  |  |  | $709.5\pm 0.2$ |
Chen et al. (1995) | 707.7 |  |  |  |
Crocombette et al. (1995) |  |  |  | 702.8 |
Schedel-Niedrig et al. (1995)ccExplicitly given in the reference and, if available, the uncertainty. |  |  |  | $711.3\pm 0.5$ |
Garvie & Buseck (1998) | 707.8 | 707.7 |  |  |
Kurata et al. (1998) |  |  |  | 709.2 |
Berling et al. (1999) | 707.5 |  |  |  |
Gloter et al. (2000) |  |  |  | 711.2 |
Kortright & Kim (2000)ddPrivate communication. | 706.9 |  |  |  |
Regan et al. (2001) |  |  |  | 709.3 |
Mitterbauer et al. (2003) |  |  |  | 708.9 |
Todd et al. (2003)ccExplicitly given in the reference and, if available, the uncertainty. |  |  |  | 710.2 |
Calvert et al. (2005)ccExplicitly given in the reference and, if available, the uncertainty. |  |  |  | 709.5 |
de Groot et al. (2005) |  | 707.9 |  | 709.5 |
Vayssieres et al. (2005) |  |  |  | 709.2 |
Wang et al. (2006) | 709.0 |  |  |  |
Braicovich et al. (2007) | 708.4 |  |  |  |
Kang et al. (2008) | 707.3 |  |  |  |
Chen et al. (2009) |  |  |  | 709.7 |
Feldhoff et al. (2009)ccExplicitly given in the reference and, if available, the uncertainty. | 708.1 |  |  | 710.8 |
Lee et al. (2009)ddPrivate communication. |  | 707.3 | 706.4 | 708.0 | $708.3$
Galakhov et al. (2010) | 708.3 |  |  |  |
Takagi et al. (2010) | 706.6 |  |  |  |
Titov et al. (2010)ccExplicitly given in the reference and, if available, the uncertainty. |  |  |  | 709.5 |
Chen et al. (2011) | 706.8 |  |  |  |
Braun et al. (2012) |  |  |  | 707.1 |
Hirsch et al. (2012a) | 708.0 |  |  |  |
Peak & Regier (2012)ccExplicitly given in the reference and, if available, the uncertainty. |  |  |  | 709.5 |
Average measurement | $707.5\pm 0.7$ | $708.4\pm 1.2$ |  | $709.1\pm 1.8$ |

Surveying the published relative and absolute Fe $L$ photoabsorption data
(most recently reviewed in Miedema & de Groot 2013), it is clear that one of
the biggest uncertainties is the accuracy of the energy scales of the data.
This can be readily seen by the comparison in Table 2 for the experimentally
measured peak energies of the $L_{3}$ feature for metallic iron,
fayalite666The fayalite $L_{3}$ feature reported by Krishnan (1990), Garvie &
Buseck (1998), and de Groot et al. (2005) is single peaked. However, Lee et
al. (2009) report an $L_{3}$ feature that consists of two peaks of nearly the
same amplitude at 706.6 and 708.3 eV. This may be due to the higher energy
resolution of their measurements. Here, we use the average energy of these two
peaks., and hematite. The published values range, respectively, from
$706.7-709.0$, $707.5-710.5$, and $702.8-711.3$ eV. These ranges are much
larger than the accuracy of the wavelength scales for RGS777See Table 8 of
https://xmm-
tools.cosmos.esa.int/external/xmm_user_support/documentation/uhb/XMM_UHB.pdf
and for the Chandra gratings888See
https://cxc.harvard.edu/proposer/POG/html/chap8.html of $\pm 5$ and $\pm 11$
mÅ, respectively. In the Fe $L$ bandpass, these correspond to energy scale
uncertainties of approximately $\pm 0.2$ and $\pm 0.4$ eV, respectively. For
lepidocrocite and ferrous sulfate, we are unaware of any measurements other
than those of Lee et al. (2009) with which to compare. In order to account for
the uncertainties in the energy scale of the laboratory data, we found that we
needed to introduced an additive shift to the energy scale for our
photoextinction modes to best match to the observational data, as is described
in Section 5.2. The average and standard deviation of the measured energy
position of the $L_{3}$ peak is reported in Table 2 merely to convey the high
variance found in the literature.

####  3.2.2 Photoextinction Model

Computing the absorption and scattering properties of a material requires
knowledge of the real and imaginary parts of the index of refraction. The
absorption cross sections derived via laboratory work provide the imaginary
part of the optical constants; the real part can be derived via the Kramers-
Kronig relations (Kronig 1926, Lucarini et al. 2005). We use the published
photoabsorption data of Kortright & Kim (2000) for metallic Fe and of Lee et
al. (2009) for crystalline fayalite, ferrous sulfate, hematite and
lepidocrocite. For each compound, we used the kkcalc software (Watts 2014) to
apply the Kramers-Kroenig transformation on the Fe $L$-shell absorption data.
Table 1 summarizes the chemical formula (i.e., stoichiometry) and mineral
densities (Roberts et al. 1990, Lide 2000) that are supplied as inputs to
kkcalc, which uses that information to model photoionization cross sections of
the remaining elements in the compound (following Henke et al. 1993), and to
extrapolate the absorption curve up to 500 keV (following Biggs & Lighthill
1988). As a result, our model optical constants span the entire wavelength
range covered by the XMM-Newton RGS observations and includes a rudimentary
model for photoabsorption from other elements in each compound (e.g., O and
Si).

With the optical constants and the Mie scattering algorithm of Bohren &
Huffman (1983), we compute the absorption and scattering cross sections for
spherical grains. Given the sub-arcminute imaging resolution of XMM-Newton, as
well as the energy range and grain sizes considered, the amount of scattered
light captured by the spectral extraction aperture is negligible (Corrales et
al. 2016). The final dust extinction cross section was computed by summing the
absorption and scattering cross sections of each grain and then integrating
over the grain size distribution. We follow the ISM dust size properties
derived by Mathis et al. (1977; hereafter MRN).

To apply the cross sections in a standard X-ray spectral fitting framework, we
build on the high resolution ISM photoextinction model, ISMdust (Corrales et
al. 2016), a local model designed for use in XSPEC (Arnaud 1996). The default
version of ISMdust uses generic extinction cross sections for graphitic and
silicate dust grains, derived from the optical constants of Draine (2003),
where Fe photoabsorption is provided solely by the silicate component. We
generated a custom version of ISMdust, dubbed SolidFeL, to investigate solid
Fe $L$-shell extinction from an MRN distribution of spherical dust particles.
The transmitted flux $F$ for each model is given by

$\frac{F}{F_{a}}=\exp\left(-\Sigma_{i}\kappa_{i}M_{{\rm d},i}\right),$ (1)

where $F_{a}$ represents the flux of the gas-phase absorbed spectrum and
$\kappa_{i}$ is the extinction opacity (in g-1 cm2) for each compound $i$
integrated over the dust grain-size distribution. The free parameter in each
model is $M_{{\rm d},i}$, which is the dust mass column (in g cm-2) of
compound $i$. The SolidFeL extinction opacities are computed following the MRN
power-law slope of $-3.5$ with dust particle radii from 5 nm to $0.3~{}\mu{\rm
m}$.

##  4 Observations

###  4.1 Cygnus X-1 and GX 339-4

The Galactic black hole X-ray binaries Cygnus X-1 and GX 339-4 make excellent
background sources by which to study the composition of the line-of-sight ISM
via X-ray absorption (e.g., Schulz et al. 2002, Juett et al. 2006, Hanke et
al. 2009, Lee et al. 2009, Gatuzz et al. 2016, Westphal et al. 2019). Their
ISM column densities lie in an ideal range, large enough to produce deep Fe
$L$-shell extinction features without completely extinguishing the X-ray count
rate in the corresponding energy range. The broad-band X-ray spectrum of the
two sources are generally described by a two-component model with a variable
intensity (e.g., Tanaka & Lewin 1995, Schulz et al. 2002) consisting of a
power-law component and a multicolor disk component. As the two sight lines
are nearly orthogonal to each other, they sample the ISM in unique directions
around the Galactic plane.

Cygnus X-1 ($l_{\rm II}=71.33^{\circ}$, $b_{\rm II}=3.07^{\circ}$)999Galactic
longitude and latitude (B1950) was discovered at the dawn of X-ray astronomy
(Giacconi et al. 1967; and reference therein) and subsequently found to
consist of an $\approx 21~{}M_{\odot}$ black hole and an $\approx
41~{}M_{\odot}$ O9.7 Iab stellar companion (Sota et al. 2011, Miller-Jones et
al. 2021) at a distance of $\sim 2.2$ kpc (Ziółkowski 2005, Gaia Collaboration
2020). It has a binary orbit of $\approx 5.6$ days (Brocksopp et al. 1999), an
inclination of $i\approx 27^{\circ}$ (Miller-Jones et al. 2021), and a radial
velocity of $-2.7\pm 3.5$ km s-1 (Gontcharov 2006). It is characterized by two
distinct spectral states with a power-law photon index $\mathit{\Gamma}_{\rm
hard}\sim 1.7$ and $\mathit{\Gamma}_{\rm soft}\sim 2.5$, respectively. The
disk component is most manifest in the hard state and difficult to detect
otherwise. This source is highly variable and displays intervals of flaring
activity.

GX 339-4 ($l_{\rm II}=339.94^{\circ}$, $b_{\rm II}=-4.33^{\circ}$) is a black
hole XRB discovered in outbursts by Markert et al. (1973), a state found to
repeat $1-3$ times per year. Although GX 339-4 is well-studied, its physical
parameters remain somewhat uncertain due to the weakness of its companion star
(Cowley et al. 2002, Hynes et al. 2004, Zdziarski et al. 2004, Parker et al.
2016, Heida et al. 2017, Sreehari et al. 2019, Zdziarski et al. 2019). The
most recent near-IR study by Heida et al. (2017) suggest a K-type donor star
and confirms a reported binary orbit of 1.76 days (Hynes et al. 2003, Levine &
Corbet 2006). Heida et al. (2017) estimate an inclination between
$37^{\circ}-78^{\circ}$ and radial velocity of $64\pm 8$ km s-1. They derived
a binary mass function $f(M)=1.91\pm 0.08\ M_{\odot}$, a factor $\sim 3$ lower
than the earlier work, and predict a black hole mass in the range of $2.3-9.5\
M_{\odot}$. The latest distance estimate to GX 339-4 is 5 kpc (Gaia DR3, Gaia
Collaboration 2020)

###  4.2 XMM-Newton RGS Data

A wealth of archival XMM-Newton observations are available for both binaries
of interest. Cygnus X-1 was observed 13 times between $2004-2016$, using the
RGS as the prime instrument, and acquired in various flux states, from which
we select a subset based on its binary orbital phase, as described below. GX
339-4 was observed 21 times in its various flux states between $2001-2015$. We
selected observation from the 2015 transition back to the hard, quiescent flux
state (Petrucci et al. 2017). Logs of these observations are presented in
Tables 3 and 4, respectively.

The flux from both sources is too bright for the nominal image spectrometry
with XMM-Newton, and they are best studied using the high spectral resolution
RGS (den Herder et al. 2001). RGS consists of two detectors, RGS1 and RGS2,
comprising a linear array of nine charge coupled devices (CCDs) operated in
single-photon-counting and frame-transfer mode. Each detector sits at the
focal plane behind an X-ray mirror / reflection grating assembly pair that
disperse about half of the incident light onto the RGS detector plane. For a
given X-ray source, the dispersed spectrum appears as a set of arcs, one for
each spectral order. For this work we restrict our analysis to the summed
first-order spectrum to maximize the signal-to-noise. RGS achieves a resolving
power of $R=E/\Delta E=150-800$, limited by the $\approx$ $6^{\prime\prime}$
full width at half maximum (FWHM) spatial resolution of the telescope mirrors.
At the Fe $L$-shell features of interest in this study, RGS has a combined a
first-order effective area of $A_{\mathrm{eff}}\sim 110$ cm2 and energy
resolution of $\Delta E\approx 2.5$ eV. The RGS energy-scale calibration
accuracy in the vicinity of the Fe $L$-shell features is $\pm 0.2$ eV, less
than the chemical shift between the ferrous and ferric Fe $L$-shell features
considered in this work.

Table 3: XMM-Newton RGS Observations of Cygnus X-1 Used in this Work

ObsID | Start Date | Exposure | Mid MJD | Mid Phase
---|---|---|---|---
|  | (ks) | (TBD) |
0745250501 | 2016 May 29 | 144.8 | 57538.74449 | 0.324
0745250601 | 2016 May 31 | 145.4 | 57540.76372 | 0.684

Note. — Only the portion of the two observations that fell within the quadrant
outside of conjunction with the O-star were considered in this work, with a
total filtered exposure time of 195 ks. Here, ObsID is Observation
Identification number, UTC is Coordinated Universal Time, MJC is Modified
Julian Date, and TBD is Barycentric Dynamical Time.

Table 4: XMM-Newton RGS Observations of GX 339-4 Used in this Work  ObsID | Start Date | Exposure
---|---|---
|  | (ks)
0760646201 | 2015 Aug 28 | 18.8
0760646301 | 2015 Sep 02 | 18.0
0760646401 | 2015 Sep 07 | 22.5
0760646501 | 2015 Sep 12 | 20.9
0760646601 | 2015 Sep 17 | 48.7
0760646701 | 2015 Sep 30 | 48.0
Figure 1: Graphical representation of the binary phase of the Cygnus X-1 black
hole during the XMM-Newton RGS observations relative to the inferior
conjunction of the O-star, as determined from the radial-velocity-derived
ephemeris of Brocksopp et al. (1999; $P=5.599829$ days at phase-zero epoch
41874.207 MJD). Each observation in Table 3 is drawn as a solid arc in phase
that covers its exposure, ordered chronological with the innermost arc for the
earliest observation. In order to avoid possible phase-dependent interaction
of the X-rays from the black hole with the wind from the companion star, we
only consider exposure intervals in the green quadrants in our analysis.

All observations were reprocessed using the latest calibrations and reduced
and analysed using the Science Analysis Software (SAS) version
xmmsas_20190531_1155-18.0.0. Nominal RGS1 and RGS2 source and background
first-order spectral files and their response matrices were generated using
the SAS rgsproc task. This task calls rgsspectrum to extracted source photons
falling on a rectangle aperture whose width covers 95% of the point spread
function (PSF). The background is extracted from the adjacent regions well
outside the target PSF. There are no known contaminating sources in the
extracted field. Intervals of high background were automatically identified
and filtered out of the event files using the default threshold applied to the
background light curves. The resulting spectra from the two detectors were
combined using the SAS task rgscombine after verifying that spectral fits to
the individual components were in agreement within the measurement
uncertainties. Not all RGS CCDs were active during all observations, but by
merging the RGS1 and RGS2 data, a continuous dispersed spectrum was obtained
over the energy range of interest for the Fe $L$-shell absorption study.

These data were acquired in HighEventRateWithSES mode (i.e., SAS “Spectroscopy
HER+SES” mode), which is appropriate for bright sources. Nevertheless, some of
the brighter Cygnus X-1 observations suffered from pile-up, when more than one
photon fell in a pixel during a readout cycle, distorting the spectrum. These
observations were identified and rejected using the fractional difference in
flux between grating orders (e.g., Costantini et al. 2012).

Furthermore, the intrinsic spectrum of Cygnus X-1 may be modified due to
periodic interactions of the X-ray emisison from the compact object with the
wind from the binary companion. To minimize these possible spectral
distortions, we restricted our analysis to data accumulated during binary
phase intervals well away from the inferior or superior conjunction of the
O-star, following the approach of Westphal et al. (2019), using the the
radial-velocity-derived ephemeris of Brocksopp et al. (1999). This is
illustrated in Figure 1, where we highlight in green suitable portion of each
observation. We truncated and combined the two long observations that fell
mostly within the quadrants of the orbital phase perpendicular to conjunction
(given in Table 3). These observations were close in time and have similar
fluxes and spectra. In contrast, the orbital ephemeris for GX 339-4 is not yet
well established and we used the observations selected for consistent spectral
state.

After reprocessing, reducing, and filtering the data sets, eliminating piled-
up observations, and applying orbital phase and/or epoch selection, a single
merged source spectrum for each source was generated, along with each
associated background spectrum and response matrix. The final Cygnus X-1
spectrum yielded a filtered exposure time of 195 ks, with a net count rate of
11.3 cps in the full RGS energy band, and similarly, 444 ks and 3.11 cps for
GX 339-4. We fit the unbinned spectra using the XSPEC spectral fitting program
(version 12.10.1f, Arnaud 1996). All spectral channels are well populated
around the Fe features of interest; however, for GX 339-4, below $0.5$ keV,
the counts in some channels are $<10$. Hence, for consistency, we used
C-Statistic, $C_{\rm stat}$, tests (Cash 1979) for all fits.

##  5 Spectral Analysis and Results

###  5.1 Baseline Continuum Model

Our aim is to identify possible contributions to the ISM by the Fe-bearing
compounds codified in the SolidFeL model. Our method is to establish the
baseline continuum model for the observed spectra of Cygnus X-1 and GX 339-4,
outlined below, and then evaluate the likely contribution to the ISM of trial
Fe compounds for the fixed continuum.

The spectral continua of Cygnus X-1 and GX 339-4 over a nominal $0.45-1.80$
keV RGS bandpass are generally characterized by a simple non-thermal power-law
model. The only evidence for X-ray reflection is from two broad emission
features that are modeled here by two Gaussians, one centered at an energy
slightly below the Fe $L$-shell photoabsorption edge, and the other at an
energy just below the Ne $K$-shell edge. Similar features have been found in
spectra of other XRBs (Madej et al. 2010; 2014, Psaradaki et al. 2020).

To account for ISM absorption in the spectra of the two sources, we used a
combination of the ISMabs and ISMdust models. With ISMabs, we accounted for
contributions from C ii, N i, O i, ii, iii, Ne i, and Mg ii. In the diffuse
ISM, these are the dominant gas-phase species for their elements, due to the
combination of their ionization potentials and the shape of the ISRF (Snow &
McCall 2006). The absorption edges of C ii and N i fall below our fitted
energy range; however, they provide an important contribution to the continuum
shape, and their column densities are estimated as a constant fraction of
hydrogen column density $N({\rm H})$, using the abundances and depletion
factors of Wilms et al. (2000). Gatuzz et al. (2016), in their analysis of
both sources, found absorption by O i and Ne i and, at a much reduced level, O
ii and O iii as well as Ne ii and Ne iii. In our spectra, we clearly identify
O ii and O iii but neither Ne ii nor Ne iii is evident. In this baseline
continuum model, we did not include absorption by gas-phase Fe, as it has been
inferred to be heavily depleted in the diffuse ISM. We used ISMdust to account
for absorption by graphite and silicate dust grains. Following the findings of
Corrales et al. (2016), we set the dust-to-hydrogen mass ratio to 1% and the
silicate-to-graphite grain-mass ratio to 60/40.

The result of fitting the baseline continuum model (power-law, Gaussian
emission features, ISMabs, and ISMdust) is shown in Figure 2 for the two
sources. Their spectral parameters are presented in Table 5. Both sources are
in their soft state and the only spectral features from their expected
surrounding warm disks are the inferred X-ray reflection features. Previous
studies have found weak spectral features from from O vii or viii that is
attributed to hot gas either in the ISM or local to the source. For example,
for GX 339-4, Pinto et al. (2013) and Gatuzz & Churazov (2018) detect hot gas,
but the $N_{\rm H}$ of the hot gas is $\lesssim 2\%$ of the cold gas. Here, we
found no statistically significant spectral features for either source,
although this could be due to orbital phase differences in the data used for
the present analysis. The ISM column densities and underlying continuum models
are consistent with previous work (e.g., Schulz et al. 2002, Juett et al.
2006, Hanke et al. 2009, Lee et al. 2009, Gatuzz et al. 2016, Westphal et al.
2019). The ISMdust model describes dust abundance using a mass column, $M_{\rm
d}$ in units of $10^{-4}$ g cm-2, from which number densities are derived
following an MRN power-law distribution of dust grains from 0.005 to 0.3
$\mu{\rm m}$. The silicate and graphite dust mass columns are denoted $M_{\rm
sil}$ and $M_{\rm gra}$, respectively, in Table 5.

Figure 2:  The RGS spectra (crosses) for Cygnus X-1 (left) and GX 339-4
(right) using the parameters given in Table 5. The vertical error bars show
the $1\sigma$ statistical uncertainty and the horizontal the energy bin
widths. In each upper panel, the best-fit continuum model is shown by the
solid lines, while the dashed lines indicate the model components. The lower
panels show the residuals from the fitted model in units of $\sigma$. The gap
in the Cygnus X-1 spectrum is due to the loss of corresponding RGS CCD
detectors covering those dispersed wavelengths. Table 5: Best-fit Spectral
Results for the Baseline Continuum ModelaaXSPEC model: ISMabs * ISMdust *
(Pow+Gau+Gau). The ISMabs parameter $N(X)$ is the column density for species
$X$. The power-law model for the flux is of the form
$F(E)=N\exp(-\mathit{\Gamma})$. The Gaussian is of the form
$F(E)=N_{E_{x}}\exp[-(E-E_{x})^{2}/(2\sigma^{2})]$. See text for details.

|  |  | Cygnus X-1 | GX 339-4 |
---|---|---|---|---|---
Model | Parameter | Units | Value | Value | Notes
Coordinates | ($l_{\rm II}$, $b_{\rm II}$) | degree | (71.334, 3.066) | (338.939, -4.326) | Galactic (B1950)
ISMabs |  $N$(H) |  $10^{22}$ cm-2 | $0.567\pm 0.004$ | $0.49\pm 0.01$ |
|  $N$(C ii) |  $10^{16}$ cm-2 | $68.04\pm 0.04$ | $59.0\pm 0.1$ |  $N$(C ii) $=1.2\times 10^{-4}$ $N$(H)
|  $N$(N i) |  $10^{16}$ cm-2 | $43.09\pm 0.03$ | $37.4\pm 0.8$ |  $N$(N i) $=7.6\times 10^{-5}$ $N$(H)
|  $N$(O i) |  $10^{16}$ cm-2 | $310\pm 5$ | $273\pm 19$ |
|  $N$(O ii) |  $10^{16}$ cm-2 | $6.4\pm 2.8$ | $18_{-11}^{+17}$ |
|  $N$(O iii) |  $10^{16}$ cm-2 | $5\pm 2$ | $6_{-4}^{+9}$ |
|  $N$(Ne i) |  $10^{16}$ cm-2 | $45\pm 3$ | $39\pm 7$ |
|  $N$(Mg ii) |  $10^{16}$ cm-2 | $15.6\pm 2.7$ | $14\pm 8$ |
Power-law | $\mathit{\Gamma}$ | unity | $2.36\pm 0.03$ | $2.63\pm 0.07$ |
| $N_{\Gamma}$ | arbitrary | $2.36\pm 0.04$ | $0.152\pm 0.006$ |
Gaussian | $E_{1}$ | keV | $0.830\pm 0.002$ | $0.821\pm 0.007$ |
| $N_{E_{1}}$ | arbitrary | $(7.8\pm 0.5)\times 10^{-2}$ | $(2.5\pm 0.7)\times 10^{-3}$ |
| $E_{2}$ | keV | $0.679\pm 0.003$ | $0.664\pm 0.008$ |
| $N_{E_{2}}$ | arbitrary | $(7.4\pm 1.1)\times 10^{-2}$ | $(5.8\pm 3.2)\times 10^{-3}$ |
| $\sigma_{1,2}$ | keV | $(3.6\pm 0.5)\times 10^{-2}$ | $(3.2\pm 0.7)\times 10^{-2}$ |  $\sigma_{1,2}$ are linked
ISMdust | $M_{\rm sil}$ |  $10^{-4}$ g cm-2 | $0.569\pm 0.004$ | $0.494\pm 0.010$ |  $=1.00357$ $N$(H)bbISMdust parameters $M_{\rm sil}$ and $M_{\rm gra}$ are normalized in units of $10^{-4}$ g cm-2 equivalent to $N$(H) in units of $10^{22}$ cm-2.
| $M_{\rm gra}$ |  $10^{-4}$ g cm-2 | $0.379\pm 0.003$ | $0.329\pm 0.007$ |  $=0.669$ $N$(H)bbISMdust parameters $M_{\rm sil}$ and $M_{\rm gra}$ are normalized in units of $10^{-4}$ g cm-2 equivalent to $N$(H) in units of $10^{22}$ cm-2.
$C_{\rm stat}$(DoF) |  |  | 4295.58 (1814) | 2580.84(2013) |

Note. — Continuum spectral fits are for the $0.45-1.8$ keV range. Quoted
uncertainties correspond to a 90% confidence level. for two interesting
parameters. The degrees of freedom (DoF) for the fit statistic is given in
parentheses.

###  5.2 Fitting Solid-Phase Fe Photoabsorption

Figure 3:  RGS spectra of the Fe feature in Cygnus X-1 fitted in the
$0.69-0.74$ keV energy range with the (left) unshifted and (right) shifted
($+1.88$ eV) fayalite model in SolidFeL (solid lines). These panels highlight
the evidence for an energy offset error in the published experimental cross
section data. Similar results are found for the other species listed in Table
1.

Having established a well-defined continuum model, we then fit for the Fe-
bearing compounds by replacing the silicate component of the ISMdust model
with one of the trial SolidFeL compounds in Table 1. First, we refit the
baseline continuum model across the full $0.45-1.80$ keV band, and then we
restricted the fit to the Fe $L$-shell features around the narrow $0.69-0.74$
keV range, allowing only the abundance of the trial compound to vary.

These SolidFeL spectral fits yielded an unexpected and critical issue for
their analysis, as the residuals displayed artifacts suggesting that the
interstellar Fe absorption features are offset in energy compared to the
laboratory measurements. An example of this is shown in Figure 3 for SolidFeL
fits for fayalite using both unshifted and shifted laboratory data. Similar
artifacts were observed with every compound tried and for both sight lines. A
notable feature of this comparison is the apparent emission at energies below
the $L_{3}$ feature, which is solely an artifact of the inaccurate laboratory
energy scale.

Motivated by these artifacts, we reviewed the published literature reporting
absolute energies for Fe $L$-shell XAFS. This is discussed in detail in
Section 3.2 and summarized in Table 2. There we found variances of $\sim 1-2$
eV among the published $L_{3}$ peak energies. In principle, such a shift could
also depend on the photon energy (Müller et al. 2021); however, over the
narrow energy range of the Fe $L$-shell features, such an effect is only a
small correction to the overall shift and, thus, can be neglected here.

To allow for an offset in the laboratory measurements, we introduced an energy
shift to the photoabsorption data as a free parameter in our fits. Our new
fitting method follows all of the procedures above, after systematically
shifting the energy scale for each laboratory template in 0.1 eV increments
for each of the five Fe-bearing compounds considered here (see Figure 4).
Because the $L$-shell features are generally asymmetric and double-peaked,
this method provided a reasonably accurate determination of the best fit for
each trial compound using the shape of the extinction features.

The spectral fits that result from using the optimal shift is presented in
Table 6 and the best-fit spectra are shown explicitly for the case of hematite
in Figure 5 for Cygnus X-1 and GX 339-4. The fits using energy-shifted
lepidocrocite and fayalite for each sight line are nearly identical to those
shown here (see, e.g., the right panel of Figure 3 for a fit to Cygnus X-1
with energy-shifted fayalite). Table 6 demonstrates that, for a given
compound, the results for both lines of sight imply the same energy shift and
ranking of the goodness-of-fit statistic for that compound, which demonstrates
that the need to shift the laboratory data for Fe $L$-shell absorption is not
due to an anomaly in the Cygnus X-1 spectrum alone, as implied by previous
work (e.g., Juett et al. 2006, Hanke et al. 2009, Gatuzz et al. 2015). For
this study, we do not fit for multiple dust species at once because the
uncertainty of the correct systematic shift for each laboratory measurement
provides a free parameter space that is too large to draw meaningful
conclusions at this time.

Lastly, we also examined the fit when gas-phase Fe ii was included. The
resulting Fe ii column density was consistent with zero with a three-sigma
upper limit of 5% of the total Fe column density for each compound. This is in
agreement with past UV and optical studies that have found that over 90% of
interstellar Fe is in the solid phase. In the following, we present our
conclusions from fits to the Fe $L$-shell photoabsorption features that
include solid Fe-bearing compounds only.

Figure 4:  Calibration of the apparent offset error in the energy scale of the
measured photoabsorption cross sections for the Fe-bearing compounds used in
this work (see text for details). Plotted is the $C_{\rm stat}$ divided by the
DoF for fits to the spectra of Cygnus X-1 (left) and GX 339-4 (right) as a
function of an energy shift in 0.1 eV increments, for each SolidFeL compound.
These fits are restricted to the $0.69-0.74$ keV bandpass around the Fe
$L$-shell features with the fitted $0.45-1.8$ keV continuum model held fixed.
Note the change of the ordinate scale between the two panels. Table 6: Narrow-
band spectral fits to the Fe $L$-shell Features in the Two ISM Sight Lines
Probed.

|  | Cygnus X-1 | GX 339-4
---|---|---|---
SolidFeL |  | $\Delta E_{\rm shift}$ | $M_{{\rm d},i}$ | $C_{\rm stat}$ | $\Delta E_{\rm shift}$ | $M_{{\rm d},i}$ | $C_{\rm stat}$
Compound $i$ | Formula | (eV) | ($10^{-4}~{}{\rm g~{}cm}^{-2}$) | (${\rm DoF}=119$) | (eV) | ($10^{-4}~{}{\rm g~{}cm}^{-2}$) | (${\rm DoF}=119$)
Lepidocrocite |  $\gamma$-FeOOH | $1.14\pm 0.26$ | $0.301\pm 0.022$ | 201.1 | $1.04\pm 0.61$ | $0.349\pm 0.047$ | 145.1
Hematite |  $\alpha$-Fe2O3 | $1.21\pm 0.26$ | $0.320\pm 0.022$ | 211.0 | $1.10\pm 0.61$ | $0.376\pm 0.050$ | 145.2
Fayalite | Fe2SiO4 | $1.88\pm 0.27$ | $0.277\pm 0.019$ | 216.3 | $1.80\pm 0.63$ | $0.318\pm 0.043$ | 143.9
Ferrous Sulfate | FeSO4 | $2.32\pm 0.26$ | $0.394\pm 0.032$ | 467.7 | $2.18\pm 0.59$ | $0.533\pm 0.082$ | 173.4
Metallic Iron | Fe | $2.51\pm 0.31$ | $0.252\pm 0.009$ | 357.6 | $2.66\pm 0.71$ | $0.290\pm 0.026$ | 155.1

Note. — The calibrated energy shift $\Delta E_{\rm shift}$ is applied in the
laboratory frame of the cross sections for each compound (see text). The mass
column is from the fits to the Fe features in the $0.69-0.74$ keV range using
the SolidFeL model with the fitted $0.45-1.8$ keV continuum model held fixed.
Uncertainties are quoted at a 90% confidence level for one interesting
parameter.

Figure 5:  RGS spectra highlighting the Fe $L$-shell extinction features of
the diffuse ISM in the $0.69-0.74$ keV energy range for (left) Cygnus X-1 and
(right) GX 339-4. The solid line shows the best-fit model for energy-shifted
hematite.

##  6 Discussion

Given the $1-2$ eV spread in the published laboratory data for the Fe $L_{3}$
photoabsorption feature, it is difficult to constrain the mineralogy of the
interstellar dust along the two sight lines based solely on the best-fit
energy shifts reported in Table 6. However, a few conclusions can be made
based on the shape of the Fe L photoabsorption features relative to the
laboratory results. First, ferrous sulfate produces the worst fit for each
sight line, leading us to conclude that this compound is unlikely to be
abundant in the diffuse ISM. The second worst fitting compound is that of an
MRN distribution of metallic Fe particles, no matter what energy shift is
applied. This leads us to conclude that metallic Fe is not likely to be the
dominant carrier of Fe in the diffuse ISM. Our study, however, does not
necessarily rule out the presence of metallic Fe nanoparticles, which we
discuss in Section 6.2.

This leaves lepidocrocite, hematite, and fayalite as the three most viable
interstellar Fe-bearing compounds in the diffuse ISM, for the sight lines and
analyses presented here. However, it is difficult to determine the most likely
of the three as their best-fit statistics are so similar. In addition, the
energy shifts for lepidocrocite and hematite are nearly identical. The
similarity in the fits for these two species is because their photoabsorption
structures are so alike in shape and magnitude that current X-ray astronomy
spectrometers cannot distinguish the subtle differences between the two
compounds. We might, however, be able to disfavor fayalite, in part because of
abundance constraints as we discuss later.

###  6.1 Abundances

The best-fit model free parameters, using the optimal energy shift for each
Fe-bearing compound, are given in Tables 7 and 8. The tables also report the
corresponding O, Fe, and Ne abundances per H atom for each of these fits,
reported as $\log A_{Z}+12$, where $A_{Z}$ is the number of atoms per H atom
for the element $Z$. To convert the mass column $M_{{\rm d},i}$ for each Fe-
bearing species $i$ into an Fe atom column density, we multiply by the value
of $N_{\rm Fe}$ given in Table 1. We found that the percent error on the free
parameters listed in both tables are identical to those in Table 5. We use
those percent errors in reporting the propagated error on the abundance
measurements in Tables 7 and 8. Below, we compare these abundances to Solar
abundances (Asplund et al. 2021) and to the canonical ISM abundances used in
X-ray spectroscopy (Wilms et al. 2000).

Note that the Mg abundances reported in Tables 7 and 8 are non-physical
because we replaced the ISMdust silicate model with the Lee et al. (2009) Fe-
bearing compounds, none of which contained Mg. This causes the best gas-phase
Mg component from ISMabs to get larger, compared to the results shown in Table
5, in order to make up for the missing extinction component that was present
in the ISMdust silicate model. Furthermore, there is no specific stoichiometry
associated with the Draine (2003) silicate opacities used in ISMdust, so a
physically relevant estimation of the Mg column from $M_{\rm sil}$ in Table 5
is not possible in this work.

Table 7: Best fit results for Cygnus X-1 with dust model energy scale shifted

Model | Parameter | Unit | Fayalite | Hematite | Lepidocrocite | Metallic Fe | Ferrous Sulfate
---|---|---|---|---|---|---|---
ISMabs |  $N$(H) | 1022 cm-2 | 0.582 | 0.596 | 0.592 | 0.592 | 0.574
|  $N$(O i) | 1016cm-2 | 366 | 371 | 363 | 383 | 354
|  $N$(O ii) | 1016cm-2 | 1.1 | 1.4 | 1.1 | 1.7 | 0.6
|  $N$(O iii) | 1016cm-2 | 10 | 10 | 9 | 12 | 8
|  $N$(Ne i) | 1016cm-2 | 57 | 56 | 54 | 59 | 54
|  $N$(Mg ii) | 1016cm-2 | 32.0 | 32.5 | 32.5 | 31.6 | 32.6
Power-law | $\mathit{\Gamma}$ | unity | 2.27 | 2.34 | 2.30 | 2.25 | 2.35
| Norm | arbitrary | 2.13 | 2.30 | 2.20 | 2.18 | 2.26
Gaussian | $E_{1}$ | keV | 0.832 | 0.832 | 0.831 | 0.834 | 0.830
| Norm1 | arbitrary | 0.059 | 0.066 | 0.067 | 0.056 | 0.073
| $E_{2}$ | keV | 0.681 | 0.678 | 0.681 | 0.700 | 0.668
| Norm2 | arbitrary | 0.078 | 0.064 | 0.072 | 0.055 | 0.106
| $\sigma_{1,2}$ | keV | 0.036 | 0.036 | 0.036 | 0.036 | 0.036
SolidFeL | $M_{{\rm d},i}$ | 10-4 g cm-2 | $0.277\pm 0.019$ | $0.320\pm 0.022$ | $0.301\pm 0.022$ | $0.252\pm 0.009$ | $0.394\pm 0.032$
$C_{\rm stat}$ |  |  | 216.3 | 211.0 | 201.1 | 357.6 | 467.7
$\log A_{\rm O(gas)}+12$ | $8.813\pm 0.009$ | $8.805\pm 0.009$ | $8.802\pm 0.009$ | $8.828\pm 0.009$ | $8.804\pm 0.008$
$\log A_{\rm O(Fe-dust)}+12$ | $7.752\pm 0.030$ | $7.781\pm 0.030$ | $7.840\pm 0.032$ | – | $8.040\pm 0.035$
$\log A_{\rm Fe}+12$ | $7.451\pm 0.030$ | $7.605\pm 0.030$ | $7.539\pm 0.032$ | $7.663\pm 0.016$ | $7.438\pm 0.035$
$\log A_{\rm Ne}+12$ | $7.995\pm 0.033$ | $7.974\pm 0.033$ | $7.964\pm 0.033$ | $8.001\pm 0.033$ | $7.975\pm 0.033$

Note. — Only free parameters are listed. When not listed, the corresponding
uncertainties are given in Tab. 5. The DOF is 119.

From our solid-phase Fe fits, we obtain an Fe/H abundance that is close to
solar ($\log A_{\rm Fe}+12=7.46\pm 0.04$, Asplund et al. 2021) and ISM ($\log
A_{\rm Fe}+12=7.43$, Wilms et al. 2000), but generally larger by $\sim
0.1-0.3$ dex ($25-100\%$). When we included gas-phase Fe ions in our modeling,
we obtained gas-phase Fe abundances that were consistent with zero. Thus we
are unable to constrain the abundance of gas-phase Fe with these X-ray data
alone. Given that $90-99\%$ of the Fe is locked up in dust, accounting for the
missing gas-phase Fe will only increase the total cold-phase Fe abundance for
both sight lines by $\sim 0.004-0.046$ dex. This is on the order of the
abundance measurement uncertainties.

Of the top three compounds that best fit the diffuse ISM data, fayalite puts
the total Fe content closest to solar, while hematite and lepidocrocite
suggest over-abundances on the $\sim 5\sigma$ level. Along the Cygnus X-1
sight line, the Fe abundance is consistent with solar, if fayalite is the
dominant carrier of Fe in the diffuse ISM; whereas for GX 339-4, the Fe
abundance is $\sim 30\%$ larger. If the iron oxides hematite and lepidocrocite
account for the majority of Fe-bearing interstellar dust, then the Fe/H
abundance toward Cygnus X-1 would be a factor of $\sim 1.4$ higher than solar
and toward GX 339-4 a factor of $\sim 2$ higher.

Table 8: Best fit results for GX339–4 with dust model energy scale shifted

Model | Parameter | Units | Fayalite | Hematite | Lepidocrocite | Metallic Fe | Ferrous Sulfate
---|---|---|---|---|---|---|---
ISMabs |  $N$(H) | 1022 cm-2 | 0.51 | 0.53 | 0.52 | 0.52 | 0.50
|  $N$(O i) | 1016 cm-2 | 316 | 323 | 314 | 335 | 299
|  $N$(O ii) | 1016 cm-2 | 6.3 | 7.3 | 6.1 | 8.6 | 4.8
|  $N$(O iii) | 1016 cm-2 | 9 | 10 | 8 | 12 | 7
|  $N$(Ne i) | 1016 cm-2 | 49 | 46 | 45 | 53 | 38
|  $N$(Mg ii) | 1016 cm-2 | 29 | 30 | 30 | 29 | 31
Power-law | $\mathit{\Gamma}$ | unity | 2.62 | 2.70 | 2.67 | 2.62 | 2.77
| Norm | arbitrary | 0.149 | 0.164 | 0.155 | 0.154 | 0.169
Gaussian | $E_{1}$ | keV | 0.824 | 0.822 | 0.821 | 0.826 | 0.818
| Norm1 | arbitrary | 0.0018 | 0.0023 | 0.0024 | 0.0015 | 0.0035
| $E_{2}$ | keV | 0.672 | 0.663 | 0.670 | 0.700 | 0.661
| Norm2 | abritrary | 0.0039 | 0.0030 | 0.0034 | 0.0014 | 0.0068
| $\sigma_{1,2}$ | keV | 0.032 | 0.032 | 0.032 | 0.032 | 0.032
SolidFeL | $M_{{\rm d},i}$ | 10-4 g cm-2 | $0.318\pm 0.043$ | $0.376\pm 0.050$ | $0.349\pm 0.047$ | $0.290\pm 0.026$ | $0.533\pm 0.082$
$C_{\rm stat}$ |  |  | 143.9 | 145.2 | 145.1 | 155.1 | 173.4
$\log A_{\rm O(gas)}+12$ | $8.813\pm 0.009$ | $8.807\pm 0.010$ | $8.800\pm 0.009$ | $8.834\pm 0.010$ | $8.793\pm 0.009$
$\log A_{\rm O(Fe-dust)}+12$ | $7.868\pm 0.059$ | $7.905\pm 0.058$ | $7.959\pm 0.059$ | – | $8.228\pm 0.065$
$\log A_{\rm Fe}+12$ | $7.566\pm 0.059$ | $7.728\pm 0.058$ | $7.658\pm 0.059$ | $7.779\pm 0.039$ | $7.626\pm 0.065$
$\log A_{\rm Ne}+12$ | $7.984\pm 0.033$ | $7.938\pm 0.033$ | $7.936\pm 0.033$ | $8.006\pm 0.033$ | $7.886\pm 0.033$

Note. — Only free parameters are listed. When not listed, the corresponding
uncertainties are given in Tab. 5. The DOF is 119.

Using these results, we can determine Fe-bearing dust-to-hydrogen mass ratio
for dust-grain compound $i$ from

$X_{{\rm d},i/{\rm H}}=\frac{M_{{\rm d},i}}{N{\rm(H)}m_{\rm p}},$ (2)

where $m_{\rm p}$ is the proton mass. Assuming the average dust-to-gas mass
ratio of the Milky Way ISM is 0.01 (Draine 2011), then $X_{\rm d/H}$ for all
dust compounds should sum to 0.014, for a He/H number ratio of 10%. The
average mass ratio of Fe-bearing dust grains from our best fitting candidates
are $X_{\rm Fe-dust/H}=0.003$ and $0.004$ for Cygnus X-1 and GX 339-4,
respectively. This suggests that Fe-bearing grains make up $\sim 20-30\%$ of
the dust-to-hydrogen mass expected in the ISM.

The total abundance of cool gas-phase O is determined here from the ISMabs
fits to the three O charge states detected. For both sight lines, we find a
gas-phase O abundance that is larger by $\sim 0.1$ dex ($\sim 30\%$) than the
solar ($\log A_{\rm O}+12=8.69\pm 0.04$, Asplund et al. 2021) and ISM ($\log
A_{\rm Fe}+12=8.69$, Wilms et al. 2000) values.

We can also infer the abundance of O locked up in dust grains. During the fit
procedure, we replaced the ISMdust silicate model with SolidFeL, which
includes only the laboratory measurements for Fe-bearing compounds, a few of
which contained O, but no Mg-bearing silicates. Thus we are only able to
consider O abundances from Fe-bearing dust in this work, reported as $A_{\rm
O(Fe-dust)}+12$ in Tables 7 and 8. For our top three dust candidates, the
abundance of O in Fe-bearing grains is about 1 dex lower than in the gas
phase.

Mg-rich silicates are suspected to account for the majority of ISM silicates
and to be an ample reservoir for solid-phase O. Assuming that 100% of
interstellar Si is depleted into the Mg-silicate end-members enstatite
(MgSiO3) or forsterite (Mg2SiO4), then $\log A_{\rm O(Mg-sil)}+12$ would be
$7.75$ and $7.87$, respectively, using the ISM abundances of Wilms et al.
(2000). These values are comparable to our reported $A_{\rm O(Fe-dust)}$
values. This implies that if Fe oxides are indeed the dominant Fe-bearing dust
compound, then the amount of ISM O depleted into Fe oxides is comparable to
that depleted into Mg silicates. Adding together the inferred abundances for
$A_{\rm O(Fe-dust)}$ and $A_{\rm O(Mg-sil)}$, the Fe-oxide models imply a
total O depletion $\sim 20\%$. Depletions of $10-30\%$ are reasonable for the
diffuse ISM (Jenkins 2009, Psaradaki et al. 2020; 2023). If we assume that the
majority of interstellar silicates are in enstatite, a requirement due to the
fact that the abundance of Si is approximately equal to that of Mg, then we
can estimate the total mass of Mg-rich silicate dust based on the mass and
mixture of each element in the compound stoichiometry and the abundance of Si
relative to H. This yields $X_{\rm Mg-sil/H}\approx 0.002$, which accounts for
$\sim 10\%$ of the interstellar dust mass. Together with $X_{\rm Fe-
dust/H}=0.003-0.004$, this would leave the next suspected dust species,
carbonaceous grains, to account for the remaining $\sim 60\%$ of the total
interstellar dust mass.

Since the abundances of Mg, Fe, and Si are close to one another, assuming that
fayalite (our third-best fitting model) is the dominant Fe-bearing dust
compound would necessarily reduce the amount of Si available to form enstatite
or forsterite. The fayalite fit suggests solid-Fe abundances that are close to
Solar (i.e., accounting for all known solid-phase Fe), implying that the
majority of Si must also be in fayalite. This is inconsistent with the current
understanding of interstellar silicate mineralogy, for which Mg/Fe ratios
$\approx 1-3$ are found (see Sections 2 and 6.3). With little Si remaining in
the budget to form Mg-rich silicates, the fayalite fit yields an O depletion
$\sim 10\%$. The fayalite model also yields $X_{\rm Mg-sil/H}\approx
0.003-0.004$, for Cygnus X-1 and GX 339–4, respectively, implying that
$20-30\%$ of interstellar dust is in Fe-rich silicates. However, with
relatively little Si remaining to form Mg-rich silicates, the remaining
$70-80\%$ are presumably in carbonaceous grains. This picture is broadly
inconsistent with the majority of mineralogical models of interstellar dust.

Since Ne is a noble gas that does not deplete into the solid phase, it
provides an excellent benchmark for abundance constraints. Our measured values
of $\log A_{\rm Ne}+12=7.94-8.00$ are slightly underabundant by $0.05-0.1$ dex
($10-30\%$) compared to studies of the local solar neighborhood ($8.09\pm
0.05$, Nieva & Przybilla 2012), the Orion Nebula ($0.05\pm 0.03$ Simón-Díaz &
Stasińska 2011), and the solar wind ($8.06\pm 0.05$ Asplund et al. 2021).
Curiously, our measurements are in better agreement with less modern values
for the Ne abundance of the ISM ($7.94$, Wilms et al. 2000; and references
therein). Note that, if we were to adopt a lower value for $N({\rm H})$ based
on the observed Ne column density, it would cause the relative over-abundances
of O and Fe reported above to be even more pronounced. However, our observed
Ne abundances are within 2$\sigma$ of the expected values, which is not
significant enough to challenge our current assumptions.

Overall, there are major systematic difficulties in determining the total ISM
abundance of H from X-ray spectroscopy. That is because H has no spectrocopic
signature in the $0.3-10$ keV band and only contributes to continuum
absorption. Thus, the $N({\rm H})$ value measured from the X-ray band is
inferred indirectly by continuum fitting from an assumed abundance table and,
in this work, an assumed dust-to-gas mass ratio of various compositions. In
the $<0.5$ keV super-soft X-ray band, continuum absorption by C is strong, and
instruments are at the limits of sensitivity. As a result, the spectroscopic
signature of C is rarely measurable. Our X-ray spectra are thereby unable to
constrain the C/H abundance or C-bearing dust mass column directly, and the
inferred H abundance will change depending on those assumptions. The estimated
abundance of carbonaceous dust relative to silicates varies widely from
approximately equal (Draine 2003, Corrales et al. 2016) to grain models that
contain almost no carbonaceous dust except for PAHs (Zubko et al. 2004,
Hensley & Draine 2023). Here, we have tied the C abundance to the H abundance
following the assumptions present in Corrales et al. (2016). However, since C
is a significant source of soft X-ray extinction, the true mass ratio of
carbonaceous grains relative to H is a source of systematic uncertainty for
all abundance determinations in the X-ray.

###  6.2 Astrophysical Implications

The ability to predict the mineralogical form of interstellar Fe from Solar
System analogs is limited, due to refractory processing by the solar
protoplanetary disk. Nonetheless, these analogs provide a good starting point
for the materials that can form in the diffuse ISM. Suspected presolar grains
and cometary dust collected from the comet Wild 2 contain Fe in the form of
FeS and Fe-Ni metal with $<2\%$ Ni by number (Bradley 1994, Zolensky et al.
2006, Leroux et al. 2008a; b) The closest analog present in our study is
ferrous sulfate (FeSO4), which is the worst fitting compound for both sight
lines. Other studies have also not found substantial signatures of iron
sulfides in the ISM (see Section 6.3). Our findings rule out ferrous sulfate
as a major reservoir of interstellar Fe.

A number of studies have proposed metallic Fe as the main carrier of
interstellar Fe, from both a theoretical and observational perspective. Pure
metallic iron nanoparticles, either embedded in silicates or later exposed as
the silicates erode in the warmer phases of the ISM, are proposed to explain
the observed Fe-depletion patterns of the ISM (Zhukovska et al. 2018).
Continuum extinction from pure metallic iron is also invoked in order to
explain the full spectrum of absorption in dusty circumstellar outflows
(Harwit et al. 2001, Kemper et al. 2002). However, recent laboratory
experiments failed to produce pure metallic Fe particles in a space-like
environment (Kimura et al. 2017). In our study here, metallic Fe with an MRN
distribution provided the second to worst fit even when we allow for different
energy scale calibrations. We conclude that the majority of interstellar Fe is
not an MRN distribution of grains in metallic form, though metallic Fe
nanoparticles may still account for a small fraction of the total ISM Fe
abundance.

From both a theoretical and observational standpoint, interstellar silicates
are expected to be Mg-rich. Enstatite and forsterite, the Mg-rich end-members
of olivines (Mg2xFe2(1-x)SiO4) and pyroxenes (MgxFe(1-x)SiO3), respectively,
condense at higher temperatures than Fe-enriched silicates (Gail 2010). Thus
for silicates formed in AGB stellar outflows, the Mg-rich silicates likely
freeze out before they can be enriched in Fe. Collected samples of suspected
presolar grains and cometary dust also contain silicates almost entirely
comprised of Mg end-members (Bradley 1994; 2010, Hanner & Zolensky 2010, Mann
& Jessberger 2010). As described in Section 6.1, a dust model where fayalite
accounts for the majority of solid interstellar Fe is physically inconsistent
with the low abundance of Fe-bearing silicates measured both in situ and from
the IR signatures in the ISM. Fitting fayalite to the observed Fe $L$-shell
photoabsorption features also requires a shift in energy scale calibration
that is quite large compared to the majority of laboratory measurements (Table
2). For these reasons, fayalite is disfavored by our analysis as the
predominant form of Fe-bearing dust.

Chemically active O is cosmically the most abundant element heavier than
chemically inert He, making it highly likely that gas-phase Fe will bind with
O in the ISM. This warrants the exploration of iron oxides, such as hematite
and lepidocrocite, here, despite both of them being absent from metereotic and
interplanetary dust particles. Unfortunately, the extinction features of these
two compounds are nearly identical when the throughput and spectral resolution
of modern day X-ray telescopes and spectrometers are considered; the compounds
cannot be distinguished from one another. Nonetheless, these are the two best-
fitting compounds that require the smallest energy shift with respect to the
published laboratory work. This leads us to hypothesize that Fe3+-bearing
compounds are most likely to be the dominant carriers of solid-phase Fe in the
ISM. If this is the case, then interstellar Fe may be more abundant relative
to solar than previously estimated. However, systematic uncertainties in
measuring $N{\rm(H)}$ accurately in the X-ray band makes it difficult to draw
definitive conclusions about iron oxides via abundance arguments.

If iron oxides truly are the most abundant component of solid-phase Fe in the
ISM, then they provide a distinct reservoir for interstellar O that has not
been highly explored. As described in Section 6.1, the amount of O depleted
into iron oxides could be comparable to that depleted into Mg silicates. This
is of interest for the oxygen problem – namely, that the observed O depletion
rate in the ISM cannot be explained entirely by its incorporation by Si, Mg,
and Fe compounds (Jenkins 2009). However, the sight lines studied here are
unlikely to resolve this issue. For a distance of 2.2 kpc to Cygnus X-1
(Arnason et al. 2021) and 5 kpc to GX 339-4 (Gaia DR3, Gaia Collaboration
2020), and using our derived $N({\rm H})$ values, the average density for each
sight line is $\left<n_{\rm H}\right>\approx 0.9$ and $0.3$ cm-3,
respectively. One molecular cloud with a radius $\sim 100$ pc and $n_{\rm
H}\sim 10^{2}$ cm-3 in the sight line would add $\sim 3\times 10^{22}$ cm-2 to
$N({\rm H})$, which is $\sim 5-6$ times higher than our observed $N{\rm(H)}$
values. Hence, our observations do not probe the state of interstellar dust
particles in the densest regions of molecular clouds, where the oxygen problem
may be solved by the formation of O-rich ices.

###  6.3 Comparison to other work

Recent laboratory measurements of over a dozen compounds have enabled new
X-ray astromineralogical studies of Fe-bearing grains through Fe extinction
directly (Rogantini et al. 2018, Westphal et al. 2019, Psaradaki et al. 2023)
or indirectly through Si, Mg, and O absorption features (Zeegers et al. 2017;
2019, Rogantini et al. 2019; 2020, Psaradaki et al. 2020). Those studies all
used MRN distributions. However, the photoabsorption cross section data from
those studies are not yet publicly available. This prevented our being able to
incorporate those results into SolidFeL for abundance determinations and also
prevented our being able to explore the calibration accuracy of the absolute
energy scale for their cross section data. Within these constraints, we
discuss here their findings relative to ours.

Comparing the astronomical results based of the above new laboratory data, a
unified, comprehensive picture of Milky Way interstellar dust has yet to
emerge. Examination of Mg $K$\- and Si $K$-shell features suggest $\sim
70-80\%$ of silicates by number are amorphous olivine with ${\rm Mg/Fe}\sim 1$
(MgFeSiO4; Rogantini et al. 2019, Zeegers et al. 2019, Rogantini et al. 2020).
On the other hand, direct measurement of the Fe $L$-shell extinction data for
five XRBs suggests that the most prevalent silicates are amorphous, Mg-rich
pyroxenes with ${\rm Mg/Fe}=3$ (Mg0.75Fe0.25SiO3), constituting by number
about 70% of the Fe- and O-bearing dust along the line of sight (Psaradaki et
al. 2023). In that study, only the sight line of GX 9+9 was best fit with a
dust column where the majority of Fe- and O-bearing dust was amorphous olivine
with ${\rm Mg/Fe}=1$, in agreement with the previous Mg $K$\- and Si $K$-shell
studies.

Our SolidFeL study included the olivine fayalite, but not a pyroxene compound.
However, both olivine and pyroxene are Fe2+-bearing minerals and are likely to
peak in their photoabsorption cross section at similar energies (as was seen
for the Fe3+ minerals lepidocrocite and hematite). Thus we suggest that
fayalite serves as a reasonable proxy for pyroxene in this work.
Unfortunately, if Fe-rich silicates are indeed the majority host material for
interstellar Fe, then there is insufficient interstellar Si available to
provide significant dust mass in Mg-rich silicates (demonstrated in Section
6.1). This picture is incongruent with the majority of astromineralogical
studies, from both the X-ray (described above) and the IR (Min et al. 2007)
perspectives. However, though this picture is generally disfavored in our
analysis, systematic uncertainties in the $N{\rm(H)}$ column means that we
cannot not completely exclude fayalite as an Fe-bearing constituent of the
diffuse ISM based on these abundance arguments alone.

Psaradaki et al. (2023) also conclude that $10-20\%$ of the dust column is
made up of metallic iron, after shifting the Kortright & Kim (2000) cross
section by $-0.3$ eV to match the position observed by Fink et al. (1985),
listed in Table 2, whereas we needed to shift the metallic Fe data by
$\approx+2.53$ eV. We note that 10% is also on the order of magnitude expected
for the abundance of gas-phase Fe in the diffuse ISM. Our fit procedures were
unable to constrain the abundance of gas-phase Fe, given its small
contribution to the absorption signal. It follows that we are likely unable to
comment on a similar abundance of metallic Fe particles. Nonetheless, the fact
that the Kortright & Kim (2000) metallic Fe cross section requires shifting in
nearly every study of interstellar Fe (e.g., Juett et al. 2006, Hanke et al.
2009, Gatuzz et al. 2015, Psaradaki et al. 2023), as well as in our work,
demonstrates that the absolute energy calibration of pure metallic iron is
essential for accurate identification and abundance determinations of
interstellar dust, and needs to be revisited.

Additional compounds examined by Psaradaki et al. (2023) include troilite
(FeS), pyrite (FeS2), and magnetite (Fe3O4). They concluded that these
compounds resulted in spectral fits that were less preferred. The oxidation
state of Fe in troilite and pyrite is Fe2+. For magnetite it is Fe2+(Fe3+)2;
hence, magnetite is likely to present XAFS that are quite different from the
other compounds in our study. Westphal et al. (2019) analyzed observations
from the sight line to Cygnus X-1 and worked to fit XAFS from FeS, metallic
Fe, and silicate samples of suspected presolar dust grains. They concluded
that less than $38.5\%$ of Fe resides in silicates; the rest is in metallic Fe
with no signatures from FeS. If their finding that the Fe in the ISM is
primarily in metallic form, then it would also dominate the observed $L$-shell
extinction features and provide one of the better fits in our analysis. We
find that this is not the case, particularly for the Cygnus X-1 sight line
that both we and Westphal et al. (2019) analyzed, which implies that metallic
Fe is not the dominant species for this sight line. This is in contradiction
to their findings. Still, all X-ray absorption studies to date agree that Fe-S
compounds do not account for an appreciable fraction of Fe-bearing dust
grains. Our two best fitting compounds, lepidocrocite and hematite, are not
present in prior works, warranting the future consideration of Fe3+ oxides for
the mineralogy of interstellar dust.

###  6.4 Comparison to other X-ray models for interstellar extinction

In this work, we have presented a local XSPEC model, SolidFeL, with dust
extinction cross sections drawn from the laboratory work of Kortright & Kim
(2000) and Lee et al. (2009). The cross sections included in SolidFeL are not
shifted from the reported laboratory energies. The solid-phase, broad-band and
photoelectric $K$-shell extinction features from elements other than Fe are
modeled following the absorption profiles drawn from Hanke et al. (2009), from
which optical constants are then derived by kkcalc (as described by Watts
2014). Below, we describe how SolidFeL differs from other XSPEC models of
interstellar extinction.

The XSPEC model ISMabs (Gatuzz et al. 2015) provides high resolution
theoretical cross sections for select interstellar metals. It models gas-phase
photoabsorption only, and would be physically accurate if all interstellar
metals were in the gas phase. The tbabs model also provides absorption-only
properties of the ISM using the simplified cross sections for neutrals from
Verner & Yakovlev (1995), but is modified for self-shielding by dust grains
and follows the abundance tables of Wilms et al. (2000). Both ISMabs and tbabs
have shifted the metallic Fe cross sections of Kortright & Kim (2000) to match
the observed position of interstellar Fe $L$-shell features (Wilms et al.
2000, Gatuzz et al. 2015). Both of these models are missing the dust
scattering component to X-ray interstellar extinction. The updated version of
ISMabs (Gatuzz et al. 2015) be found at
https://github.com/efraingatuzz/ISMabs.

The XSPEC model xscat (Smith et al. 2016) solely models the scattering
component of extinction from interstellar dust. This model contains the most
comprehensive set of dust grain size distributions currently available in the
X-ray, but does not account for dust grain absorption and shielding effects.
The ISMdust model provides the full extinction profile – absorption plus
scattering – from a power-law distribution of interstellar dust grains only,
and is most accurate for telescopes that are able to resolve dust scattering
halo images (Corrales et al. 2016). It uses the empirical broad-band dust
opacities of Draine (2003). ISMdust is best used in combination with an
absorption-only model like tbvarabs or ISMabs to capture the gas-phase
component of interstellar absorption, the latter of which was used in this
work.

##  7 Summary and Conclusions

The majority of interstellar Fe is injected in the gas phase by Type Ia SNe,
implying that Fe must condense into the solid phase in the diffuse environment
of interstellar space (Dwek 2016). This makes it difficult to draw conclusions
about Fe-bearing interstellar dust from the mineralogical identifications of
circumstellar dust. Additionally, there are no well-established solid
interstellar Fe features in the IR. X-ray spectroscopy is thereby the only
direct method for studying the mineralogy of interstellar iron.

We have examined archival XMM-Newton RGS spectra of Cygnus X-1 and GX 339-4,
which exhibit strong Fe $L$-shell extinction features around 0.7 keV. The
spectra were fit with a non-thermal power-law model undergoing gas-phase
absorption using ISMabs and dust-phase extinction using our SolidFeL model,
which contains Fe $L$-shell extinction cross sections derived from Kortright &
Kim (2000) and Lee et al. (2009). We found that none of our available dust
compounds provided a good fit to the spectra, unless we shifted the absolute
energy scale of the laboratory data. We have determined the energy shift
required for each species to best fit the observations.

Of the five compounds tested, three provide equally good fits for different
absolute energy shifts: the iron oxides lepidocrocite and hematite, and the
iron end-member of olivine silicates (fayalite). Lepidocrocite and hematite
are Fe3+ minerals and have nearly identical XAFS cross sections that are
unresolvable with modern day X-ray satellite spectrometers. Fayalite is an
Fe2+ mineral, and requires a larger energy calibration shift than the Fe3+
oxides. We note that the energy shift required to fit the iron oxides to the
data is within the standard deviation of the laboratory literature, whereas
the shift required to fit fayalite requires a more significant departure from
literature values. More importantly, a model in which fayalite accounts for
the majority of Fe-bearing dust is physically inconsistent with the large
abundance of Mg-rich silicates inferred from IR observations of astronomical
sources and from in situ measurements of interstellar dust candidates captured
in our Solar System. This makes Fe3+ iron-oxide compounds our top choice as a
major reservoir for interstellar Fe. Examination of other Fe3+ minerals is
warranted for future study.

The oxidation state of interstellar Fe is important, as it defines the
starting conditions of Fe in protostellar clouds. If iron oxides truly
represent the majority of Fe-bearing compounds in the ISM, that does not match
the state of trace iron found in presolar and suspected interstellar grains
collected in the Solar System. This conclusion would imply that interstellar
dust must be significantly processed through the collapse of the molecular
cloud that eventually forms a protostar and protoplanetary disk. To verify
this hypothesis, careful calibration of absolute energy scale for metallic Fe
and various oxidation states of solid Fe compounds is required, as that is the
chief source of systematic uncertainty in this work.

Other conclusions from this study are summarized as follows.

  * •

Including gas-phase Fe as one of our model parameters resulted in best-fit
abundances that were consistent with zero. These findings support that Fe is
heavily depleted from the gas phase in the diffuse ISM.

  * •

The large parameter space of possible energy shifts and combinations of Fe-
bearing dust species prevents us from drawing meaningful conclusions with
multi-species fits at this time.

  * •

The total abundance of O and Fe seem over-abundant compared to solar values,
while Ne appears slightly under-abundant. However, most results are within
3$\sigma$ of the uncertainties reported in Asplund et al. (2021).

  * •

The depletion and total dust mass fraction of carbonaceous grains is a strong
source of systematic uncertainty for determining the total H column, which
might explain some of the discrepancies in our abundance estimates.

  * •

From Fe $L$-shell absorption alone, we estimate that Fe-bearing materials
account for $20-30\%$ of the total mass of interstellar dust. If this is the
form of Fe oxides, then Mg-rich silicates would account for 13% of the
interstellar dust mass, leaving $\approx 60\%$ for carbonaceous grains. These
conclusions are predicated on the hypothesis that there are distinct chemical
populations of interstellar dust.

We thank the anonymous reviewer for their constructive comments. E.V.G.,
T.R.K., F.P., and D.W.S. were supported, in part, by the NASA Astrophysics
Data Analysis Program (ADAP) grant 80NSSC19K0571. The work by L.R.C. and I.P.
was supported through NASA ADAP grant 80NSSC20K0883. M.M. is grateful for
funding of this project by the Federal Ministry of Education and Research
(BMBF; 05K10GUB, 05K16GUC, and 05K19GUC) and by the German Research Foundation
(DFG, project grant No. 510114039). S.S. acknowledge financial support by the
German Federal Ministry for Education and Research (BMBF, grant Nos. 05K19GU4
and 05K19RG3, respectively). XMM

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