# Inelastic Processes in Strontium-Hydrogen Collisions and Their Impact on Non-LTE Calculations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Atomic Data Calculations

#### 2.1. Processes in Sr${}^{+}$ + H and Sr${}^{2+}$ + H${}^{-}$ Collisions

#### 2.2. Processes in Sr + H and Sr${}^{+}$ + H${}^{-}$ Collisions

## 3. Non-LTE Calculations for Sr ii

#### 3.1. Model Atom of Sr ii and Method of Calculations

#### 3.2. Non-LTE Effects for Lines of Sr ii

- Grid I:
- ${T}_{\mathrm{eff}}$ = 5000 to 6500 K, with a step of 250 K; $\mathrm{log}\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}$ = 3.0 to 5.0, with a step of 0.5; [Fe/H] = $-2.0$, $-2.5$, $-3.0$, and $-4.0$; [Sr/Fe] = $-1.0$ to +1.0, with a step of 0.5.
- Grid II:
- ${T}_{\mathrm{eff}}$ = 4000 to 5000 K, with a step of 250 K; $\mathrm{log}\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}$ = 0.5 to 2.5, with a step of 0.5; [Fe/H] = $-2.0$, $-2.5$, $-3.0$, and $-4.0$; [Sr/Fe] = $-1.5$ to +0.5, with a step of 0.5.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LTE | Local thermodynamic equilibrium |

NLTE | Non-local thermodynamic equilibrium |

SN | Supernova |

dSph galaxy | dwarf spheroidal galaxy |

## Appendix A

**Table A1.**SrH${}^{+}$ (k 0${}^{+}$) molecular states (in the J-J representation), the corresponding scattering channels, their asymptotic energies with respect to the ground-state level (taken from NIST [31]).

k | Scattering Channels | Asymptotic |
---|---|---|

Energies (eV) | ||

1 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}5s\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{S}}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 0.0 |

2 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}4d\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{D}}_{{}^{3}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 1.8047016 |

3 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}4d\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{D}}_{{}^{5}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 1.8394593 |

4 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}5p\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{P}}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 2.9403088 |

5 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}5p\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{P}}_{{}^{3}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 3.0396772 |

6 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}6s\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{S}}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 5.9185754 |

7 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}5d\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{D}}_{{}^{3}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 6.6066604 |

8 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}5d\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{D}}_{{}^{5}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 6.6174048 |

9 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}6p\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{P}}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 6.91456 |

10 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}6p\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{P}}_{{}^{3}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 6.95029 |

11 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}4f\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{F}}_{{}^{7}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 7.561801 |

12 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}4f\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{F}}_{{}^{5}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 7.561962 |

13 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}7s\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{S}}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 8.0545218 |

14 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}6d\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{D}}_{{}^{3}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 8.3717688 |

15 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}6d\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{D}}_{{}^{5}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 8.3767629 |

16 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}7p\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{P}}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 8.515153 |

17 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}7p\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{P}}_{{}^{3}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 8.532235 |

18 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}5f\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{F}}_{{}^{5}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 8.811036 |

19 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}5f\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{F}}_{{}^{7}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 8.811036 |

20 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}5g\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{G}}_{{}^{7}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 8.847240 |

21 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}5g\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{G}}_{{}^{9}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 8.847240 |

22 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}8s\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{S}}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 9.080243 |

23 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}7d\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{D}}_{{}^{3}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 9.251862 |

24 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}7d\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{D}}_{{}^{5}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 9.254565 |

25 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}8p\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{P}}_{{}^{3}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 9.337473 |

26 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}6f\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{F}}_{{}^{7}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 9.491412 |

27 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}6f\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{F}}_{{}^{5}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}^{\circ}$ | 9.491412 |

28 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}6g\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{G}}_{{}^{7}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 9.514262 |

29 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}6g\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{G}}_{{}^{9}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 9.514262 |

30 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}9s\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{S}}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}{/}_{2}}$ | 9.653112 |

31 | ${\mathrm{Sr}}^{2+}\left(\right)open="("\; close=")">4{p}^{6}\phantom{\rule{3.33333pt}{0ex}}{}^{1}{\mathrm{S}}_{0}$ | 10.2762764 |

**Table A2.**SrH (k ${}^{2}{\Sigma}^{+}$) molecular states, the corresponding scattering channels, their asymptotic energies with respect to the ground-state level (taken from NIST [31]).

k | Scattering Channels | Asymptotic |
---|---|---|

Energies (eV) | ||

1 | $\mathrm{Sr}\left(\right)open="("\; close=")">5{s}^{2}\phantom{\rule{3.33333pt}{0ex}}{}^{1}\mathrm{S}$ | 0.0 |

2 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s5p\phantom{\rule{3.33333pt}{0ex}}{}^{3}{\mathrm{P}}^{\circ}$ | 1.8228877 |

3 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s4d\phantom{\rule{3.33333pt}{0ex}}{}^{3}\mathrm{D}$ | 2.2631734 |

4 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s4d\phantom{\rule{3.33333pt}{0ex}}{}^{1}\mathrm{D}$ | 2.4982425 |

5 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s5p\phantom{\rule{3.33333pt}{0ex}}{}^{1}{\mathrm{P}}^{\circ}$ | 2.6902652 |

6 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s6s\phantom{\rule{3.33333pt}{0ex}}{}^{3}\mathrm{S}$ | 3.600349 |

7 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s6s\phantom{\rule{3.33333pt}{0ex}}{}^{1}\mathrm{S}$ | 3.7929029 |

8 | $\mathrm{Sr}\left(\right)open="("\; close=")">4d5p\phantom{\rule{3.33333pt}{0ex}}{}^{3}{\mathrm{F}}^{\circ}$ | 4.1725763 |

9 | $\mathrm{Sr}\left(\right)open="("\; close=")">4d5p\phantom{\rule{3.33333pt}{0ex}}{}^{1}{\mathrm{D}}^{\circ}$ | 4.1940009 |

10 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s6p\phantom{\rule{3.33333pt}{0ex}}{}^{3}{\mathrm{P}}^{\circ}$ | 4.2061469 |

11 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s6p\phantom{\rule{3.33333pt}{0ex}}{}^{1}{\mathrm{P}}^{\circ}$ | 4.2276633 |

12 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s5d\phantom{\rule{3.33333pt}{0ex}}{}^{1}\mathrm{D}$ | 4.3056546 |

13 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s5d\phantom{\rule{3.33333pt}{0ex}}{}^{3}\mathrm{D}$ | 4.3431318 |

14 | $\mathrm{Sr}\left(\right)open="("\; close=")">5{p}^{2}\phantom{\rule{3.33333pt}{0ex}}{}^{3}\mathrm{P}$ | 4.4051164 |

15 | $\mathrm{Sr}\left(\right)open="("\; close=")">4d5p\phantom{\rule{3.33333pt}{0ex}}{}^{3}{\mathrm{D}}^{\circ}$ | 4.5181299 |

16 | $\mathrm{Sr}\left(\right)open="("\; close=")">5s7s\phantom{\rule{3.33333pt}{0ex}}{}^{3}\mathrm{S}$ | 4.6400683 |

17 | ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">5s\phantom{\rule{3.33333pt}{0ex}}{}^{2}\mathrm{S}$ | 4.9408674 |

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## Notes

1 | http://kurucz.harvard.edu/atoms/3801/ (accessed on 1 October 1996). |

2 | http://marcs.astro.uu.se (accessed on 1 August 2019). |

3 | https://physics.nist.gov/PhysRefData/ASD (accessed on 1 October 1996). |

4 | http://www.inasan.ru/~lima/ (accessed on 21 August 2021). |

**Figure 1.**The graphical representation for the inelastic processes rate coefficients in ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">4{p}^{6}nl\phantom{\rule{3.33333pt}{0ex}}{}^{2}{\mathrm{L}}_{\mathrm{j}}$ and ${\mathrm{Sr}}^{2+}\left(\right)open="("\; close=")">4{p}^{6}\phantom{\rule{3.33333pt}{0ex}}{}^{1}{\mathrm{S}}_{0}$ collisions at the temperature T = 6000 K. The labels for the initial and final states are given in Table A1.

**Figure 2.**The rate coefficients (in cm${}^{3}$/s) at T = 6000 K for the neutralization processes in ${\mathrm{Sr}}^{2+}\left(\right)open="("\; close=")">4{p}^{6}\phantom{\rule{3.33333pt}{0ex}}{}^{1}{\mathrm{S}}_{0}$ collisions. Empty diamonds correspond to the calculations with the account for the fine structure; filled circles, to the calculations without fine structure; and dashed line shows the general dependence of the rate coefficient according to the simplified quantum model [30].

**Figure 3.**The graphical representation for the inelastic processes rate coefficients in $\mathrm{Sr}\left(\right)open="("\; close=")">{}^{1,3}\mathrm{L}$ and ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">5s\phantom{\rule{3.33333pt}{0ex}}{}^{2}\mathrm{S}$ collisions. The labels for the initial and final states are given in Table A2.

**Figure 4.**The rate coefficients (in cm${}^{3}$/s) at T = 6000 K for the mutual neutralization processes in ${\mathrm{Sr}}^{+}\left(\right)open="("\; close=")">5s\phantom{\rule{3.33333pt}{0ex}}{}^{2}\mathrm{S}$ collisions. Filled circles correspond to the processes due to one-electron transitions; stars, to two-electron transitions; dashed line shows the general dependence of the rate coefficient according to the simplified quantum model [32].

**Figure 5.**Non-LTE abundance corrections for Sr ii 4077 Å in the model atmospheres of giants with common ${T}_{\mathrm{eff}}$ = 4500 K and varied $\mathrm{log}\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}$ and [Sr/Fe] (two left panels) and dwarfs with common ${T}_{\mathrm{eff}}$ = 5250 K and varied $\mathrm{log}\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}$ and [Sr/Fe] (two right panels) as a function of metallicity.

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Yakovleva, S.A.; Belyaev, A.K.; Mashonkina, L.I.
Inelastic Processes in Strontium-Hydrogen Collisions and Their Impact on Non-LTE Calculations. *Atoms* **2022**, *10*, 33.
https://doi.org/10.3390/atoms10010033

**AMA Style**

Yakovleva SA, Belyaev AK, Mashonkina LI.
Inelastic Processes in Strontium-Hydrogen Collisions and Their Impact on Non-LTE Calculations. *Atoms*. 2022; 10(1):33.
https://doi.org/10.3390/atoms10010033

**Chicago/Turabian Style**

Yakovleva, Svetlana A., Andrey K. Belyaev, and Lyudmila I. Mashonkina.
2022. "Inelastic Processes in Strontium-Hydrogen Collisions and Their Impact on Non-LTE Calculations" *Atoms* 10, no. 1: 33.
https://doi.org/10.3390/atoms10010033