# Dielectronic satellite emission from a solid-density Mg plasma: relationship
to models of ionisation potential depression

G. Pérez-Callejo<EMAIL_ADDRESS>Departamento de Física Teórica
Atómica y Óptica, Universidad de Valladolid, 47011 Valladolid, Spain  T. Gawne
Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road,
Oxford OX1 3PU, UK  T. R. Preston  European XFEL GmbH, Holzkoppel 4,
Schenefeld, 22869, Germany  P. Hollebon  Department of Physics, Clarendon
Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK  O. S.
Humphries  European XFEL GmbH, Holzkoppel 4, Schenefeld, 22869, Germany  H.-K.
Chung  Korea Institute of Fusion Energy (KFE), Daejeon, 34133, South Korea  G.
L. Dakovski  Linac Coherent Light Source, SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  J. Krzywinski  Linac Coherent Light
Source, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA  M. P.
Minitti  Linac Coherent Light Source, SLAC National Accelerator Laboratory,
Menlo Park, CA 94025, USA  T. Burian  Department of Radiation and Chemical
Physics, Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 182
21 Prague 8, Czech Republic  Plasma Physics Department, Institute of Plasma
Physics, Czech Academy of Sciences, Za Slovankou 3, 182 00 Prague 8, Czech
Republic  J. Chalupský  Department of Radiation and Chemical Physics,
Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 182 21 Prague
8, Czech Republic  V. Hájková  Department of Radiation and Chemical Physics,
Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 182 21 Prague
8, Czech Republic  L. Juha  Department of Radiation and Chemical Physics,
Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 182 21 Prague
8, Czech Republic  V. Vozda  Department of Radiation and Chemical Physics,
Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 182 21 Prague
8, Czech Republic  U. Zastrau  European XFEL GmbH, Holzkoppel 4, Schenefeld,
22869, Germany  S. M. Vinko  Department of Physics, Clarendon Laboratory,
University of Oxford, Parks Road, Oxford OX1 3PU, UK  Central Laser Facility,
STFC Rutherford Appleton Laboratory, Didcot OX11 0QX, UK  S. J. Rose  Plasma
Physics Group, The Blackett Laboratory, Imperial College London, Prince
Consort Road, London, SW7 2AZ, UK  Department of Physics, Clarendon
Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK  J. S. Wark
<EMAIL_ADDRESS>Department of Physics, Clarendon Laboratory,
University of Oxford, Parks Road, Oxford OX1 3PU, UK

###### Abstract

We report on experiments where solid-density Mg plasmas are created by heating
with the focused output of the Linac Coherent Light Source x-ray free-
electron-laser. We study the K-shell emission from the Helium and Lithium-like
ions using Bragg crystal spectroscopy. Observation of the dielectronic
satellites in Lithium-like ions confirms that the M-shell electrons appear
bound for these high charge states. An analysis of the intensity of these
satellites indicates that when modelled with an atomic-kinetics code, the
ionisation potential depression model employed needs to produce depressions
for these ions which lie between those predicted by the well known Stewart-
Pyatt and Ecker-Kroll models. These results are largely consistent with recent
Density Functional Theory calculations.

##  I Introduction

The focused output of hard x-ray free-electron-lasers (FELs), such as the
Linac Coherent Light Source (LCLS), with peak spectral brightnesses many
orders of magnitude greater than those of any synchrotron, provides a means to
create hot (many hundreds of eV) plasmas at exactly solid density. Each pulse
created by the FEL can, when it is operating in self-amplified spontaneous
emission (SASE) mode, contain of order a few mJ of energy, which can be
focused to micron-scale spots with Be lenses or Kirkpatrick-Baez mirrors [1].
The short duration of the pulses (typically sub-100 fs) ensures that the x-ray
energy is deposited in the solid target in the focal plane on a timescale
short compared with its disassembly time. The combination of energy, spot
size, and pulse duration noted above corresponds to intensities on target of
order at least 1017 Wcm-2.

If the photon energy of the incoming FEL radiation exceeds that of the K-edge
of the atoms in the cold target, and the subsequent ions created, the
photoionization by the FEL results in copious K-shell core holes being
created, which are subsequently filled by radiative decay from the upper
levels or via the Auger process. Both the photoionized and Auger electrons are
ejected into the continuum, rapidly thermalising via further collisional
ionisation and electron-electron scattering, heating the electrons in the
system to many tens or even hundreds of eV. The filling of the core holes
created by the photoionisation typically occurs on a femtosecond timescale,
short compared with the FEL pulse duration (which is typically several tens of
femtoseconds), and certainly much shorter than the target disassembly time,
effectively ensuring that the resultant K-shell emission comes from the target
when it is still at solid density. Thus K-shell spectroscopy of the solid
density plasma produced provides detailed information on the charge states
produced.

There have now been several studies of the x-ray spectra produced from solid
targets in the manner described above [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13], which have allowed a wealth of information to be gleaned concerning
various properties and processes occuring in these dense plasmas, such as
opacity [9], collisional rates [7, 12], and the phenomenon of saturable
absorption in the x-ray regime [8]. Of particular relevance here are those
studies which have looked at ionization potential depression (IPD) [3, 6, 11].
The lowering of the energy of the start of the continuum, or IPD, is a
fundamental process that occurs in dense plasmas as a result of the
electrostatic interactions between the atom or ion and the surrounding charged
particles [14, 15, 16, 17], leading to a reduction in the atomic binding
energies. A knowledge of where the continuum lies for ions has a direct impact
on many important plasma processes, such as the ionization and ion charge
state distribution, the equation of state, and the opacity and transport
properties.

In the work cited above, the IPD was ascertained experimentally for a
particular charge state by noting the photon energy of the FEL for which
copious Kα radiation for the relevant ion was produced, this being interpreted
as the photon energy needed to create a K-shell core hole by photoionization
into the continuum. This method was applied to elements of relatively low $Z$
(12-14), and relied on the fact that for the electron temperatures produced –
typically less than 200 eV – the atomic energies and plasma temperatures are
such that the vast majority of the resultant $K$-shell emission cannot be
produced via thermal processes, but only due to photoionization, and thus
recorded x-ray spectra, although time-integrated over each individual pulse,
are effectively gated by the pulse duration. The results were compared with
predictions from simple semi-classical models often employed in atomic-
kinetics calculations, for example the Stewart-Pyatt (SP) [16] and Ecker-Kröll
(EK) models [17], whose mathematical expressions are discussed later in the
text.

In the first of these x-ray heating experiments, studying Al, it was found
that the EK model gave better agreement width the data [3]. As a result of
matching the K-shell binding energy, no M-shell electrons were found to be
bound. In that particular experiment the authors note that the spectra from
some of the highest charge states (up to Helium-like) were complicated by the
presence of Kβ radiation from the cold solid, and in this original work no
firm conclusion about the degree of IPD for them was made. However, it was
subsequently noted that were the EK model to be applied for the higher charge
states as well, the M-shell would also not be bound [5]. This latter
conclusion is at odds with the work of Hoarty and co-workers [18], who
observed the presence of radiation from the He-$\beta$ transition of Al in
experiments using optical lasers for heating and compression, and for which
the spectra could not be reconciled with the EK model. At the same time, the
SP model was also found to be inadequate for modelling both experiments.

The difficulty of finding a single, semi-classical IPD model that captures the
pertinent physics that such an approach purports to describe is perhaps not
surprising, given that the plasmas under study are dense quantum systems.
Indeed, the authors of ref. [3] note in their original work that the EK and SP
models are“ultimately both unlikely to capture fully the complex physics of
atomic systems embedded within dense plasma environments over wide ranges of
plasma conditions and charge states.” The veracity of the above statement has
been given further credence by recent extensive computational studies based on
Density Functional Theory (DFT) [19]. These calculations reveal that one of
the main underlying issues in such dense systems is that a binary distinction
between electrons that are free, and in the so-called continuum, and those
that are bound to an atom or ion, cannot definitively be made – the problem is
inherently quantum mechanical in nature.

Nevertheless, owing to the manner in which the majority of standard atomic-
kinetics-based calculations are constructed, their use requires that such a
distinction between bound and free electrons is made and thus, at least for
the foreseeable future, simple IPD models are likely to continue to be
adopted. As a result, within the work encompassed by the DFT calculations
cited above, the use of the technique of the inverse participation ratio was
employed to give a measure of the degree of boundness of the Kohn-Sham
wavefunctions [19], and hence deduce where the energy of the continuum ought
best to be placed, were one forced to make such an artificial division between
bound and free. Within the above constraints and caveats, it was found that
for solid density Al and Mg plasmas, for the highest charge states the most
appropriate energies at which to consider the electrons to be free would
correspond to a position lying somewhere between those predicted by the SP and
EK models, although the EK model would still give a better fit for the lower
charge states. In particular, it was noted that for Li and He-like ions, it
would appear that the M-shell should be treated as being bound, and radiation
from the M-shell states should be experimentally observable. In fact, the
authors of [19] report the direct observation of He- and Li-like Mg Kβ
emission.

It is in the above context that we present here an analysis of the emission
spectrum from the Helium and Lithium-like ions in an FEL-generated solid-
density, optically-thin, Mg plasma. We specifically investigate the intensity
of the dielectronic satellites, where the spectator electron lies in the $n=3$
principal quantum number, i.e. the M-shell. Studies of the intensity of such
satellites, be they due to L or M-shell spectator electrons, has a long
history, and has previously been shown to be of use in determining plasma
conditions in both astrophysical [20, 21, 22, 23, 24] and laboratory based
plasmas [25, 26, 27, 28, 29, 30, 31, 32, 33].

In analysing the intensity of these M-shell satellites, our main finding is
that the Li-like states for which the $n=3$ level is occupied by one electron
are in good thermal contact with the equivalent He-like state without the
$n=3$ electron, and as a result the intensity of these satellites is sensitive
to the IPD. Furthermore, for best agreement with the experimental data given
the parameters of the FEL, we need to place the IPD for these charge states
between the EK and SP limits, as advised by the DFT studies, and to this
degree the data is consistent with them.

##  II Experimental set-up

(a)  (b)

Figure 1: (1a) Schematic drawing of the experimental set-up. (1b) Experimental
data of Mg K-shell emission spectrum showing a whole set of emission lines
from different ionization stages. The lines have been labelled according to
the number of electrons of the emitting ion.

The experiment was performed at the Soft X-Ray (SXR) end station [34] of the
LCLS, using a set-up which has been thoroughly discussed in previous
publications [10, 9, 35, 13].

Targets consisted of 54 -nm-thick Mg foils. These were irradiated using 100 fs
x-ray pulses with a photon energy of 1540 eV. The nominal pulse energy was 1.7
mJ, which is reduced to 0.51 mJ after transmission through the beamline optics
[35, 9, 10]. The size of the focal spot on target was measured ex situ using
imprint measurements on PbI2 [36], obtaining an effective focal spot area of
$8.5\,\mathrm{\micro m^{2}}$, which corresponds to a maximum irradiance of
$\sim 10^{17}\,\mathrm{W~{}cm^{-2}}$.

The targets were irradiated at a 45 ° angle with respect to their normal, and
their x-ray emission was collected at an angle of 20 ° to that normal by means
of a flat-crystal Bragg spectrometer. We employed a Beryl $(10\bar{1}0)$
crystal, whose lattice spacing ($2d=$15.96 Å) corresponds to a diffraction
angle of $\sim 35\,\mathrm{\degree}$ for the Heα line of Mg. The diffracted X
rays were then recorded with a Princeton Instruments Charge Coupled Device
(CCD) camera. A schematic drawing of the experimental set-up is shown in Fig.
1a.

Figure 1b shows a typical example of the K-shell emission spectra from the
solid Mg plasma that was obtained in the experiment. Owing to the time-
integrated nature of the measured spectra, emission from the different
ionization species present in the plasma as it heats up is present in the
data. The lines are labelled according to the number of bound electrons in the
core of the emitting ion (the label ‘He’ corresponds to two bound electrons,
‘Li’ to three, and so on). As mentioned in the introduction, in this work we
will focus on the emission from He-like Mg, and the associated Li-like
satellites which lie in the energy range of $1340-1370\,\mathrm{eV}$.

##  III Modelling

###  III.1 Atomic kinetics

The plasma evolution and resultant spectra were modelled using a combination
of two codes: the atomic kinetics non-LTE (Local Thermodynamic Equilibrium)
code SCFLY [10], and a separate, stand-alone, LTE Saha-Boltzmann code. As we
shall explain in more detail below, we use SCFLY to determine the overall
evolution of the plasma in terms of superconfigurations, and also to confirm
that the system is very close to LTE. We then use a Saha-Boltzmann approach,
with more detailed atomic physics (but now assuming LTE) to model the spectra
of the satellites for comparison with the experimental results.

SCFLY is based upon the commonly used FLYCHK code [37], adapted to treat x-ray
laser problems. SCFLY is based on a superconfiguration approach – i.e. it
provides the populations for states defined solely by the number of electrons
with specific principal quantum numbers. Thus the ground state of a lithium-
like ion, with two electrons in the K-shell, and one in the L-shell, is
denoted (210), whereas in its first excited state, with the L-shell electron
excited to the M-shell, it would be denoted (201). It takes as its input the
x-ray laser intensity as a function of time, and solves for the evolution of
ground and excited-state superconfiguration populations. The electrons in the
continuum are assumed to obey classical statistics, and to instantaneously
thermalise to a temperature dictated by their overall energy content. In
contrast with the electrons, we assume that on the timescale of typical FEL
pulses the ions remain at room temperature throughout the calculation, given
that the timescale for electron-ion equilibriation, in terms of their
temperatures, is several picoseconds [38, 39].

Within both the SCFLY and Saha-Boltzmann solver we take into account the
degree of IPD by considering two widely used models, namely the EK model and
the SP model. The levels of continuum lowering predicted by these models in
the high-density limit, where the SP model is effectively equivalent to the
Ion-Sphere model [16], are given by [5]

$\Delta
I_{EK}=C_{EK}\cdot\frac{(Z+1)e^{2}}{4\pi\varepsilon_{0}}\cdot\left[\frac{4\pi\left(n_{e}+n_{i}\right)}{3}\right]^{1/3}$
(1)

and

$\Delta
I_{SP}=C_{SP}\cdot\frac{3}{2}\cdot\frac{(Z+1)e^{2}}{4\pi\varepsilon_{0}}\cdot\left(\frac{4\pi
n_{e}}{3\cdot(Z+1)}\right)^{1/3},$ (2)

where, for each model, $C_{EK}=C_{SP}=1$, $Z$ is the charge of the ion ($0$
for the neutral atom), $e$ is the electron charge, $n_{e}$ and $n_{i}$ are the
electron and ion number density respectively and $\varepsilon_{0}$ is the
electric permittivity of vacuum. Note that in order to explore models that lie
between SP and EK we will also in what follows show results for which these
two constants $C_{EK}$ and $C_{SP}$ differ from unity.

As noted in the introduction, the main result we will be studying in this work
is the intensity of the lithium-like dielectronic satellites. Within SCFLY,
the transition between superconfigurations that corresponds to satellites
associated with an upper state containing an L-shell electron is the
(120)-(210) transition, and for M-shell satellites is the (111)-(201)
transition. At the more detailed level, taking into account the various
configurations and fine-structure effects, these two superconfigurational
transitions encompass the gamut of satellites that are well known to accompany
the helium-like resonance line, and which, for the L-shell satellites, are
usually labelled according to the notation proposed by Gabriel [20]. It is
these detailed transitions that are modelled by the Saha-Boltzmann code, under
its LTE assumption, and which we justify in the next section.

Figure 2: Temperature evolution of solid density Mg, as predicted by SCFLY for
different IPD models, upon irradiation with an FEL pulse with irradiance
$3\times 10^{17}\,\mathrm{W~{}cm^{-2}}$ at 1540 eV. The plot is shown up to 10
ps for illustrative purposes, but it should be noted that disassembly of the
target will take place on a timescale of order picoseconds.

To model the plasma kinetics, we first calculated the temperature evolution of
the plasma with SCFLY. It is even at this early stage that the choice of an
IPD model affects our results, since the electron temperature of the plasma is
dependent on the level of ionization and therefore on the level of continuum
lowering. We show this effect in Fig. 2, where the temperature evolution for
the peak laser irradiance ($\sim 3\times 10^{17}\,\mathrm{W~{}cm^{-2}}$) is
shown for the SP and EK IPD models. In these simulations, and all that follow,
we assume that the incident FEL pulse is gaussian in time with a FWHM of 100
fs, which peaks at a time of 100 fs. In the simulations shown here, which
assume constant ion density, the cooling of the plasma after the FEL pulse is
only due to radiation while in practice, disassembly of the target will take
place on a timescale of order picoseconds.

The main difference between these IPD models can be seen in Figs. 3a and 3b,
where we show the ionization energy of the M-shell for He- and Li-like ions as
a function of time. The SP model predicts the M-shell to be always bound, with
a binding energy around $\sim 100\,\mathrm{eV}$, which is of the same order as
the electron energy. This means that a large proportion of the free electrons
are able to collisionally ionize these states, losing part of their thermal
energy. In contrast, in the EK model the M-shell becomes completely free when
the plasma heats up, so collisional ionization loses are somewhat reduced.
Also shown in Fig. 3 are the results of scaled SP and EK models, such that the
M-shell remains bound (in the bound-free picture of atomic kinetics codes),
but with a lower ionization energy, which increases the rate of collisional
ionization, thus increasing the energy losses and reducing the overall plasma
temperature.

(a)  (b)

Figure 3: Time-evolution of the level of the continuum edge with respect to
the M-shell of He-like (3a) and Li-like (3b) ions, for the different IPD
models mentioned in the text. In this picture, when the continuum energy is
positive, it means that the M-shell is bound and when it is negative, the
M-shell becomes free. Note how, while for the usual SP and EK models the
M-shell is respectively bound or free for both ion species (when the
temperature is sufficiently high), for the case of 88% EK and 130% SP, the
M-shell is barely bound for both charge states and much closer to the
continuum edge.

###  III.2 Spectrum

Having used SCFLY to determine the temperature evolution of the plasma, the
more detailed atomic kinetics were solved using an iterative Saha-Boltzmann
LTE code which treats Ne-like to Be-like ions using configuration averaged
levels, while explicitly including the configurations and fine structure
levels of Li-like to H-like ions. The energy of the different atomic states
and the transition probabilities were obtained from the Los Alamos National
Laboratory (LANL) atomic codes [40].

The particular region of the spectrum in which we are interested is around the
Helium-like resonance line, along with its associated satellites, that is, the
region between $1340$ and 1370 eV shown in Fig. 1b. Since the targets used in
this experiment were 54 nm-thick, the peak optical depth of the plasma in the
spectral range of the Heα emission (determined by the resonance line) is
$\tau<0.2$ [9]. All of the radiation in this region, between 1340 and 1370 eV,
is produced by radiative transitions from the three superconfigurations
(110)-(200), (120)-(210), and (111)-(201).

As noted in the introduction, for the solid-density plasmas produced here, the
electron temperatures are such that very few K-shell holes are produced
thermally for the He and Li ionisation stages, compared with the number that
are produced due to photoionisation by the incident FEL radiation. Indeed,
during the FEL pulse, the photoionisation production of such inner core holes
exceeds that due to thermal collisional processes by about two orders of
magnitude. However, and importantly for our analysis here, the electron
density of the system is so high, and thus electron collisional processes so
fast, that the superconfigurations (and also configurations and fine structure
levels within them) are extremely close to LTE, apart from the deviation in
the population of the K-shell induced by the photoionisation process. However,
importantly, the photoionization due to the FEL does not significantly disturb
the other aspects of LTE relationships within the system. This is best
illustrated by example. Consider two superconfigurations that do not have
K-shell holes: (210) and (211). Photoionisation by the FEL would, from these
superconfigurations, produce (110) and (111) respectively, and indeed towards
the peak of the pulse the fractional populations of ions with such K-shell
holes is of order a few percent. However, the collisional processes are so
fast that the ratio of (110) to (111) remains almost identical to that which
would pertain in the circumstances of no FEL, but the same electron
temperature (i.e. that given by the Saha-Boltzmann equation at this
temperature). Note also, that given that the (111) superconfiguration can
undergo Auger decay, collisional processes are also much more important than
this effect.

(a)  (b)

Figure 4: (4a) Population ratio between the 110 and 111 states obtained from
SCFLY compared with those obtained by assuming LTE for the SP IPD model as a
function of time. (4b) Fractional population of the He-like 110 states (red)
and the Li-like 111 satellite states (blue) as a function of time as predicted
by SCFLY (solid lines) and pure LTE (dashed lines).

As an example of this we plot the ratio of the populations of the (110)
superconfiguration to that of the (111) superconfiguration, as predicted by
SCFLY, firstly with the FEL on and running the simulation in full non-LTE
mode, and then secondly, switching the FEL off (so no photoionisation occurs),
but assuming LTE, following the time-dependent temperature from the first non-
LTE simulation. As can be seen in Fig. 4a, the ratio between the populations
of the superconfigurations is almost identical at each point in time, and
whether the FEL is on or off. This is despite the fact that the absolute value
of the populations of the superconfigurations is quite different between the
non-LTE calculation including the FEL, and the one that assumes LTE: this can
be seen in Fig. 4b, where we plot the number density of ions in the (110) and
(111) superconfigurations as a function of time for the non-LTE and LTE case.
Note that the populations are about two orders of magnitude higher whilst the
FEL is on, due to photoionisation of the K-shell, in the non-LTE case.

As LTE between superconfigurations has been shown to be maintained in the
presence of an FEL drive, it is therefore reasonable to use the time-dependent
temperature from SCFLY as a basis of the model, assuming LTE between
configurations, and the fine structre levels within them in order to construct
a detailed spectrum. In this limit, as the plasma is optically thin, the
intensity emitted at a given time from a given detailed transition depends
only on its population and spontaneous emission rate. We use the Saha-
Boltzmann solver, with the time-dependent temperature provided by SCFLY, to
determine the time-dependent populations of all of the relevant levels, and
thus the detailed emission. For the Helium-like line and its satellites we
included the resonance ($1s2p~{}\leftidx{{}^{1}}P_{1}\rightarrow
1s^{2}~{}\leftidx{{}^{1}}S_{0}$) and intercombination
($1s2p~{}\leftidx{{}^{3}}P_{1}\rightarrow 1s^{2}~{}\leftidx{{}^{1}}S_{0}$)
lines from the Heα complex and all Li-like L- and M-shell satellite
transitions with energies above 1330 eV and with spontaneous emission rates
$A$ above $0.08\times 10^{13}\,\mathrm{s^{-1}}$. This corresponds to $13$
L-shell satellite lines and $32$ M-shell satellite lines, which are shown in
Table 1. Note that, although the spectral region of interest for this work
lies between $1340$ and 1370 eV, we set the lower limit of the energy of the
lines considered to be 10 eV below the region of interest, to include the
emission from the wings of the lines.

(a)  (b)

Figure 5: Shape of the Heα emission obtained by the (5a) Steward-Pyatt and
(5b) Ecker-Kröll models (red lines) compared with the experimental data (blue
line) and the associated $1\sigma$ uncertainty (grey area). Below the main
line, the total contribution from the satellite lines with an M or an L-shell
spectator electron, as well as that of the Heα emission are indicated with the
green, black and brown solid lines respectively. The green and black dotted
lines correspond to the emission from individual fine structure states.

The intensity of each transition is distributed across its lineshape. Given
that, for the duration of the emission, the ions are static, the contribution
of the Doppler effect to the broadening of the line can be considered
negligible. By studying the mechanisms affecting line broadening in the
experimental conditions, using the code ALICE [41], we observed that, under
these conditions, the lineshape is mostly determined by the rate of
collisional ionization and recombination, and thus, the lineshape is well
modelled by a Lorentzian curve. The experimental width of the isolated Heα
resonance line was determined by fitting the high-energy side of the line,
where no satellite emission is present. The obtained FWHM of the Lorentzian
was $\Delta E=4.4\pm 0.5\,\mathrm{eV}$. We observed that the shape of the
total spectrum was not strongly dependent on the width of the individual
satellite lines -given the numerous satellite transitions, the shape of the
individual features is lost. For this reason, and given that the rate of
collisional recombination to the M-shell for a He-like ion is the same as the
rate of collisional ionization of an M-shell electron for a Li-like ion, for
simplicity we considered all the individual transitions to have the same
Lorentzian width as the Heα line. The lineshapes are then modelled as

$I(E)=\frac{I_{\text{total}}}{\pi}\frac{\Delta E/2}{(E-E_{0})^{2}+(\Delta
E/2)^{2}},$ (3)

where $I_{\text{total}}$ is the total intensity of a transition defined by the
populations and spontaneous rate, and $E_{0}$ is the transition energy.

Table 1: List of He-like transitions and Li-like satellites included in the spectra calculation. Type |  Transition |  Energy (eV) |  A-rate ($10^{13}\,\mathrm{s^{-1}}$) |  Gabriel’s notation
---|---|---|---|---
|  |  |  |
He-like |  $1s2p~{}^{3}P_{1}\rightarrow 1s^{2}~{}^{1}S_{0}$ |  1343.1 |  $1.21\times 10^{-3}$ |  y
He-like |  $1s2p~{}^{1}P_{1}\rightarrow 1s^{2}~{}^{1}S_{0}$ |  1352.5 |  2.0469 |  w
L-shell satellite |  $1s2p^{2}~{}^{2}D_{3/2}\rightarrow 1s^{2}2p^{1}~{}^{2}P_{3/2}$ |  1331.3 |  0.0499 |  l
L-shell satellite |  $1s2p^{2}~{}^{2}D_{5/2}\rightarrow 1s^{2}2p^{1}~{}^{2}P_{3/2}$ |  1331.3 |  0.9596 |  j
L-shell satellite |  $1s2p^{2}~{}^{2}D_{3/2}\rightarrow 1s^{2}2p^{1}~{}^{2}P_{1/2}$ |  1331.8 |  0.9270 |  k
L-shell satellite |  $1s2p^{2}~{}^{2}P_{1/2}\rightarrow 1s^{2}2p^{1}~{}^{2}P_{3/2}$ |  1333.4 |  0.9147 |  c
L-shell satellite |  $1s2p^{2}~{}^{2}P_{3/2}\rightarrow 1s^{2}2p^{1}~{}^{2}P_{3/2}$ |  1333.8 |  2.6167 |  a
L-shell satellite |  $1s2p^{2}~{}^{2}P_{1/2}\rightarrow 1s^{2}2p^{1}~{}^{2}P_{1/2}$ |  1333.8 |  2.0883 |  d
L-shell satellite |  $1s2p^{2}~{}^{2}P_{3/2}\rightarrow 1s^{2}2p^{1}~{}^{2}P_{1/2}$ |  1334.3 |  0.3683 |  b
L-shell satellite |  $1s2s2p~{}^{2}P_{1/2}\rightarrow 1s^{2}2s^{1}~{}^{2}S_{1/2}$ |  1335.3 |  1.7592 |  r
L-shell satellite |  $1s2s2p~{}^{2}P_{3/2}\rightarrow 1s^{2}2s^{1}~{}^{2}S_{1/2}$ |  1335.6 |  1.8214 |  q
L-shell satellite |  $1s2s2p~{}^{2}P_{1/2}\rightarrow 1s^{2}2s^{1}~{}^{2}S_{1/2}$ |  1339.9 |  0.2026 |  t
L-shell satellite |  $1s2s2p~{}^{2}P_{3/2}\rightarrow 1s^{2}2s^{1}~{}^{2}S_{1/2}$ |  1340.4 |  0.1406 |  s
L-shell satellite |  $1s2p^{2}~{}^{2}S_{1/2}\rightarrow 1s^{2}2p^{1}~{}^{2}P_{3/2}$ |  1343.0 |  0.7842 |  m
L-shell satellite |  $1s2p^{2}~{}^{2}S_{1/2}\rightarrow 1s^{2}2p^{1}~{}^{2}P_{1/2}$ |  1343.2 |  0.2636 |  n
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{3}P)^{2}P_{3/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{3/2}$ |  1337.4 |  0.0807 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{3}P)^{2}P_{1/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{1/2}$ |  1337.5 |  0.0854 |
M-shell satellite |  $1s^{1}2s^{1}3d^{1}~{}(^{1}S)^{2}D_{5/2}\rightarrow 1s^{2}3p^{2}~{}^{2}P_{3/2}$ |  1341.9 |  0.2530 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{3}P)^{2}D_{5/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{3/2}$ |  1341.9 |  0.1543 |
M-shell satellite |  $1s^{1}2s^{1}3d^{1}~{}(^{1}S)^{2}D_{3/2}\rightarrow 1s^{2}3p^{2}~{}^{2}P_{1/2}$ |  1342.0 |  0.2760 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{3}P)^{2}F_{5/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{3/2}$ |  1342.6 |  0.3080 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{3}P)^{2}F_{7/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{5/2}$ |  1343.1 |  0.3324 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{3}P)^{2}S_{1/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{3/2}$ |  1343.8 |  0.3058 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{3}P)^{2}S_{1/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{1/2}$ |  1344.0 |  0.0910 |
M-shell satellite |  $1s^{1}2p^{1}3s^{1}~{}(^{3}P)^{2}P_{1/2}\rightarrow 1s^{2}3s^{1}~{}^{2}S_{1/2}$ |  1344.9 |  0.2500 |
M-shell satellite |  $1s^{1}2p^{1}3s^{1}~{}(^{3}P)^{2}P_{3/2}\rightarrow 1s^{2}3s^{1}~{}^{2}S_{1/2}$ |  1345.4 |  0.1850 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{1}P)^{2}D_{5/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{3/2}$ |  1347.4 |  1.8410 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{1}P)^{2}D_{3/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{3/2}$ |  1347.4 |  0.2166 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{1}P)^{2}D_{3/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{1/2}$ |  1347.5 |  1.5855 |
M-shell satellite |  $1s^{1}2p^{1}3s^{1}~{}(^{1}P)^{2}P_{1/2}\rightarrow 1s^{2}3s^{1}~{}^{2}S_{1/2}$ |  1348.1 |  1.8705 |
M-shell satellite |  $1s^{1}2p^{1}3s^{1}~{}(^{1}P)^{2}P_{3/2}\rightarrow 1s^{2}3s^{1}~{}^{2}S_{1/2}$ |  1348.1 |  1.8226 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{1}P)^{2}P_{1/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{3/2}$ |  1348.4 |  0.4731 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{1}P)^{2}P_{1/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{1/2}$ |  1348.6 |  1.4666 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{1}P)^{2}P_{3/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{3/2}$ |  1348.6 |  1.6772 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{1}P)^{2}D_{3/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{5/2}$ |  1349.4 |  0.1720 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{1}P)^{2}D_{3/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{3/2}$ |  1349.5 |  1.8304 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{1}P)^{2}D_{5/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{5/2}$ |  1349.5 |  1.7970 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{1}P)^{2}D_{5/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{3/2}$ |  1349.5 |  0.2028 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{1}P)^{2}S_{1/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{3/2}$ |  1350.0 |  1.2453 |
M-shell satellite |  $1s^{1}2p^{1}3p^{1}~{}(^{1}P)^{2}S_{1/2}\rightarrow 1s^{2}3p^{1}~{}^{2}P_{1/2}$ |  1350.1 |  0.4075 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{1}P)^{2}F_{7/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{5/2}$ |  1350.6 |  1.6623 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{1}P)^{2}F_{5/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{5/2}$ |  1350.7 |  0.1690 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{1}P)^{2}F_{5/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{3/2}$ |  1350.7 |  1.4779 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{1}P)^{2}P_{1/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{3/2}$ |  1351.8 |  1.9741 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{1}P)^{2}P_{3/2}\rightarrow 1s^{2}3d^{1}~{}^{2}D_{5/2}$ |  1351.8 |  1.7840 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{3}P)^{2}P_{3/2}\rightarrow 1s^{2}3s^{1}~{}^{2}S_{1/2}$ |  1351.9 |  0.1761 |
M-shell satellite |  $1s^{1}2p^{1}3d^{1}~{}(^{3}P)^{2}P_{1/2}\rightarrow 1s^{2}3s^{1}~{}^{2}S_{1/2}$ |  1352.4 |  0.1594 |

###  III.3 Effect of the focal spot

In order to take into account the spatial distribution of the focussed FEL
x-rays, for each pulse we model 25 intensity bins spanning six orders of
magnitude, as detailed by Ciricosta et al. [10]. The intensity of the $n$-th
bin was obtained as

$I_{n}(\omega,t)=I_{0}(\omega,t)\cdot e^{-2(n\cdot 0.1)^{2}}.$ (4)

It is worth mentioning that not all the bins contribute to the Heα emission,
since for $n\gtrsim 11$ the temperature is not sufficiently high to ionize the
the plasma up to neither Li nor He-like state. This spatial distribution was
measured experimentally by the imprint method, as described in more detail in
[10]. The full spectrum was then obtained as a sum of the time-integrated
spectra calculated for each intensity bin, weighted by the fluence scan of the
laser spot.

##  IV Results

Figure 5 shows both the experimental data, and simulated spectra, for the
Helium-like emission and associated Li-like satellites. We show in red the
simulated spectra of the Heα region for both the SP (Fig. 5a) and the EK (Fig.
5b) IPD models. The brown line corresponds to the contribution from the Heα
emission, while the green and black solid lines correspond to the collective
emission from the Li-like satellites with an L- and M-shell spectator electron
respectively (the dotted green and black lines correspond to each individual
satellite transition). The red band corresponds to the total spectrum, with
the width of the band corresponding to the uncertainty introduced by the error
in the fit to the width of the Lorentzian. When comparing with the
experimental data, shown in blue, it can be seen that the SP model
overestimates the intensity of the M-shell satellites, thus predicting a Heα
feature $\sim 2\,\mathrm{eV}$ wider than the experimental result. The opposite
happens with the EK model, where there is almost no M-shell satellite
contribution, and therefore the line appears narrower than observed.

(a)  (b)

Figure 6: (6a) Comparison of the shapes of the Heα emission obtained for both
130% of the SP model and 88% of the EK model (green-dotted and red-dashed
regions respectively) compared with the experimental data. The width of the
error regions corresponds to the uncertainty in the line widths. (6b) Line
profile obtained using the SP IPD model scaled to 130% (red) compared to the
experimental data (blue line and grey area). The contribution from individual
lines is shown following the same colour convention as Fig. 5. The black arrow
marks the shoulder-like feature caused by the M-shell satellite emission.

It is worth noting that the brown line labeled He-$\alpha$ in the figures
includes the contribution from the resonance line
$1s2p~{}\leftidx{{}^{1}}P_{1}\rightarrow 1s^{2}~{}\leftidx{{}^{1}}S_{0}$
(labeled $w$ following Gabriel’s notation [20]), centered at
$E_{w}=1352.5\,\mathrm{eV}$; and the intercombination line
$1s2p~{}\leftidx{{}^{3}}P_{1}\rightarrow 1s^{2}~{}\leftidx{{}^{1}}S_{0}$
($y$), centered at $E_{y}=1343.1\,\mathrm{eV}$. The contribution of the
intercombination line, however, can barely be resolved, since its intensity is
$\sim 0.2\%$ that of the resonance line. Whilst the intercombination line is a
well-known intense feature of spectra from plasmas produced by irradiation
with optical lasers, it is almost completely absent here. This is due to the
very high electron densities present in these experiments, which contrast with
the critical electron densities in optical experiments. With optically
produced laser plasmas, which are far from LTE, significant population can
build up in the $1s2p~{}^{3}P_{1}$ state, giving rise to a large
intercombination line intensity, despite the very low transition rate to the
lower $1s^{2}~{}^{1}S_{0}$ level. However, here the collisional effects cause
a much lower LTE-dictated population of $1s2p~{}^{3}P_{1}$, leading to almost
complete absence of the transition.

From these figures, it seems clear that the intensity of the M-shell
satellites seen experimentally cannot be fitted by either the full SP or EK
models, at least for the FEL intensity used. However, if we consider the
intensity of these satellites to indicate in some way where the IPD should
actually lie, it would be somewhere between the two. We thus reran the SCFLY
and Saha-Boltzmann solver for the two IPD models, but changed the value of $C$
within them from unity, while keeping the functional form of the model the
same (see Eqns. 1 and 2). The best fits were obtained for $C_{EK}=0.88$ and
$C_{SP}=1.30$, cases for which both models predict the M-shell to remain just
bound for the whole duration of the emission (see Figs. 3a and 3b
respectively).

The resulting spectra are shown in Fig. 6, where 6a shows the total spectra
for both cases and Fig. 6b corresponds to the individual contribution of each
line for the 130% SP IPD model, following the same color convention as for
Fig. 5. It can be seen that the emission from the M-shell satellites is
reponsible for the shape of the low-energy wing of the line (where it creates
a shoulder-like feature that we are indicating with an arrow), whereas the Heα
emission is the only contribution on the high-energy end.

In the context of the recent DFT calculations by Gawne et al. [19], it is
encouraging that to obtain the best fit to the satellite spectra we need to
invoke an IPD value that lies between the SP and EK limits, as that was
precisely the conclusion of that work. Furthermore, the required scaling of
both the EK and SP models to match the experimental data is in excellent
agreement with the required scaling to match the DFT results (Gawne et al.
found an IPD value corresponding to 133% that predicted by the SP model and
89% the prediction of the EK model). However, before stating definitively that
this is the case, care should be taken to note that the simulated intensity of
the satellites compared with the resonance line will depend on accurate
modelling of the temperature of the system, as the satellite intensity is
determined by the ratio of the ionisation energy of their upper levels to the
temperature. This in turn, as well as any inherent limitations of the model,
entails accurately knowing the experimental x-ray FEL intensity incident upon
the target: a figure that is generally quoted to be known within about 30%
[10]. It is therefore important to also investigate the sensitivity of the
results to the FEL intensity on target.

(a)  (b)

Figure 7: Spectra obtained for the EK (7a) and SP (7b) IPD models for the
nominal laser intensity (yellow squares), and the results obtained by
modifying the laser intensity to 150% (red circles) and 50% the nominal value
(brown diamonds). The results obtained by scaling the IPD models, as presented
in Fig. 6, are also shown for comparison (green crosses).

To this end we ran several simulations with FEL intensities that differed from
those measured experimentally by up to a factor of two lower and up to 50%
higher than the experimental value. As described previously in subsection
III.3, these simulations were integrated over the shape of the focal spot, in
order to be directly comparable with the current results. The results are
shown in Fig. 7a for the full EK model and in Fig. 7b for the full SP model.
In both figures, the current result for an IPD level of 130% of the SP model
is included for comparison. As expected, in the spectral region of the M-shell
satellites, the predicted line width differs very little in the case of the EK
model, as the M-shell electrons are not bound in any event. The main changes
to the spectrum appear in the region of the L-shell satellites, which become
more intense with respect to the Heα emission as the laser intensity is
decreased.

However, in the case of using the full SP model, a reduction in the laser
intensity does lead to an increase in satellite intensity predicting an even
wider line. On the other hand, an increase in the intensity slightly reduces
the satellite contribution from both M- and L-shell satellites. We find that
increasing the laser intensity by $50\%$ results in a line width that is still
$\sim 1\,\mathrm{eV}$ wider than the experimental result, while the emission
around 1340-1345 eV starts deviating from the data as well. Furthermore,
looking in more detail we see the shoulder asymmetry mentioned before in the
lower energy side to the main Heα peak ($\sim 1348\,\mathrm{eV}$) appearing
for the scaled-IPD models. This feature is not present in the full SP
simulations, but does appear in the data. We thus conclude that given we
believe we know the incident FEL flux to within about 30%, the simulations
still indicate that the satellite intensities are more consistent with an IPD
model that lies between the EK and SP limits once the He and Li-like ion
stages are reached.

##  V Conclusions

In this work we have investigated the sensitivity of the intensity of M and
L-shell satellites to the Heα emission from a solid-density plasma to the IPD
model used. By adjusting the IPD level in an atomic-kinetics code, in
conjunction with an LTE Saha-Boltzmann solver, we find that both the SP and EK
IPD models fail to reproduce the experimental spectra, but obtain best
agreement with the experimental data by employing a degree of IPD that lies
between these two extremes, (88% of EK and 130% of SP). These values are in
good agreement with recent results obtained using first principles
simulations, and indicate that the M-shell of Li-like and He-like Mg under
these conditions lies very close to the continuum edge. Whilst the intensity
of the M-shell satellites does depend on the intensity of the FEL, our
knowledge of the incident intensity, and the observation of asymmetry in the
Heα peak, does lead credence to the conclusion that the best fit IPD lies
between the EK and SP limits.

As outlined in the introduction, it should be borne in mind that for such
dense quantum systems the distinction between bound and free states is
somewhat artificial, yet in the mode in which current atomic-kinetics
calculations are performed, we are often forced to make this division. The
fact that the experimental spectra can be reasonably reproduced within this
simple framework, using IPD values guided by first-principle simulations,
gives confidence that they are still of use in the modelling of the spectra of
hot dense plasmas.

## Acknowledgements

T.G., J.S.W. and S.M.V. acknowledge support from AWE via the Oxford Centre for
High Energy Density Science (OxCHEDS). S.M.V. acknowledges support from the
Royal Society. J.S.W. and S.M.V. acknowledge support from the UK EPSRC under
grants EP/P015794/1 and EP/W010097/1. G.P.-C. acknowledges support from
Spanish Ministry of Science and Innovation under Research Grant No.
PID2019-108764RB-I00. S.M.V. is a Royal Society University Research Fellow.
T.B., J.Ch., V.H., L.J., and V.V. appreciate financial support from the Czech
Ministry of Education (LG15013 and CZ.02.1.01/0.0/0.0/16-013/0001552—ERDF) and
Czech Science Foundation (grant No. 20-08452S)

Use of the Linac Coherent Light Source (LCLS), SLAC National Accelerator
Laboratory, is supported by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515.

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