# The Semiclassical Limit of the Gailitis Formula Applied to Electron Impact Broadening of Spectral Lines of Ionized Atoms

## Abstract

**:**

## 1. Introduction

## 2. Brief Recall: Stark Broadening Widths of Isolated Lines of Ionized Atoms in the Impact Approximation

## 3. The Gailitis Formula for the Feshbach Resonances which Increase the Elastic Cross-Section $\mathit{\sigma}(\mathit{k}\mathit{\alpha}\mathit{L}\leftrightarrow \mathit{k}\mathit{\alpha}\mathit{L})$

## 4. The Semiclassical Limit of the Galitis Formula for the Elastic Cross-Section Entering the Stark Width

- N.B. In addition, if results of the Gailitis resonances between J-levels would be needed, Equations (5), (7) and (8), must be modified. For that, an additional step, again using standard angular algebra calculations, has to be conducted. We only give the result there: the $\alpha $ and the L and ${L}^{\prime}$ which enter Equation (8) must be simply replaced by the $\gamma $, J and ${J}^{\prime}$.

## 5. Examples of Results

#### 5.1. Case between J-Levels: Example of the Two Fine Structure Components of P XIII 3s-3p

## 6. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Results of the SCP code for CII $3d-4f,\lambda =4267$ Å, electron collisions. Angular frequency units, density ${N}_{p}={10}^{18}$${\mathrm{cm}}^{-3}$, temperatures T in Kelvin.

T | $0.50\phantom{\rule{0.166667em}{0ex}}\times {10}^{4}$ | $0.1\phantom{\rule{0.166667em}{0ex}}\times {10}^{5}$ | $0.50\phantom{\rule{0.166667em}{0ex}}\times {10}^{5}$ |

Full width at half maximum | $0.225\phantom{\rule{0.166667em}{0ex}}\times {10}^{13}$ | $0.178\phantom{\rule{0.166667em}{0ex}}\times {10}^{13}$ | $0.122\phantom{\rule{0.166667em}{0ex}}\times {10}^{13}$ |

Inelastic collision contribution from the upper level | $0.124\phantom{\rule{0.166667em}{0ex}}\times {10}^{13}$ | $0.106\phantom{\rule{0.166667em}{0ex}}\times {10}^{13}$ | $0.841\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Inelastic collision contribution from the lower level | $0.176\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.137\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.114\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Feshbach resonances contribution from the upper level | $0.835\phantom{\rule{0.166667em}{0ex}}\times {10}^{9}$ | $0.481\phantom{\rule{0.166667em}{0ex}}\times {10}^{9}$ | $0.718\phantom{\rule{0.166667em}{0ex}}\times {10}^{8}$ |

Feshbach resonances contribution from the lower level | $0.108\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.733\phantom{\rule{0.166667em}{0ex}}\times {10}^{10}$ | $0.161\phantom{\rule{0.166667em}{0ex}}\times {10}^{9}$ |

Elastic collisions contribution (without resonances) | $0.829\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.581\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.261\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

**Table 2.**Results of the SCP code for CIV $2s-2p,\lambda =1549$ Å, electron collisions. Angular frequency units, density ${N}_{p}={10}^{18}$${\mathrm{cm}}^{-3}$, temperatures T in Kelvin.

T | $0.10\phantom{\rule{0.166667em}{0ex}}\times {10}^{6}$ | $0.2\phantom{\rule{0.166667em}{0ex}}\times {10}^{6}$ | $0.40\phantom{\rule{0.166667em}{0ex}}\times {10}^{6}$ |

Full width at half maximum | $0.396\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.291\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.221\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Inelastic collision contribution from the upper level | $0.465\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.382\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.346\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Inelastic collision contribution from the lower level | $0.547\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.663\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.668\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Feshbach resonances contribution from the upper level | $0.299\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.184\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.944\phantom{\rule{0.166667em}{0ex}}\times {10}^{10}$ |

Feshbach resonances contribution from the lower level | $0.823\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0367\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.149\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Elastic collisions contribution (without resonances) | $0.183\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.131\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.951\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

**Table 3.**Results of the SCP code for CIV $3s-3p,\lambda =5805$ Å, electron collisions. Angular frequency units, density ${N}_{p}={10}^{17}$${\mathrm{cm}}^{-3}$, temperatures T in Kelvin.

T | $0.50\phantom{\rule{0.166667em}{0ex}}\times {10}^{4}$ | $0.1\phantom{\rule{0.166667em}{0ex}}\times {10}^{5}$ | $0.30\phantom{\rule{0.166667em}{0ex}}\times {10}^{5}$ |

Full width at half maximum | $0.106\phantom{\rule{0.166667em}{0ex}}\times {10}^{13}$ | $0.70\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.417\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Inelastic collision contribution from the upper level | $0.133\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.15\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.132\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Inelastic collision contribution from the lower level | $0.20\phantom{\rule{0.166667em}{0ex}}\times {10}^{10}$ | $0.17\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.615\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Feshbach resonances contribution from the upper level | $0.187\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.922\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.27\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Feshbach resonances contribution from the lower level | $0.347\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.226\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.805\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Elastic collisions contribution (without resonances) | $0.395\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.214\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.116\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

**Table 4.**Results of the SCP code for Ar XV $2{s}^{2}\phantom{\rule{0.277778em}{0ex}}{}^{1}{S}_{0}-2s\phantom{\rule{1.0pt}{0ex}}3p\phantom{\rule{0.277778em}{0ex}}{}^{1}{P}_{1}^{0},\lambda =24.7$ Å, electron collisions. Angular frequency units, density ${N}_{p}={10}^{20}$${\mathrm{cm}}^{-3}$, temperatures T in Kelvin.

T | $0.50\phantom{\rule{0.166667em}{0ex}}\times {10}^{6}$ | $0.1\phantom{\rule{0.166667em}{0ex}}\times {10}^{7}$ | $0.2\phantom{\rule{0.166667em}{0ex}}\times {10}^{7}$ |

Full width at half maximum | $0.14\phantom{\rule{0.166667em}{0ex}}\times {10}^{14}$ | $0.101\phantom{\rule{0.166667em}{0ex}}\times {10}^{14}$ | $0.727\phantom{\rule{0.166667em}{0ex}}\times {10}^{13}$ |

Inelastic collision contribution from the upper level | $0.663\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.573\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.488\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Inelastic collision contribution from the lower level | $0.199\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.287\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.304\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Feshbach resonances contribution from the upper level | $0.173\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.779\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.332\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Feshbach resonances contribution from the lower level | $0.582\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.284\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.126\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Elastic collisions contribution (without resonances) | $0.132\phantom{\rule{0.166667em}{0ex}}\times {10}^{14}$ | $0.920\phantom{\rule{0.166667em}{0ex}}\times {10}^{13}$ | $0.648\phantom{\rule{0.166667em}{0ex}}\times {10}^{13}$ |

**Table 5.**Results of the SCP code for P XIII $3s\phantom{\rule{0.166667em}{0ex}}1/2-3p\phantom{\rule{0.166667em}{0ex}}1/2,\lambda =1741.3$ Å, electron collisions. Angular frequency units, density ${N}_{p}={10}^{18}$${\mathrm{cm}}^{-3}$, temperatures T in Kelvin.

T | $0.10\phantom{\rule{0.166667em}{0ex}}\times {10}^{6}$ | $0.5\phantom{\rule{0.166667em}{0ex}}\times {10}^{6}$ | $0.2\phantom{\rule{0.166667em}{0ex}}\times {10}^{7}$ |

Full width at half maximum | $0.511\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.243\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.140\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Inelastic collision contribution from the upper level | $0.716\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.462\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.318\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Inelastic collision contribution from the lower level | $0.436\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.462\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.337\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Feshbach resonances contribution from the upper level | $0.200\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.368\phantom{\rule{0.166667em}{0ex}}\times {10}^{10}$ | $0.768\phantom{\rule{0.166667em}{0ex}}\times {10}^{9}$ |

Feshbach resonances contribution from the lower level | $0.624\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.840\phantom{\rule{0.166667em}{0ex}}\times {10}^{10}$ | $0.127\phantom{\rule{0.166667em}{0ex}}\times {10}^{10}$ |

Elastic collisions contribution (without resonances) | $0.313\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.139\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.725\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

**Table 6.**Results of the SCP code for P XIII $3s\phantom{\rule{0.166667em}{0ex}}1/2-3p\phantom{\rule{0.166667em}{0ex}}3/2,\lambda =1645.8$ Å, electron collisions. Angular frequency units, density ${N}_{p}={10}^{18}$${\mathrm{cm}}^{-3}$, temperatures T in Kelvin.

T | $0.10\phantom{\rule{0.166667em}{0ex}}\times {10}^{6}$ | $0.5\phantom{\rule{0.166667em}{0ex}}\times {10}^{6}$ | $0.2\phantom{\rule{0.166667em}{0ex}}\times {10}^{7}$ |

Full width at half maximum | $0.511\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.244\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.141\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ |

Inelastic collision contribution from the upper level | $0.727\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.468\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.321\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Inelastic collision contribution from the lower level | $0.436\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.462\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.337\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

Feshbach resonances contribution from the upper level | $0.191\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.364\phantom{\rule{0.166667em}{0ex}}\times {10}^{10}$ | $0.774\phantom{\rule{0.166667em}{0ex}}\times {10}^{9}$ |

Feshbach resonances contribution from the lower level | $0.624\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ | $0.840\phantom{\rule{0.166667em}{0ex}}\times {10}^{10}$ | $0.127\phantom{\rule{0.166667em}{0ex}}\times {10}^{10}$ |

Elastic collisions contribution (without resonances) | $0.314\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.139\phantom{\rule{0.166667em}{0ex}}\times {10}^{12}$ | $0.727\phantom{\rule{0.166667em}{0ex}}\times {10}^{11}$ |

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**MDPI and ACS Style**

Sahal-Bréchot, S.
The Semiclassical Limit of the Gailitis Formula Applied to Electron Impact Broadening of Spectral Lines of Ionized Atoms. *Atoms* **2021**, *9*, 29.
https://doi.org/10.3390/atoms9020029

**AMA Style**

Sahal-Bréchot S.
The Semiclassical Limit of the Gailitis Formula Applied to Electron Impact Broadening of Spectral Lines of Ionized Atoms. *Atoms*. 2021; 9(2):29.
https://doi.org/10.3390/atoms9020029

**Chicago/Turabian Style**

Sahal-Bréchot, Sylvie.
2021. "The Semiclassical Limit of the Gailitis Formula Applied to Electron Impact Broadening of Spectral Lines of Ionized Atoms" *Atoms* 9, no. 2: 29.
https://doi.org/10.3390/atoms9020029