# Lowest-Order Thermal Correction to the Hydrogen Recombination Cross-Section in Presence of Blackbody Radiation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermal Vertex Correction to the Recombination Process

## 3. Recombination and Ionization Coefficients

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Feynman diagrams representing the thermal correction to the thermal interaction potential. A wavy line ($\gamma $) indicates the photon emission process; and a dashed line (${\gamma}_{T}$) corresponds to the thermal Coulomb photon exchange of a bound electron with a nucleus. The double-solid line denotes the bound electron in the nucleus field (the Furry picture). Notations i and f represent the initial and final states of a bound electron, respectively, and m corresponds to the intermediate state represented in the electron propagator. Subfigures (

**a**,

**b**) represent the accompanying Feynman graphs and, as usual, differ from each other in the order of time for the emission and interaction vertices.

**Figure 2.**Integration contour ${C}_{1}$ in ${k}_{0}$ plane. Arrows on the contour define the pole-bypass rule. The poles $\pm {\omega}_{k}$ are denoted with × marks.

**Table 1.**Numerical values of energy shifts $\mathsf{\Delta}{E}_{A}^{\beta}=\langle A|{V}^{\beta}\left(r\right)|A\rangle $ for different atomic states A at temperatures $T=300$ K (upper line) and $T=3000$ K (lower line) in a hydrogen atom. The first column shows the considered state $({n}_{A},{l}_{A})$. In the second column, the energy shift is calculated with the approximate potential ${V}^{\beta}\left(r\right)$ given by Equations (38) and (52) in [6]. In the third column, the energy shift is calculated with the potential ${V}^{\beta}\left(r\right)$ given by Equation (51) in [6]. All values are in Hz.

$({\mathit{n}}_{\mathit{A}},{\mathit{l}}_{\mathit{A}})$ | $\mathsf{\Delta}{\mathit{E}}_{{\mathit{n}}_{\mathit{A}}{\mathit{l}}_{\mathit{A}}}^{\mathit{\beta}}$, Equation (38) | $\mathsf{\Delta}{\mathit{E}}_{{\mathit{n}}_{\mathit{A}}{\mathit{l}}_{\mathit{A}}}^{\mathit{\beta}}$, Equation (51) |
---|---|---|

(1,0) | $-3.36$ | $-3.36$ |

$-3.36\times {10}^{3}$ | $-3.36\times {10}^{3}$ | |

(2,0) | $-46.98$ | $-46.98$ |

$-4.698\times {10}^{4}$ | $-4.698\times {10}^{4}$ | |

(10,0) | $-2.80\times {10}^{4}$ | $-2.80\times {10}^{4}$ |

$-2.80\times {10}^{7}$ | $-2.80\times {10}^{7}$ | |

(10,9) | $-1.29\times {10}^{4}$ | $-1.29\times {10}^{4}$ |

$-1.29\times {10}^{7}$ | $-1.29\times {10}^{7}$ | |

(20,0) | $-4.48\times {10}^{5}$ | $-4.48\times {10}^{5}$ |

$-4.48\times {10}^{8}$ | $-4.47\times {10}^{8}$ | |

(20,19) | $-1.93\times {10}^{5}$ | $-1.93\times {10}^{5}$ |

$-1.93\times {10}^{8}$ | $-1.93\times {10}^{8}$ | |

(100,0) | $-2.80\times {10}^{8}$ | $-2.78\times {10}^{8}$ |

$-2.80\times {10}^{11}$ | $-2.78\times {10}^{11}$ | |

(100,99) | $-1.14\times {10}^{8}$ | $-1.13\times {10}^{8}$ |

$-1.14\times {10}^{11}$ | $-9.171\times {10}^{10}$ | |

(200,0) | $-4.47\times {10}^{9}$ | $-4\times {10}^{9}$ |

$-4.47\times {10}^{12}$ | $-3.72\times {10}^{11}$ | |

(200,99) | $-1.80\times {10}^{9}$ | $-1.73\times {10}^{9}$ |

$-1.80\times {10}^{12}$ | $-5.06\times {10}^{11}$ |

**Table 2.**Recalculated transition rates and thermal corrections at $T=300$ K to one-photon electric dipole transitions between highly excited states due to the thermal energy shift, see Equations (53) and (54) in [6] for details. All values are given in s${}^{-1}$.

${\mathit{n}}_{\mathit{i}},{\mathit{l}}_{\mathit{i}}$ | ${\mathit{n}}_{\mathit{f}},{\mathit{l}}_{\mathit{f}}$ | ${\mathit{W}}_{\mathbf{if}}$ | $\mathsf{\Delta}{\mathit{W}}_{\mathbf{if}}^{\mathbf{ind}}$ | $\mathsf{\Delta}{\mathit{W}}_{\mathbf{if}}^{\mathbf{v}}$ | $\mathsf{\Delta}{\mathit{W}}_{\mathbf{if}}^{\mathbf{v},\mathbf{ind}}$ |
---|---|---|---|---|---|

$(10,9)$ | $(9,8)$ | $1.320\times {10}^{4}$ | $5.419\times {10}^{3}$ | $2.213\times {10}^{-5}$ | $2.811\times {10}^{-6}$ |

$(50,1)$ | $(49,0)$ | $2.682$ | $3.077\times {10}^{2}$ | $1.998\times {10}^{-4}$ | $1.524\times {10}^{-2}$ |

$(50,49)$ | $(49,48)$ | $7.137\times {10}^{-1}$ | $81.861$ | $2.190\times {10}^{-5}$ | $1.671\times {10}^{-3}$ |

$(70,1)$ | $(69,0)$ | $4.840\times {10}^{-1}$ | $1.541\times {10}^{2}$ | $2.759\times {10}^{-4}$ | $5.852\times {10}^{-2}$ |

$(70,69)$ | $(69,68)$ | $9.369\times {10}^{-2}$ | $29.830$ | $2.186\times {10}^{-5}$ | $4.636\times {10}^{-3}$ |

$(100,1)$ | $(99,0)$ | $7.953\times {10}^{-2}$ | $74.387$ | $3.858\times {10}^{-4}$ | $2.407\times {10}^{-1}$ |

$(100,99)$ | $(99,98)$ | $1.093\times {10}^{-2}$ | $10.221$ | $2.175\times {10}^{-5}$ | $1.356\times {10}^{-2}$ |

**Table 3.**Thermal corrections to the partial recombination and ionization coefficients for spontaneous and stimulated processes for the $1s$ and $2s$ states at different temperatures. The coefficients ${\alpha}_{nl}$ are calculated using Equation (14), the first, second and third contributions are denoted as $\mathsf{\Delta}{\alpha}_{1s}^{\left(1\right)}$, $\mathsf{\Delta}{\alpha}_{1s}^{\left(2\right)}$, $\mathsf{\Delta}{\alpha}_{1s}^{\left(3\right)}$, respectively. Values with index $\beta $ denote corresponding stimulated recombination corrections. Summation over m in Equation (14) was performed in the range $m\in [1,100]$, which guarantees the given numbers in the table. The correction to the partial ionization coefficient $\mathsf{\Delta}{\beta}_{nl}$ is given as a total contribution and coincides with the sum of $\mathsf{\Delta}{\alpha}_{nl}^{\left(1\right)}$, $\mathsf{\Delta}{\alpha}_{nl}^{\left(2\right)}$, $\mathsf{\Delta}{\alpha}_{nl}^{\left(3\right)}$, $\mathsf{\Delta}{\alpha}_{nl}^{\beta ,\left(1\right)}$, $\mathsf{\Delta}{\alpha}_{and}^{\beta ,\left(2\right)}$ and $\mathsf{\Delta}{\alpha}_{and}^{\beta ,\left(3\right)}$, as it should be according to the detailed balance. All values are given in ${\mathrm{m}}^{3}{\mathrm{s}}^{-1}$.

T = 300 K | T = 1000 K | T = 3000 K | T = 5000 K | T = 10,000 K | T = 20,000 K | |
---|---|---|---|---|---|---|

${\alpha}_{1s}$ | $9.4939\times {10}^{-19}$ | $5.1848\times {10}^{-19}$ | $2.9688\times {10}^{-19}$ | $2.2812\times {10}^{-19}$ | $1.5819\times {10}^{-19}$ | $1.0787\times {10}^{-19}$ |

${\alpha}_{1s}^{\beta}$ | $0.0$ | $6.9968\times {10}^{-88}$ | $2.0781\times {10}^{-42}$ | $2.2263\times {10}^{-33}$ | $1.1211\times {10}^{-26}$ | $2.0858\times {10}^{-23}$ |

$\mathsf{\Delta}{\alpha}_{1s}^{\left(1\right)}$ | $-3.3362\times {10}^{-29}$ | $-6.6434\times {10}^{-28}$ | $-1.0148\times {10}^{-26}$ | $-3.5689\times {10}^{-26}$ | $-1.9282\times {10}^{-25}$ | $-1.0049\times {10}^{-24}$ |

$\mathsf{\Delta}{\alpha}_{1s}^{\beta ,\left(1\right)}$ | $0.0$ | $-8.9930\times {10}^{-97}$ | $-7.1673\times {10}^{-50}$ | $-3.5334\times {10}^{-40}$ | $-1.4032\times {10}^{-32}$ | $-2.0339\times {10}^{-28}$ |

$\mathsf{\Delta}{\alpha}_{1s}^{\left(2\right)}$ | $-1.4502\times {10}^{-24}$ | $-1.0971\times {10}^{-23}$ | $-6.3683\times {10}^{-23}$ | $-1.4172\times {10}^{-22}$ | $-4.1445\times {10}^{-22}$ | $-1.1997\times {10}^{-21}$ |

$\mathsf{\Delta}{\alpha}_{1s}^{\beta ,\left(2\right)}$ | $0.0$ | $-2.6489\times {10}^{-92}$ | $-8.3235\times {10}^{-46}$ | $-2.6099\times {10}^{-36}$ | $-5.5717\times {10}^{-29}$ | $-4.3721\times {10}^{-25}$ |

$\mathsf{\Delta}{\alpha}_{1s}^{\left(3\right)}$ | $-3.3162\times {10}^{-29}$ | $-6.4671\times {10}^{-28}$ | $-9.4444\times {10}^{-27}$ | $-3.2380\times {10}^{-26}$ | $-1.6872\times {10}^{-25}$ | $-8.5139\times {10}^{-25}$ |

$\mathsf{\Delta}{\alpha}_{1s}^{\beta ,\left(3\right)}$ | $0.0$ | $-8.9365\times {10}^{-97}$ | $-6.8742\times {10}^{-50}$ | $-3.3163\times {10}^{-40}$ | $-1.2731\times {10}^{-32}$ | $-1.7797\times {10}^{-28}$ |

$\mathsf{\Delta}{\beta}_{1s}$ | $-1.4503\times {10}^{-24}$ | $-1.0972\times {10}^{-23}$ | $-6.3702\times {10}^{-23}$ | $-1.4178\times {10}^{-22}$ | $-4.1479\times {10}^{-22}$ | $-1.2019\times {10}^{-21}$ |

${\alpha}_{2s}$ | $1.3919\times {10}^{-19}$ | $7.6117\times {10}^{-20}$ | $4.3716\times {10}^{-20}$ | $3.3664\times {10}^{-20}$ | $2.3419\times {10}^{-20}$ | $1.5998\times {10}^{-20}$ |

${\alpha}_{2s}^{\beta}$ | $5.02703\times {10}^{-77}$ | $2.7449\times {10}^{-37}$ | $4.2385\times {10}^{-26}$ | $6.3229\times {10}^{-24}$ | $2.3283\times {10}^{-22}$ | $1.2711\times {10}^{-21}$ |

$\mathsf{\Delta}{\alpha}_{2s}^{\left(1\right)}$ | $-1.9237\times {10}^{-29}$ | $-3.8311\times {10}^{-28}$ | $-5.6857\times {10}^{-27}$ | $-1.9496\times {10}^{-26}$ | $-1.0001\times {10}^{-25}$ | $-4.8333\times {10}^{-25}$ |

$\mathsf{\Delta}{\alpha}_{2s}^{\beta ,\left(1\right)}$ | $-6.9737\times {10}^{-87}$ | $-1.3982\times {10}^{-45}$ | $-5.6939\times {10}^{-33}$ | $-3.8470\times {10}^{-30}$ | $-1.0792\times {10}^{-27}$ | $-4.3657\times {10}^{-26}$ |

$\mathsf{\Delta}{\alpha}_{2s}^{\left(2\right)}$ | $-1.8429\times {10}^{-26}$ | $-1.3955\times {10}^{-24}$ | $-8.1121\times {10}^{-24}$ | $-1.8065\times {10}^{-23}$ | $-5.2879\times {10}^{-23}$ | $-1.5317\times {10}^{-22}$ |

$\mathsf{\Delta}{\alpha}_{2s}^{\beta ,\left(2\right)}$ | $-1.0928\times {10}^{-82}$ | $-9.0071\times {10}^{-42}$ | $-1.4704\times {10}^{-29}$ | $-6.4156\times {10}^{-27}$ | $-1.0055\times {10}^{-24}$ | $-2.4067\times {10}^{-23}$ |

$\mathsf{\Delta}{\alpha}_{2s}^{\left(3\right)}$ | $-2.6134\times {10}^{-28}$ | $-5.0962\times {10}^{-27}$ | $-7.4361\times {10}^{-26}$ | $-2.5487\times {10}^{-25}$ | $-1.3296\times {10}^{-24}$ | $-6.7418\times {10}^{-24}$ |

$\mathsf{\Delta}{\alpha}_{2s}^{\beta ,\left(3\right)}$ | $-3.5101\times {10}^{-52}$ | $-1.8839\times {10}^{-44}$ | $-7.5179\times {10}^{-32}$ | $-5.0406\times {10}^{-29}$ | $-1.4108\times {10}^{-26}$ | $-5.7965\times {10}^{-25}$ |

$\mathsf{\Delta}{\beta}_{2s}$ | $-1.8458\times {10}^{-25}$ | $-1.4010\times {10}^{-24}$ | $-8.1921\times {10}^{-24}$ | $-1.8346\times {10}^{-23}$ | $-5.5330\times {10}^{-23}$ | $-1.8508\times {10}^{-22}$ |

**Table 4.**The corrections to the total recombination and ionization coefficients for spontaneous and stimulated processes for case A at different temperatures. All values are given in ${\mathrm{m}}^{3}{\mathrm{s}}^{-1}$.

T = 300 K | T = 700 K | T = 1000 K | T = 3000 K | T = 5000 K | T = 10,000 K | T = 20,000 K | |
---|---|---|---|---|---|---|---|

${\alpha}_{A}$ | $4.32385\times {10}^{-18}$ | $2.52126\times {10}^{-18}$ | $2.00071\times {10}^{-18}$ | $9.63800\times {10}^{-19}$ | $6.78908\times {10}^{-19}$ | $4.16397\times {10}^{-19}$ | $2.50652\times {10}^{-19}$ |

${\alpha}_{A}^{\beta}$ | $2.15163\times {10}^{-18}$ | $1.72895\times {10}^{-18}$ | $1.56064\times {10}^{-18}$ | $1.10529\times {10}^{-18}$ | $9.29960\times {10}^{-19}$ | $7.28372\times {10}^{-19}$ | $5.65045\times {10}^{-19}$ |

$\mathsf{\Delta}{\alpha}_{A}$ | $-2.29004\times {10}^{-20}$ | $-1.41894\times {10}^{-20}$ | $-1.16107\times {10}^{-20}$ | $-6.26605\times {10}^{-21}$ | $-5.04184\times {10}^{-21}$ | $-2.74662\times {10}^{-21}$ | $-2.64351\times {10}^{-21}$ |

$\mathsf{\Delta}{\alpha}_{A}^{\beta}$ | $2.56355\times {10}^{-21}$ | $1.50305\times {10}^{-21}$ | $1.16062\times {10}^{-21}$ | $4.99707\times {10}^{-22}$ | $8.75126\times {10}^{-23}$ | $2.46134\times {10}^{-22}$ | $-6.53582\times {10}^{-23}$ |

${\beta}_{A}$ | $6.47549\times {10}^{-18}$ | $4.25021\times {10}^{-18}$ | $3.56135\times {10}^{-18}$ | $2.06909\times {10}^{-18}$ | $1.60887\times {10}^{-18}$ | $1.14477\times {10}^{-18}$ | $8.15697\times {10}^{-19}$ |

$\mathsf{\Delta}{\beta}_{A}$ | $-2.03369\times {10}^{-20}$ | $-1.26864\times {10}^{-20}$ | $-1.04501\times {10}^{-20}$ | $-5.76635\times {10}^{-21}$ | $-4.95433\times {10}^{-21}$ | $-2.50048\times {10}^{-21}$ | $-2.70887\times {10}^{-21}$ |

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**MDPI and ACS Style**

Triaskin, J.; Zalialiutdinov, T.; Anikin, A.; Solovyev, D.
Lowest-Order Thermal Correction to the Hydrogen Recombination Cross-Section in Presence of Blackbody Radiation. *Atoms* **2021**, *9*, 80.
https://doi.org/10.3390/atoms9040080

**AMA Style**

Triaskin J, Zalialiutdinov T, Anikin A, Solovyev D.
Lowest-Order Thermal Correction to the Hydrogen Recombination Cross-Section in Presence of Blackbody Radiation. *Atoms*. 2021; 9(4):80.
https://doi.org/10.3390/atoms9040080

**Chicago/Turabian Style**

Triaskin, Jaroslav, Timur Zalialiutdinov, Aleksei Anikin, and Dmitrii Solovyev.
2021. "Lowest-Order Thermal Correction to the Hydrogen Recombination Cross-Section in Presence of Blackbody Radiation" *Atoms* 9, no. 4: 80.
https://doi.org/10.3390/atoms9040080