MATHEMATICA MONTISNIGRI
Vol LVIII (2023)

SIMULATION OF DENSE ALUMINUM PLASMAUNDER INTENSE X-RAYS
D.A. KIM1*, I.YU. VICHEV1, A.D. SOLOMYANNAYA1, AND A.S. GRUSHIN1
1

Keldysh Institute of Applied Mathematics, Miusskaya sq. 4, Moscow 125047, Russia,*Corresponding author. E-mail: kimda@kiam.ru

DOI: 10.20948/mathmontis-2023-58-5
Summary. We used the THERMOS code to calculate properties of hot dense aluminumplasma created and probed by ultrashort intense X-ray pulses generated by a free-electron
laser at the LCLS facility. In our simulations we accented on taking into account theionization potentials depression and considered both the stationary and non-stationary
approaches. The calculation results reproduced the main experimental observations
qualitatively and were in good agreement with other published simulations. As it turned out,
despite the shortness of the X-ray impact, the non-stationary effects were rather negligible incontrast to those of the ionization potentials depression.
1 INTRODUCTION
Modern high-energy facilities such as X-ray free electron laser (XFEL) allow to investigate
the properties of matter at extreme conditions [1 – 6]. The impact of high-intensity ultrashort(tens to hundreds of femtoseconds) pulses of X-rays on a sample target leads to the formationof a photoionized plasma with the presence of empty inner electron shells in ions of different
charge multiplicity (so called hollow ions [7]). At the Linac Coherent Light Source (LCLS)
facility at Stanford (USA), solid-density aluminum plasma was studied under the influence of
X-ray radiation with photon energies in the range of 1500 – 1900 eV [1, 2]. This spectral
range embraces the K-shell (main quantum number n = 1 ) photoionization threshold as one of
the goals for the experiment was to provide data for verification of ionization potentials
depression (IPD) models. Experimental diagnostics included emission spectra of the plasma,
which contained distinct sets of Kα lines signing the presence of the hollow of various
multiplicity. From theoretical point of view, in order to simulate this experiment one has to1) Properly take into account the IPD effects. The incident X-ray photons were either beable to ionize directly the K-shell electron or not, depending on the ion charge [2]. If
the former scenario is implemented, it starts a sequence of atomic processes which
results in very recognizable emission spectra – the set Kα lines.
2) As the aforementioned sequence of atomic processes takes place, it is necessary tocorrectly estimate rates of the dominant atomic processes. In this case, these are thephotoionization of K-shell electrons with the following Auger decay of resulting
excited state; as the temperature of plasma rises, the collisional ionization processes
begin to take over thus diminishing the role of other processes.
The experimental data are of great value for verification the theoretical models of radiativeplasma. Numerical study of nonequilibrium plasma is usually carried out using the2020 Mathematics Subject Classification: 82D10, 81-10, 85-10.
Key words and Phrases: Non-stationary plasma, collisional-radiative equilibrium, photoionization, hollow ions,
X-ray free-electron laser.

68

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushincollisional-radiative (CR) models, in which the system of rate equations is solved that
determines the populations of ion states. The model is sensitive to the set of energy levels,
values of oscillator strengths, cross sections of collisional and radiative processes [8]. Density
effects such as ionization potential depression (IPD) and spectral line broadening should also
be taken into account. It might be preferable to use nonstationary CR model in case ofultrashort X-ray impact, because characteristic times of elementary atomic processes can becomparable to the pulse duration.
The THERMOS code has been developed at the Keldysh Institute of Applied Mathematicsof the Russian Academy of Science. THERMOS allows one to calculate states populations,
thermodynamic and optical properties of plasma in various approximations, equilibrium andnon-equilibrium [9, 10]. In this paper, we demonstrate the capabilities of THERMOS formodeling dense nonstationary photoionized aluminum plasma using one of the cases
considered at the 10th NLTE Code Comparison Workshop (hereinafter NLTE10) [11]. The
case is relevant to the measurements obtained at the LCLS facility but has simplified
formulation that does not take into account certain physical processes responsible for plasma
heating and expansion. This approach makes it possible to compare the CR codes developedby different scientific groups, thus outlining the path for further models improvement [12 –19].
2 PROBLEM STATEMENT AND THE MODEL DESCRIPTION
As one of the NLTE10 cases, it was proposed to simulate the evolution of level
populations and the emission spectrum in solid-density aluminum plasma under high-intensity
monochromatic ultrashort X-ray impact. The X-ray pulse has exposure duration of ∆t = 80 fs
with flat temporal profile and constant spectral intensity I 0 = 21.23 PW/cm2/eV. Photon
energy is in the thin bandwidth from 1648 to 1652 eV. The parameters are close to theexperimental ones [1, 2]. The workshop organizers provided the time dependence of theelectron temperature T ( t ) obtained using the SCFLY code [20] for the given parameters (seeFig. 1). The ion density is constant and equal to ni = 6 × 1022 cm-3.
In order to determine the populations xks of ion with charge k in the state s at each
moment of time t , it is necessary to solve the system of nonstationary rate equations:
dxks
dt
 xks

= ψ ks ( x ) − ϕ ks × xks , kxks = Z 0 , k = 0… Z ,

= 1,

ks

ψ ks ( x )

ks

/ abs
dex / em=  α ksex′→
ks × xks′ +  α ks′→ks × xks′ +s′< sci / phi / ai
k −1s′→ks

+ α
s′

ϕ ks

(1)

s′> s

/ dc
× xk −1s′ +  α kir+/1phrs′→ks × xk +1s′ ,s′/ phi / ai
ir / phr / dc
= α ksci→
k +1s′ +  α ks →k −1s′ +
s′

+ α

s′

dex / emks →ks′s′< s+  α ksex→/ abs
ks′ ,
s′> s

69

(2)

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushin

where Z 0 stands for plasma mean charge, Z is atomic number, α are the rates of radiative or
collisional processes. The latter can be divided into two groups, the ones that result in ion
charge variation and the ones that don't. Considered processes can be described schematicallyas follow:1a) collisional ionization (ci) and three-body recombination (ir):e + xkse + e '+ xk +1s ' ,1b) photoionization (phi) and photorecombination (phr):ℏω + xks
e + xk +1s ' ,1c) autoionization (Auger effect) (ai) and dielectronic capture (dc):ℏω + xks 'xks ''
e + xk +1s ,2a) collisional excitation (ex) and de-excitation (dex):
e + xkse '+ xks ' ,2b) photoabsorption (abs) and emission (em):ℏω + xks
xks ' .
We also consider processes that do not affect populations, but affect the emission spectrum:3a) bremsstrahlung and inverse bremsstrahlung:e + ℏω + xks
e '+ xks .
Here e and e ' are electrons with different energies, ℏω stands for photon, s ' and s '' are
excited and doubly excited states.

Temperature, eV

120

80

40

0
0

20

40

60

80

100

120

140

160

Time, fs

Figure 1. Time dependence of the electron temperature in dense aluminum plasma exposed to X-rays
with photon energy of 1650 eV and duration of 80 fs. The calculation was carried out using the
SCFLY code [20]The rates for collisional processes were calculated by the Lotz formula [21] (collisional
ionization and recombination) and the Van Regemorter formula [22, 23] (collisional

70

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushin

excitation and de-excitation). The Kramers approximation [24, 25] was employed to calculate
the photoionization and photorecombination rates. The autoionization and dielectronic capture
rates were calculated in the dipole approximation with the formula from Ref. [26].
System (1) is a system of ordinary differential equations of the 1st order. Its matrix ofcoefficients has a dimension equal to the number of ion configurations Nc taken into
account. Moreover the system (1) is stiff, since its coefficients (rates of processes) differ by
up to several tens of orders, which makes it similar to the Robertson example [27].
Often when modeling radiative plasma, it is sufficient to apply a stationary approximation( dxks / dt = 0 ) of collisional-radiative equilibrium (CRE). This gives a significant gain incomputation time. However, if the rate of change of external radiation or thermodynamic
parameters is greater than or comparable to the rates of the aforementioned atomic processes,
it is necessary to solve the system (1) in nonstationary approximation [28]. In this paper, weuse both stationary and nonstationary approximations of CR model, to understand how strong
the nonstationary effects are for the considered case.
A special numerical scheme was used to solve the nonstationary system (1):xˆksr +1 =xks + τψ ks ( x r ) (1 + τϕ ks / 2 )2

1 + τϕ ks + (τϕ ks ) / 2, x r = ( x + xˆ r ) / 2, xˆ 0 = x , r = 0,1.

(3)

Here x = { xks ,1 ≤ ks ≤ Nc} is a configuration population vector, xˆks is a value of population atnext time moment tˆ = t + τ , where τ is a time step, r is a iteration index. Scheme (3) was
proposed in Ref. [29]. This is a second-order explicit-implicit scheme with only two
iterations, which is well applicable to solving stiff systems.
For known populations xks at each moment of time t , it is possible to determine the timeintegrated spectrum of radiation losses for photons with energy ω by the formulaJ ω = 4π  j (ω , t )dt ,(4)

where spectral emissivity isj (ω , t ) = niℏω 3{ xks′ (t )σ ksbb′,ks (ω ) +4π 3c 2 ks′ks+  xks′ (t )σ (ω ) + e
fb
ks′

−ω /T

σ (ω )},

(5)

ff

ks′

ℏ = h 2π is the reduced Planck constant, c is the speed of light, σ bb (ω ) , σ fb (ω ) , σ ff (ω ) arebound-bound (line emission), free-bound (photorecombination), and free-free (bremsstrahlung
emission) cross-sections, respectively.
The influence of the external radiation field can be described in terms of rates of boundbound and bound-free radiative processes as following:

α ksabs→ks′ (t ) ~ W (ω, t ) |ω =ω

ks → ks ′α ksem′→ks (t ) ~ (1 + W (ω, t ) ) |ω =ω

,ks ′→ ks71,

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushin∞α (t ) ~ phinℓW (ω, t )ε nIPD
ℓ
∞

α

phr
nℓ

(t ) ~ ε nIPDℓe − ω /T ( t )

ω

ω

dω,

(6)

(1 + W (ω, t ) ) d ω,

where ε nIPD
is the depressed due to the density effects ionization potential for the level nℓ inℓconfiguration ks , W (ω , t ) is a function of photon distribution, that can be expressed in termsof radiation intensity I ω ( t ) :W (ω , t ) =π5
I ω (t ),15σω 3(7)

σ is the Stefan–Boltzmann constant.
In the considered case I ω (t ) = I 0θ1 (t )θ2 (ω ) , where1 , 0 ≤ t ≤ 80 fs,
0 , t ≥ 80 fs

θ1 ( t ) = 

1 ,1648 ≤ ω ≤ 1652 eV.
θ 2 (ω ) = 
0 , ω < 1648 eV, ω > 1652 eV(8)

One of the most important aspects of the considered problem is the strong influence ofhigh density. This leads to ionization potentials depression and broadening of spectral lines
and photoionization thresholds. We use two well-known approximations, Ecker-Kroll (EK)[30] and Stewart-Pyatt (SP) [31], to determine the magnitude of ionization potential
depression. Both approximations are quite simple and give an estimation ∆I IPD ~ 1 r0 , where
1/3

r0 = 1.388 ( A ρ ) is the radius of an electrically neutral atomic cell, A is atomic weight, ρis density.
At high densities, the energy levels are broadened, therefore, statistical accounting forphotoionization processes can be approached with simple approximation in which theprobability of transition from a bound state into continuum is described by the Gaussianfunction. Then the broadening of the photoionization thresholds is described by the
cumulative distribution function [9]Φ (ω ) =where erf( z ) =2 ω − ε nIPD
1
ℓ 
+
1
erf
,


IPD  
2
 ∆I


(9)

z

e dt is the error function, ∆I
π 
−t2

IPD

acts as broadening parameter. The

0photoionization threshold is described by the formula:σɶ nfbℓ (ω ) = σ nfbℓ (ω ) × Φ (ω ) .

72

(10)

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushin

Fig. 2 shows the photoionization cross-section for the ion Al3+ at temperature T = 12 eV anddensity ρ = 2.7 g/cm3. The calculation was performed by the THERMOS code with andwithout broadening. Obviously, higher density leads to more distinct broadening.
At the considered conditions, broadening of spectral lines is mostly due to collisional
broadening by electron impact and the Stark effect, the latter is calculated in nearest neighborapproximation [32]. We use the Voigt profile [9] to form the shape of a spectral line. The
Doppler effect and natural broadening are also included.

Bound-free cross section, Mb

10

broadened
w/o broadening

1
Al

3+

K-edge

0.1

10-2

-3

10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Photon energy, keV

Figure 2. Photoionization cross-section for the ion Al3+ at temperature T = 12 eV and density
ρ = 2.7 g/cm3

The wave functions, energy spectrum, ionization potentials, oscillator strengths, and crosssections were calculated using the THERMOS code applying the Hartree-Fock-Slater modelin the isolated ion approximation, accounting for states with unfilled K-shell was explicitly
included. Preliminary calculated database contained the information for Nc = 5925
configurations of aluminum ions of various multiplicity with principal quantum numbersn ≤ 6 . The obtained data were compared and corrected if necessary with data from the FAC
code [33] and the NIST database [34]. The ionization potentials depression is in accordancewith the models mentioned above. At the given density, both IPD approximations predict
disappearance of states with principal quantum number n > 2 for ions Al3+ – Al8+. For ionswith charge higher than 8+ states with n = 3 do exist, but their populations are negligible anddo not affect the final spectrum. In considered case such a reduction in configurations leads toa reduction in the number of equations Nc to be solved in system (1) by 1-2 orders ofmagnitude.
In the considered simplified formulation of the problem, the plasma has no size,
accordingly, the effects associated with spatial inhomogeneity and nonlocality of photon
reabsorption are not taken into account. The complex processes that determine the heating of
the plasma by X-rays and the influence of hot electrons are also beyond the scope of this73

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushinwork. Full-fledged simulation of the experiment [1, 2] is not the goal of this work. The main
objective is to verify existing CR models using a simplified problem given at NLTE10 andevaluate the effects of density and nonstationarity.

3 RESULTS AND DISCUSSIONSThe evolution of level populations and the emission spectrum was calculated with several
variants of the CR model implemented in the THERMOS code. It includes stationary andnonstationary models, with and without IPD approximation. Calculation results showedsignificant difference between EK and SP models at the given conditions. Fig. 3 shows
ionization potentials for aluminum ions, calculated using the SP and EK models andexcluding IPD at the end of the pulse. The magnitude of ∆I IPD for the K-shell varies from54 to 175 eV for ions Al3+ – Al8+. Moreover, for the EK and SP models, there is a rather
distinct discrepancy, e.g. for the Al6+ the difference is about 40 eV. Thus, X-ray radiation witha photon energy of 1650 eV can lead to direct photoionization of the K-electron for the Al3+,
Al4+ and Al5+ ions in the case of SP model, while in the case of EK, the decrease of ionization
potential is stronger, hence photoionization of the K-electron of the Al6+ ion is also possible.
1900

Ionization potential, eV

1850
1800

w/o IPD
SP
EK
exp
X-ray

1750
1700
1650
1600
1550
1500
3

4

5

6

7

8

Ion charge
Figure 3. Ionization potentials of the K-electron for aluminum ionsFig. 4 shows the time dependence of mean charge Z 0 . It is presented in comparison withthe depersonalized results of other NLTE10 participants (gray curves). The major group starts
from Z 0 (t = 0) ≈ 3 . THERMOS with IPD also gives Z 0 (t = 0) ≈ 3 for both EK and SP. At thegiven density, the ionization potential depression is so high that only K and L shells remain74

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushin

populated, which can accommodate only 10 out of 13 electrons in total, thus the mean charge
cannot be less than 3. As the time increases there is a slight discrepancy between Z 0 values,
which can be explained by differences in the rates and sets of atomic data. There are two graycurves with initial value of Z0 (t = 0) ≈ 1 , possibly due to underestimation of density effects isin the corresponding codes.
One of the gray curves starts from Z 0 (t = 0) ≈ 0 . It is obvious that the corresponding code
does not take into account the effect of IPD. Further values of Z 0 (t > 0) in this case are much
lower than those of the main group. The THERMOS code without IPD produces similar
results (green curve). This is a physically incorrect result, it is given solely to demonstrate theinfluence of high density.

Mean charge

8

6

4
SP
EK
w/o IPD
SP (stationary)
EK (stationary)

2

0
0

20

40

60

80

100

120

140

160

Time, fs

Figure 4. Time dependences of the mean charge calculated with the THERMOS code using stationary
and nonstationary CR models with and without IPD in comparison with similar calculations of the
NLTE10 workshop participants (gray curves)

The dots in Fig. 4 show values of the mean charge obtained with the THERMOS codeusing the stationary CR model. Each dot corresponds to density, radiation field, andtemperature at a given time moment. The results for the mean charge calculated in thestationary approximation are close to the nonstationary ones. The total ionization rates fordifferent charge states have the order of magnitude 1014 – 1015 s-1, i.e. the characteristic timesof these processes do not exceed tens of femtoseconds. Thus, the mean charge increases by∆Z 0 = 1 in about 10 – 20 fs, which is confirmed by the data in Fig. 4. However, due to thepresence of some slower processes, the mean charge calculated in the nonstationary approachturns out to be approximately 0.1 – 0.5 lower than the stationary approximation value.

75

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushin

At the given conditions with high density and intense external X-ray radiation, the total
rates of collisional ionization, autoionization, and photoionization compete with each other.
Over the time, the collisional ionization begins to prevail due to increase of the free electrons(see Tab. 1). For aluminum ions Al6+ and higher in the case of SP and Al7+ and higher in thecase of EK, the total photoionization rate drops sharply, since the photoionization of
K-electron (the most probable photoionization process) becomes impossible at the considered
X-ray photon energy ω = 1650 eV. The process of K-electron photoionization is followed byeither Auger decay or radiative decay with the electron transition from L-shell to K-shell.
Fig. 5 shows the time-integrated spectra of radiation losses calculated by the THERMOS code(colored) in comparison with other codes participating in the NLTE10 workshop (gray). The
solid lines correspond to the nonstationary approximation, and the dashed lines correspond to
the stationary one. All the presented spectra are normalized to unity for clarity. In contrast to
the mean charge results, the spectra calculated using various codes significantly differ fromeach other in positions, widths and relative brightness of spectral lines. The main reason
behind this discrepancy is the use of different sets of atomic data in the codes.
Rate×1013, s-1time, fs
21.7
42.4
59.6
76.9
80.3

ci
8.6
32
49
61
62

EK
phi ai5.8 4.4
5.9 1.3
6
0.77
6
0.6
0
0.0056

ci
2.3
13
22
30
31

SP
phi
ai
0.16 5.9
0.13 1.5
0.15 0.8
0.15 0.57
00.0019

Table 1. Total rates of collisional ionization, photoionization, and autoionization for
ion Al6+ calculated with the THERMOS code at given parameters
As we can see from Fig. 5 all codes predict the presence of bright Kα lines for ions withcharge 4+, 5+, 6+, and some codes, where apparently the EK model is used, predict thepresence of bright Kα line for the Al7+ ion as well. As for the THERMOS code results themain difference in the spectra is observed for the Kα line corresponding to the ion of Al7+. In
the case of EK model, direct photoionization of the 1s-electron is possible (and mostprobable) for the ion Al6+, therefore, bright Kα line corresponding to the Al7+ is observed in
the spectrum. In the case of SP model direct photoionization of the 1s-electron is no longer
possible for Al6+ ion (see Fig. 3), but nevertheless this spectral line is also present. However,its relative brightness is significantly lower being a result of low rate processes, for example
the following sequence of transitions: 1s2s22p4 (Al6+*) → [collisional ionization of an electronfrom a 2p-shell] → 1s2s22p3 (Al7+*) → [radiative decay] → 1s22s22p2 (Al7+).

76

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushin

10
Al

4+

Al

5+

Qtot, arb.u.

1

Al

6+

Al

SP
EK
SP (stationary)
EK (stationary)

7+

0.1

-2

10

-3

10

-4

10

1500

1550

1600

1650

Photon energy, eV
Figure 5. Time-integrated radiation losses calculated using the THERMOS code in various
approximations compared with the results of the NLTE10 workshop participants (gray)

The spectra calculated using the THERMOS code in the stationary approximation are close
to those calculated in the nonstationary approximation, both in case of SP and EK. It turned
out that in this high density case, the effects of nonstationarity are rather insignificant, despite
such short X-ray pulse.

4 CONCLUSION
The comparative analysis shows that most of the codes provide similar results for the timedependence of the mean charge. Both stationary and nonstationary models with and without
IPD approximation are implemented in the THERMOS code. The results for the mean charge
obtained using the THERMOS code for all variants are in good agreement with most of theother codes presented at the NLTE10 workshop. Taking the nonstationarity into account gives
insignificant difference in the mean charge values in comparison with the stationary approachsince the total ionization rates are comparable to the characteristic rate of temperature change
and X-ray pulse duration. This conclusion allows calculation of the states populations andemission spectra under given conditions using simplified stationary approximation, withsignificant acceleration of computation at sufficient accuracy.
For near solid-density plasma the choice of IPD model turns out to be decisive.
Calculations without IPD in this case are simply inapplicable, since they produce physicallyincorrect result. This work considers two IPD models, Stewart-Pyatt and Ecker-Kroll, which,despite their similarity, give significant differences in emission spectra. In particular, for the77

D.A. Kim, I.Yu. Vichev, A.D. Solomyannaya and A.S. Grushin

Al6+ ion the difference between EK and SP in the shift of the K-electron ionization potential isabout 40 eV, which is decisive in modeling the emission spectrum – the Kα line of the Al7+
ion is an order of magnitude brighter in the case of EK since direct photoionization of theK-electron of the Al6+ ion is possible in contrast to the SP model.
A comparison of the emission spectra presented at the NLTE10 workshop shows
significant differences in the results from code to code. CR models are sensitive to the set of
considered ion states and their details, which is determined, among other things, by the level
splitting due to the type of coupling (LS-, jj-coupling, or intermediate).
This work shows that for a full-fledged simulation of experiments similar to the one carriedout at the LCLS facility, it is required to refine the collisional-radiative model by selecting the
optimal set of states, as well as correctly describing the transition rates between them.
In this paper it is shown that the THERMOS code can calculate the properties of matter inextreme states with high density and high-intensity X-ray exposure, but preliminary it isrequired to compile a correct atomic database. Based on the presented results, further
improvement of the code is planned, which will allow complex calculations to be carried out
in-line with radiative hydrodynamics, where radiation transfer, level kinetics, matter flows,
and thermodynamic quantities will be consistent.

Acknowledgements: The calculations were performed on the hybrid supercomputer K100
installed in the Supercomputer Center of Collective Usage of KIAM RAS and MVS-10P(JSCC RAS).
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Received, September 14, 2022

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