Collisional–Radiative Modeling (FLYCHK)

• Collisional–radiative modeling is a computational framework that uses coupled kinetic equations to determine atomic and ionic level populations in plasmas subject to both collisional and radiative processes.
• FLYCHK models rely on high-fidelity atomic data obtained via methods such as relativistic configuration interaction and distorted-wave approximations to accurately predict emission spectra.
• These models are essential for diagnostic applications in laboratory and fusion plasmas, where spectral line intensity ratios are used to infer electron density and temperature.

Collisional–Radiative (CR) modeling is a fundamental computational framework for describing the population kinetics of atomic and ionic energy levels in plasmas subject to both collisional (typically electron impact) and radiative processes. In practical applications, CR models are crucial for interpreting and predicting emission spectra, extracting plasma parameters, and enabling diagnostics in laboratory, astrophysical, and fusion plasmas. The FLYCHK family of codes and related models are widely utilized for such tasks, especially under non-local thermodynamic equilibrium (non-LTE) conditions, where atomic level populations deviate strongly from Boltzmann equilibrium due to low densities or strong external fields. The development and implementation of advanced CR models require accurate atomic data, sophisticated rate equation solvers, treatments of electron energy distribution functions, and diagnostic strategies. This article provides a detailed reference on the principles, methodologies, and applications of collisional–radiative modeling, with particular focus on FLYCHK-style models and high-fidelity atomic kinetic calculations as evidenced in recent literature.

1. Fundamental Principles of Collisional–Radiative Modeling

At the core of a CR model is a set of coupled kinetic rate equations describing the time evolution of level populations $n(p)$ for atomic or ionic state $p$: $$\frac{d}{dt} n(p) = \sum_{q<p} C(q,p)\, n_e\, n(q) - \left\{ \sum_{q<p}[F(p,q)\, n_e + A(p,q)] + \sum_{q>p} C(p,q)\, n_e \right\} n(p) + \sum_{q>p}[F(q,p)\, n_e + A(q,p)] n(q).$$ Here, $C(q,p)$ and $F(p,q)$ are electron-impact excitation and deexcitation rate coefficients, $A(p,q)$ are spontaneous radiative transition probabilities, and $n_e$ denotes electron density. This framework incorporates both collisional and radiative processes, extending beyond the assumptions of LTE. Under steady-state (quasi-steady-state, QSS) conditions, population derivatives vanish ($d n(p)/dt = 0$), yielding algebraic relations that can be solved for $n(p)$.

The calculated population distributions serve as the basis for emission modeling. The spectral intensity for a transition $p\rightarrow q$ can be written as: $$I_{pq}(\lambda) \propto n(p) A(p,q) \varphi(\lambda),$$ with $\varphi(\lambda)$ describing the normalized line profile, commonly taken as a Doppler-broadened Gaussian incorporating instrumental and natural width contributions.

CR modeling assumes the plasma is optically thin unless otherwise specified and typically includes all energetically relevant levels and configurations to capture electron correlation and configuration interaction effects.

2. Atomic Data Requirements and Computational Methods

Reliable CR modeling necessitates high-accuracy atomic data: level energies, radiative transition probabilities, and electron-impact (de)excitation cross sections. In the case of complex, high-$Z$ ions such as W$^{26+}$, as studied in (Ding et al., 2015), data are obtained via full-relativistic configuration interaction (CI) approaches implemented in codes such as FAC (Flexible Atomic Code). All important configurations—both ground and excited, including those resulting from single or double electron substitutions—are considered to account for electron correlation and interaction effects. Transition rates for various multipole orders (E1, M1, etc.) and electron-impact excitation/deexcitation cross sections are calculated; the latter typically employ the distorted-wave (DW) approximation.

Table: Major atomic data components and computational methods in CR modeling

Quantity	Method	Tool/Algorithm
Energy levels	Relativistic CI, MCDF, RMBPT	FAC, MCDHF, RMBPT
Radiative transition rates (A)	Relativistic CI, MCDF, EOL, TS	FAC, OPAMCDF
Electron-impact (de-)excitation	Distorted-Wave, IPIRDW	FAC, RDW, IPIRDW

A rigorous CR model integrates this atomic data directly, eschewing empirical corrections except where high-level calculations are unavailable. The importance of configuration interaction and proper level mixing is particularly underscored for heavy and/or open-shell ions.

3. Implementation for Laboratory and Fusion-Relevant Plasmas

In experimental contexts—such as electron beam ion trap (EBIT) devices and magnetic confinement fusion (MCF) reactors—the CR model provides a direct means to interpret measured spectra and diagnose plasma parameters. For instance, in (Ding et al., 2015), a CR model was constructed for W$^{26+}$, including ground 4d$^{10}$4f$^2$ and multiple excited configurations (4d$^{10}$4f$^1$nl for $n=5$–8, various 4d$^{10}$nl n'l', and configurations with a 4d$^9$ core), to simulate the visible range (332–392 nm) M1 spectrum observed in EBIT experiments.

Fully synthetic spectra are generated by combining transition rates with the steady-state populations and convolving with line profiles. Agreement with observed spectra—after minor wavelength alignment—is used to validate the atomic data and CR model assumptions. Consistency between theory and experiment demonstrates both the necessity and adequacy of including extensive configuration mixing, accurate rate coefficients, and all dominant collisional and radiative channels.

4. Sensitivity to Plasma Parameters and Diagnostic Applications

A central application of CR modeling is plasma parameter diagnostics, made possible by the dependence of line intensity ratios on electron density and, in some cases, electron temperature. Pairs of transitions close in wavelength and differing in sensitivity to $n_e$ are identified as diagnostic pairs. The intensity ratio $I(a)/I(b) = [n(a)A(a)]/[n(b)A(b)]$ is computed as a function of $n_e$ by solving the QSS equations under experimental or reactor parameters (e.g., EBIT with mono-energetic beams, MCF with Maxwellian $T_e$ distributions).

In the EBIT case, the CR model uses a delta-function electron energy distribution; under these conditions, diagnostic line ratios vary strongly for $n_e\sim 10^9$–$10^{11}$ cm$^{-3}$. For MCF plasmas (with $T_e \sim 1.5$ keV, $n_e\sim10^{13}$–$10^{15}$ cm$^{-3}$), some line intensity ratios change by two orders of magnitude with $n_e$, establishing their usefulness for electron density diagnostics.

5. Extensions and Theoretical Evaluations

Analysis extends to varying plasma conditions, comparing EBIT monoenergetic and reactor Maxwellian electron energy distributions. The same set of kinetic rate equations is solved, substituting the appropriate electron distribution function in the computation of population and collisional rates.

Theoretical evaluations (e.g., Figs. 2 and 3 in (Ding et al., 2015)) provide the electron density ($n_e$) dependence of line ratios, offering sensitivity curves that allow practitioners to invert measured intensity ratios directly to infer $n_e$ in real time. The interplay between dominant collisional rates and competition between radiative and collisional level population/depopulation underpins the density and, by extension, temperature sensitivity of different transitions.

6. Mathematical Framework and Key Model Equations

The collisional–radiative model is quantitatively specified by the master equation for level populations, which, in QSS form, reads: $$0 = \sum_{q < p} C(q,p) n_e n(q) - \left\{ \sum_{q < p} [F(p,q) n_e + A(p,q)] + \sum_{q > p} C(p,q) n_e \right\} n(p) + \sum_{q > p} [F(q,p) n_e + A(q,p)] n(q).$$ The emission intensity is computed as: $$I_{p,q}(\lambda) \propto n(p) A(p,q) \varphi(\lambda).$$ Population assignments in the synthetic spectrum are validated by reproducing experimental line positions and relative intensities.

The atomic physics inputs (energies, $A$-values, $C$ and $F$ collisional rates) are calculated at each configuration level, ensuring robustness against parameter manipulation or undetected configuration mixing.

7. Impact and Significance

The CR modeling approach detailed in (Ding et al., 2015) has several important consequences:

• It demonstrates that reliable reproduction of experimental spectra, particularly for W$^{26+}$ and similar highly charged ions, is contingent upon comprehensive atomic data and proper collisional–radiative balance.
• It validates the selection of certain M1 transition pairs as high-sensitivity, real-time diagnostics for $n_e$ in low- and high-density plasma regimes.
• It establishes a computational and methodological template for further paper of tungsten and other key fusion-relevant impurities, facilitating the monitoring and control of impurity behavior in fusion environments.
• It highlights the necessity of advanced computational tools (e.g., FAC) capable of large-scale, relativistic configuration interaction calculations as the foundation for credible CR modeling.

The formalism and methodology provide a rigorous, reproducible pathway from ab initio atomic physics data to diagnostic plasma spectroscopy and underpin major advances in interpreting spectroscopic signatures in both laboratory and fusion plasmas.