# Relativistic B-Spline R-Matrix Calculations for Electron Collisions with Ytterbium

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## Abstract

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## 1. Introduction

## 2. Numerical Method

^{+}in the basis configurations $4{\mathrm{f}}^{14}6\mathrm{s}$, $4{\mathrm{f}}^{14}6{\mathrm{p}}_{1/2,3/2}$, and $4{\mathrm{f}}^{14}5{\mathrm{d}}_{3/2,5/2}$. With these orbitals in hand, we carried out a bound-state close-coupling calculation, i.e., we looked for bound-state solutions of the $\mathrm{e}-{\mathrm{Yb}}^{+}$ scattering problem [13]. This method has the advantage that it enables us to systematically increase the number of states ultimately included in the close-coupling collision calculation for $\mathrm{e}-\mathrm{Yb}$ scattering. Specifically, so-called pseudo-states can easily be generated by forcing the orbitals to vanish at the boundary of the R-matrix box. These pseudo-states represent a finite discretization of the infinite number of high-lying Rydberg states as well as the target continuum. Excitation cross sections for the pseudo-states then provide estimates for excitation of the Rydberg series as well as the ionization cross section. In order to further improve the structure description, a few “perturbers”, i.e., doubly-excited states, were added to the configurations ultimately used for the neutral Yb atom.

## 3. Results and Discussion

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Sample Availability

## Abbreviations

BSR | B-Spline R-matrix |

DBSR | Dirac B-Spline R-matrix |

BPRM | Breit-Pauli R-matrix |

## References

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**Figure 1.**Cross section for elastic e-Yb scattering obtained in the DBSR-20, BPRM-10, and BPRM-28 models. The experimental data are from Predojević et al. [6].

**Figure 2.**Cross section for electron impact excitation of the ${\left(4{\mathrm{f}}^{14}6{\mathrm{s}}^{2}\right)}^{1}{\mathrm{S}}_{0}\to {\left(4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{p}\right)}^{3}{\mathrm{P}}_{1}$ transition in Yb. The DBSR-20 results are compared with the BPRM-10 and BPRM-28 predictions and the experimental data of Predojević et al. [7].

**Figure 3.**Cross section for electron impact excitation of the ${\left(4{\mathrm{f}}^{14}6{\mathrm{s}}^{2}\right)}^{1}{\mathrm{S}}_{0}\to {\left(4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{p}\right)}^{3}{\mathrm{P}}_{2}$ (solid lines) and ${\left(4{\mathrm{f}}^{14}6{\mathrm{s}}^{2}\right)}^{1}{\mathrm{S}}_{0}\to {\left(4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{p}\right)}^{3}{\mathrm{P}}_{0}$ (dashed lines) transitions in Yb. The DBSR-20 results are compared with the BPRM-10 and BPRM-28 predictions.

**Figure 4.**Cross section for electron impact excitation of the ${\left(4{\mathrm{f}}^{14}6{\mathrm{s}}^{2}\right)}^{1}{\mathrm{S}}_{0}\to {\left(4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{p}\right)}^{1}{\mathrm{P}}_{1}$ transition in Yb. The DBSR-20 results are compared with the BPRM-10 and BPRM-28 predictions and the experimental data of Predojević et al. [6].

**Figure 5.**Cross section for electron impact excitation of the ${\left(4{\mathrm{f}}^{14}5{\mathrm{p}}^{6}6{\mathrm{s}}^{2}\right)}^{1}{\mathrm{S}}_{0}\to {\left(4{\mathrm{f}}^{13}5{\mathrm{p}}^{6}5\mathrm{d}6{\mathrm{s}}^{2}\right)}^{1}{\mathrm{P}}_{1}$ transition in Yb. The DBSR-20 results are compared with the experimental data of Predojević et al. [7]. The BPRM models did not include this state.

**Figure 6.**Cross section for electron impact excitation of the ${\left(4{\mathrm{f}}^{14}6{\mathrm{s}}^{2}\right)}^{1}{\mathrm{S}}_{0}\to {\left(4{\mathrm{f}}^{14}5\mathrm{d}6\mathrm{s}\right)}^{1}{\mathrm{D}}_{2}$ transition in Yb. The DBSR-20 results are compared BPRM-28 predictions (BPRM-10 did not include this state) and the experimental data of Predojević et al. [7].

**Figure 7.**DBSR-20 predictions of cross sections for electron-induced transitions between the states indicated in the legend.

Configuration | State | DBSR-20 | NIST | Difference |
---|---|---|---|---|

$4{\mathrm{f}}^{14}6{\mathrm{s}}^{2}$ | ${}^{1}{\mathrm{S}}_{0}$ | 0.00000 | 0.00000 | 0.00000 |

$4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{p}$ | ${}^{3}{\mathrm{P}}_{0}^{\mathrm{o}}$ | 2.00952 | 2.14349 | 0.13397 |

$4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{p}$ | ${}^{3}{\mathrm{P}}_{1}^{\mathrm{o}}$ | 2.08836 | 2.23072 | 0.14236 |

$4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{p}$ | ${}^{3}{\mathrm{P}}_{2}^{\mathrm{o}}$ | 2.27080 | 2.44378 | 0.17298 |

$4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{p}$ | ${}^{1}{\mathrm{P}}_{1}^{\mathrm{o}}$ | 2.94029 | 3.10806 | 0.16777 |

$4{\mathrm{f}}^{14}5\mathrm{d}6\mathrm{s}$ | ${}^{3}{\mathrm{D}}_{1}$ | 3.16487 | 3.03626 | −0.12861 |

$4{\mathrm{f}}^{14}5\mathrm{d}6\mathrm{s}$ | ${}^{3}{\mathrm{D}}_{2}$ | 3.18188 | 3.06889 | −0.11299 |

$4{\mathrm{f}}^{14}5\mathrm{d}6\mathrm{s}$ | ${}^{3}{\mathrm{D}}_{3}$ | 3.20878 | 3.13319 | −0.07559 |

$4{\mathrm{f}}^{13}5\mathrm{d}6{\mathrm{s}}^{2}$ | ${}^{1}{\mathrm{P}}_{1}^{\mathrm{o}}$ | 3.36630 | 3.57781 | 0.21151 |

$4{\mathrm{f}}^{14}5\mathrm{d}6\mathrm{s}$ | ${}^{1}{\mathrm{D}}_{2}$ | 3.38142 | 3.43159 | 0.05017 |

$4{\mathrm{f}}^{14}6\mathrm{s}7\mathrm{s}$ | ${}^{3}{\mathrm{S}}_{1}$ | 3.92405 | 4.05363 | 0.12958 |

$4{\mathrm{f}}^{14}6\mathrm{s}7\mathrm{s}$ | ${}^{1}{\mathrm{S}}_{0}$ | 4.14867 | 4.25894 | 0.11027 |

$4{\mathrm{f}}^{14}6\mathrm{s}7\mathrm{p}$ | ${}^{3}{\mathrm{P}}_{0}^{\mathrm{o}}$ | 4.56381 | 4.72265 | 0.15884 |

$4{\mathrm{f}}^{14}6\mathrm{s}7\mathrm{p}$ | ${}^{3}{\mathrm{P}}_{1}^{\mathrm{o}}$ | 4.58505 | 4.73299 | 0.14794 |

$4{\mathrm{f}}^{14}6\mathrm{s}7\mathrm{p}$ | ${}^{3}{\mathrm{P}}_{2}^{\mathrm{o}}$ | 4.61358 | 4.77983 | 0.16625 |

$4{\mathrm{f}}^{14}6\mathrm{s}7\mathrm{p}$ | ${}^{1}{\mathrm{P}}_{1}^{\mathrm{o}}$ | 4.62774 | 5.02929 | 0.40155 |

$4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{d}$ | ${}^{3}{\mathrm{D}}_{1}$ | 4.77436 | 4.93565 | 0.16129 |

$4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{d}$ | ${}^{3}{\mathrm{D}}_{2}$ | 4.80083 | 4.93929 | 0.13846 |

$4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{d}$ | ${}^{3}{\mathrm{D}}_{3}$ | 4.80516 | 4.95516 | 0.15000 |

$4{\mathrm{f}}^{14}6\mathrm{s}6\mathrm{d}$ | ${}^{1}{\mathrm{D}}_{2}$ | 4.83792 | 4.96699 | 0.12907 |

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**MDPI and ACS Style**

Hamilton, K.R.; Bartschat, K.; Zatsarinny, O. Relativistic B-Spline R-Matrix Calculations for Electron Collisions with Ytterbium. *Atoms* **2021**, *9*, 47.
https://doi.org/10.3390/atoms9030047

**AMA Style**

Hamilton KR, Bartschat K, Zatsarinny O. Relativistic B-Spline R-Matrix Calculations for Electron Collisions with Ytterbium. *Atoms*. 2021; 9(3):47.
https://doi.org/10.3390/atoms9030047

**Chicago/Turabian Style**

Hamilton, Kathryn R., Klaus Bartschat, and Oleg Zatsarinny. 2021. "Relativistic B-Spline R-Matrix Calculations for Electron Collisions with Ytterbium" *Atoms* 9, no. 3: 47.
https://doi.org/10.3390/atoms9030047