# Ion Dynamics Effect on Stark-Broadened Line Shapes: A Cross-Comparison of Various Models

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## Abstract

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## 1. Introduction

## 2. Theory, Models and Simulations

- to find the time evolution of ${U}_{l}\left(t\right)$ for a given microfield configuration, which means solving the following equation:$$\frac{d{U}_{l}\left(t\right)}{dt}=-i[{L}_{0}-\mathbf{d}\xb7{\mathbf{F}}_{l}\left(t\right)]\phantom{\rule{3.33333pt}{0ex}}{U}_{l}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{U}_{l}\left(0\right)=1$$
- and to average it over a statistical ensemble of the microfields ${\left\{\phantom{\rule{3.33333pt}{0ex}}\right\}}_{l\in F}$.

#### 2.1. The Numerical Simulations

**ER-simulation**, the simulated plasma is an electrically neutral ensemble of statistically independent charged particles made of ${N}_{i}$ ions and ${N}_{e}$ electrons moving along straight line trajectories within a spherical volume. An emitter is assumed to be placed at the center of such a box. The temporal evolution of the whole system is measured along a discrete time axis from zero to a definite number of times of a fixed increment. Every temporal state is given by the set of values of the positions and velocities of the particles in the system. At every time step, the electric field produced by ions and electrons is calculated using Coulomb’s law or a Debye-screened field. This electric field is an input to the Schrödinger equation that computes the emitter time evolution operator. For hydrogen and when the no-quenching approximation is considered, the atom state is described with the Euler–Rodrigues parameters [24].

**HSTRK**and

**HSTRK_FST**codes also use the Gigosos–Cardeñoso approach [25]. Both codes rely on the Hegerfeld–Kesting–Seidel method of collision-time statistics [26] and compute $C\left(t\right)$. Depending on the appropriate option, HSTRK can do an electron only, ion only or joint simulation, but one can also do combinations, e.g., electron simulation and quasi-static ions or impact electrons and ion simulation. For the Fourier transform, if a long-time exponential behavior is detected for times $t>\tau $, then the contribution to the Fourier transform of the ($\tau ,\infty $) region is computed analytically using the detected exponential decay and added to ${\int}_{0}^{\tau}dtC\left(t\right){e}^{\u0131\omega t}$. τ is determined via start-up runs, e.g., a run with a small number of configurations is done to obtain a rough idea of the HWHM and τ is adjusted to cover at least a number of inverse HWHMs. The integral is done by Filon’s rule [27].

**SimU**is a combination of two codes: a molecular dynamics (MD) simulation of variable complexity and a solver for the evolution of an atomic system with the MD field history used as a (time-dependent) perturbation. A technical difference from other numerical simulation methods is the way the spectrum is calculated. Instead of employing the dipole autocorrelation function via Equation (1), SimU calculates the Fourier transform of the dipole matrix:

**Xenomorph**, is based on the models of Gigosos and González [30], where a straight line assumption is made. A general Schrödinger solver described in [31] is used to obtain the eigenvalues ${E}_{n}\left(t\right)$ and eigenvectors $\left|n\right(t)\rangle $ at every time step of the simulation. The emitter time evolution operator is then evaluated:

#### 2.2. The Models

**QuantSt.MMM**, MMM (for ions) is combined with a quantum-statistical approach to calculate pressure broadening due to plasma electrons. The perturbation by electrons is considered to second order in the potential [34,35].

**MELS**and

**MERL**are based upon the

**BID**model. The latter derives from the MMM, but its formulation is based on statistical mechanics [36] and provides a unified description of radiative and transport properties for charged emitters [20]. The stochastic line shape is written as:

**FFM**), on which the

**PPP**code and, recently, the

**QC-FFM**code rely. The latter is a hybrid model using the quasi-contiguous approximation [37] for H-like transitions and the FFM for modeling the microfield dynamics effect. The FFM relies on a different idealization of the stochastic process than MMM and BID. Here, the quantum system perturbed by a time-dependent microfield behaves like a set of field-dressed two-level transitions (SDT) subject to a collision-type mixing process. More precisely, the fluctuation mechanism of these SDT obeys a stationary Markov process defined by the instantaneous probability of states ${p}_{j}={a}_{j}/{\sum}_{k}{a}_{k}$ (${a}_{j}$ being the intensity of the SDT, j) and the transition rates between these states ${\mathbf{W}}_{k,j}=-{\Gamma}_{j}{\delta}_{k,j}+{W}_{k,j}$, where ${\Gamma}_{k,j}=\nu {\delta}_{i,j}$ and ${W}_{k,j}=\nu {p}_{j}$.

**ST-PST**model is based on the standard theory with a number of options. Specifically, apart from the pure ST results, ST-PST can (and by default does) also compute the results of ST with penetrating collisions correctly accounted for analytically [18]. In addition, an

**FST-FFM**calculation is also done [8]: first, an Ω is determined, exactly as described above for HSTRK_FST. Next, the FFM is applied to the field that excludes the fast, impact part. Last, the two profiles are convolved. As a result, the impact limit is correctly built in and recovered, hence extending the FFM validity without sacrificing its speed. Note, however, that with the current FST implementation, which uses the completed collision assumption for the impact phase space, the far wings are not accurate, as already discussed.

**UTPP**code is devoted to the calculation of hydrogen line shapes in regimes where the impact approximation for ions is reasonably accurate. Such a regime is attained for lines with a low principal quantum number in magnetic fusion experiments in the absence of Doppler broadening (Doppler-free line shape models were required for radiation transport simulations, e.g., [40]). In UTPP, a line shape is calculated using the following formula:

## 3. Comparisons and Discussion

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

**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

**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 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

**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).

**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

**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 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 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 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).

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 |

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Ferri, S.; Calisti, A.; Mossé, C.; Rosato, J.; Talin, B.; Alexiou, S.; Gigosos, M.A.; González, M.A.; González-Herrero, D.; Lara, N.;
et al. Ion Dynamics Effect on Stark-Broadened Line Shapes: A Cross-Comparison of Various Models. *Atoms* **2014**, *2*, 299-318.
https://doi.org/10.3390/atoms2030299

**AMA Style**

Ferri S, Calisti A, Mossé C, Rosato J, Talin B, Alexiou S, Gigosos MA, González MA, González-Herrero D, Lara N,
et al. Ion Dynamics Effect on Stark-Broadened Line Shapes: A Cross-Comparison of Various Models. *Atoms*. 2014; 2(3):299-318.
https://doi.org/10.3390/atoms2030299

**Chicago/Turabian Style**

Ferri, Sandrine, Annette Calisti, Caroline Mossé, Joël Rosato, Bernard Talin, Spiros Alexiou, Marco A. Gigosos, Manuel A. González, Diego González-Herrero, Natividad Lara,
and et al. 2014. "Ion Dynamics Effect on Stark-Broadened Line Shapes: A Cross-Comparison of Various Models" *Atoms* 2, no. 3: 299-318.
https://doi.org/10.3390/atoms2030299