# Radiative Recombination Plasma Rate Coefficients for Multiply Charged Ions

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## Abstract

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## 1. Introduction

## 2. Theory Furthermore, Computations

#### 2.1. Radiative Recombination Cross Sections

#### 2.2. Temperature-Dependent Plasma Rate Coefficients

#### 2.3. Radiative Recombination Cascades

#### 2.4. Implementation of Plasma Rate Coefficients into the Jac Toolbox

`Atomic.Computation`, if the proper settings are provided by the user for the

`PhotoRecombination`module. These

`computations`are organized analog to calculating most other atomic processes within Jac, and where all the requested (photorecombination) amplitudes are always obtained from the associated module. The implementation of a radiative-recombination cascade scheme, in contrast, includes fewer correlations in the representation of the atomic levels but supports a much greater flexibility in dealing with the capture into high-n shells, the discretization of the electron continuum or the desired temperature range of the plasma.

#### 2.5. Data Types for Radiative Recombination Cascades

`Orbital`for specifying the quantum numbers and radial components of single-electron orbital functions, or a

`Level`for the full representation of a single level $\alpha \phantom{\rule{0.166667em}{0ex}}\mathbb{J}$, which then encompasses all information about the radial wave functions, the coupling of the angular momenta as well as the mixing (and number) of configuration state functions within the given basis. In total, there are at present about ∼250 of these data structures in Jac, though most of them remain hidden to the user of the toolbox. As usual, this set of data structures needs to be enlarged by a few items if the code is expanded toward new features. While we shall not discuss details of the current implementation, we here explain the data (structure) that has been designed in order to deal with general RR cascade schemes.

`Cascade.Radiative`

`RecombinationScheme`in Jac, which helps perform all the RR cascade computations below. Apart from the multipoles of the radiation field, which are to be included into the photorecombination process, this cascade scheme enables one to select the partial waves of the free electron, e.g., the orbital angular momentum values

`(lValues)`as well as the maximum and number of the free-electron energies in the discretization of the electron continuum. Whereas the integration ${\int}_{0}^{\infty}d\epsilon \dots $ in the rate coefficient (3) runs formally over all positive kinetic energies $\epsilon >0$, we internally make use of a Gauss–Legendre integration in order to keep the energy discretization simple and independent of the temperature ${T}_{e}$, for which plasma rate coefficients are to be evaluated in subsequent simulations. Often a maximum energy ${\epsilon}_{\phantom{\rule{0.166667em}{0ex}}\mathrm{max}}\phantom{\rule{0.166667em}{0ex}}\approx \phantom{\rule{0.166667em}{0ex}}5\times {k}_{B}\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{e},\phantom{\rule{0.166667em}{0ex}}\mathrm{max}}$ and an energy mesh of about 12–24 points have been found sufficient in order to obtain plasma rate coefficients with 15–20% accuracy. For ions with a complex (valence) shell structure, further work is, however, needed to better understand how these rate coefficients depend on details of the energy discretization. The structure

`Cascade.RadiativeRecombinationScheme`also requires the user to provide a list of nonrelativistic subshells

`(intoShells)`into which the electrons are to be placed, in addition to those shells that already belong to the ionic core. The specification of the core occupation (i.e., of the initial levels ${\alpha}_{i}{\mathbb{J}}_{i}$ of the ions) in terms of one or several reference configurations is common to all cascade schemes and is provided independently to the cascade computations. Indeed, such a flexible but powerful description of the shell occupation as well as all further input have been found relevant and necessary for covering a wide range of possible applications of RR plasma rate coefficients.

## 3. Radiative Reombination Plasma Rate Coefficients for Multiply Charged Ions

#### 3.1. Plasma Rate Coefficients for Initially Fe${}^{\phantom{\rule{0.166667em}{0ex}}24+}$ Ions

`name`of the computations, the specification of the grid, the nuclear charge of the ion $(Z=26)$ and the initial $1{s}^{2}$ configuration, all that has to be provided by the user is the list of partial waves of free electrons in the continuum

`(lValues)`and the shells

`(intoShells)`, into which an electron can be captured eventually. In addition, a few parameters, specific to the radiative-recombination scheme (cf. Figure 3), help control the size of the computations. These parameters comprise the maximum energy and the number of free-electron energies in the integration of the plasma rate coefficients, a minimum photon energy for evaluating the many-electron amplitudes as well as a few others, which can be provided by the user in order to overwrite existing default values. Indeed, this input provides a rather flexible control for most open-shell ions without either the final-electron configurations nor the relevant capture amplitudes needing to be selected or specified explicitly.

`[Cascade.SCA()]`is used by default to deal, in turn, with each final configuration of the electron capture. These configurations are built from the initial configuration(s) and an additional electron in one of the

`intoShells`. In these cascade computations, the major effort refers to the calculation of the fine-structure levels for each configuration as well as to the numerical evaluation of the RR amplitudes (1). Despite its simplicity, however, the input to these computations still requires some physical insight into both the electronic structure of the initial ion and the intricacy of the associated many-electron continuum. This involvement refers, for instance, to the partial waves, the energies of the free electrons, the choice of multipoles or some prior insight, which of the excited shells are in fact relevant for the given charge state and temperatures. In these cascade computations, no attempt is internally made in order to control that all important configurations are taken into account for the representation of the final levels ${\alpha}_{f}{\mathbb{J}}_{f}$ (owing to the coupling of the multipoles and partial waves), nor that all the computed configurations are relevant for the rate coefficients of interest. Obviously, this insight of the user into the RR process cannot be completely formalized in advance.

`Cascade.Simulation`(lower panel) for extracting the desired RR plasma rate coefficients. Such cascade simulations generally combine the amplitudes and cross sections from above and (numerically) perform the integration over the relevant part of the electron continuum. A Gauss–Legendre integration has been implemented here, in contrast to a more natural Gauss–Laguerre integration scheme for plasma rate coefficients, in order to enable internally the use of the same many-electron amplitudes but for different temperatures of the plasma. These cascade simulations are typically much faster than the prior computations, and often quite similar simulations are performed repeatedly for different temperatures, initial levels, subshells of the captured electron, or other choices for the capture process, while still using the same transition data. For the computation of the total RR plasma rate coefficients, we just list the multipoles and the plasma temperatures in the input by specifying an instance of the data structure

`Cascade.RrRateCoefficients(..)`as the requested property of our study. We also provide here the initial level number of the ground level as well as the temperatures of the plasma. Besides the choice of multipoles, the rate coefficients can also be restricted to selected fine-structure levels or subshells using a few optional arguments to

`Cascade.RrRateCoefficients()`. Although, as mentioned before, such a separation into cascade computations and simulations includes fewer correlations in the representation of the fine-structure levels [54], it makes the estimation of plasma rate coefficients more flexible for atoms and ions with different shell structure. However, further work is likely needed in order to accelerate and, perhaps, parallelize these computations for ions with complex shell structure of open d- and f-shell elements [55].

#### 3.2. Trends for Multiply and Highly Charged Ions

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Selected features of the Jac toolbox [21] that help compute the level structure, processes and cascades of most atoms and ions across the periodic table. Here, emphasis has been placed on the radiative recombination of multiply charged ions and several associated plasma rate coefficients. Apart from the recombination of ions, this toolbox supports several further (relativistic) computations as briefly brought up in this jigsaw puzzle.

**Figure 2.**Definition of the data structure

`Cascade.RadiativeRecombinationScheme`in Jac that helps perform the cascade computations in Section 3 below. It is one of the supported cascade schemes of a

`Cascade.Computation`and enables the user to specify the multipole components of the radiation field in the photorecombination of the ion, the partial waves of the incoming free electrons, the discretization of the free-electron continuum, as well as the shells into which electrons are to be captured. See text for further explanations.

**Figure 3.**Input to the Jac toolbox for the cascade computation (upper panel) and simulation (lower panel) of RR plasma rate coefficients of initially helium-like Fe${}^{\phantom{\rule{0.166667em}{0ex}}24+}$ ions. The total plasma rate coefficients are calculated for plasma temperatures ${10}^{\phantom{\rule{0.166667em}{0ex}}5}\phantom{\rule{0.166667em}{0ex}}\lesssim \phantom{\rule{0.166667em}{0ex}}{T}_{e}\phantom{\rule{0.166667em}{0ex}}\lesssim \phantom{\rule{0.166667em}{0ex}}{10}^{\phantom{\rule{0.166667em}{0ex}}7}$ K, and by restricting the electron capture into $n\ell $ shells with $n=2\phantom{\rule{0.166667em}{0ex}}\dots \phantom{\rule{0.166667em}{0ex}}12$ and $\ell \le f$. While the cascade computations just generate all associated transition amplitudes, the subsequent simulation combines these data and extract the desired rate coefficients for a given set of temperatures. See the text for further discussion.

**Figure 4.**Total RR plasma rate coefficients as function of the (plasma) temperature ${T}_{e}$ for initially helium-like Fe${}^{\phantom{\rule{0.166667em}{0ex}}24+}$ ions. Results from this work with electron capture into the $n\ell $ shells with $n=2\phantom{\rule{0.166667em}{0ex}}\dots \phantom{\rule{0.166667em}{0ex}}12$ and $\ell \le f$ in length (red line) and velocity gauge (blue line) are compared with previous computations by Verner and Ferland ([13]; green line), the parametrization by Arnaud and Raymond ([10]; orange line) and the scaled analytical rates by Kotelnikov and Milstein ([42]; violet line).

**Figure 5.**The same as in Figure 4 but for the fine-structure and shell-resolved RR plasma rate coefficients. Left panel: Capture into the $2s\phantom{\rule{0.277778em}{0ex}}J=1/2$ (red line), the $2p\phantom{\rule{0.277778em}{0ex}}J=1/2$ (blue line) and $2p\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}J=3/2$ level (green line). Right panel: Capture into all levels with $n=2$ (red line), $n=3$ (blue line), and $n=4$ (green line).

**Figure 6.**The same as in Figure 4 but for the capture into the initially excited $i\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1s2s{\phantom{\rule{0.277778em}{0ex}}}^{3}{S}_{1}$ and $1s2s{\phantom{\rule{0.277778em}{0ex}}}^{1}{S}_{0}$ levels. Data are shown for the total RR plasma rate coefficients ${\alpha}^{\text{}\left(\mathrm{RR}\right)}({T}_{e},i)$ for these two levels (left panel) as well as for just the capture into the $1{s}^{2}2s$ and $1s2{s}^{2}$ configurations (right panel).

**Figure 7.**The same as in Fig. 4 but for the capture into selected helium-like ions. Left panel: RR plasma rate coefficients for Ne${}^{\phantom{\rule{0.166667em}{0ex}}8+}$ (red line), Fe${}^{\phantom{\rule{0.166667em}{0ex}}24+}$ (blue line), We${}^{\phantom{\rule{0.166667em}{0ex}}72+}$ (green line), and by applying the electric-dipole (E1) approximation. Right panel: Comparison of the ratio of total plasma rate coefficients, normalized on the E1 approximation, for W${}^{\phantom{\rule{0.166667em}{0ex}}72+}$ within the E1 (red line), E1+M1 (blue line), E1+M1+E2 (green line) and the E1+M1+E2+M2 approximation (orange line), respectively.

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**MDPI and ACS Style**

Fritzsche, S.; Maiorova, A.V.; Wu, Z.
Radiative Recombination Plasma Rate Coefficients for Multiply Charged Ions. *Atoms* **2023**, *11*, 50.
https://doi.org/10.3390/atoms11030050

**AMA Style**

Fritzsche S, Maiorova AV, Wu Z.
Radiative Recombination Plasma Rate Coefficients for Multiply Charged Ions. *Atoms*. 2023; 11(3):50.
https://doi.org/10.3390/atoms11030050

**Chicago/Turabian Style**

Fritzsche, Stephan, Anna V. Maiorova, and Zhongwen Wu.
2023. "Radiative Recombination Plasma Rate Coefficients for Multiply Charged Ions" *Atoms* 11, no. 3: 50.
https://doi.org/10.3390/atoms11030050