mail:stierhofe-mail<EMAIL_ADDRESS>
11institutetext:  Dr. Karl Remeis-Observatory and Erlangen Centre for
Astroparticle Physics Friedrich-Alexander-Universität Erlangen-Nürnberg,
Sternwartstr. 7, 96049 Bamberg, Germany 22institutetext: Max-Planck-Institut
für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany 33institutetext:
Institute of Theoretical Physics, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany 44institutetext: Université
Grenoble Alpes, CNRS, Institut NEEL, 25 rue des Martyrs BP 166, 38042 Grenoble
Cedex 9, France 55institutetext: CERN, 1211 Geneva 23, Switzerland
66institutetext: Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germany 77institutetext: NASA Goddard Space Flight Center, 8800
Greenbelt Rd., Greenbelt, MD 20771, USA 88institutetext: Lawrence Livermore
National Laboratory, 7000 East Ave., Livermore, CA 94550, USA 99institutetext:
Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität Jena,
Max-Wien-Platz 1, 07743 Jena, Germany 1010institutetext: Department of
Astronomy, University of Maryland, College Park, MD 20742 1111institutetext:
Institut de Recherche en Astrophysique et Planétologie, 9, avenue du Colonel
Roche BP 44346, 31028 Toulouse Cedex 4, France 1212institutetext: Ulsan
National Institute of Science and Technology, 50 UNIST-gil, Ulsan, South Korea
1313institutetext: Institute for Methods and Instrumentation in Synchrotron
Radiation Research G-ISRR, Helmholtz-Zentrum Berlin für Materialien und
Energie, Albert-Einstein-Strasse 15, 12489 Berlin, Germany 1414institutetext:
GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, 64291
Darmstadt, Germany 1515institutetext: Helmholtz-Institut Jena, Fröbelstieg 3,
07743 Jena, Germany

A key requirement for the correct interpretation of high-resolution X-ray
spectra is that transition energies are known with high accuracy and
precision. We investigate the K-shell features of $\mathrm{Ne}$,
$\mathrm{CO_{2}}$, and $\mathrm{SF_{6}}$ gases, by measuring their photo ion-
yield spectra at the BESSY II synchrotron facility simultaneously with the
1s–$n$p fluorescence emission of He-like ions produced in the Polar-X EBIT.
Accurate _ab initio_ calculations of transitions in these ions provide the
basis of the calibration. While the $\mathrm{CO_{2}}$ result agrees well with
previous measurements, the $\mathrm{SF_{6}}$ spectrum appears shifted by
$\sim$0.5 eV, about twice the uncertainty of the earlier results. Our result
for $\mathrm{Ne}$ shows a large departure from earlier results, but may suffer
from larger systematic effects than our other measurements. The molecular
spectra agree well with our results of time-dependent density functional
theory. We find that the statistical uncertainty allows calibrations in the
desired range of 1–10 meV, however, systematic contributions still limit the
uncertainty to ${\sim}$40–100 meV, mainly due to the temporal stability of the
monochromator energy scale. Combining our absolute calibration technique with
a relative energy calibration technique such as photoelectron energy
spectroscopy will be necessary to realize its full potential of achieving
uncertainties as low as 1–10 meV.

# A new benchmark of soft X-ray transition energies of $\mathrm{Ne}$,
$\mathrm{CO_{2}}$, and $\mathrm{SF_{6}}$: paving a pathway towards ppm
accuracy

J. Stierhofinst:remeis,mail:stierhof  S. Kühninst:mpik  M.
Winterinst:fauphys,inst:neel  P. Mickeinst:cern  R. Steinbrüggeinst:desy  C.
Shahinst:gsfc,inst:mpik,inst:llnl  N. Hellinst:llnl  M. Bissingerinst:remeis
M. Hirschinst:remeis  R. Ballhauseninst:remeis  M. Langinst:remeis  C.
Gräfeinst:remeis  S. Wipfinst:ujena  R. Cumbeeinst:gsfc,inst:umaryland  G. L.
Betancourt-Martinezinst:irap  S. Parkinst:unist  J. Niskaneninst:hzb  M.
Chunginst:unist  F. S. Porterinst:gsfc  T.
Stöhlkerinst:ujena,inst:gsi,inst:hij  T. Pfeiferinst:mpik  G. V.
Browninst:llnl  S. Bernittinst:ujena,inst:hij,inst:gsi,inst:mpik  P.
Hansmanninst:fauphys  J. Wilmsinst:remeis  J. R. Crespo López-Urrutiainst:mpik
M. A. Leuteneggerinst:gsfc

(Received 30 November 202/Accepted 23 January 2022)

††journal: Eur. Phys. J. D

##  1 Introduction

High-resolution astrophysical X-ray spectroscopy has become routine in the
last 20 years, with diffraction grating spectrometers on Chandra and XMM-
Newton providing resolving powers of $\Delta\lambda/\lambda\sim 1000$
(brinkman2000a, ; denherder:2001, ; canizares:2005, ; devries:2015, ). These
instruments have enabled the measurements of the conditions in the emitting
plasmas, e.g., through observations of the triplets from He-like ions,
precision Doppler velocity and line shape measurements in a variety of
astrophysical plasmas, including stellar coronae and winds, cataclysmic
variables, X-ray binaries containing neutron stars and black holes, supernova
remnants, or outflows in active galactic nuclei (e.g., paerels:2003, ;
steenbrugge:2005, ; ishibashi:2006, ; miskovicova:2016, ; 2014Sci…345…64K, ;
drake:2019, ; nowak:2019, ). Due to the success of these measurements, future
astrophysical X-ray observatories such as XRISM, Athena, Arcus, or Lynx,
envision spectral resolving powers as high as 5000, implying the ability to
accurately determine centroids to 10 ppm, or $3\,\mathrm{km}\,\mathrm{s}^{-1}$
absolute Doppler velocity (mcentaffer:2013, ; Heilmann2017, ; Barret2018, ;
Bandler2019, ; Tashiro2020, ; Smith2020, ). These instruments will open up the
field of spatially resolved, high-resolution X-ray spectroscopy, and will
allow scientists to access techniques that are currently not available to
X-ray astronomy such as X-ray Fine Structure Absorption measurements for
solids (lee:2009a, ), the imaging of velocity fields in galaxy clusters
(cucchetti:2018, ), or diagnosing the properties of the Warm and Hot
Intergalactic Medium (walsh:2020, ).

The ground and on-orbit calibration of existing and future instruments as well
as the interpretation of the existing and future observations require
accurately calibrated atomic transition energies (e.g., devries:2015, ;
plucinsky:2017, ). In one- and two-electron ions, these energies are
calculable with part per million (ppm) accuracy for the astrophysically
relevant atomic numbers less than 30 (e.g., JohnsonSoff1985, ; Drake1988, ;
Artemyev2005, ; yerokhin2019, ), and theory has been experimentally
benchmarked with precision as good as 10 ppm (e.g., Kubicek2014, ;
beiersdorfer2015, ).

Inner shell transition energies in less-ionized species, neutral atoms,
molecules, and solids, are far more challenging to calculate accurately, and
thus must be obtained experimentally. These experiments, however, rely on
existing soft X-ray calibration standards, which have limitations to their
accuracy. We recently found a discrepancy in the extensively used standard of
the Rydberg transitions of molecular oxygen of almost 0.5 eV (leutenegger2020,
), thus resolving a tension between astrophysical and laboratory measurements
of transitions of atomic oxygen (McLaughlin2013, ), which had been calibrated
against this molecular standard (WightBrion1974_O2, ). Such discrepancies
raise the question of whether other commonly-used soft X-ray standards may
have errors of comparable magnitude, given that many such standards are based
on similar experimental techniques using electron energy loss spectroscopy
(EELS).

Even if the error in the earlier molecular oxygen standard is an outlier, the
typical experimental precision of soft X-ray standards obtained with EELS is
still of order 0.1 eV (or 100 ppm at 1 keV), which is far too large to fully
exploit the capabilities of current and future X-ray astronomical and ground
based facilities, and not precise enough for the calibration needs of many
future instruments. Modern synchrotron facilities are capable of sufficient
photon fluxes and resolving powers that determining centroids of peaks with
statistical precision of 1–10 ppm is routine in a variety of experimental
disciplines (e.g., mueller2018, ; epp2015, ), so to the extent that scientific
results depend on the absolute transition energies, calibration will often be
the limiting factor.

The anticipated high precision of line energy measurements enabled by high
spectral resolution coupled with large photon fluxes in future space-based
observatories, as well as in high-performance synchrotron beamlines, implies a
need to reevaluate soft X-ray transition energies of common elements and
materials that have been used for energy calibration using the same accurate
standards used by leutenegger2020 : highly charged ions (HCI) with one or two
electrons. To further illustrate the capabilities of these methods, in this
paper we present measurements of photoion yield spectra for $\mathrm{CO_{2}}$
around the oxygen K-edge, $\mathrm{SF_{6}}$ around the F K-edge, and Ne around
its K-edge. These are calibrated using K-shell transitions of He-like N, O,
and F, respectively. The remainder of this paper is structured as follows: In
Sect. 2 we describe our experimental setup, which combines a synchrotron
beamline with an electron beam ion trap (EBIT) to generate the calibrating
ions and a gas cell, and discuss the energy calibration and systematic
limitations from this setup. In Sect. 3 we present the results of our
calibration of the photoionization spectra for neon, $\mathrm{SF_{6}}$, and
$\mathrm{CO_{2}}$. In order to understand the structure of the molecular edges
in greater detail, in Sect. 4 we then compare the experimental results for the
molecules with theoretical simulations. We summarize the paper in Sect. 5.

##  2 Experimental setup and data analysis

###  2.1 Experimental Setup

Figure 1: Our scheme for simultaneous measurement of neutral gas
photoionization and HCI fluorescence (adapted from leutenegger2020, ).
Monochromatic linearly polarized X-rays produced by the synchrotron beamline
U49-2/PGM-1 enter the PolarX-EBIT endstation from the left, and excite the
HCI. Subsequently, the fluorescence is detected by two silicon drift
detectors. The off-axis electron gun allows the synchrotron X-ray beam to pass
through to our second endstation, a low-pressure gas cell using a channeltron
for detection of photoions.

Our experimental setup is depicted in Fig. 1. Monochromatic X-rays from a
synchrotron beamline pass through an EBIT, where they interact with HCI.
Fluorescence emission from these ions provides the basis for the absolute
calibration of the monochromator energy scale in our experiment. The
synchrotron radiation passes through the low density plasma in the EBIT with
virtually no attenuation, and then enters a gas photoionization cell
containing the atoms or molecules under investigation. A channeltron inside
the gas cell detects the ion yield due to the interaction of the X-ray beam
with the gas. The gas cell with injected gases was operated with a pressure of
few $10^{-}7$ Torr.

Our setup used the PolarX-EBIT (micke2018, ), which features an off-axis
electron gun, enabling the photon beam to pass through the EBIT. The electron
beam is tuned to an energy sufficent to ionize atoms entering the trap up to
the He-like charge state, but staying below the threshold for K-shell
excitations, and also avoiding dielectronic-recombination resonances. The
X-ray photons interacting with the ions thus produce a K-shell fluorescence
signal that is uncontaminated by X-rays following collisional excitation. We
measured this fluorescence signal with silicon-drift detectors (SDDs) that are
mounted perpendicular to the electron beam axis.

For H-like and He-like systems it is possible to calculate the transition
energies with uncertainties of $\lesssim$1 meV (yerokhin2019, ). This _ab
initio_ provides the absolute calibration reference for our measurements.
Since our experiment allows us to measure the fluorescence in the EBIT
simultaneously with the ion yield in the gas cell, we avoid problems that are
intrinsic to non-simultaneous energy calibrations.

For $\mathrm{Ne}$, $\mathrm{CO_{2}}$ and $\mathrm{SF_{6}}$ investigation, we
measure the fluorescence of He-like fluorine, nitrogen, and oxygen,
respectively. We induce it with soft X-ray photons provided by the BESSY II
plane-grating monochromator (PGM) beamline U49-2/PGM-1 (sawhney2001, ).
Because of the linear polarization of the beam and the dipolar character of
the resonances, there is a strong dependency of the fluorescence on the
viewing angle (bernitt2013, ). Therefore, we used two SDDs, one aligned
parallel to and the other perpendicular to the polarization axis. Polarization
also slightly affects the ion-yield measured in the gas cell. The channeltron
was aligned parallel to the polarization axis, but because it was close to the
photon beam, it has a finite angular acceptance. The acquired photoion spectra
showed features excited by both polarization axes, albeit with a stronger
contribution from the parallel axis.

Individual scans for each of the three gases were performed in equidistant
energy steps from low to high energies, scanning the photon energy in ranges
of 866–871 eV for $\mathrm{Ne}$, 533–540 eV for $\mathrm{CO_{2}}$, and 684–705
eV for $\mathrm{SF_{6}}$. At each scan step, the integrated ion-production
rate and HCI-fluorescence rates were recorded together with the nominal energy
of the beam line. To achieve the highest possible accuracy and minimize
uncertainty, the calibration line must lie within the scan range. This was
possible for the $\mathrm{CO_{2}}$ and $\mathrm{SF_{6}}$ scans. For the
$\mathrm{Ne}$ scan, the chosen calibration line was 10 eV lower in energy and
had to be recorded in a separate scan. The details of data recording and
processing are described in (leutenegger2020, ), where the same setup was
used.

###  2.2 Energy Calibration

The nominal calibration of the beamline wavelength scale uses the grating
equation

$mN\lambda=\cos\alpha-\cos\beta$ (1)

where $N$ is the line density of the grating and $m$ the diffraction order. In
our experiment, $m=1$ and $N=602.4\,\mathrm{mm}^{-1}$. The incident and
reflection angles $\alpha$ and $\beta$ are measured with respect to the plane
of reflection. These angles are determined from the rotation angles of the
mirror and grating using high-precision rotation encoders. Typically, the true
wavelength of the beamline has a slight offset from the nominal value derived
using the encoder positions. This offset can be corrected using the
calibration lines. In many experiments it is common practice to apply a linear
offset to wavelength or energy based on a calibration feature. However, since
the grating equation is nonlinear, this introduces a systematic error that
increases with separation from the calibration feature. Specifically, in
energy space the grating equation is given by

$E=\frac{hcmN}{\cos\alpha-\cos\beta}$ (2)

where $h$ is Planck’s constant and $c$ is the speed of light. We used the
defined CODATA 2018 value $hc=1\,239.841\,984\,\mathrm{eV}\,\mathrm{nm}$
(newell2018, ; bipm2018, )111The nominal energies reported by the beamline use
$hc=1\,239.86\,\mathrm{eV}\,\mathrm{nm}$..

The angles comprise two parameters while the selection of energy fixes only
one degree of freedom. The remaining degree of freedom is fixed by $\alpha$
and $\beta$ adhering to the fixed-focus condition (follath2001, , equation 2,
converted to our angle convention)

$\sin\beta=c_{\textnormal{ff}}\sin\alpha,$ (3)

with $c_{\textnormal{ff}}$ set to 2.25 for U49-2/PGM-1. This fixed focus
condition ensures that the image of a source at a fixed distance to the
grating is projected to a fixed point behind the grating with a scaling,
$c_{\textnormal{ff}}$, that is independent of the energy.

Throughout our campaign we found a discrepancy of more than 3 eV at the energy
of the $\mathrm{O}^{6+}\,\rm 1s^{2}\,{}^{1}S_{0}\rightarrow 1s2p\,{}^{1}P_{1}$
transition. We will call this line O${}_{\textnormal{w}}$ in the following
(gabriel1972, ); its theoretical energy is 573.961 eV (Drake1988, ). Assuming
that the accuracy of the angle steps is stable (at least over single scans,
containing 100–1000 steps each), this discrepancy must be due to an offset in
the angles such that $\alpha=\alpha^{\prime}+\Delta\alpha$,
$\beta=\beta^{\prime}+\Delta\beta$, where $\alpha^{\prime}$ and
$\beta^{\prime}$ are the incident and reflected angles of the photons as
reported by the beamline.

A single calibration feature only permits to determine either $\Delta\alpha$
or $\Delta\beta$. A natural choice for their relation is to have the corrected
values fulfill the fixed-focus condition, which can be approximately achieved
through

$\Delta\alpha
c_{\textnormal{ff}}=\frac{\cos\beta^{\prime}}{\cos\alpha^{\prime}}\Delta\beta.$
(4)

Although we emphasize that this choice is not a priori theoretically
motivated, we found that the derived energy scale is not sensitive to a
particular relation between $\Delta\alpha$ and $\Delta\beta$ if both are
sufficiently small, and the calibration is applied to an appropriately small
energy range. The reason is that small changes of $\alpha$ or $\beta$ have the
same effect, that is, shifting the energy scale. It is only for large changes
of $\alpha$ or $\beta$ that the slope of the calibration changes.

The final calibration of the energy of the gas cell measurements is achieved
through a simultaneous fit to both the photoion yield spectrum in the gas cell
and the fluorescence spectrum in the EBIT. Using theoretical values for the
energies of the fluorescence lines, the free parameters of the fit are the
angular shifts of $\alpha$ and $\beta$, the energies of the photoionization
resonances in the gas cell spectrum, and their respective widths. Each
fluorescence line and photoionization resonance is modeled with a Voigt
profile, with the Gaussian $\sigma$ and Lorentzian $\Gamma$ parameters
representing a combination of instrument profile, natural linewidth, and
thermal Doppler broadening; both spectra also include background which we
model with a energy independent constant. In the fluorescence spectrum, it is
mainly caused by the high-energy tail of the pulse height distribution of low-
energy photons detected by the SDDs. In the photoionization spectra, it
results from residual gases that do not have resonant features in the bands of
interest, and can thus be treated as a constant contribution to the detected
signal.

All of the fits are evaluated using the Cash statistic (cash1979, ), a version
of the likelihood ratio test that is appropriate for Poisson distributed data.
Since we model the calibration and ionization data simultaneously, it is
possible to estimate confidence intervals for our parameters of interest by
confidence search (cash1976, ). These intervals describe the total statistical
uncertainty for each emission line in the photoionization spectrum, including
the one from the calibration measurement (but excluding systematic
uncertainties, as discussed below). The confidence intervals derived using
this approach cover the 90% uncertainty interval and are typically in the
range between 1 to 10 meV.

###  2.3 Systematic limitations

Figure 2: Measured position of the Ow resonance in terms of the angle
correction at different times during the campaign. Black crosses and red
triangles respectively indicate the reported energy after and before the
photon measurement, and blue squares their mean (see text for details). The
right scale shows the translated energy spread around the mean of all
measurements at the O${}_{\textnormal{w}}$ line (inner) and the Ne 1s–3p
transition (outer). Measurements of this line were mainly used to benchmark
the X-ray beam at different settings causing a variation in the uncertainty.

The calibration uncertainty of the photoionization spectra is dominated by
systematic terms. As discussed in Sect. 2 and 2.2, the statistical
uncertainties on the calibration are typically smaller than 10 meV, while the
theoretical uncertainties in our calibration line energies are smaller than 1
meV.

In our current setup, a contribution to the uncertainty much larger than those
comes from the stability of the beamline. We can estimate the long-term
variability of the beamline from scans measuring the same transition
throughout the measurement campaign. The fluctuation of the shift parameter,
$\Delta\beta$, is shown in Fig. 2 for repeated measurements of the Ow. On the
right-hand $y$-axis, we display the corresponding effect of such an angular
shift on the energy calibration at the Ow line (574 eV) and the neon K-edge
(870 eV). In our experiment we requested the reported monochromator energy and
angle settings twice for each scan step; once before data acquisition with the
SDD and once after. We found that the reported energy values before the SDD
acquisition often showed unreasonably high fluctuations, probably attributable
to the relaxation of the beamline to the selected energy immediately after
moving the monochromator, even after the allowed settling time. For this
reason we only used the values reported after each scan step for our further
analysis. As more extensively discussed in the supplemental material of
(leutenegger2020, ), based on repeated scans of multiple closely-spaced
photoionization lines in the gas cell, and also on studies of the shapes of
single fluorescence lines in the SDD, we conclude that such large shifts do
not occur in single scans; however, shifts of up to 40 meV can be expected
near Ow. Given that the energy shifts for a fixed angular shift become larger
at higher energies, we estimate that the systematic energy shift at the Ne
K-edge can be as high as 100 meV; we discuss this further in Sect. 3.1.

##  3 Energy calibration of $\mathrm{Ne}$, $\mathrm{CO_{2}}$, and
$\mathrm{SF_{6}}$

We now discuss the results of modeling each of the photoionization spectra
measured for neon, $\mathrm{SF_{6}}$ and $\mathrm{CO_{2}}$.

###  3.1 $\mathrm{Ne}$ Rydberg series

We calibrated our scan of atomic neon using the F${}^{7+}\,K_{\beta}$
transition ($E_{\textnormal{F}\,K_{\beta}}$ = 857.5108(7) eV, yerokhin2019, ).

This line was scanned before and after the actual ionization measurement of
neon and, therefore, not simultaneously. As discussed above, this adds an
additional uncertainty which can, in principle, be as large as 150 meV (Fig.
2). The angular shift corrections measured for the two F${}^{7+}\,K_{\beta}$
scans differ by $0.2^{\prime\prime}$, corresponding to about 30 meV at
$E_{\textnormal{F}\,K_{\beta}}$. Instead of using the averaged shift
correction as obtained from both calibrations, we weigh the shift correction
of the neon data with Student’s $t$ distribution (student1908, ). In this way
we can estimate the statistical uncertainty due to the variation of the
calibration by assuming that these are drawn from a normal distribution. Just
accounting for statistical variations, the resulting 90% confidence interval
for the energy of the neon lines is $\pm 15$ meV. The systematic uncertainty
can not be quantified directly but can be deduced by comparison to previous
experiments. Overall we estimate a 100 meV calibration uncertainty.

Figure 3: a: Neon spectrum (blue points) calibrated using measurements of F
$K_{\beta}$ in scans before and after the photoionization measurements. The
neon emission lines are modeled by a sequence of Voigt profiles to determine
the peak positions (red solid line, components red dashed lines). The green
vertical bars indicate the line positions as reported in (mueller2017, ) and
the gray solid line outlines their data (scaled to the 1s–3p transition). b:
The ratio between the data and the model. The calibration line was modeled
with Voigt parameters $\sigma=0.172$ eV and $\Gamma<0.001$ eV.

The Rydberg series (Fig. 3) is modeled by a set of five Voigt profiles without
constraints on the line shape parameters. The scan range does not reach up to
the series limit such that it is not possible to include a component for the
edge without constraints on its location. The model used by mueller2018 ,
where the positions are constrained by a Rydberg series modified by a quantum
defect, is not describing our data to a satisfying level. Therefore we did not
constrain the line positions and we also did not include a component for the
ionization edge. Ignoring contributions of the ionization edge to the high
energy part of the scan causes the fifth Voigt profile to model all
contributions from higher Rydberg transitions and the ionization edge. This
behavior of the model can have an effect on the position of the 1s-6p line,
but we expect that the effect on the lower transitions is only marginal and
below the uncertainty. The resulting model is shown in Fig. 3 and has only a
few residual patterns left. A part of these residuals can be attributed to the
uncertainty (or jitter) of the reconstructed energy grid. We verify this by
fitting the same model to the data using the nominal energy grid. Here the
residuals cluster around the wings of the model lines since a jitter in the
energy grid has a larger impact at energies the derivative of the model has a
larger absolute value. For the reconstructed energy grid these residual
patterns are stretched over a wider energy range.

Table 1: Measured $\mathrm{Ne}$ Rydberg transitions lines calibrated against
the F $K_{\beta}$ line.

| Energy [eV]
---|---
Transition | This work(a) | Müller et al.(mueller2017, )(b)
1s–3p | 867.278 | 867.290
1s–4p | 868.980 | 868.928
1s–5p | 869.620 | 869.530
1s–6p | 869.920 | 869.815

(a) Calibrated against F $K_{\beta}$ (857.5108(7) eV, yerokhin2019, )
(b) Calibrated against neon 1s–3p (wuilleumier1971, )

Statistical uncertainties of the peak positions are $\pm 15$ meV but are
largely exceeded by systematic variations of up to 100 meV (see text). Recent
high resolution measurements of the neon Rydberg series are given for
comparison (mueller2017, ).

Our determined line positions are given in Table 1. We compare these results
with those found in (mueller2017, ), which have been calibrated using the
measured 1s–3p transition of (wuilleumier1971, ). The agreement of the first
line is very good, while the subsequent lines diverge more with higher energy.
The difference of order 50-100 meV in the higher-$n$ lines is consistent with
the amplitude of drift observed in the energy scale of U49-2/PGM-1, as
discussed in the supplemental material of (leutenegger2020, ).

###  3.2 $\mathrm{CO_{2}}$ Oxygen K-edge

Figure 4: a: Calibrated $\mathrm{CO_{2}}$ spectrum (blue points) based on
simultaneous measurements of N $K_{\epsilon}$. Emission lines are modeled by
Voigt profiles (red solid line, components red dashed lines). Model components
may represent multiple unresolved transitions. The green lines indicate the
transition energies in the Rydberg complex reported in (adachi2005, ) for the
two resolved symmetry directions 0∘ (solid) and 90∘ (dashed). Solid gray and
dashed-dotted gray line indicated their data, for 0∘ and 90∘, respectively.
The dotted gray vertical line indicates the location of the calibration line
for our data. b: Residual water vapor in the gas cell adds additional spectral
features. The background was estimated from data from a second gas cell (black
points); The corrected spectrum (orange points) shows the difference between
the data of the two gas cells. The uncorrected data is again given here for
reference (blue points). c: Residuals between model and data as ratio. d: Sum
of the fluorescence spectra produced in the EBIT measured by the two SDDs. The
calibration line was modeled with Voigt parameters $\sigma=0.082$ eV and
$\Gamma<0.001$ eV.

We measured the photoionization yield of $\mathrm{CO_{2}}$ in the range 533 to
540 eV. The calibration of the energy grid is based on the theoretical
predictions of the He-like nitrogen transition N $K_{\epsilon}$
($E_{\textnormal{N}\,K_{\epsilon}}=538.4924(3)$ eV, yerokhin2019, ).

This line was measured simultaneously with the ionization spectrum, thus
significantly reducing the overall uncertainty. The measured $\mathrm{CO_{2}}$
spectrum is rich, featuring (in our spectrum) unresolved vibrational
structure, and showing a mixture of lines from both polarizations (adachi2005,
; okada2002, ). To determine the transition energies, we empirically modeled
the spectrum with a set of 10 Voigt profiles, which in many cases represent
blends of unresolved emission lines. The choice of 10 lines is only supported
by the number of features which are identifiable by eye. The background is
modeled with an energy independent constant. Fig. 4 shows the calibrated data
and best fit model. This best fit model is reasonably good, with residuals
comparable to the neon measurement. On closer inspection, the spectrum shows a
rich structure which is only barely resolved in our data but clearly visible
in recent resonant inelastic X-ray scattering (RIXS) measurements
(soderstrom2020, ).

This large number of parameters in the empirical model poses a difficult
problem for classical fit algorithms, especially with the addition of the
calibration function itself. To find the minimum of this function, we made use
of the Markov Chain Monte Carlo (MCMC) algorithm proposed in (foreman2013, ).
It explores the probabilistic parameter space, and additionally gives the
parameter uncertainty. The resulting 90% confidence for the first 9 line
profiles is $\pm 3$ meV. The tenth line is only partly covered by the scan,
and therefore not well constrained.

Table 2: $\mathrm{CO_{2}}$ measured transitions in our calibration.

| Energy [eV] |
---|---|---
Transition | This work(a) | Okada (okada2002, )(b) | Adachi (adachi2005, )
$\left.\parbox{16.0pt}{{{$\pi^{*}$\\\ 3s$\sigma$}}}\mkern 9.0mu\right\\}$ |  535.334 535.582 | 535.4(c) | 535.4(c)
contam. |  537.069 | — | —
contam. |  537.937 | — | —
3p$\pi_{u}$ |  538.487 |  538.53 |  538.53
$\left.\parbox{16.0pt}{{{3p$\sigma_{u}$\\\ 3p$\pi_{u}$}}}\mkern 9.0mu\right\\}$ |  538.720 | $\left\\{\mkern 9.0mu\textnormal{\parbox{0.0pt}{538.78\\\ 538.83}}\mkern 9.0mu\right.$ | $\left\\{\mkern 9.0mu\textnormal{\parbox{0.0pt}{538.78\\\ 538.82}}\mkern 9.0mu\right.$
$\left.\parbox{16.0pt}{{{4s$\sigma_{g}$\\\ 3p$\sigma_{u}$}}}\mkern 9.0mu\right\\}$ |  538.908 | $\left\\{\mkern 9.0mu\textnormal{\parbox{0.0pt}{538.93\\\ 539.04}}\mkern 9.0mu\right.$ | $\left\\{\mkern 9.0mu\textnormal{\parbox{0.0pt}{538.91\\\ 539.06}}\mkern 9.0mu\right.$
$\left.\parbox{16.0pt}{{{4s$\sigma_{g}$\\\ 3p$\sigma_{u}$}}}\mkern 9.0mu\right\\}$ |  539.197 | $\left\\{\mkern 9.0mu\textnormal{\parbox{0.0pt}{539.18\\\ 539.30}}\mkern 9.0mu\right.$ |  539.20
3d$\pi_{g}$ |  539.595 |  539.67 |  539.64

(a) Calibrated against N $K_{\epsilon}$ ($538.4924(3)$ eV, yerokhin2019, ).
(b) calibrated against $\mathrm{CO_{2}}$ transitions from (prince1999, ).
(c) Unresolved blend of $\pi^{*}$ and 3s$\sigma$, reported in (wight1974, ).

For comparison the experimental values of (okada2002, ; adachi2005, ) are
listed. Assignments are based on the assignments of (okada2002, ). Line
blending and mixing is indicated by braces. We estimate the uncertainty of our
energy scale to 40 meV (see text).

We list the resulting line positions in Table 2, where the assignments are by
strongest contribution to our model based on the measurement of (okada2002, ).
The resonance peak shows two main features (e.g., adachi2005, ) generally
associated to the valence orbital and contribution from the 3s$\sigma$ state.
A small emission line is visible at the shoulder of the resonance peak
together with an excess of events between the resonance and the Rydberg
complex compared to recent high resolution RIXS measurements (soderstrom2020,
). This excess can be attributed to residual water vapor in the gas cell.

An estimate of the residual gas is obtained from a second gas cell operated
upstream of the first cell and separated from it by a thin SiN window. The
second cell was operated with no sample gas injection, and therefore all
photoions detected are from background gases. In principle, the background gas
composition in the two cells may be different. However, due to insufficient
bakeout, the residual gas in our vacuum chambers was dominated by water vapor,
as can be seen by comparing the features in the background gas spectrum to
previously published measurements of water vapor (WIGHT1974_H2O_etal). As
indicated in Fig. 4, we see that the residual gas spectrum explains the
feature on the high energy side of the $\pi^{*}$ resonance as well as the
unexpectedly high amplitude of the continuum between the resonance and the
Rydberg complex. Because we could not be certain that the amplitude of the
background spectrum was the same in both cells, we cannot use the second cell
to correct the first. However, we can try modeling the background in the first
cell based on the spectrum of the second and assess the impact on our fit
results. We found that the energy of the $3s\sigma$ peak moved to slightly
higher energy, while the other peak energies were unaffected. We attribute the
remaining residuals in the $\pi^{*}$ resonance to a combination of unresolved
vibrational structure (ljungberg2017, ; vazdacruz2021, ) and a possible non-
ideal instrument lineshape. The dominant uncertainty in the transition energy
determination is drift in the monochromator energy scale. Based on the
analysis in (leutenegger2020, ) we estimate this uncertainty to be 40 meV for
these lines.

Direct comparison of the results with the literature is in general not
possible due to the blending of transitions. However, the 3p$\pi_{u}$
transition is easily identifiable in our scan as well as in recent
measurements (soderstrom2020, ; adachi2005, ; okada2002, ). Additionally, its
energy is very close to our calibration line and, hence it has much smaller
shifts due to drift. Using this as a reliable reference energy, we see that
the Rydberg complex from (okada2002, ) appears at slightly higher energies.
The result of (okada2002, ), calibrated using $\mathrm{CO_{2}}$ measurements
from (prince1999, ), which in turn are calibrated against 02 measuremnts from
(hitchcock1980, )222For the calibration with respect to O2, (prince1999, )
references (sodhi1984, ). However, the given value is actually obtained from
(hitchcock1980, ).. Given the uncertainties of 100-200 meV in their work, we
conclude that our measurements of the transition energy of 3p$\pi_{u}$ agree.
Similar EELS measurements (eustatiu2000b, ) also place the $\pi^{*}$ resonance
at a higher energy, but comparison of the Rydberg complex is difficult due to
their limited energy resolution and lack of polarization selectivity.

###  3.3 $\mathrm{SF_{6}}$ Fluorine K-edge

We scanned the fluorine K-edge of $\mathrm{SF_{6}}$ in the range from 685 to
705 eV measuring the photoionization yield of the gas in the gas cell. The
calibration is based on the O $K_{\gamma}$ transition of He-like oxygen
($E_{\textnormal{O\,K}_{\gamma}}=697.7859(5)$ eV, yerokhin2019, ).

Figure 5: a: $\mathrm{SF_{6}}$ photionization spectrum (blue points)
calibrated by simultaneous measurement of the O $K_{\gamma}$ transition and
modeled by five Voigt profiles and one error function (red solid line,
components red dashed lines). The position of the calibration line is
indicated by the dotted vertical line. The solid green lines indicate the
measured peak positions from (hudson1993, ) and the edge as measured in
(siegbahn1969, ) (indicated with diagonal marks). The gray curve outlines the
measurements of (hudson1993, ) scaled to match the present results. b: Ratio
between the data and the best-fit model. c: Sum of fluorescence spectra
measured with the two SDDs. The calibration line is modeled with the Voigt
parameters $\sigma=0.079$ eV and $\Gamma=0.031$ eV

Following (hudson1993, ), we describe the spectrum by a sequence of five Voigt
profiles together with an error function to account for the photoelectric
absorption edge at the Rydberg series limit. In (hudson1993, ) an arctangent
function convolved with a Gaussian was used to describe the edge, but given
our resolution we cannot discriminate between these choices. Hence, for
simplicity we used only an error function to model the edge. The calibrated
data and resulting best fit are displayed in Fig. 5. Similarly to (hudson1993,
), we have to add a line around 696 eV, otherwise an excess of events remains
visible in the data in comparison with the model. Further justification for
emission at this energy is given by theoretical predictions (see Sect. 4).
From the confidence calculations, we estimate the 90% uncertainty to $\pm 3$
meV for the line positions; the edge energy has an uncertainty of $+2$ meV.
The systematic uncertainty is dominated by drift in the beamline energy scale,
and based on (leutenegger2020, ), we estimate it to be 60 meV.

Table 3: Calibrated $\mathrm{SF_{6}}$ transitions and comparison to other
publications.

| Energy [eV]
---|---
This work |  | Other experiments
Symmetry | Experiment(a) | Theory |  | Eustatiu(b) | Francis(c) | Hudson(d) | Hitchcock(e)
a1g | 688.448 | — |  | 687.9 | 688.0 | 689.0 | 688.0
$\left.\parbox{10.0pt}{{{\text@underline{a\textsubscript{1g}}\\\ t\textsubscript{1u}}}}\mkern 9.0mu\right\\}$ | 692.082 | $\left\\{\mkern<EMAIL_ADDRESS>692.23}}\mkern 9.0mu\right.$ |  | 691.4 | 692.4 | 692.9 | 692.6
$\left.\parbox{10.0pt}{{{t\textsubscript{1u}\\\ e\textsubscript{g}\\\ t\textsubscript{2g}\\\ \text@underline{t\textsubscript{1u}}}}}\mkern 9.0mu\right\\}$ | 694.217 | $\left\\{\mkern 9.0mu\textnormal{\parbox{0.0pt}{693.67\\\ 693.81\\\ 693.95\\\<EMAIL_ADDRESS>9.0mu\right.$ |  | 693.5 | 694.0 | 694.7 | 694.6
$\left.\parbox{10.0pt}{{{t\textsubscript{1u}\\\ t\textsubscript{1u}\\\ \text@underline{t\textsubscript{2u}}}}}\mkern 9.0mu\right\\}$ | 696.296 | $\left\\{\mkern 9.0mu\textnormal{\parbox{0.0pt}{695.11\\\ 695.43\\\<EMAIL_ADDRESS>9.0mu\right.$ |  | — | — | 696.3 | —
I.P. | 696.998 | — |  | 694.6(f) | 694.6(f) | — | 694.6(f)
t2g | 699.446 |  699.51 |  | 698.8 | 698.9 | 699.9 | 699.1

(a) Calibrated against O $K_{\gamma}$ ($697.7859(5)$ eV, yerokhin2019, ). |  (b) Eustatius et al. (eustatiu2000a, ).
---|---
(c) Francis et al. (francis1995, ). |  (d) Hudson et al. (hudson1993, ).
(e) Hitchcock & Brion (hitchcock1978, ). |  (f) Determined from XPS (siegbahn1969, ).

Theory values obtained by TDDFT calculations (see Sect. 4. The first
excitation (a${}_{\textnormal{1g}}$) is chosen to align with the experiment.
Assignments are based on these calculations, where the largest contribution to
each spectral feature is underlined. Line blends are indicated by braces. For
comparison, the results of selected publications are listed. We estimate the
uncertainty of our energy scale to 60 meV.

Overall, the empirical model describes the data very well. However, the lowest
energy line has significant residuals. These residuals may originate from a
combination of unresolved vibrational structure (ljungberg2017, ;
vazdacruz2021, ) and a possibly non-ideal instrument lineshape. The spectrum
has been measured several times in the past with varying results (e.g.,
siegbahn1969, ; gelius1974, ; hitchcock1978, ; hudson1993, ; francis1995, ;
eustatiu2000a, ). Many of the EELS measurements use the measurements in
(hitchcock1978, ) for calibration, which itself is based on measurements from
(hitchcock1977, ). The EELS measurements have a difference of
${\sim}500\,\mathrm{meV}$ with our result, two to three times more than their
claimed uncertainty, but in agreement with the discrepancy in
(leutenegger2020, ). The photoionization measurements from (hudson1993, ) also
show a shift to higher energies, but their calibration was provided only by
the used beamline. The results of this work together with selected previous
results are given in Table 3. It is evident that the often used value of the
edge energy as reported in (siegbahn1969, ) is not compatible with our results
(indicated in Fig. 5). A similar observation can be made from the spectrum
given in (hudson1993, ); however, their edge location is not quantified.

##  4 Modeling K-edge absorption spectra from first principles

In order to assist in the interpretation of the experimental data we performed
_ab initio_ TDDFT simulations of the oxygen K-edge excitations in
$\mathrm{CO_{2}}$ (shown in Fig. 6) and fluorine K-edge in $\mathrm{SF_{6}}$
(shown in Fig. 7). The computation of the molecular orbitals associated with
the excited states allows us to attach symmetry labels to the experimental
peaks. Moreover, the arbitrarily high resolution of the simulated spectra can
help to understand whether observed peaks originate from single broadened
transition lines or if there is a richer spectral structure which cannot be
resolved experimentally.

###  4.1 Calculation details

For the simulation we employ Time-Dependent Density Functional Theory (TDDFT)
as implemented in the ORCA quantum chemistry code (ORCA_main, ). We used a
minimally augmented diffuse quadruple zeta basis set ma-def2-QZVPP
(Weigend2005_basis, ; Zheng2011_basis, ) and the Coulomb fitting auxiliary
basis def2/J in combination with the hybrid functional PBEh$\alpha$
(Perdew1996_functional, ). All calculations employed the RIJCOSX approximation
(Neese2009_rijcosx, ). Moreover, we considered only purely electronic effects
and neglected vibrational modes of the molecules. The only free parameter of
the DFT simulation is the mixing factor $\alpha$ of the hybrid functional, for
which we found the best agreement with the present experiments at a value of
$\alpha=35\,\%$. The infinitely sharp transitions of the TDDFT simulation were
subsequently broadened (convolved) for comparison with experimental data.
Following common procedure, we employed an energy-independent Gaussian of 1.8
eV (i.e., accounting for measurement effects) and an energy-dependent
Lorentzian (i.e., life time broadening) of the order of 0.10–4.47 eV. Since
the energy offset of the TDDFT spectra is known to be unreliable, following
common practice the simulated spectra were shifted to align the lowest lying
excitation with the experimental data.

###  4.2 General remarks

While our TDDFT calculation includes core-hole effects (Petersilka1996, ;
Bunau2012, ) beyond a simple mean-field limit, excitonic multiplet splittings
of the excited states are negligible in K-edges (as opposed to, e.g.,
transition metal L- or rare-earth M-edges). Therefore, a single-particle
picture can be used to interpret the excitations as the promotion of an oxygen
or fluorine 1s core electron into “unoccupied” molecular orbitals. We can
exploit this single-particle nature and associate to each peak in the spectrum
a corresponding single electron molecular-orbital computed from the self-
consistent DFT (and plotted with the Avogadro program (Avogadro2012, )). Its
symmetry (and degeneracy) then allows us to classify the excitations in terms
of irreducible representations of the molecules point-group. For more details
on approximation and simulation strategies for (especially oxygen) K-edge
absorption in atoms, molecules, and solids, we refer the interested reader to
a recent review (Frati2020, ).

###  4.3 The oxygen K-edge of $\mathrm{CO_{2}}$

In Fig. 6 we show the comparison of simulation (orange and green lines) and
experiment together with the earlier EELS data (eustatiu2000b, ). The first
part of the K-edge is dominated by the well known transition into the
$\pi_{u}$ orbital at around 535.4 eV. The peaks at higher energies are
typically assigned to Rydberg transitions. In this energy region we get a
satisfactory agreement in terms of the overall relative spectral weight at a
low resolution (see broadened simulation vs. EELS in Fig. 6). However, the
simulation misses the splitting of peaks picked up by higher resolution
experiments. An explanation might be our neglect of vibrational modes. Indeed,
$\mathrm{CO_{2}}$ as a linear ($D_{\infty h}$) molecule, is a prime candidate
even for irregular vibrational structure due to the Renner-Teller effect
(Frati2020, ). Moreover, we point out that the simulated spectra correspond to
absorption with unpolarized light and can thus only be directly compared to
the EELS data (eustatiu2000b, ). We do not account for matrix elements in the
transition that account for polarization dependence in the new experimental
data.

The summary of our _ab initio_ symmetry classification of the peaks can be
found in Table 4.

Figure 6: $\mathrm{CO_{2}}$ X-ray absorption spectrum from TDDFT calculations for oxygen K-edge. Highly resolved peaks were numerically broadened to visualize agreement of simulation and experiment. Experimental data are shown after subtraction of contamination. Additional data extracted from eustatiu2000b  with aligned first peak are depicted for comparison. The baseline of all of these spectra has been unified. Corresponding molecular orbitals are plotted below. Table 4: Assignment of irreps for transition orbitals with corresponding transition energy in K-edge excitation of $\mathrm{CO_{2}}$. irrep | energy [eV]
---|---
$e_{1u}$ ($\pi_{u}$) | 535.4
$a_{1g}$ ($\sigma_{g}^{+}$) | 536.07
$e_{1u}$ ($\pi_{u}$) | 538.11
$a_{2u}$ ($\sigma_{u}^{-}$) | 538.16
$a_{1g}$ ($\sigma_{g}^{+}$) | 539.19
$e_{1g}$ ($\pi_{g}$) | 539.88

###  4.4 The fluorine K-edge in $\mathrm{SF_{6}}$

Figure 7: SF6 X-ray absorption spectrum from TDDFT calculations for fluorine
K-edge. Highly resolved peaks are numerically broadened to visualize agreement
of simulation and experiment. Experimental data are shifted to the baseline.
The corresponding molecular orbitals are plotted below the $x$-axis.

In Fig. 7 we compare our simulation to the experiment. In contrast to the
oxygen K-edge of $\mathrm{CO_{2}}$, the fluorine K-edge in octahedral
($O_{h}$) $\mathrm{SF_{6}}$ is not dominated by a single transition and has a
comparable spectral weight in three main structures between 685 eV and 705 eV.
With the same calculation parameters that we used for $\mathrm{CO_{2}}$, we
find an overall satisfactory agreement with experiment. The comparison reveals
that particularly the central double-peak structure around 693.5 eV might
originate from a variety of transitions which are, however, not resolved in
experiment. In Table 3 we provide a comprehensive list of energies and
symmetry character of the transitions.

##  5 Conclusions

We used a newly introduced experimental setup to provide precise calibration
references in the soft X-ray regime. A careful statistical analysis shows that
the resulting energy calibration can in principle provide an accuracy of 1–10
meV (at energies in the 500–800 eV range). The resulting calibrations have no
dependence on previous measurements and therefore do not carry any legacy
uncertainty present in other measurements. The achieved accuracy is limited by
significant systematic uncertainties that exceed the statistical uncertainties
by almost an order of magnitude. We performed several measurements of
molecular absorption spectra that are commonly used for energy calibration.
The results for $\mathrm{CO_{2}}$ show relatively good agreement with previous
ones; for $\mathrm{SF_{6}}$, we see a shift similar to that found in
(leutenegger2020, ). Significant differences appearing in the $\mathrm{Ne}$
measurement compared to earlier works might be partly an artefact of the non-
simultaneous measurement of the calibration, and require further
investigation. Our theoretical simulations of the $\mathrm{SF_{6}}$ spectrum,
although consisting of numerous features, also show fairly good qualitative
agreement with the data. We are able to attribute a much richer structure
underlying the measured spectral weight at 691–697 eV, supporting (hudson1993,
) in contrast to other works (eustatiu2000a, ; francis1995, ; hitchcock1978,
). For $\mathrm{CO_{2}}$, the experimental spectra exhibit several features
not captured by the simulations. We attribute these differences to us
neglecting polarization dependence (dichroism) and vibrational effects. Since
such vibrational and symmetry-resolving effects do not influence the relative
positions of the peaks due to optical excitations, explicitly correlated
methods from many-body perturbation theory (e.g., Bethe-Salpeter formalism;
Onida2002, ) may improve predictions from theory.

Despite the systematic effects still present in our current experiment, we
have reduced the overall uncertainty in comparison with various previous
measurements. For further investigations aiming at reaching a statistically
dominated accuracy, it is necessary to follow in time small relative energy
shift of the photon beam energy selected by the monochromator. This could be
achieved by, e.g., photoemission spectroscopy simultaneously performed with
the photoionization measurements, and would remove any dependency caused by
the beamline.

The accuracy of theoretical calculations for few-electron HCI surpasses that
of any other soft X-ray standards, and thus our method can in principle
provide references at the level of $\sim$50 meV for this range. Such
references will find various applications in different fields of research, and
help, as shown in this work, assessing the accuracy of calculation for
molecular systems. Crucially, our calibration method and the present results
address essential needs of upcoming X-ray astrophysics missions.

###### Acknowledgements.

Financial support was provided by the Max-Planck-Gesellschaft (MPG) and
Bundesministerium für Bildung und Forschung (BMBF) through project 05K13SJ2.
We thank HZB for the allocation of synchrotron radiation beamtime at BESSY II.
C.S. acknowledges the support by an appointment to the NASA Postdoctoral
Program at the NASA Goddard Space Flight Center, administered by Universities
Space Research Association under contract with NASA, by the Lawrence Livermore
National Laboratory (LLNL) Visiting Scientist and Professional Program
Agreement No. VA007036 and VA007589, and by MPG. Work by UNIST was supported
by the National Research Foundation of Korea (No. NRF-2016R1A5A1013277). Work
by LLNL was performed under the auspices of the U. S. Department of Energy
under Contract No. DE-AC52-07NA27344 and supported by NASA grants to LLNL.
M.A.L. and F.S.P. acknowledge support from NASA’s Astrophysics Program. Work
by G.B. was supported by a NASA Space Technology Research Fellowship. Work by
R.C. was supported by NASA under award number 80GSFC21M0002 and by an
appointment to the NASA Postdoctoral Program at the NASA Goddard Space Flight
Center. Work by M.W. was supported by the ERASMUS+ traineeship program.

## Author contributions

J.S., M.W., N.H., P.H., J.W., and M.A.L. wrote the initial draft of the paper.
The project was initiated by M.A.L., J.R.C.L-U., S.B., and G.V.B. Theoretical
calculations were carried out by M.W. and P.H. Data reduction and analysis was
done by J.S. and M.A.L. Everyone, except M.W. and P.H., was involved preparing
the project or conducting the experiment.

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