# The Third and Fourth Workshops on Spectral Line Shapes in Plasma Code Comparison: Isolated Lines

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## Abstract

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## 1. Introduction

## 2. Presentation of the Studied Cases

## 3. Short Description of the Methods and Codes

`GRASP/DARC`, and three semi-classical codes:

`SCP`, and

`STARCODE`(two variants, with and without penetrating collisions), and

`SimU`. In addition, an improvement of the

`SCP`method has been made, only for the inelastic neutral–electron cross-section. It is denoted hereafter as

`SCPVB`.

`GRASP/DARC`[12] calculates electron-impact broadening and shifts. They are calculated in the frame of relativistic quantum mechanics.`GRASP`[13] obtains the energy levels and the electronic orbitals of the N-electron target ion radiator, with the fine structure included. Then the Dirac Atomic R-matrix Code (`DARC`) is used to construct and solve the $(N+1)$-electron colliding system (the target ion + one free electron). Thus, there is no multipole expansion of the interaction potential, which cannot be limited e.g., to only the dipole one. Solving this system of coupled equations leads to the scattering S-matrix. The widths and shifts are obtained through an adequate sum over the l quantum numbers of the perturber electron, and through an average over the Maxwell distribution of the electrons.`SCP`[14], and earlier references therein, is a semi-classical perturbation method based on the standard impact approximation, both for electron and ion projectiles. It applies for isolated lines only. The atomic structure is an external input to the code. The classical perturber moves on a straight path for neutrals, and a hyperbola for ions. The long range Coulomb interaction potential is expanded up to second order. Then, the S-matrix is obtained through a perturbation expansion in the frame of the evolution operator in interaction representation. Dipolar and quadrupole interactions are taken into account for the widths, and the dipole interaction is exclusively taken into account for the shifts. Thanks to adequate cut-offs, symmetrization and unitarity of the S-matrix are fulfilled [15]. Consequently, excitation cross-sections are zero under the threshold. Debye shielding is taken into account. Feshbach resonances for electron projectiles and ion lines are included in the code by use of the semi-classical limit of the Gailitis formula [16], but has been removed for SLSP calculations.`SCPVB`[17] improves the calculation of the electron–neutral-atom`SCP`inelastic cross-sections by accounting for the momentum and energy transfer during the collision. The momentum transfer is accounted for by considering the angular distribution of the velocity variation of the projectile electron in the course of the inelastic collision. This enables different weights associated to the m quantum numbers of the kinetic momentum projection along the collision $Oz$ axis, which is also the propagation direction of the incoming electron. The symmetrization procedure is also improved, by treating the first half of the collision under the initial conditions, whereas the second half is modeled under the final conditions.`STARCODE`[18] (with and without penetration) uses the impact approximation and the complete collision approximation. The classical perturber (electron or ion) moves along a classical path: a straight line for neutral atoms, and a hyperbola for ion radiators. The atomic structure of the quantum atom can be perturbed by interactions: thus, the possibility of penetration of the perturber, which modifies (softens) the interaction, can be taken into account. The calculations with and without the penetration effect taken into account will be designated below as`STARCODE-P`and`STARCODE-NP`, respectively. Long-range and short-range terms are taken into account in the expansion of the Coulomb interaction potential in the monopole, dipole and quadrupole terms. The S matrix along with accurate estimates (based on the non-semiclassical and nonperturbative terms) is first obtained through the second-order perturbation expansion in the frame of the evolution operator in the interaction representation; since the error estimates are invariably unacceptably large, a full numerical solution of the Schrödinger equation follows, so that all results are based on this fully numerical solution. The only exception is ion broadening (not considered here), where rigorous bounds can usually show that the ion broadening is unimportant.`SimU`[19] is based on an N-body numerical simulation of the motion of the interacting plasma particles (both ions and electrons, but for the present calculations, only electrons were modeled). The interaction potential includes dipole and quadrupole terms.`SimU`treats isolated and overlapping lines, and dipole-allowed and dipole-forbidden radiative transitions as well. Trajectories are affected by the charge of the radiating ion and the perturbers. Since a Debye potential is assigned to the radiator, the trajectories of the perturbers are not hyperbolic for ionized elements. The evolution of the radiator is obtained from the resulting microfield histories by solving the (time-dependent) Schrödinger equation. Then, the line profile is obtained through the Fourier transform of the resulting radiator time-dependent dipole function.

## 4. Li I $\mathbf{2}\mathbf{s}-\mathbf{2}\mathbf{p}$

#### 4.1. Width and Shift

`SCP`, on one hand, and

`STARCODE`and

`SimU`, is expected. Indeed, all SC codes consider electrons as classical projectiles ignoring the back-reaction due to changes in the internal quantum degrees of freedom of the radiator. As a result, the under-threshold excitation process remains allowed and, more generally, the direct and inverse processes are related through the detailed-balance relation corresponding to $T=\infty $. This is clearly nonphysical with respect to cross-sections, and in

`SCP`a symmetrization procedure [15] is used as a workaround. Due to the symmetrization, a correct microreversibility of the cross-sections in excitation and de-excitation is obtained. Consequently, the excitation cross-sections are zero under the threshold. Penetrating collisions become more important at higher energies. Comparing the left and right part of Figure 1, the effect of the quadrupole interaction is rather small for all codes (between 5% and 25%).

`SCP`does not presently account for the quadrupole interaction when calculating the Stark shift; this will be enabled in a future version of the code. Therefore, the

`SCP`results in the left and right panels of Figure 2 are identical.

#### 4.2. Fractional Inelastic Width

`SCP`and

`STARCODE-NP`(no penetration) show a minor decrease or saturation at higher energies.

#### 4.3. Cross-Sections

#### 4.3.1. Total Cross-Section

#### 4.3.2. Inelastic Excitation Cross-Section

`SCPVB`method improves the results of the ordinary

`SCP`inelastic cross-sections [17], in particular due to the improvement of the symmetrization procedure, such that the

`SCPVB`results become rather close to the experimental values. Also, we note that

`SCP`and

`SimU`apparently have the same high-E asymptote, which exceeds the experimental values by about one third.

#### 4.3.3. Partial Cumulative Excitation Cross-Section

#### 4.3.4. Elastic Pseudo Cross-Section

`SCP`and

`STARCODE`attribute higher relative importance to the quadrupole effect than

`SimU`. This explains the difference in the high-E tails observed in Figure 3.

## 5. B III $\mathbf{2}\mathbf{s}-\mathbf{2}\mathbf{p}$

#### 5.1. Width and Shift

`SCP`and other SC can be explained by the symmetrization issue. The excitation part of the

`SCP`width is zero under the threshold. If Feshbach resonances were included as usual in the

`SCP`method for ions, the behavior of the

`SCP`width would be similar to the others and probably of the same order of magnitude.

`GRASP/DARC`cannot be calculated with dipole interaction only. Consequently, it appears only on the right part of Figure 8. It agrees with the SC codes.

`STARCODE-NP`is of the same order of magnitude as SC ones, and, as expected,

`STARCODE`with penetration is smaller. The penetrating collisions are important for all energies. Similarly to the Li I results, the effect of the quadrupole interaction is not very important.

`SCP`shifts are significantly smaller than the shifts calculated by other SC codes. A source of this discrepancy remains unclear. Again,

`GRASP/DARC`is rather close to the other SC codes. Similar to the width, the effect of quadrupoles is minor for all codes.

#### 5.2. Fractional Inelastic Width

`GRASP/DARC`values significantly lower than the others, especially at lower energies. In addition,

`SCP`shows a behavior which qualitatively differs; the inclusion of Feshbach resonances, which were not taken into account in these calculations, might modify this comment, since it would increase the contribution of the dipole interaction under the excitation threshold. We should also recall that the inelastic quadrupole part is zero for

`SCP`, while

`GRASP/DARC`cannot separate the dipole part. We note that inclusion of the quadrupole interaction also worsens agreement between SC codes in the case of elastic pseudo cross-section of Li $2s-2p$ (Figure 7).

#### 5.3. Cross-Sections

#### 5.3.1. Total Cross-Section

`GRASP/DARC`cannot separate dipole and quadrupole contributions. Therefore,

`GRASP/DARC`only appears on the right part of the figure. As for Li I,

`SCP`is zero under the threshold (6 eV for B III $2s-2p$), and the

`SCP`quadrupole part is zero. Its increasing behavior for low energies is explained by symmetrization and microreversibility.

#### 5.3.2. Inelastic Excitation Cross-Section

`GRASP/DARC`, are higher than the experimental ones by a factor varying between 1.5 to 2. The results of

`GRASP/DARC`reach the bottom of the error bar of the experimental result. This comparison shows that the origin of the puzzle between experimental and theoretical line widths is probably elsewhere.

#### 5.3.3. Partial Cumulative Excitation Cross-Section

#### 5.3.4. Elastic Pseudo Cross-Section

`SCP`behavior begins to increase with energy at relatively high energies, but remains small.

`STARCODE-P`results are the smallest, which is consistent with earlier results [8].

`GRASP/DARC`results are the highest. This may explain why the

`GRASP/DARC`widths [12] are not very different from those of approached methods. For comparison, fully quantum-mechanical results calculated by the convergent close coupling (

`CCC`) code [22], as appeared in Ref. [6], are shown. As noted in the previous SLSP workshops [5], there are significant discrepancies between

`CCC`and

`GRASP/DARC`. In fact,

`CCC`seems to agree well with

`STARCODE`with penetration at high energies and

`STARCODE`without account of penetration for the lowest energy.

## 6. Conclusions

## Author Contributions

`GRASP/DARC`—B.D.;

`SCP`—S.S.-B. and M.S.D.;

`SCPVB`—V.B.;

`STARCODE`—S.A.;

`SimU`—E.S. All authors participated in the discussions and in the preparation of the manuscript.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Li I $2s-2p$ FWHM as a function of the energy E. (

**Left**) the dipole interaction is only taken into account; (

**Right**) the dipole and quadrupole interaction are both taken into account.

**Figure 2.**Li I $2s-2p$ shift as a function of the energy E. (

**Left**) the dipole interaction is only taken into account; (

**Right**) the dipole and quadrupole interaction are both taken into account.

**Figure 3.**Li I $2s-2p$ fractional inelastic width as a function of energy E. (

**Left**) only the dipole interaction is taken into account; (

**Right**) the dipole and quadrupole interactions are both taken into account.

**Figure 4.**Li I $2s-2p$ total cross-section as a function of the energy E. (

**Left**) the dipole interaction is only taken into account; (

**Right**) the dipole and quadrupole interaction are both taken into account.

**Figure 5.**Li I $2s-2p$ excitation cross-section as a function of the energy E: comparison with experiment [20].

**Figure 6.**Li I $2s-2p$ cumulative partial excitation cross-section as a function of L, for an energy equal to 5 eV.

**Figure 7.**Li I $2s-2p$ elastic pseudo cross-section as a function of energy E. (

**Left**) only the dipole interaction is taken into account; (

**Right**) the dipole and quadrupole interaction are both taken into account.

**Figure 8.**B III $2s-2p$ FWHM as a function of the energy E. (

**Left**) only the dipole interaction is taken into account; (

**Right**) the dipole and quadrupole interaction are both taken into account.

**Figure 9.**B III $2s-2p$ shift as a function of the energy E. (

**Left**) only the dipole interaction is taken into account; (

**Right**) the dipole and quadrupole interaction are both taken into account.

**Figure 10.**B III $2s-2p$ fractional inelastic width as a function of energy E. (

**Left**) only the dipole interaction is taken into account; (

**Right**) the dipole and quadrupole interaction are both taken into account.

**Figure 11.**B III $2s-2p$ total cross-section as a function of the energy E. (

**Left**) only the dipole interaction is taken into account; (

**Right**) the dipole and quadrupole interaction are both taken into account.

**Figure 12.**B III $2s-2p$ cumulative excitation cross-section as a function of L, for an energy equal to 7 eV.

**Figure 13.**B III $2s-2p$ elastic pseudo cross-section as a function of the energy E. (

**Left**) only the dipole interaction is taken into account; (

**Right**) the dipole and quadrupole interaction are both taken into account.

**Table 1.**Atomic data used for the calculations: transition energies $\Delta E$, absorption oscillator strengths f, and the reduced quadupole matrix elements $\left(\right|Q\left|\right)$.

Species | Transition | $\mathbf{\Delta}\mathit{E}$ (${\mathbf{cm}}^{-1}$) | f | $\left(\right|\mathit{Q}\left|\right)$ |
---|---|---|---|---|

Li I | $2s-2p$ | 14,903.89 | 0.7472 | $-30.48$ |

B III | $2s-2p$ | 48,381.07 | 0.3629 | $-3.328$ |

**Table 2.**B III $2s-2p$ excitation cross-sections ${\sigma}_{\mathrm{ex}}$ for $E=7\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$.

Code | ${\mathit{\sigma}}_{\mathbf{ex}}$ (${10}^{-15}\phantom{\rule{0.166667em}{0ex}}{\mathbf{cm}}^{2}$) | Code/Experiment [21] Ratio |
---|---|---|

GRASP/DARC | 0.56 | 0.66 |

SCP | 1.19 | 1.40 |

STARCODE-NP | 1.57 | 1.84 |

STARCODE-P | 1.37 | 1.57 |

SimU | 1.90 | 2.20 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sahal-Bréchot, S.; Stambulchik, E.; Dimitrijević, M.S.; Alexiou, S.; Duan, B.; Bommier, V.
The Third and Fourth Workshops on Spectral Line Shapes in Plasma Code Comparison: Isolated Lines. *Atoms* **2018**, *6*, 30.
https://doi.org/10.3390/atoms6020030

**AMA Style**

Sahal-Bréchot S, Stambulchik E, Dimitrijević MS, Alexiou S, Duan B, Bommier V.
The Third and Fourth Workshops on Spectral Line Shapes in Plasma Code Comparison: Isolated Lines. *Atoms*. 2018; 6(2):30.
https://doi.org/10.3390/atoms6020030

**Chicago/Turabian Style**

Sahal-Bréchot, Sylvie, Evgeny Stambulchik, Milan S. Dimitrijević, Spiros Alexiou, Bin Duan, and Véronique Bommier.
2018. "The Third and Fourth Workshops on Spectral Line Shapes in Plasma Code Comparison: Isolated Lines" *Atoms* 6, no. 2: 30.
https://doi.org/10.3390/atoms6020030