---
abstract: 'Collisional–radiative (CR) models based on *ab initio* atomic structure calculation have been utilized over 20 years to analyze many-electron atomic and ionic spectra. Although the population distribution of the excited states in plasmas and their emission spectra are computed using the CR models, systematic and analytical understanding of the population kinetics are still lacking. In this work, we present a continuous CR model (CCRM), in which we approximate the dense energy structure of complex many-electron atoms by a continuum. Using this model, we predict asymptotic population distributions of many-electron atoms in plasmas and their electron-density and temperature dependence. In particular, the CCRM shows that the population distribution of highly excited states of many-electron atoms in plasmas resembles a Boltzmann distribution but with an effective excitation temperature. We also show the existence of three typical electron-density regions and two electron-temperature regions where the parameter dependence of the excitation temperature is different. Analytical representations of the effective excitation temperature and the boundaries of these phases are also presented.'
author:
- Akira Nishio
- 'Julian C. Berengut'
- Masahiro Hasuo
- Keisuke Fujii
bibliography:
- 'refs.bib'
title: ' Population kinetics of many-electron atoms in ionizing plasmas studied using a continuous collisional radiative model '
---

Introduction
============

The spectra of many-electron atomic ions can be seen in various optically thin plasmas. In the stellar atmosphere, neutral and singly charged iron (Fe) are the dominant components in the absorption spectra in terms of the number of lines [@Tousey1988]. Many of Fe absorption lines have been identified to study the stellar atmosphere [@Nave1994; @Castelli2010; @Peterson2017]. Highly charged tin and actinoide ions play an important role in realizing ultraviolet light sources [@Osullivan2015; @Suzuki2012; @Torretti2020], in which quasi-continuum emission in laser-produced plasmas is used. Since the radiative power should be concentrated into a particular energy region for the commercial light source realization, many works have been carried out to understand the population dynamics in plasmas [@Osullivan2015; @Torretti2020]. In fusion tokamak plasmas, highly charged tungsten ions convert electron kinetic energy to strong radiation and therefore need to be controlled [@Putterich2008; @Murakami2015]. The thermalization process of the nuclei in kilonovae, which has recently been probed from the emission of neutral transition metals, is yet to be understood [@Pian2017; @Tanaka2018].

The collisional–radiative (CR) model is the key tool to study the population kinetics of many-electron atoms in plasmas and their emission and absorption spectra. This model solves the steady state equation of the excited state population of ions by taking into account the rates of elementary processes in plasmas. In order to perform accurate predictions, accurate atomic data are required, i.e., energy levels and transition rates of many elementary processes, including electron-impact excitations and spontaneous transitions. Therefore, many works have been dedicated to develop and improve *ab initio* calculation of these atomic data [@Gu2008; @amusia1997computation; @Bar-Shalom2001]. Although this first-principles approach has been successful in many cases [@TheVenin1999; @Dodin2014; @Torretti2020; @Murakami2015], accurate calculations are still difficult and computationally demanding, particularly for many-valence-electron atoms and ions. This difficulty comes from their strong wavefunction mixing, which requires an unacceptably large Hilbert basis space to represent their wavefunctions. Due to the complexity and difficulty of the first-principle computation of the atomic structure, it is difficult to understand and validate the CR model result.

A probabilistic model may provide a complementary approach to the first-principle calculation. For a system with sufficiently strong mixing of basis states, i.e., systems exhibiting many-body quantum chaos, it is known that some of the properties of their atomic structure can be represented using a statistical theory [@Flambaum1994]. Although its applications to plasma diagnostics are very limited, we have recently shown that the intensity statistics of many-electron atoms can be understood from this structure, and can be used to measure electron temperature in plasmas [@Fujii2020]. Since the use of probabilistic nature of many-electron atoms requires only a small amount of atomic data, this approach is not only robust against possible numerical errors, but also gives us a systematic insight of the population kinetics in plasmas.

In this work, we develop a continuous CR model (CCRM), in which we approximate the dense energy levels of many-electron atoms by a continuum based on the statistical theory. For our simplified theory, only two atomic parameters are used to represent the spectrum: the energy scale of the level density growth; and another energy scale that describes the decay of transition strengths. Based on this model, we will show that the population distribution of highly excited states of many-electron atoms is similar to Boltzmann’s distribution but with an effective excitation temperature $T_\mathrm{ex}$. The dependence of this excitation temperature on electron density ($n_\mathrm{e}$) and temperature ($T_\mathrm{e}$) are then studied using the CCRM, revealing the existence of three typical $n_\mathrm{e}$ regions and two $T_\mathrm{e}$ regions. In particular, it is shown that in low $T_\mathrm{e}$ regions, the excitation temperature becomes almost $T_\mathrm{e}$ even in low $n_\mathrm{e}$ plasmas. This property indicates much wider applicability of the Boltzmann’s method, which is a well-known method to estimate $T_\mathrm{e}$ values from the emission spectra in high $n_\mathrm{e}$ plasmas. It also indicates the wider applicability of the new temperature diagnostics based on the line intensity statistics, in which the Boltzmann population distribution is assumed [@Fujii2020]. The population kinetics is also compared with that of hydrogen (H)-like ions, which has been extensively studied as shown in Fig. \[fig:diagram\] (a) (see also Appendix \[sec:hydrogen\] for details) [@Fujimoto].

In section \[sec:crmodel\], we briefly describe the principle of the CR model and show some simulation results obtained using an *ab initio* calculation code for several many-electron atomic ions. In section \[sec:model\], we present our CCRM to study the population kinetics of many-electron atoms and compare it with the *ab initio* simulation result. In section \[sec:discussions\], we discuss its parameter dependence.

![image](diagram){width="13.5cm"}

Summary
=======

In this work, we studied the population kinetics of many-electron atoms in plasmas. From the statistical theory of the many-electron-atom structure, we constructed a continuous CR model that has only two atom-specific energy scales as parameters, $\epsilon_0$ and $\sigma$. From this model, the population distribution in highly excited states was found to be Boltzmann-like, but with excitation temperature sometimes smaller than the electron temperature. We also clarified that there are different phases depending on values of $n_\mathrm{e}$ and $T_\mathrm{e}$ and derived analytical representations of the boundaries.

Some of our findings can be directly used for plasma diagnostics. For example, the Boltzmann method has been frequently used to estimate $T_\mathrm{e}$, based on the slope of the population distribution and the assumption of the saturation phase, and therefore, the applicability of this method has been limited only to high-density plasmas. However, as can be seen in Fig. \[fig:Tex\], if $T_\mathrm{ex} < \epsilon_0 / 2k$, then $T_\mathrm{ex} \approx T_\mathrm{e}$ can be inferred. This clearly shows much wider applicability of the Boltzmann method for low-temperature plasmas. This property also enables us to use a new temperature diagnostics using line intensity statistics, which has been proposed in Ref. [@Fujii2020]. By contrast, if $T_\mathrm{ex} \gtrsim \epsilon_0 / 2k$, then the inference of $T_\mathrm{e}$ may be difficult without knowing $n_\mathrm{e}$.

For highly charged ions in low-density and high-temperature plasmas, such as heavy ions in tokamak core plasmas or in electron-beam ion traps, the population is mostly concentrated in the low excited states, to which our model is not applicable. However, our finding [Eq. (\[eq:Tex\_highT\_lowN\])]{} may still be useful to estimate the cascade contribution from very highly excited states, which is difficult to consider from first-principles owing to the enormous requirement of the computation resources.

In this work, we only compared our model with another simulation model, FAC. Comparison with experimental observation is desirable; however, because of the difficulty in the level identification and accurate computation of the transition rates for highly excited states, it is not available at the current stage. We leave it for future studies.

In principle, our CCRM could be further developed to include additional atomic structure data, such as more sophisticated line-strength functions, based on individual orbitals within the statistical theory of many-body quantum chaos [@Flambaum1998]. While this would not add significant computational overhead, the simplicity of our current formulation, [Eq. (\[eq:strength-function\])]{}, allows for analytical exploration of the effective excitation temperature through phase space.

This work was partly supported by JSPS KAKENHI Grant Number 19K14680, the grant of Joint Research by the National Institutes of Natural Sciences (NINS program No, 01111905), and partly by the Max-Planck Society for the Advancement of Science. JCB is supported by the Alexander von Humboldt Foundation. We thank José Crespo Lopéz-Urrutia for useful discussions.
