Computer simulations of the Stark effect in the helium-beta complex of krypton in ICF conditions

Technical Summary: Computer Simulations of the Stark Effect in the Helium-β Complex of Krypton in ICF Conditions

Problem Statement
Spectroscopy of inertial confinement fusion (ICF) experiments relies on dopants, such as krypton, to diagnose the temperature and density of imploding plasmas. Specifically, the K-shell transitions of He-like species, particularly the He-β line (transition between $n=3$ and $n=1$), are critical for density measurements. However, the He-β line is accompanied by Li-like satellite transitions ($1sn'\ell'3\ell \to 1s^2n'\ell'$), which partially overlap with the main line. Modeling the entire He-β complex under extreme ICF conditions (electron densities $n_e = 10^{24}$ to $10^{25}$ cm$^{-3}$ and temperatures $T_e = 3$ keV) is computationally prohibitive due to the sheer number of atomic levels and non-zero electric dipole matrix elements involved. While computer simulation models (CSMs) are the most advanced tools for calculating Stark profiles, their application to such complex lineshapes has been challenging. Furthermore, previous studies using "standard theory" approximations (assuming static ions and impact electrons) suggested that interference terms in the electron impact operator were negligible for these transitions, a finding that required verification under full simulation conditions.

Methodology
The authors present a comparative study using multiple semi-classical computer simulation codes to model the Stark broadening of the Kr He-β line and its $n=2$ and $n=3$ Li-like satellites. The codes employed include:

• SIMULA and SIMULAm: Models where particles follow straight (rectilinear) trajectories, treating perturbers as independent. SIMULAm is a hybrid code that incorporates the electron broadening operator from standard theory at each timestep to reduce computational cost.
• SimU and SimUSP: Models that explicitly solve the interaction between perturbers and the emitter. SimU uses hyperbolic trajectories (accounting for Debye screening), while SimUSP enforces straight paths to isolate the effect of the emitter-perturber interaction.
• DinMol: A full molecular dynamics code where all electrons and ions interact via a complex multibody Coulomb field, allowing for the detection of recombination events (recombination broadening).

The study evaluates four levels of particle interaction complexity: independent particles, interaction with the emitter, full $N$-body interaction, and $N$-body interaction with recombination broadening. Additionally, the authors introduce a novel concept: Simulation-Equivalent Interference Terms (SEIT). This approach isolates the contribution of interference terms (off-diagonal elements in the impact operator) within the framework of full computer simulations by selectively omitting specific terms in the trace of the dipole correlation function.

Key Results

1. Code Consistency: Codes with the same underlying physics (independent particle trajectories) yield identical results, regardless of numerical implementation differences. SIMULAm, the hybrid code, shows remarkable agreement with full CSMs (SIMULA, SimUSP), validating its use to speed up simulations by a factor of $\sim$50.
2. Effect of Particle Interaction: When interaction with the radiator is included (SimU) or full molecular dynamics are used (DinMol), the spectral lines become narrower compared to independent particle models. This is attributed to the repulsion of perturbing ions by the charged emitter, which reduces the ionic microfield. The inclusion of recombination broadening in DinMol counteracts this narrowing slightly, resulting in a profile intermediate between SIMULA and SimU.
3. Density Dependence: At lower densities ($10^{24}$ cm$^{-3}$), the main He-β components are resolved. At higher densities ($10^{25}$ cm$^{-3}$), Stark mixing intensifies, causing forbidden transitions (e.g., $1D_2 \to 1S_0$ and $3P_2 \to 1S_0$) to dominate the spectrum, while unperturbed lines contribute only to the wings. All models follow the expected $n_e^{2/3}$ scaling for line widths.
4. Interference Terms (SEIT): The study finds that SEIT have a negligible effect on the shape of $n=2$ satellites. However, for $n=3$ satellites, particularly at the highest density ($10^{25}$ cm$^{-3}$), SEIT significantly modify the line shape. Removing SEIT causes the satellite emission to lose structural features (such as shoulders on the high-energy wing) and results in broadening. This contradicts recent findings by Gallardo-Diaz et al. [21], which suggested interference terms were minor. The authors attribute this discrepancy to the fact that in the previous study, electron impact broadening (where interference terms act) was a minor contribution compared to ion broadening and atomic structure; under conditions where electron broadening is isolated or dominant, the interference terms are critical.
5. Synthetic Spectra: When combined with atomic kinetics (ABAko) and thermal Doppler broadening, the simulation results produce synthetic spectra that differ from previous "standard theory" predictions (MERL code), particularly in the high-energy wing of the He-β line, due to the inclusion of ion dynamics.

Significance and Claims
The paper claims to provide the first computer simulation results for the Li-like satellite emission of the Kr He-β complex under ICF-relevant conditions. The work demonstrates the robustness of current simulation codes and validates the use of hybrid approaches (like SIMULAm) for efficient diagnostics. A primary contribution is the definition and application of SEIT, which clarifies the role of interference terms in complex satellite transitions. The authors conclude that while omitting interference terms may be acceptable for $n=2$ satellites or when electron broadening is negligible, it leads to significant inaccuracies for $n=3$ satellites at high densities. The study emphasizes that future diagnostics must account for these effects to avoid errors in density inference, particularly as experimental conditions move toward higher densities or involve transitions from higher principal quantum numbers. The work also highlights the necessity of moving beyond the dipole approximation in future studies to capture plasma polarization shifts observed in experiments.