To test the accuracy of the different numerical codes based either on stochastic and collisional models or numerical simulations, calculations for standardized case problems were carried out and analyzed [

5]. A preselected set of transitions on a grid of electron densities (

${n}_{e}$) and temperatures (

$T={T}_{e}={T}_{i}$) have been proposed, and for each case, the atomic and plasma models have been specified. In this way, various contributions that can affect the Stark broadened line shape, such as the influence of particle correlations on electric microfields, the effects of external fields, the high-n merging with continuum or the satellite broadening, have been investigated. For the present purpose, we will only focus on cases where the ion dynamics effect was studied.

#### 3.1. Hydrogen Lyman-α and Lyman-β Lines

The following examples consider the hydrogen Lyman-

α and Lyman-

β lines in an ideal plasma consisting of protons for electron densities

${n}_{e}={10}^{17}-{10}^{19}$ cm

${}^{-3}$ and temperatures

$T=1-100$ eV. These cases are not necessarily practical, but permit basic comparisons to assess the influence of ion dynamics on the line profiles. Here, only pure ionic linear Stark effect is considered (

$\Delta n\ne 0$ interactions are ignored) and the fine structure is not taken into account. The concept of ideal plasma means that unscreened particles moving along straight path trajectories are considered in the numerical simulations, and the Holtsmark static-field distribution function [

22] is used in the models.

An overall comparison of the results is presented in

Figure 1. For each subcase (determined by a combination of

$({n}_{e},T)$) and for each code, ratios between the full-width at half-maximum (FWHM) and an average of FWHM of all submitted results have been evaluated [

5]:

The graph is divided in two regions: the left side corresponds to results for the Lyman-α line and the right side to the Lyman-β line. Each region is divided into three sub-regions that correspond to the three densities chosen. Finally, in each sub-region, each set of results corresponds to the temperatures, $T=1,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}100$ eV, respectively. For the Lyman-α case, the results present a large dispersion, deviating from the average by more than a factor of five in each direction. In contrast, the scatter for the Lyman-β shows a rather good agreement between the codes. In fact, these two lines present a completely different behavior concerning the ion dynamics effect.

**Figure 1.**
Overall comparison of the workshop results of the ion dynamics effect on Lyman-α and -β hydrogen lines. For each subcase, i.e., different pairs of $({n}_{e},T)$, the scatter of ratios between the different results and an average value is plotted. The different symbols correspond to: (black dot) SimU; (red square) UTPP; (blue triangle) PPP; (blue asterisk) Xenomorph; (cyan open triangle) HSTRK; (cyan triangle) HSTRK_FST; (red diamond) ER-simulation; (green circle) QuantST.MMM; (black cross) QC-FFM.

**Figure 1.**
Overall comparison of the workshop results of the ion dynamics effect on Lyman-α and -β hydrogen lines. For each subcase, i.e., different pairs of $({n}_{e},T)$, the scatter of ratios between the different results and an average value is plotted. The different symbols correspond to: (black dot) SimU; (red square) UTPP; (blue triangle) PPP; (blue asterisk) Xenomorph; (cyan open triangle) HSTRK; (cyan triangle) HSTRK_FST; (red diamond) ER-simulation; (green circle) QuantST.MMM; (black cross) QC-FFM.

#### 3.1.1. The Lyman-α Line

The static Stark effect of the Lyman-α line (as all the $\Delta n=n-{n}^{\prime}=1$ lines, where n and ${n}^{\prime}$ are the principal quantum number of the upper and lower states, respectively) features a strong unshifted component that is highly sensitive to the ion dynamics effect. Thus, even though the Lyman-α line is the simplest case from the atomic structure point of view, it presents a non-trivial Stark-broadening behavior.

In

Figure 2, only results from the numerical simulations are plotted for the sake of clarity. One sees that in the range of 1 to 100 eV, the simulations either predict that the width increases when the plasma temperature increases (for the fixed density

${n}_{e}={10}^{19}$ cm

${}^{-3}$, they present a temperature dependency as ∼

${T}^{1/3}$) or predict that the width is mostly insensitive to the temperature’s rise (for the fixed density

${n}_{e}={10}^{17}$ cm

${}^{-3}$). Concerning the dependence on the plasma density, the width, which is mainly due to the width of the central component for

$T=1$ eV, increases as

${n}_{e}^{1/3}$. For

$T=100$ eV, the results show a

${n}_{e}^{2/3}$ dependence, corresponding to the quasi-static behavior of the lateral components [

2]. We mention, however, that the cutoff of the Coulomb interaction at a finite box size may not accurately reproduce an ideal plasma [

42].

**Figure 2.**
Lyman-α ion FWHMs as a function of (**a**) of T at fixed densities and (**b**) of ${n}_{e}$ at fixed temperatures. The ideal, one-component plasma consisting of protons is assumed. Only results from numerical simulations are presented: (red circle) ER-simulation; (blue square) HSTRK; (black dot) SimU; (green asterisk) Xenomorph.

**Figure 2.**
Lyman-α ion FWHMs as a function of (**a**) of T at fixed densities and (**b**) of ${n}_{e}$ at fixed temperatures. The ideal, one-component plasma consisting of protons is assumed. Only results from numerical simulations are presented: (red circle) ER-simulation; (blue square) HSTRK; (black dot) SimU; (green asterisk) Xenomorph.

In general, the highest temperature results in the best agreement among simulation codes, for all densities. For lower temperature differences are more discernible, with the most discernible being the appearance of shoulders in ER-simulation and SimU for the highest density and lowest temperature and the lack of such shoulders in HSTRK. This is a general trend at the lowest temperature of 1 eV for all densities, with HSTRK producing significantly larger widths than both ER-simulation and SimU.

The dispersion of the results of the various models demonstrates the difficulties in accurately treating the ion dynamics effect (see

Figure 3). In every studied case, the PPP displays a weaker ion dynamics effect on this line, probably due to an incomplete description of this effect on the central component. The FFM mixes the unshifted components with the Stark-shifted components with a unique fluctuation rate. Yet, the unshifted components are not sensitive to the microfield intensity, but only to its rotation, whereas the Stark-shifted components are sensitive to the microfield vibration [

43]. A more detailed discussion on the influence of the microfield directionality in the line shape is presented in a separate study [

44].

Concerning the description of the ion dynamics effect in terms of microfields mixing, the QuantST.MMM results compare less favorably to the simulations, especially in the far wings.

As already discussed, the far wings of HSTRK_FST are not reliable in this version, due to the complete collision assumption used in the computation of the impact part. This is an artifact of this assumption rather than an inherent limitation of the method.

The UTPP code yields a line width systematically larger than the results obtained from other codes or models and, in particular, the results from numerical simulations. If the latter give reference profiles, this result is expected in general, because the plasma conditions are such that static effects with simultaneous strong collisions are important. However, the application of the UTPP to the electron broadening (not presented here) also indicates a significant discrepancy, with an overestimate of the numerical simulation results by a factor of two. It has been suggested that this discrepancy stems from the fact that the simulations that use a box actually miss a significant contribution to the line broadening, due to the far perturbers, namely, those inside the

$v/\overline{\gamma}$ sphere, but outside the simulation box. It is quite difficult to test this argument by enlarging the simulation box up to

$v/\overline{\gamma}$, because this would imply a very large number of particles (up to several billions). An adaptation of UTPP able to account for a plasma of finite size has been performed and has led to a line shape in good agreement with the simulations [

42]. This could suggest that an artificial setting of an infinite Debye length in the numerical simulations able to work with an infinite Debye length requires a careful interpretation of the results.

**Figure 3.**
Lyman-α line shape in ideal ionic one component plasma (OCP) calculated for (**a**) the more dynamical regime (${n}_{e}={10}^{17}$ cm${}^{-3}$ and $T=100$ eV) and (**b**) the more static regime (${n}_{e}={10}^{19}$ cm${}^{-3}$ and $T=1$ eV): SimU (black dash); ER-simulation (red dash); HSTRK_FST (blue dot-dash); PPP (solid cyan); QuantST.MMM (solid purple); UTPP (solid green).

**Figure 3.**
Lyman-α line shape in ideal ionic one component plasma (OCP) calculated for (**a**) the more dynamical regime (${n}_{e}={10}^{17}$ cm${}^{-3}$ and $T=100$ eV) and (**b**) the more static regime (${n}_{e}={10}^{19}$ cm${}^{-3}$ and $T=1$ eV): SimU (black dash); ER-simulation (red dash); HSTRK_FST (blue dot-dash); PPP (solid cyan); QuantST.MMM (solid purple); UTPP (solid green).

#### 3.1.2. The Lyman-β Line

The static profile of the Lyman-

β line (as all of the

$\Delta n=2$ lines) normally shows a dip at the line center. One sees in

Figure 4 that, due to the ion dynamics effect, the simulations fill this dip, and the width increases with increasing temperature. This trend is seen for plasma conditions that correspond to typical microfield fluctuation rate values (see Equation (

5)) smaller than the splitting of the two Stark components measured in the static case. Here, for

${n}_{e}={10}^{17}$ cm

${}^{-3}$, the Stark splitting of the static line shape is equal to

$5.9\times {10}^{-3}$ eV, and the typical fluctuation rate is equal to

$\hslash \nu =6.8\times {10}^{-4}$ eV and

$\hslash \nu =2.5\times {10}^{-3}$ eV for

$T=1$ eV and

$T=10$ eV, respectively. For

$T=100$ eV,

$\hslash \nu =2.2\times {10}^{-2}$ eV,

i.e., three-times greater that the Stark splitting in the static case. The two components merge, leading to a line shape that is narrower than the one calculated for

$T=10$ eV, as is seen in

Figure 4a. Note that for an infinite fluctuation rate, the line shape becomes the Dirac

δ-function.

The agreement between the Lyman-

β FWHM results of different codes is much better than that for Lyman-

α, as is shown in

Figure 1. Nevertheless, the concept of FWHM is not really adequate for such a line with a dip in the center. A better way to discuss the ion dynamics effect on a Lyman-

β line would be the measure of the relative dip given by:

where

${I}_{max}$ and

${I}_{{\omega}_{0}}$ are the maximum intensity and the intensity at the center of the line, respectively.

Table 1 shows the relative dip from the different codes for

${n}_{e}={10}^{17}$ cm

${}^{-3}$, while the line shapes are shown in

Figure 5.

Obviously, the QC approximation, and, hence, the QC-FFM method, is inherently unable to reproduce the central structure (a peak or a dip) of a low-

n spectral line. However, the wings of such lines, as well as entire profiles of higher-

n transitions, show a very good agreement with numerical simulations [

13].

**Figure 4.**
The ion dynamics effect on the Lyman-β line for different values of T obtained by SimU: (solid red $T=1$ eV; (green dash) $T=10$ eV and (blue dot-dash) $T=100$ eV at a fixed (**a**) ${n}_{e}={10}^{17}$ cm${}^{-3}$ and (**b**) ${n}_{e}={10}^{19}$ cm${}^{-3}$. The ideal one-component plasma consisting of protons is assumed.

**Figure 4.**
The ion dynamics effect on the Lyman-β line for different values of T obtained by SimU: (solid red $T=1$ eV; (green dash) $T=10$ eV and (blue dot-dash) $T=100$ eV at a fixed (**a**) ${n}_{e}={10}^{17}$ cm${}^{-3}$ and (**b**) ${n}_{e}={10}^{19}$ cm${}^{-3}$. The ideal one-component plasma consisting of protons is assumed.

**Figure 5.**
Lyman-β line for ${n}_{e}={10}^{17}$ cm${}^{-3}$ and $T=10$ eV: SimU (black dash); ER-simulation (red dash); Xenomorph (blue dot-dash); PPP (solid cyan); QC-FFM (solid orange); QuantST. MMM (solid purple); UTPP (solid green).

**Figure 5.**
Lyman-β line for ${n}_{e}={10}^{17}$ cm${}^{-3}$ and $T=10$ eV: SimU (black dash); ER-simulation (red dash); Xenomorph (blue dot-dash); PPP (solid cyan); QC-FFM (solid orange); QuantST. MMM (solid purple); UTPP (solid green).

**Table 1.**
The relative dip (%) measured on the Lyman-β line from the different codes for ${n}_{e}={10}^{17}$ cm${}^{-3}$.

**Table 1.**
The relative dip (%) measured on the Lyman-β line from the different codes for ${n}_{e}={10}^{17}$ cm${}^{-3}$.
T (eV) = | 1 | 10 | 100 |

ER-simulation | 75 | 44 | 10 |

SimU | 56 | 19 | 0 |

Xenomorph | 56 | 14 | / |

PPP | 70 | 31 | 0 |

QuantSt.MMM | 71 | 55 | 32 |

UTPP | 0.6 | 0.6 | 0 |

#### 3.2. Argon He-α and He-β Lines

The argon H- and He-like lines are observed in inertial confinement fusion implosion core plasmas when a tracer amount of argon is added to the deuterium gas fill to diagnose the plasma conditions [

45]. Such a diagnostic relies on the temperature sensitivity of the satellite line;s relative intensity to the resonance one and the density dependence in the Stark broadening of both satellite and resonance line profiles [

46]. Moreover, they are sensitive to the ion dynamics effects and present a challenge for theoretical models [

47]. We only focus here on the He-

α and He-

β lines. A specific study of the effect of satellite line shapes on the He-

β line can be found elsewhere [

48].

Two electron densities, ${n}_{e}=5\times {10}^{23}$ cm${}^{-3}$ and ${n}_{e}=2\times {10}^{24}$ cm${}^{-3}$, and a plasma temperature of $T=1$ keV were selected for this comparison. Plasma ions are deuterons with $0.1\%$ Argon XVII. The MELS and MERL (BID) and the PPP (FFM) models submitted results, and the numerical simulation, SimU, was recently extended to describe such lines. Here, the simulation accounts for all interactions; no artificial cutoff arises as for the ideal case conditions. We consider it as a reference.

Figure 6 displays the He-

α profiles calculated with the PPP code within the quasi-static approximation. For clarity, results from MELS are not plotted here, but the agreement between the two codes is very good. The small differences observed between both codes are explained by the difference in the electron broadening treatment (the impact approximation is used in PPP, while a frequency dependent collision operator is used in MELS). The quasi-static profile is the superimposition of a strong intensity component, which corresponds to the

$1s2p{\phantom{\rule{3.33333pt}{0ex}}}^{1}{P}_{1}-1{s}^{2}{\phantom{\rule{3.33333pt}{0ex}}}^{1}{S}_{0}$ resonance transition, and a weak intensity component, which corresponds to the

$1s2p{\phantom{\rule{3.33333pt}{0ex}}}^{3}{P}_{1}-1{s}^{2}{\phantom{\rule{3.33333pt}{0ex}}}^{1}{S}_{0}$ intercombination transition. The pure electron broadened profiles are plotted for each component for a better understanding of the ionic Stark effect on the line shapes. Both components display a pronounced quadratic Stark effect in their “blue” wing, and forbidden lines appear on top of their “red” wing.

Concerning the ion dynamics effect, BID and FFM show a different behavior. BID profiles present a more pronounced deviation relative to the static calculation than the FFM.

Figure 7 and

Figure 8 illustrate this for the two densities. These discrepancies cannot be explained by the use of a different fluctuation rate in both models. A specific study using the same fluctuation rate for both models shows that the BID and FFM are in good agreement for varying values of this parameter for the resonance line, but not for the intercombination line [

39].

Figure 9 shows this difference using both models with the same fluctuation rate. The difference seen on the forbidden component of the intercombination line might be due to a numerical inaccuracy, because of the very weak value of its intensity.

Moreover, numerical simulation results from the SimU code do not discriminate between the stochastic models. For example, in

Figure 9, both models agree with the simulation on the allowed transitions, but not on the forbidden transitions. This might be due to a different dynamics between strong microfields, which are emphasized by the quadratic Stark effect of the allowed transitions, and weak microfields, which are the cause of the linear Stark effect of the forbidden transitions.

**Figure 6.**
The He-α line calculated within the quasi-static approximation for $T=1$ keV and ${n}_{e}=2\times {10}^{24}$ cm${}^{-3}$. (Black line) the entire profile; (blue line) resonant line profile; (red line) intercombination line profile. The pure electron-broadened profiles are plotted in dashed lines for each component.

**Figure 6.**
The He-α line calculated within the quasi-static approximation for $T=1$ keV and ${n}_{e}=2\times {10}^{24}$ cm${}^{-3}$. (Black line) the entire profile; (blue line) resonant line profile; (red line) intercombination line profile. The pure electron-broadened profiles are plotted in dashed lines for each component.

**Figure 7.**
The He-α line for $T=1$ keV and ${n}_{e}=5\times {10}^{23}$ cm${}^{-3}$: static ions MELS (grey dash); ion dynamics BID (solid red); FFM (solid blue) and SimU (black dot).

**Figure 7.**
The He-α line for $T=1$ keV and ${n}_{e}=5\times {10}^{23}$ cm${}^{-3}$: static ions MELS (grey dash); ion dynamics BID (solid red); FFM (solid blue) and SimU (black dot).

**Figure 8.**
The He-α line for $T=1$ keV and ${n}_{e}=2\times {10}^{24}$ cm${}^{-3}$: static ions (grey dash); ion dynamics BID (solid red); FFM (solid blue); and SimU (black dot).

**Figure 8.**
The He-α line for $T=1$ keV and ${n}_{e}=2\times {10}^{24}$ cm${}^{-3}$: static ions (grey dash); ion dynamics BID (solid red); FFM (solid blue); and SimU (black dot).

**Figure 9.**
The He-α line for $T=1$ keV and ${n}_{e}=2\times {10}^{24}$ cm${}^{-3}$: (**a**) resonance line and (**b**) intercombination line. Static ions MELS (red dash) and PPP (blue dash); ion dynamics BID (solid red); FFM (solid blue); and SimU (black dot).

**Figure 9.**
The He-α line for $T=1$ keV and ${n}_{e}=2\times {10}^{24}$ cm${}^{-3}$: (**a**) resonance line and (**b**) intercombination line. Static ions MELS (red dash) and PPP (blue dash); ion dynamics BID (solid red); FFM (solid blue); and SimU (black dot).

In order to explain these differences, a specific study on the pure ion-broadened profiles was carried out. As both resonance and intercombination lines present similar atomic systems, we will only focus the discussion of the resonance line.

Figure 10 shows FFM profiles for different fluctuation rates and the SimU profile. It seems that different values of

ν are needed to reproduce different portions of the simulated profile. A lower fluctuation rate has to be used to fit the forbidden component, whereas a higher

ν is needed to reproduce the allowed component. This can be interpreted as weak and strong microfields not producing the same dynamics effect on the line shape.

**Figure 10.**
The He-α line, the strong component for $T=1$ keV and ${n}_{e}=5\times {10}^{23}$ cm${}^{-3}$: SimU (black circles); FFM with $\nu =3$ eV (solid blue); $\nu =5.62$ eV (solid red); and $\nu =8$ eV (solid black).

**Figure 10.**
The He-α line, the strong component for $T=1$ keV and ${n}_{e}=5\times {10}^{23}$ cm${}^{-3}$: SimU (black circles); FFM with $\nu =3$ eV (solid blue); $\nu =5.62$ eV (solid red); and $\nu =8$ eV (solid black).

Finally, the He-

β line is presented in

Figure 11. At the chosen plasma conditions, as the Stark splitting of the He-

β quasi-static line shape is greater than the fluctuation rate and the electron width is larger, the ion dynamics effect is less pronounced than on the He-

α.

Figure 11 shows SimU, BID and FFM in rather good agreement relative to the discrepancies of their quasi-static profiles. The measure of the dynamics-to-static relative depth is defined by:

There is a fairly good agreement between the BID and the FFM (see

Table 2).

**Figure 11.**
The He-β line for $T=1$ keV and (**a**) ${n}_{e}=5\times {10}^{23}$ cm${}^{-3}$; (**b**) ${n}_{e}=2\times {10}^{24}$ cm${}^{-3}$. Static ions: MERL (red dot), PPP (blue dot); SimU (black dot); BID (solid red); FFM (solid blue).

**Figure 11.**
The He-β line for $T=1$ keV and (**a**) ${n}_{e}=5\times {10}^{23}$ cm${}^{-3}$; (**b**) ${n}_{e}=2\times {10}^{24}$ cm${}^{-3}$. Static ions: MERL (red dot), PPP (blue dot); SimU (black dot); BID (solid red); FFM (solid blue).

**Table 2.**
Dynamics-to-static relative dip (%) measured on the argon He-β line for $T=1$ keV.

**Table 2.**
Dynamics-to-static relative dip (%) measured on the argon He-β line for $T=1$ keV.
Models | BID | FFM |
---|

${N}_{e}=5\times {10}^{23}$ cm${}^{-3}$ | 58 | 57 |

${N}_{e}=1\times {10}^{24}$ cm${}^{-3}$ | 50 | 51 |

${N}_{e}=2\times {10}^{24}$ cm${}^{-3}$ | 47 | 48 |