# Photoionization and Electron-Ion Recombination of n = 1 to Very High n-Values of Hydrogenic Ions

## Abstract

**:**

## 1. Introduction

#### 1.1. Existing Values for Photoionization Cross Sections

#### 1.2. Existing Values for Electron-Ion Recombination of Hydrogenic Ions

## 2. Theory of Photoionization and Electron-Ion Recombination of Hydrogenic Ions

#### 2.1. Photoionization of Hydrogenic Ions

#### 2.1.1. Recurrence Relations for the g Bound-Free Transition Integral

#### 2.1.2. Ground State Photoionization Cross Sections of Hydrogen

#### 2.2. Electron-Ion Recombination of Hydrogenic Ions

#### 2.2.1. Top-Up Contribution from Very High-n Recombination

#### 2.2.2. Recombination with Respect to Photoelectron Energy

#### 2.3. Relation between Hydrogen and Hydrogenic Ions

#### 2.4. Relativistic Fine Structure Splitting of ${\sigma}_{PI}$ and ${\sigma}_{RC}$ from LS Coupling

## 3. Programs for Photoionization and Electron-Ion Recombination, and the Data Files

## 4. Results and Discussions

#### 4.1. Photoionization Cross Sections

#### 4.2. Electron-Ion Recombination

## 5. Conclusions

- Study of the two inverse processes of photoionization and electron-ion recombination are presented with a brief review of theory that treat them precisely and more accurately compared to all existing atomic structure codes that use other approximation, mainly distorted wave approximation.
- Detailed features of both the processes are illustrated. Although hydrogen and hydrogenic ions do not have any resonant features, accurate characteristic variation with energy and temperature are crucial for precise astrophysical spectroscopy and modeling.
- The present work provides atomic data files containing ${\sigma}_{PI}\left(E\right)$ for all l-levels of n from 1 to 800, and ${\alpha}_{RC}\left(T\right)$ of all values of n from 1 to 800. It also provides total ${\alpha}_{RC}(H,T)$ with temperature, and total ${\alpha}_{RC}(H,E)$, and total ${\sigma}_{RC}\left(E\right)$ with photoelectron energy. This is the first time that all these data with very high n have been made available for applications.
- Use of precise theory, numerical methods, high precision computers, as explained at the end of the sections of photoionization, and electron-ion recombination, can predict the accuracy of the present results is within 5% for most energy and temperature ranges.
- The work provides the FORTRAN program “hpxrrc.f” which can generate all these values. It also computes l-level specific ${\alpha}_{RC}(nl,T)$. The program “hpxrrc.f” can also compute ${\alpha}_{RC}(H,T)$ for any hydrogenic ion of charge Z using the data of hydrogen.
- Importance of relativistic effects and how to obtain fine structure components for photoionization and electron-ion from their values in the present LS coupling approximation have been discussed.
- All atomic data and the program will be available online at database, NORAD-Atomic-Data ([24], http://norad.astronomy.ohio-state.edu (accessed on 1 September 2007)).

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Photoionization cross sections (${\sigma}_{PI}$) of (L) ns and (R) nd levels of hydrogen demonstrating rising trend at threshold and decreasing trend of the background at higher energy with increase of n.

**Figure 2.**Photoionization cross sections (${\sigma}_{PI}$) of the five l-values (0–4) of shell n = 5. They demonstrate rising trend of ${\sigma}_{PI}$ for various l-values compared to l = 0 (black curve) at the same ionization threshold.

**Figure 3.**Level specific electron-ion recombination rate coefficients (${\alpha}_{RC}\left(nl\right)$) of the 10 individual l-levels (0–9) of shell n = 10, and the n-total (red curve with highest values) which is the summed total of all 10 individual levels. They demonstrate ${\alpha}_{RC}\left(nl\right)$ peak does not follow values of l-order at low temperature bu follows at higher temperature.

**Figure 4.**Total electron-ion recombination rate coefficients (${\alpha}_{RC}(n,T)$) of shells n = 1–10, which are the summed total of all individual l-levels,

**Left**) in linear scale to see the general trend at low temperature and

**Right**) in log scale for logarithmic features. The curves demonstrate ${\alpha}_{RC}(n,T)$ decreasing with increasing n.

**Figure 5.**(

**a**) Total electron-ion recombination rate coefficients ${\alpha}_{RC}(H,T)$, summed contributions from n = 1 to infinity over a wide temperature range. These values are used to obtain (

**b**) ${\alpha}_{RC}(FeXXVI,T)$ of Fe XXVI using the Z-scale formula.

**Figure 6.**The total recombination cross sections, ${\sigma}_{RC}\left(E\right)$, and recombination rate coefficients ${\alpha}_{RC}\left(E\right)$ of hydrogen with respect to the photoelectron energy E.

**Table 1.**Total recombination rate coefficients of hydrogen, ${\alpha}_{RC}(H,T)$, at a wide range of temperatures.

log${}_{10}$T | ${\mathit{\alpha}}_{\mathbf{RC}}(\mathit{H},\mathit{T})$ | log${}_{10}$T | ${\mathit{\alpha}}_{\mathbf{RC}}(\mathit{H},\mathit{T})$ | log${}_{10}$T | ${\mathit{\alpha}}_{\mathbf{RC}}(\mathit{H},\mathit{T})$ |
---|---|---|---|---|---|

−3.00 | 4.779 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 0.73 | 4.974 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.60 | 1.479 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ |

−2.67 | 3.254 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 0.75 | 4.842 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.70 | 1.232 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ |

−2.33 | 2.214 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 0.77 | 4.714 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.80 | 1.022 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ |

−2.00 | 1.504 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 0.79 | 4.588 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.90 | 8.447 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

−1.67 | 1.020 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 0.81 | 4.466 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.00 | 6.949 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

−1.33 | 6.877 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 0.83 | 4.347 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.10 | 5.691 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

−1.00 | 4.606 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 0.85 | 4.231 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.20 | 4.637 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

−0.67 | 3.057 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 0.87 | 4.118 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.30 | 3.755 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

−0.33 | 2.009 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 0.89 | 4.008 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.40 | 3.027 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

0.00 | 1.308 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 0.91 | 3.901 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.50 | 2.424 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

0.01 | 1.291 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.00 | 3.452 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.60 | 1.930 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

0.03 | 1.258 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.10 | 3.012 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.70 | 1.528 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

0.05 | 1.226 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.20 | 2.627 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.80 | 1.202 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

0.07 | 1.194 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.30 | 2.290 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.90 | 9.396 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

0.09 | 1.163 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.40 | 1.995 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 6.00 | 7.303 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

0.11 | 1.133 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.50 | 1.737 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 6.10 | 5.643 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

0.13 | 1.104 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.60 | 1.511 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 6.20 | 4.333 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

0.15 | 1.075 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.70 | 1.314 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 6.30 | 3.311 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

0.17 | 1.048 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.80 | 1.142 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 6.40 | 2.516 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

0.19 | 1.020 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.90 | 9.922 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 6.50 | 1.902 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

0.21 | 9.939 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.00 | 8.613 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 6.60 | 1.431 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

0.23 | 9.680 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.10 | 7.471 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 6.70 | 1.071 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

0.25 | 9.429 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.20 | 6.476 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 6.80 | 7.989 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ |

0.27 | 9.183 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.30 | 5.609 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 6.90 | 5.932 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ |

0.29 | 8.944 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.40 | 4.855 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.00 | 4.389 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ |

0.31 | 8.711 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.50 | 4.199 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.10 | 3.232 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ |

0.33 | 8.484 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.60 | 3.628 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.20 | 2.374 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ |

0.35 | 8.262 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.70 | 3.132 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.30 | 1.739 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ |

0.37 | 8.046 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.80 | 2.702 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.40 | 1.270 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ |

0.39 | 7.835 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.90 | 2.328 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.50 | 9.251 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ |

0.41 | 7.630 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.00 | 2.004 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.60 | 6.718 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ |

0.43 | 7.430 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.10 | 1.723 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.70 | 4.866 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ |

0.45 | 7.235 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.20 | 1.480 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.80 | 3.518 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ |

0.47 | 7.045 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.30 | 1.269 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 7.90 | 2.538 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ |

0.49 | 6.860 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.40 | 1.087 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 8.00 | 1.828 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ |

0.51 | 6.680 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.50 | 9.304 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.10 | 1.315 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ |

0.53 | 6.504 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.60 | 7.949 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.20 | 9.440 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-18}$ |

0.55 | 6.332 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.70 | 6.781 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.30 | 6.769 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-18}$ |

0.57 | 6.165 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.80 | 5.774 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.40 | 4.848 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-18}$ |

0.59 | 6.003 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.90 | 4.909 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.50 | 3.468 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-18}$ |

0.61 | 5.844 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.00 | 4.165 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.60 | 2.478 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-18}$ |

0.61 | 5.844 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.00 | 4.165 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.60 | 2.478 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-18}$ |

0.63 | 5.690 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.10 | 3.527 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.70 | 1.769 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-18}$ |

0.65 | 5.539 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.20 | 2.980 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.80 | 1.262 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-18}$ |

0.67 | 5.392 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.30 | 2.511 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 8.90 | 8.997 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-19}$ |

0.69 | 5.249 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.40 | 2.111 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 9.00 | 6.409 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-19}$ |

0.71 | 5.110 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.50 | 1.770 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ |

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Nahar, S.N.
Photoionization and Electron-Ion Recombination of *n* = 1 to Very High *n*-Values of Hydrogenic Ions. *Atoms* **2021**, *9*, 73.
https://doi.org/10.3390/atoms9040073

**AMA Style**

Nahar SN.
Photoionization and Electron-Ion Recombination of *n* = 1 to Very High *n*-Values of Hydrogenic Ions. *Atoms*. 2021; 9(4):73.
https://doi.org/10.3390/atoms9040073

**Chicago/Turabian Style**

Nahar, Sultana N.
2021. "Photoionization and Electron-Ion Recombination of *n* = 1 to Very High *n*-Values of Hydrogenic Ions" *Atoms* 9, no. 4: 73.
https://doi.org/10.3390/atoms9040073