# Plasma screening in mid-charged ions observed by K-shell line emission

M. Šmíd<EMAIL_ADDRESS>Helmholtz Zentrum Dresden Rossendorf, Bautzner
Landstraße 400, 01328 Dresden, Germany  O. Humphries  European XFEL,
Holzkoppel 4, 22869 Schenefeld, Germany  C. Baehtz  European XFEL, Holzkoppel
4, 22869 Schenefeld, Germany  E. Brambrink  European XFEL, Holzkoppel 4, 22869
Schenefeld, Germany  T. Burian  Institute of Physics, Academy of Sciences of
the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic  M. S. Cho
Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore,
California 94550, USA  T.E. Cowan  Helmholtz Zentrum Dresden Rossendorf,
Bautzner Landstraße 400, 01328 Dresden, Germany  L. Gaus  Helmholtz Zentrum
Dresden Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany  V.
Hájková  Institute of Physics, Academy of Sciences of the Czech Republic, Na
Slovance 2, 182 21 Prague 8, Czech Republic  L. Juha  Institute of Physics,
Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8,
Czech Republic  Z. Konopkova  European XFEL, Holzkoppel 4, 22869 Schenefeld,
Germany  H. P. Le  Lawrence Livermore National Laboratory, 7000 East Avenue,
Livermore, California 94550, USA  M. Makita  European XFEL, Holzkoppel 4,
22869 Schenefeld, Germany  X. Pan  Helmholtz Zentrum Dresden Rossendorf,
Bautzner Landstraße 400, 01328 Dresden, Germany  Technische Universität
Dresden, 01062 Dresden, Germany  T. Preston  European XFEL, Holzkoppel 4,
22869 Schenefeld, Germany  A. Schropp  Centre for X-ray and Nano Science CXNS,
Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany
H. A. Scott  Lawrence Livermore National Laboratory, 7000 East Avenue,
Livermore, California 94550, USA  R. Štefaníková  Helmholtz Zentrum Dresden
Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany  J. Vorberger
Helmholtz Zentrum Dresden Rossendorf, Bautzner Landstraße 400, 01328 Dresden,
Germany  W. Wang  Centre for X-ray and Nano Science CXNS, Deutsches
Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany  U.
Zastrau  European XFEL, Holzkoppel 4, 22869 Schenefeld, Germany  K. Falk
Helmholtz Zentrum Dresden Rossendorf, Bautzner Landstraße 400, 01328 Dresden,
Germany  Institute of Physics, Academy of Sciences of the Czech Republic, Na
Slovance 2, 182 21 Prague 8, Czech Republic

###### Abstract

Dense plasma environment affects the electronic structure of ions via
variations of the microscopic electrical fields, also known as _plasma
screening_. This effect can be either estimated by simplified analytical
models, or by computationally expensive and to date unverified numerical
calculations. We have experimentally quantified plasma screening from the
energy shifts of the bound-bound transitions in matter driven by the x-ray
free electron laser (XFEL). This was enabled by identification of detailed
electronic configurations of the observed K$\upalpha$, K$\upbeta$ and
K$\upgamma$ lines. This work paves the way for improving plasma screening
models including connected effects like ionization potential depression and
continuum lowering, which will advance the understanding of atomic physics in
Warm Dense Matter regime.

††preprint: APS/123-QED

### Introduction

Electrons bound in atoms are held at specific levels – shells and subshells.
The energy of these levels is determined by the electric potential of the ion,
which is influenced by the presence of other electrons, whether bound within
the atom or freely moving in its immediate vicinity. The simplest way to
measure this influence is through radiative atomic transitions, i.e.,
processes in which a bound electron moves from one level to another,
accompanied by the emission or absorption of an x-ray photon with a wavelength
exactly corresponding to the energy difference of the levels. Such
transitions, including the Cu K$\upalpha$ line whose behaviour is studied in
this work, were observed and characterized already in 1909 [1]. Its wavelength
(energy) was first measured in 1913 [2] with an accuracy better than 3% to
today’s values. Those observations actually lead to the discovery of
electronic structure of ions. Even today, observing changes in the energies of
these transitions remains an excellent method for revealing the structure of
atoms and their sensitivity to the surrounding environment. Among other
effects, we can speak of line shifts due to two factors: the influence of
bound electrons (electron configuration) and the influence of free electrons.

The understanding and quantitative analysis of both shifts relies on complex
modelling, which depends on several approximations. One of those is the
_plasma screening_ , describing how the free electron environment affects the
potential of the emitting ion. Most used models of plasma screening are based
on the calculations from about 60 years ago. Only recently, the advent of
x-ray free electron lasers (XFELs) enabled to experimentally challenge those
models, to observe plasma screening via a shift of the emission lines or
absorption edges as a function of plasma conditions. Still, it is typically
not straightforward to extract the _continuous_ effect of screening, since the
line shifts are at the same time influenced by the _discrete_ changes of the
electronic configuration, which is, similarly as screening, affected by plasma
temperature.

The change of line position due to bound electrons, or - in other words -
electronic configuration, can be well illustrated on the $1s-2p$ transition.
Its energy is mostly influenced by the K-shell occupation, electrons in the
L-shell have a smaller effect, and the influence of M- and N- shell electrons
can be often neglected: The addition of a K-shell electron actually changes
the name of the transition: so $1s^{1}-2p^{1}$ is called Lyα, while additional
electron leads to $1s^{2}-1s2p$ which is the Heα. Adding an electron to the
L-shell can produce $1s^{2}2p-1s2p^{2}$, which could be called either Li-like
satellite of Heα, or Li-like K$\upalpha$. Transitions from ions with more
electrons can then be called K$\upalpha$ satellites. Those were first
calculated in 1969 [3] as a function of the L-shell occupation, and
experimentally observed in 1975 [4]. However, the exact description of many-
electron systems like Cu is computationally challenging, as the number of
possible configurations of available electrons (29 in copper) is vast, and the
lines are mostly indistinguishable. Such modelling of non-LTE plasmas was
attempted with super-configuration codes, but the results still show large
deviations from observations [5]. For example, in the atomic model presented
in this paper, the number of K shell transitions is more then 2 million, out
of which about 1.5 million represents $1s-2p$ lines. In experiment, typically
only 9 emission lines defined by the occupation of the L-shell are resolvable.
These _heated_ K$\upalpha$ satellite lines have vast applications at Warm
Dense Matter (WDM) and plasma diagnostics, as identified already in 1981 [6],
their recent applications are shown e.g. in [7, 8, 9, 10, 11]. The advantage
of those lines is that their emission is produced by highly charged ions
present in temperatures of hundreds to thousands of eV, while their emission
energies lie in a narrow range well resolvable by high-resolution crystal
spectrometers, therefore providing a unique insight into the plasma
conditions.

The second effect altering the line position is the _plasma screening_ [12],
occuring when the free electrons surrounding the ion alter its electric field.
The screening has several consequences: The change of transition energy is
often called _Stark shift_. The _ionization potential depression_ (IPD)
describes the decrease of energy needed to remove an bound electron into
continuum, most often manifested in the form of shift of the absorption edge.
_Continuum lowering_ (CL) [13] shows that the boundary between free electrons
(continuum) and bound ones is decreasing, and therefore outer shells are
effectively disappearing – merging into continuum. These effects are most
often described by the Stewart-Pyatt model (SP) [14] from 1966. Some of recent
experiments identifying the shifts of the K-edge indicated that the modified
Ecker-Kröll model [15] fit the measurements better, spurring further model
development [12, 16, 17].

Pioneering experiment studying continuum lowering in atoms in dense plasma
environment isochorically heated by XFEL beam have been shown in 2012 [18].
The shift of absorption edges was measured in low-Z materials, and later
calculated by using Density functional theory (DFT) [19], with results in
perfect agreement to the experimental data with charge states 3–7. One recent
approach to quantify IPD was shown in [16], where the electron distribution
was modelled by classical molecular dynamics. The averaged effect over an
ensemble of configurations was than calculated, and shown to agree well with
the previous experimental data up till charge state 9. However, the spectral
simulations can completely skip the concept of IPD, as shown in [12]. Here,
the DFT-based multi-band kinetic model (VERITAS) explicitly accounts for the
interactions among ions and the dense plasma environment. Energy band shifting
and ionization balance are therefore self-consistently calculated, without
invocation of an ad hoc CL or IPD model. Such models are extremely
computationally expensive and therefore it might be difficult to apply them to
the complex atomic structures like those presented in this paper.

In 2012, Hu et al. pointed out that _“Detailed spectroscopic measurements at
warm dense matter conditions are rare, and traditional collisional-radiative
equilibrium models, based on isolated-atom calculations and ad hoc continuum
lowering models, have proved questionable at and beyond solid density.”_ [12]
In this paper, we provide the missing experimental data and extract the
measurement of plasma screening in high energy density regime. We show x-ray
emission spectra from copper driven by narrow bandwidth XFEL pulses, with
sufficient intensity to heat and ionize the material, generating an array of
transitions within the pulse length, including double core hole states. The
tunability of the XFEL photon energy is used to resonantly pump transitions
with known electronic configuration. By careful analysis of those resonances
and comparison to a detailed model of K-shell transitions calculated by the
Flexible Atomic Code [20], we connect measured emission lines with the charge
state and L-shell occupancy, and consequently measure their shift compared to
calculated values. The same transitions are also calculated while applying
plasma screening by the SP model [21] to show its difference to the
experiment. The unique feature of this dataset is that the measurement
contains the K$\upalpha$, K$\upbeta$, and K$\upgamma$ transitions, therefore
describing the modification of all electronic shells present in the material.
The observed shifts are shown to _not_ agree well with the Stewart-Pyatt
model. The data show the complex structure of the K-shell emission in highly
charged ions, and aim to guide the future development and verification of new,
more precise, models.

Figure 1:  Schematic depiction of observed transition chains: emission above K
edge (a), K$\upalpha$ emission driven by K$\upbeta$ absorption (b), and
K$\upalpha$ and K$\upalpha_{\mathrm{h}}$ driven by K$\upbeta_{\mathrm{h}}$
(c). K$\upalpha$ transitions calculated by the FAC code (d). Each circle is a
single transition with size corresponding to its oscillator strength; the
transitions are grouped according to charge state (y-axis) and L-shell
occupancy (color), and a weighted mean for each group is shown by a vertical
marker. Black stars shows the result of empirical formula introduced in this
paper.

### K-shell transitions in calculations

The K-shell energies and oscillator strengths of the emission lines were
calculated by the FAC code [20]. The $1s-2p$ transitions (K$\upalpha$ lines)
are shown in Fig. 1 (d). A mean energy weighted by the oscillator strength is
calculated for each group of transitions given by charge state and L-shell
occupancy.

In experiments, typically only 9 groups of transitions can be distinctly
resolved; often labelled as H-like (Ly$\alpha$), He-like …F-like, based on the
number of bound electrons. However, the dominant factor affecting the energy
shift is the occupancy of the L-shell, therefore we label the groups as
K$\upalpha$ L$x$, where $x$ is the L-shell occupancy in the upper state. Such
a description, however, is insufficient to comprehend the full dynamics, as
shifts of each of those lines as a function of charge state (or occupancy of
M-shell) are resolved. Therefore using a nomenclature K$\upalpha$ L$x$ M$y$
might be necessary in specific cases.

The lines with various M shell occupancies within given K$\upalpha$ L$x$
transition are typically unresolvable in experimental data, as their shift is
smaller than their widths. In the data shown in this paper, the distinction
was made possible by selectively pumping various excitation and ionization
states via the K$\upbeta$ transitions, whose sensitivity to M shell occupancy
is significantly higher.

Similar data as in Fig. 1 (d) are also calculated for other transitions of
interest, namely the K$\upalpha$ transition in so-called hollow ions (ion with
a hole in K-shell in the initial state) [22], labelled further as
K$\upalpha_{\mathrm{h}}$ ($1s^{1}-2p^{\mathrm{N}}$), K$\upbeta$
($1s^{2}-3p^{\mathrm{N}}$), K$\upbeta_{\mathrm{h}}$
($1s^{1}-3p^{\mathrm{N}}$), and K$\upgamma$ ($1s^{2}-4p^{\mathrm{N}}$),
corresponding figures are shown in appendix. Further FAC calculations were run
with the plasma potential modelled by the SP model, assuming solid density and
various plasma temperatures.

In order to demonstrate the basic scaling of those line shifts, an empirical
formula is designed to approximate the simulated line positions. This says
that the energy of transition can be approximated as

$E=E_{0}-k_{\mathrm{K}}K-k_{\mathrm{L}}L-k_{\mathrm{M}}M-c/Te,$

where $K$, $L$, and $M$ are the occupancies of respective shells,
$T_{\mathrm{e}}$ is the plasma electron temperature in eV, and $E_{0}$,
$k_{\mathrm{K}}$, $k_{\mathrm{L}}$,$k_{\mathrm{M}}$, and $c$ are constants
summarized in Tab.1. The $k_{\mathrm{x}}$ constants indicate how much the
emission line shifts with addition of one electron into given shell. Addition
of an electron into the L shell introduces a shift of about 44 and 98 eV for
K$\upalpha$ and K$\upbeta$, respectively, while the M shell electron causes
shifts of only 4, respectively 15 eV. Those shifts are slightly increased for
ions with close-to-full L shells. The sensitivity to temperature is
significantly higher for K$\upbeta$ compared to the K$\upalpha$ transition.
The fit is valid only for charge state discussed in this work, i.e. between 13
and 27. The approximation by this formula is shown by black stars in Fig.
1(d).

Case | $E_{0}$ | $k_{\mathrm{K}}$ | $k_{\mathrm{L}}$ | $k_{\mathrm{M}}$ | $c$
---|---|---|---|---|---
K$\upalpha$ ($L\leq 6$) |  9025 | 300 |  44 |  4 |  200
K$\upalpha$ ($L\geq 7$) |  9025 | 300 | 43 | 6 | 100
K$\upbeta$ ($L\leq 6$) |  10630 | 380 | 98 | 15 | 500
K$\upbeta$ ($L\geq 7$) |  10630 | 380 | 96 | 19 | 300
K$\upgamma$ |  11110 | – | 118 | 30 | —
Table 1: Fit parameters for simple formula of K$\upalpha$ and K$\upbeta$
energies.

A similar dependence of the transition energy on the presence of electrons in
the M-shell (often called spectator electrons) was shown e.g. for Al Heα in
[23]. To our knowledge, however, such shifts were not quantitatively resolved
and described before for ions with more than 3 electrons.

Figure 2:  Experimental spectra for beam energy density 110 kJ/cm2(a) with
identified resonances (stars with color corresponding to L-shell occupancy),
edges (white bars), and elastic scattering (white circles). The fitted
intensity and position of K$\upalpha$ L6 is shown in (b) and (c).

### Experiment

The experiment was performed at the HED instrument of the European XFEL
laboratory [24]. The 25 fs long x-ray beam was focused down to a sub-$\upmu$m
focal spot reaching intensities up to $7\text{\times}{10}^{18}$ W/cm2,
corresponding to irradiation of $180$ kJ/cm2, and its photon energy was varied
in the wide range above Cu K edge (8.9 – 9.9 keV). X-ray emission of the 3
$\upmu$m thick Cu foil was measured by crystal spectrometers covering the
range from neutral K$\upalpha$ till highly charged K$\upbeta$ transitions.
Spectra were measured for a variable incident pulse energy, and the knowledge
of the focal spot distribution allowed to perform the _focal spot inversion_
(see appendix II.3) to extract the spectra for a given irradiation (areal
energy density). The spectra extracted for irradiation of 110 kJ/cm2 and
various xfel photon energies are shown in Fig. 2 (a). The markers are showing
the three key features, which are subject of further analysis: X-ray Thomson
scattering (XRTS), resonances, and edges.

### Resonances

The resonant processes mean a chain of two or more electronic transitions,
where the first one is driven by photoexcitation, ended by a radiative de-
excitation. As the de-excitation follows typically on a few fs timescale (as
observed in our simulations), the probability of another process modifying the
configuration in between is low in the present cases, therefore we assume the
state of the ion is otherwise unchanged. Two such processes are depicted in
Fig. 1 (b, c).

The resonant processes are identified as emission peaks in the spectra for a
particular driving energy, whose intensity is decreasing if the driving energy
is changed. Measures of the emission were constructed via fitting a family of
Gaussian peaks to the measured spectrum, giving an estimate of yield and
position. By applying the central limit theorem to the combinatorial nature of
spectator electrons, these peaks should be well represented by a Gaussian
given the approximately linear dependence of the spectator electron
perturbations. The intensity and position of the fits of K$\upalpha$ L6 line
are shown in Fig. 2 (b) and (c). The two maxima in the intensity plot show the
resonant driving via K$\upbeta$ (at $\approx~{}9250$ eV) and
K$\upbeta_{\mathrm{h}}$ (at $\approx 9500$ eV). In both, the emission energy
shifts with the change of the driving energy, because different charge states
of K$\upbeta$, respectively K$\upbeta_{\mathrm{h}}$, are being pumped,
therefore producing K$\upalpha$ emission of ions with corresponding charge
states. Once the driving energy moves above $\approx 9600$ eV, both the
intensity and position of the line is not changing significantly, because it
is emitted from ions with K-hole made by photoionization - Fig. 1 (a) - not by
photoexcitation.

To identify the electronic configurations of the observed transitions, the
observations were compared to a model, as shown in Fig. 3. Each experimentally
observed point from Fig. 2 is interpreted as a pair of ’driving line —
emission line’, where the driving line is represented by the horizontal bar
(with width of 25 eV showing the uncertainty given by XFEL bandwidth) and the
emission by the black-outlined circles. To each measured pair of emission —
absorption channels, the charge state and L-shell occupancy is assigned by
identifying the theoretical pair with best matching energies.

### Edges

Identification of the absorption edges has a long tradition in this type of
data [18], and is typically the easiest measurement that could be done. In
copper as well as in other materials with similar Z, however, the edge
position is sharing similar energies as the K$\upbeta_{\mathrm{h}}$ line,
undermining the capability to estimate the edge position precisely. Still, we
have identified a relatively broad range of driving photon energies within
which the absorption edge can lie, those ranges are shown as white vertical
lines in Fig. 2 (a). For driving photon energies below this range, the
emission of the corresponding K$\upalpha$ line is significantly weaker than
above it. The upper boundary is the lowest energy above which the emission
does not show significant intensity or energy fluctuations. As mentioned, the
ranges overlap with the K$\upbeta_{\mathrm{h}}$ – K$\upalpha$ resonances (Fig.
2(a)), which is the main reason why this data does not allow a more precise
estimation of the edge position. The rebinding of states from the continuum,
which retain broad energy bands due to the extent of their wavefunction, has
been found to make clear identification of an edge position challenging,
resulting in different conclusions on the required IPD [25, 26, 23].

Figure 3:  Map of spectral lines in Cu. Black outlined stars show emission
lines and bars absorption energies observed in the experiment. Translucent
symbols are calculated by the FAC code for isolated atom. Color of symbols
corresponds to L shell occupancy. Figure 4: Observed line (a…e) and edge (f)
shifts as a function of charge state for energy density 110 kJ/cm2. Stars are
measured by emission, triangles by absorption. Grey lines are predictions by
SP model in FAC for various temperature assumptions. Yellow points in (f) are
experimental data from [25]. Orange bands in (a..f) are linear approximations
to the experimental data. The potential shifts of atomic shells calculated
from those approximations are shown in (g), dotted lines are inferred from the
hollow transitions.

### Charge-state dependence of plasma screening

In this work, plasma screening is measured as a difference between the
observed transition energy and calculation of its energy for isolated atom.
The found values are plotted in Fig. 4 for each line or edge separately as a
function of charge state. Plasma screening calculated by the FAC code using
the Stewart-Pyatt model is shown in grey lines, assuming solid density and
different plasma temperatures.

The trends between the model and observed shifts agree. However, for
quantitative agreement, the models would have to assume temperatures between 5
and 20 eV, which is significantly lower than the expected values of more than
a hundred eV, similar to other x-ray isochoric heating investigations for
similar states [23]. The K edge is shown in Fig. 4(f), where the significantly
higher experimental uncertainty stems from the overlap of the edge with the
K$\upbeta_{\mathrm{h}}$ lines. The SP model of the edge is largely insensitive
to temperature, with large discrepancies to observations (in the order of 200
eV), clearly demonstrating its insufficiency in modelling such dense plasmas.
The yellow dots are data from previous experiments on various elements [25],
which have indicated the trend confirmed by our measurement for higher charge
states.

This measurement shows that the Stewart-Pyatt model underestimates plasma
screening for the range of charge states investigated here of 13 – 26. Such
discrepancy is present both in the edge lowering (IPD) and in the shifts of
the emission lines (Stark shifts). Measured screening factors roughly follow
linear trends, indicated by thick orange bands in Fig. 4(a..f). The energy
shifts of various shells with respect to continuum are derived from those
approximations and plotted in Fig. 4(g). This screening effect can be
interpreted as modification of the shape of highly charged atoms (ions) due to
the plasma environment.

### Thermal conditions

There are several ways to assess the thermal conditions in the target. First,
a set of simulations with the SCFLY collisional radiative (CR) code [27] was
performed, see Appendix LABEL:app-scfly. The inherent disadvantage of that
code is that it assumes thermal distribution of electrons, which in general
might not be true, as the dominant mechanism of energy absorption is
photoionization, producing electrons with very non-thermal energies. However,
as shown in [28] and confirmed by our simulations with the non-thermal version
of the code Cretin, the electron distribution in this case is close to
Maxwellian due to rapid thermalization via frequent electron–electron
collisions, allowing the temperature to be used as a suitable metric. The
Cretin code was run with comparable conditions, and the temperatures obtained
from both codes are presented in Fig. 5 (a). The Cretin simulations model the
evolution of the electron distribution and do not assume a thermal
distribution, see appendix LABEL:cretin. The non-equilibrium factor, which is
used to characterize deviation from a thermal distribution, is decreasing
below 0.1 towards the end of XFEL pulse, showing the plasma is sufficiently
collisional to be modelled as thermalized.

The experimental approach analyses the elastic scattering data (XRTS) using
the approach shown in [29]. The temporal integration of the signal, overlap of
XRTS with emission, and overall signal to noise ratio, however, limits its
accuracy. The results are shown in Fig. 5 (a) for four different energy
densities, and are in reasonable agreement to the simulated values for
temperatures during the peak irradiation.

Figure 5: Temperatures of the plasma calculated by the CR codes SCFLY and
Cretin (a). Solid lines are temperatures during the peak of the XFEL beam,
dotted lines are the maximal temperatures, reached toward the end of the
pulse. Stars indicate temperatures estimated from the XRTS data. Observed (b)
and predicted (c) Stark shifts as a function of energy density and plasma
temperature, respectively. Blue band in (b) is a linear fit to the data, and
is transferred into (c) by using the measured temperature-energy density
relation in (a).

### Temperature dependence of plasma screening

All experimental spectra shown until this point were obtained with an
irradiation of 110 kJ/cm2. Investigations of line positions from different
heating conditions can reveal the Stark shifts as a function of temperature.
The shifts of the K$\upalpha$ Lx emission extracted from spectra with various
XFEL energy densities is shown in Fig. 5 (b). All transitions show very
similar trend – about 13 eV shift between 40 and 120 kJ/cm2. Such shift
corresponds to the theoretical model of plasma screening (Fig. 5c) with
temperature of about 20 eV. Note, that in contrast to common intuition, the
higher temperatures correspond to more equilibrated conditions in this
situation: We observe lines with charge state 22 or more, which would be
present in equilibrated plasmas only at much higher temperatures (one or few
keV); The lower the temperature we observe those transitions at, the further
the conditions are from any kind of equilibria, and the stronger the Stark
shift is.

Having a reliable Stark shift model for those conditions, those shifts could
be used to infer the electron temperature of the plasma. Yet, an inverse
process can be applied here: Observed line shifts are ascribed to plasma
temperature from the XRTS measurements, and therefore an empirical curve of
Stark shift as a function of temperature is plotted in Fig. 5(c), with broad
blue line. This agrees reasonably well with the theoretical prediction.

### Summary

We have mapped the K shell transitions in intermediate to highly charged
matter by using resonant pumping by the high intensity XFEL beam. The
connection between the absorbing and emitting line allowed identification of
the L and M shell occupancy, and therefore resolution of the transitions with
unprecedented details, i.e. to distinguish its emission for various
occupancies of K, L _and_ M shells. The FAC atomic calculations of those
transitions in isolated ions, as well as with plasma screening by the Stewart-
Pyatt model with various plasma temperature assumptions were compared to the
experimental observation to identify the discrepancies. The trends of the
experimental data qualitatively agree with the SP model for unreasonably low
temperatures, therefore confirming previously stated observation that this
model underestimates the plasma screening. This statement was so far not
observed for charge states above 15. Our data confirmed that the trend
continues up to charge state 26.

The presented experimental measurements of shifts of K$\upalpha$, K$\upbeta$,
K$\upgamma$ transitions and their hollow partners are shown as a function of
charge state. That provides a complex information about the modification of
the ion electronic potential in a well defined plasma environment. Those
measurements shall stipulate development and verification of novel codes to
model the potential and improve our understanding of precise atomic physics in
Warm Dense Matter.

### Data Availability

In alignment with the open data policy of the European XFEL, we are publishing
the x-ray spectra as well as the identified transitions in order to stipulate
the development of further plasma screening models which would better match
the experimental reality. The raw data are published by the XFEL laboratory,
see [30], the processed data (spectra and table with identified resonances)
[31] and the analysis scripts [32] and are published in Rodare repository.

### Acknowledgments

We acknowledge European XFEL in Schenefeld, Germany, for provision of X-ray
free-electron laser beamtime at Scientific Instrument HED (High Energy Density
Science) and would like to thank the staff for their assistance. The work was
also supported by the Helmholtz Association under the grant no. VH-NG-1338.
The work of M.S.C., H.P.L. and H.A.S. was performed under the auspices of the
U.S. Department of Energy by Lawrence Livermore National Laboratory under
Contract DE-AC52-07NA27344.

## References

  * [1] Charles Glover Barkla and Ch A Sadler.  Lxix. the absorption of röntgen rays.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 17(101):739–760, 1909.
  * [2] H.G.J. Moseley.  Xciii. the high-frequency spectra of the elements.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 26(156):1024–1034, 1913.
  * [3] Lewis L. House.  Theoretical Wavelengths for Ka-TYPE X-Ray Lines in the Spectra of Ionized Atoms (carbon to Copper).  The Astrophysical Journal Supplement Series, 18:21, February 1969\.
  * [4] K. A. Jamison, C. W. Woods, Robert L. Kauffman, and Patrick Richard.  $k\alpha$ satellite x rays in al, sc, and ti following bromine-ion bombardment.  Phys. Rev. A, 11:505–508, Feb 1975.
  * [5] Hae Ja Lee, Sam Vinko, Oliver Humphries, Eric Galtier, Ryan Royle, Muhammad Kasim, Shenyuan Ren, Roberto Alonso-Mori, Phillip Heimann, Mengning Liang, Matt Seaberg, Sébastien Boutet, Andrew A Aquila, Shaughnessy Brown, Mikako Makita, and European Xfel.  Driving iron plasmas to stellar core conditions using extreme x-ray radiation.  Preprint, 2023.
  * [6] E. Nardi and Z. Zinamon.  Ka satellite spectra as diagnostics for particle beam‐target interaction.  Journal of Applied Physics, 52(12):7075–7079, 12 1981.
  * [7] S. N. Chen, P. K. Patel, H.-K. Chung, A. J. Kemp, S. Le Pape, B. R. Maddox, S. C. Wilks, R. B. Stephens, and F. N. Beg.  X-ray spectroscopy of buried layer foils irradiated at laser intensities in excess of 10 20 w/cm2.  Physics of Plasmas, 16(6):062701, 06 2009.
  * [8] M. Šmíd, O. Renner, A. Colaitis, V. T. Tikhonchuk, T. Schlegel, and F. B. Rosmej.  Characterization of suprathermal electrons inside a laser accelerated plasma via highly-resolved ka-emission.  Nature Communications, 10(1):4212, Sep 2019.
  * [9] P. et al. Ordyna.  isualizing plasmons and ultrafast kinetic instabilities in laser-driven solids using x-ray scattering.  2024\.
  * [10] T. Kluge, M. Bussmann, H.-K. Chung, C. Gutt, L. G. Huang, M. Zacharias, U. Schramm, and T. E. Cowan.  Nanoscale femtosecond imaging of transient hot solid density plasmas with elemental and charge state sensitivity using resonant coherent diffraction.  Physics of Plasmas, 23(3):033103, 03 2016.
  * [11] B. Deschaud, O. Peyrusse, and F. B. Rosmej.  Simulation of xfel induced fluorescence spectra of hollow ions and studies of dense plasma effects.  Physics of Plasmas, 27, 6 2020.
  * [12] Suxing Hu, David Bishel, David Chin, Philip Nilson, Valentin Karasiev, Igor Golovkin, Ming Gu, Stephanie Hansen, Deyan Mihaylov, Nathaniel Shaffer, Shuai Zhang, and Timothy Walton.  Probing atomic physics at ultrahigh pressure using laser-driven implosions.  Nature Communications, 13, 11 2022.
  * [13] Richard M. More.  Electronic energy-levels in dense plasmas.  Journal of Quantitative Spectroscopy and Radiative Transfer, 27(3):345–357, 1982.
  * [14] John C. Stewart and Jr. Pyatt, Kedar D.  Lowering of Ionization Potentials in Plasmas.  Astrophys. J. , 144:1203, June 1966.
  * [15] G. Ecker and W. Kröll.  Lowering of the Ionization Energy for a Plasma in Thermodynamic Equilibrium.  The Physics of Fluids, 6(1):62–69, 01 1963.
  * [16] Djamel Benredjem, Jean-Christophe Pain, Annette Calisti, and Sandrine Ferri.  Plasma density effects on electron impact ionization.  Phys. Rev. E, 108:035207, Sep 2023.
  * [17] Xiaolei Zan, Chengliang Lin, Yong Hou, and Jianmin Yuan.  Local field correction to ionization potential depression of ions in warm or hot dense matter.  Physical Review E, 104, 8 2021.
  * [18] O. Ciricosta, S. M. Vinko, H. K. Chung, B. I. Cho, C. R.D. Brown, T. Burian, J. Chalupský, K. Engelhorn, R. W. Falcone, C. Graves, V. Hájková, A. Higginbotham, L. Juha, J. Krzywinski, H. J. Lee, M. Messerschmidt, C. D. Murphy, Y. Ping, D. S. Rackstraw, A. Scherz, W. Schlotter, S. Toleikis, J. J. Turner, L. Vysin, T. Wang, B. Wu, U. Zastrau, D. Zhu, R. W. Lee, P. Heimann, B. Nagler, and J. S. Wark.  Direct measurements of the ionization potential depression in a dense plasma.  Physical Review Letters, 109, 8 2012.
  * [19] SM Vinko, O Ciricosta, and JS Wark.  Density functional theory calculations of continuum lowering in strongly coupled plasmas.  Nature communications, 5(1):3533, 2014.
  * [20] M F Gu.  The flexible atomic code.  Canadian Journal of Physics, 86(5):675–689, 2008.
  * [21] M. F. Gu and P. Beiersdorfer.  Stark shift and width of x-ray lines from highly charged ions in dense plasmas.  Phys. Rev. A, 101:032501, Mar 2020.
  * [22] Rosmej, F. B. and Lee, R. W.  Hollow ion emission driven by pulsed intense x-ray fields.  EPL, 77(2):24001, 2007.
  * [23] G. Pérez-Callejo, T. Gawne, T. R. Preston, P. Hollebon, O. S. Humphries, H. K. Chung, G. L. Dakovski, J. Krzywinski, M. P. Minitti, T. Burian, J. Chalupský, V. Hájková, L. Juha, V. Vozda, U. Zastrau, S. M. Vinko, S. J. Rose, and J. S. Wark.  Dielectronic satellite emission from a solid-density mg plasma: relationship to models of ionisation potential depression, 2023.
  * [24] Ulf Zastrau, Karen Appel, Carsten Baehtz, Oliver Baehr, Lewis Batchelor, Andreas Berghäuser, Mohammadreza Banjafar, Erik Brambrink, Valerio Cerantola, Thomas E. Cowan, Horst Damker, Steffen Dietrich, Samuele Di Dio Cafiso, Jörn Dreyer, Hans-Olaf Engel, Thomas Feldmann, Stefan Findeisen, Manon Foese, Daniel Fulla-Marsa, Sebastian Göde, Mohammed Hassan, Jens Hauser, Thomas Herrmannsdörfer, Hauke Höppner, Johannes Kaa, Peter Kaever, Klaus Knöfel, Zuzana Konôpková, Alejandro Laso García, Hanns-Peter Liermann, Jona Mainberger, Mikako Makita, Eike-Christian Martens, Emma E. McBride, Dominik Möller, Motoaki Nakatsutsumi, Alexander Pelka, Christian Plueckthun, Clemens Prescher, Thomas R. Preston, Michael Röper, Andreas Schmidt, Wolfgang Seidel, Jan-Patrick Schwinkendorf, Markus O. Schoelmerich, Ulrich Schramm, Andreas Schropp, Cornelius Strohm, Konstantin Sukharnikov, Peter Talkovski, Ian Thorpe, Monika Toncian, Toma Toncian, Lennart Wollenweber, Shingo Yamamoto, and Thomas Tschentscher.  The High Energy Density Scientific Instrument at the European XFEL.  Journal of Synchrotron Radiation, 28(5):1393–1416, Sep 2021.
  * [25] O. Ciricosta, S. M. Vinko, B. Barbrel, D. S. Rackstraw, T. R. Preston, T. Burian, J. Chalupský, B. I. Cho, H. K. Chung, G. L. Dakovski, K. Engelhorn, V. Hájková, P. Heimann, M. Holmes, L. Juha, J. Krzywinski, R. W. Lee, S. Toleikis, J. J. Turner, U. Zastrau, and J. S. Wark.  Measurements of continuum lowering in solid-density plasmas created from elements and compounds.  Nature Communications, 7, 5 2016.
  * [26] Thomas Gawne, Thomas Campbell, Alessandro Forte, Patrick Hollebon, Gabriel Perez-Callejo, Oliver S. Humphries, Oliver Karnbach, Muhammad F. Kasim, Thomas R. Preston, Hae Ja Lee, Alan Miscampbell, Quincy Y. van den Berg, Bob Nagler, Shenyuan Ren, Ryan B. Royle, Justin S. Wark, and Sam M. Vinko.  Investigating mechanisms of state localization in highly ionized dense plasmas.  Physical Review E, 108:035210, 9 2023.
  * [27] H. K. Chung, B. I. Cho, O. Ciricosta, S. M. Vinko, J. S. Wark, and R. W. Lee.  Atomic processes modeling of x-ray free electron laser produced plasmas using scfly code.  volume 1811. American Institute of Physics Inc., 3 2017.
  * [28] Shenyuan Ren, Yuanfeng Shi, Quincy Y. van den Berg, Muhammad F. Kasim, Hyun-Kyung Chung, Elisa V. Fernandez-Tello, Pedro Velarde, Justin S. Wark, and Sam M. Vinko.  Non-thermal evolution of dense plasmas driven by intense x-ray fields.  Communications Physics, 6(1):99, May 2023.
  * [29] Tobias Dornheim, Maximilian Böhme, Dominik Kraus, Tilo Döppner, Thomas R. Preston, Zhandos A. Moldabekov, and Jan Vorberger.  Accurate temperature diagnostics for matter under extreme conditions.  Nature Communications, 13(1):7911, Dec 2022.
  * [30] Katerina Falk, Šmíd Michal, and Baehtz Carsten.  Spectroscopic investigation of atomic physics in isochorically heated cu.  https://in.xfel.eu/metadata/doi/10.22003/XFEL.EU-DATA-002806-00, 2022\.
  * [31] Šmíd Michal.  Spectra from 2806 experiment at european xfel.  https://doi.org/10.14278/rodare.2789, 2024.
  * [32] Šmíd Michal et al.  Analysis scripts for processing the 2806 experiment at european xfel.  https://doi.org/10.14278/rodare.2791, 2024.
  * [33] T.R. Preston, S. Göde, J.-P. Schwinkendorf, K. Appel, E. Brambrink, V. Cerantola, H. Höppner, M. Makita, A. Pelka, C. Prescher, K. Sukharnikov, A. Schmidt, I. Thorpe, T. Toncian, A. Amouretti, D. Chekrygina, R.W. Falcone, K. Falk, L.B. Fletcher, E. Galtier, M. Harmand, N.J. Hartley, S.P. Hau-Riege, P. Heimann, L.G. Huang, O.S. Humphries, O. Karnbach, D. Kraus, H.J. Lee, B. Nagler, S. Ren, A.K. Schuster, M. Smid, K. Voigt, M. Zhang, and U. Zastrau.  Design and performance characterisation of the hapg von hámos spectrometer at the high energy density instrument of the european xfel.  Journal of Instrumentation, 15(11):P11033, nov 2020.
  * [34] X. Pan, M. Šmíd, R. Štefaníková, F. Donat, C. Baehtz, T. Burian, V. Cerantola, L. Gaus, O. S. Humphries, V. Hajkova, L. Juha, M. Krupka, M. Kozlová, Z. Konopkova, T. R. Preston, L. Wollenweber, U. Zastrau, and K. Falk.  Imaging x-ray spectrometer at the high energy density instrument of the European x-ray free electron laser.  Review of Scientific Instruments, 94(3):033501, 03 2023.
  * [35] J. Chalupský, P. Boháček, T. Burian, V. Hájková, S. P. Hau-Riege, P. A. Heimann, L. Juha, M. Messerschmidt, S. P. Moeller, B. Nagler, M. Rowen, W. F. Schlotter, M. L. Swiggers, J. J. Turner, and J. Krzywinski.  Imprinting a focused x-ray laser beam to measure its full spatial characteristics.  Phys. Rev. Appl., 4:014004, Jul 2015.
  * [36] Tobias Dornheim, Maximilian P. Böhme, David A. Chapman, Dominik Kraus, Thomas R. Preston, Zhandos A. Moldabekov, Niclas Schlünzen, Attila Cangi, Tilo Döppner, and Jan Vorberger.  Imaginary-time correlation function thermometry: A new, high-accuracy and model-free temperature analysis technique for x-ray Thomson scattering data.  Physics of Plasmas, 30(4):042707, 04 2023.
  * [37] Hai P. Le, Mark Sherlock, and Howard A. Scott.  Influence of atomic kinetics on inverse bremsstrahlung heating and nonlocal thermal transport.  Phys. Rev. E, 100:013202, Jul 2019.
  * [38] Stefan P. Hau-Riege.  Nonequilibrium electron dynamics in materials driven by high-intensity x-ray pulses.  Phys. Rev. E, 87:053102, May 2013.

##  I Author contributions

M.Š. wrote the manuscript, performed most of the data analysis and run the FAC
and SCFly simulations. O.H. aided the data analysis and writing, R.Š., X.P.,
H.P.L., A.S., J.V and T.E.C. edited and reviewed the manuscript. The
experiment was lead by M.Š., K.F., O.H. and C.B., and performed by E.B., L.G.,
Z.K., M.M. X.P., T.P., R.Š. and U.Z.; T.B., V.H, L.J. performed and analysed
the focal spot via the imprint technique. A.S. and W.W. helped with the x-ray
focusing. The support with atomic simulations was provided by M.S.Ch. and
H.A.S.; H.P.L. provided the cretin simulations. J.V. did the XRTS analysis.

##  II Appendix

### Experimental details

The XFEL beam was operated in the SASE regime, providing about 25 eV
bandwidth, and was focused with a stack of 20 Beryllium lenses with radius of
curvature 50 $\upmu$m, providing a focal length of $\approx$ 30 - 40 cm, based
on the XFEL photon energy. The photon energy was scanned in the range 8750 -
9900 eV with 25 eV steps. The x-ray emission was measured by three crystal
spectrometers. Two of them employed the HAPG crystal and were aligned to the
range 7900 - 8800 eV and 8950 - 9750 eV, respectively [33]. The third one
employed a Germanium crystal and observed the details of Cu K$\upbeta$
emission in the range 8950 - 9400 eV with higher resolution [34].

The target consists of simple foils held in a 6 mm $\times$ 30 mm window,
allowing a continuous shooting with repetition frequency 10 Hz, when the speed
of the target holder was adjusted to keep the spacing between shots 20
$\upmu$m. The data in this paper are from shots where target was 3 $\upmu$m
thick Cu foil.

### Focusing

The focal length of the lenses is changing due to its chromacity by
approximately 1 mm per 25 eV. Therefore, the focusing had to be optimized
after each change of energy. This was done via continuous data acquisition
while the XFEL was on and the lens position was changing. From such data, the
position where strongest emission of ionized K$\upalpha$ lines was observed
was identified as the focused one. The quality and characteristics of the
focus was analyzed by the imprinting technique [35] at three photon energies
(8900, 9400, and 9900 eV). The analysis of the imprints provided the encircled
energy curves, Fig. 6. At all three cases it had shown comparable results,
that the distribution of inner 50% of energy was resembling 0.4 $\upmu$m
diameter Gaussian spot. Such size of the focus was mainly limited by the
bandwidth of the beam.

Figure 6: Focal spot characteristics (encircled energy) from the imprint
measurement. Thin lines are calculations of 2D Gaussian spots for comparison.

### Energy and intensity calibration

The beam energy on the target was measured by a diode detector coupled to a
diamond screen, located between the last focusing element and the target. This
detector was absolutely calibrated with the x-ray gas monitors. The last lens
has an aperture of 300 $\upmu$m, and was intentionally overfilled, i.e. the
beam size on its entrance was kept in the range of approx. 350 - 500 $\upmu$m.
This overfilling converted the spatial jitter of the XFEL beam into an energy
jitter, which means that in each data run, there was a large fluctuation of
energy in individual shots, ranging between 30 to 280 $\upmu$J. Each data run
consists of typically 3000 shots, with nominally identical conditions. Due to
the energy jitter, those shots have been grouped based on the measured energy
providing energy resolved spectra.

On those data, the so called _focal spot inversion_ was performed. This relies
on the fact that experimental spectra for each beam energy, $S_{E}$, are
integration of constituent spectra from various intensities, with ratios given
by the focal spot distribution $f_{I}$,

$S_{E}(E)=\int S_{I}(I)f_{I}(I)dI,$

where $I$ is the XFEL beam intensity, and $S_{I}$ is the emitted spectrum for
given intensity. When considering a Gaussian focal spot profile,

$I(r)=I_{0}\exp\left(-\frac{r^{2}}{2\sigma^{2}}\right),$ (1)

and considering the area enclosing each intensity contour $A=\pi r^{2}$, it is
possible to construct the areal density

$\displaystyle I(A)$ $\displaystyle=I_{0}e^{-\frac{A}{2\pi\sigma^{2}}}$ (2)
$\displaystyle A(I)$
$\displaystyle=\left\\{\begin{array}[]{lr}2\pi\sigma^{2}\ln\left(I_{0}/I\right)&0<I<I_{0}\\\
0&\mathrm{otherwise}\end{array}\right.$ (5) $\displaystyle\frac{dA}{dI}$
$\displaystyle=\left\\{\begin{array}[]{lr}-\frac{2\pi\sigma^{2}}{I}&0<I<I_{0}\\\
0&\mathrm{otherwise}\end{array}\right.$ (8)

from which it is possible to construct the inverse mapping

$\displaystyle S_{E}(E)$
$\displaystyle=\int_{0}^{I_{0}}S_{I}(I)\left|\frac{dA}{dI}\right|dI$ (10)
$\displaystyle=\int_{0}^{I_{0}}\frac{2\pi\sigma^{2}S_{I}(I)}{I}dI$ (11)
$\displaystyle I_{0}$ $\displaystyle=\frac{E}{f}{2\pi\sigma^{2}}$ (12)
$\displaystyle\frac{dS_{E}(E)}{dE}$ $\displaystyle=\frac{S_{I}(I)}{I}.$ (13)

The discretized form of the differential was used to obtain the spectra
presented in this paper

$S_{I}(I)/I=\frac{dS_{E}}{dE},$

where we retain the normalization by $I$ to yield the emitted spectrum for a
given incident power at the specified intensity, so that line emission
intensity is comparable.

While, as shown in Fig. 6, the focal spot is not purely Gaussian, the most
intense parts of the distribution – which is responsible for the heated
emission – is well described by a Gaussian profile. This could therefore be
expected to induce errors on reconstructed emission from ground state emission
lines, K$\upalpha$ and K$\upbeta$, which have contributions from the large
wings of the focal spot profile – however does not affect the conclusions of
this work on the plasma screening within solid density HED plasmas.

### Resonances

The identified resonances are written as the pairs of driving energy —
emission energy in Tab.LABEL:resonances, along with the identification of the
charge state $Z$ and number of electrons in the $L$ shell. Those data were
identified in experimental spectra for three energy densities: 70, 110 and 150
kJ/cm2.

Identification | Emission | Absorption
---|---|---
I | Z | L | K$\upalpha$ | K$\upalpha_{\mathrm{h}}$ | K$\upbeta$ | K$\upbeta_{\mathrm{h}}$ | K$\upgamma$ | K$\upbeta$ | K$\upbeta_{\mathrm{h}}$ | K$\upgamma$
[head to column names]figures/Resonances_04.csv  |  |  |  |  |  |  |  |  |  |

### XRTS

The XRTS spectrum measured at scattering angle 170° (see Fig. LABEL:figxrts),
was analysed by the imaginary time thermometry method [29, 36]. The XRTS
spectrum and the source and instrument function are subjected to a two-sided
Laplace transformation. The resulting quotient of the two quantities is
symmetric in imaginary time $\tau$-space around the inverse temperature
$k_{B}T/2$ due to detailed balance. This allows to extract the temperature
model-free.

### Collisional radiative simulations

The SCFLY code was used to model the interaction in a 0D, time-dependent
collisional-radiative (CR) simulations with zero initial temperature, and with
heating by XFEL beam with super Gaussian temporal profile. We have performed a
set of simulations with various beam and photon energies, to simulate the
whole set of experimental data, see Fig. LABEL:scfly. We can observe the
simulation matches the experiment qualitatively very well, but such approach
is not suitable for quantitative analysis as the atomic model lacks the
necessary details.

### Cretin non-thermal CR simulations

The non-thermal CR simulations evolves the electron energy distribution self-
consistently with the atomic populations. The electron distribution evolution
is governed by a kinetic Boltzmann-Fokker-Planck equation which includes all
elastic (electron – electron collision) and inelastic collisional and
radiative processes [37]. The atomic model used in the Cretin simulations has
the same set of energy levels and transitions as SCFly. Since the model does
not assume thermal distribution, the temperature shown in Fig. 5(a) is defined
as 2/3 of the kinetic energy of the electrons. Deviation from a thermal
distribution can be calculated using the non-equilibrium factor defined by
Hau-Riege[38]. First, the energy is decomposed into a low-energy and a high
energy component. The non-equilibrium factor is defined as the ratio of the
kinetic energy of the high energy component relative to the total kinetic
energy, i.e., 0 means thermalized distribution and 1 means complete non-
equilibrium distribution.

### Atomic simulations

The FAC code was used to calculate the energies of x-ray lines and edges. In
this case, we have very higher control of the atomic model, and using the new
functionality of FAC [21], we can vary the plasma screening model and see its
effect on those. Results of those calculations are seen in Fig.2 and described
in the main text.

Figure LABEL:FAC2 in this appendix shows calculations for further lines with
no plasma screening. Figure LABEL:FAC-SP shows calculations for K$\upalpha$
line with plasma screening following the Stewart-Pyatt model for various
assumptions of electron temperature.