A numerical study on plasma acceleration processes with ion dynamics at the sub-nanosecond timescale

Plasma wakefield acceleration is a groundbreaking technique for accelerating particles, capable of sustaining gigavolt-per-meter accelerating fields. Understanding the physical mechanisms governing the recovery of plasma accelerating properties over time is essential for successfully achieving high-repetition-rate plasma acceleration, a key requirement for applicability in both research and commercial settings. In this paper, we present numerical simulations of the early-stage plasma evolution based on the parameters of the SPARC_LAB hydrogen plasma recovery time experiment (Pompili et al., Comm. Phys. 7, 241 (2024)), employing spatially resolved Particle-in-Cell and fluid models. The experiment reports on a non-monotonic dependence of the plasma recovery time on the initial plasma density, an effect for which ion motion has been invoked as a contributing factor. The simulations presented here provide further insight into the role of ion dynamics in shaping this behavior. Furthermore, comparing Particle-in-Cell and fluid approaches allows us to assess the quality of fluid models for describing this class of plasma dynamics.

💡 Research Summary

The paper investigates the early‑time (< 0.1 ns) plasma dynamics that govern the recovery of accelerating fields in plasma wakefield accelerators, with a focus on the non‑monotonic dependence of recovery time on the initial plasma density observed in the SPARC_LAB hydrogen‑plasma recovery experiment (Pompili et al., 2024). In that experiment a “pump‑and‑probe” configuration was used: a high‑charge (500 pC), 84 MeV electron bunch (the pump) creates a wake in a hydrogen plasma of density n₀, and a second identical bunch (the probe) arrives after a delay Δt (0.7–13 ns). The probe energy downstream, E_P, is compared with the energy obtained when the pump is absent, E_U; the difference ΔE = E_P − E_U serves as a proxy for the plasma’s state. The measured ΔE shows a pronounced dip (ΔE ≈ −4 MeV) around n₀ ≈ 10¹⁴ cm⁻³, while for higher densities ΔE≈0, indicating a rapid recovery. The authors hypothesized that ion pinching along the pump trajectory, possibly enhanced by the ponderomotive force of the plasma wave, is responsible, but no direct ion diagnostics were available.

To explore this hypothesis, the authors performed two complementary numerical studies: (i) fully kinetic, collisionless Particle‑in‑Cell (PIC) simulations using the quasi‑3D FBPIC code, and (ii) fluid simulations based on the cold‑plasma moment equations solved with a Lattice‑Boltzmann Method (LBM) coupled to a finite‑difference time‑domain (FDTD) Maxwell solver. Both models assume a completely ionized hydrogen plasma (proton mass ions) and a cold background (1–10 eV). The pump bunch is modeled as an axially symmetric Gaussian with the same charge, rms longitudinal size σ_z≈50 µm and transverse size σ_r≈43 µm as in the experiment. The computational domain is a cylinder of length L_z = 0.78 cm (≈0.025 ns light‑travel time) and radius L_r = 1 mm, discretized with Δz = Δr = 0.005 k_p (k_p = ω_p/c). Time steps are Δt = 0.0025/ω_p for the fluid code and Δt = 0.005/ω_p for PIC; PIC runs used 25–64 macro‑particles per cell after a convergence study.

The simulations were repeated for a range of background densities n₀ from 2 × 10¹³ cm⁻³ up to 1 × 10¹⁶ cm⁻³. In each case the pump propagates at the speed of light from left to right, and the plasma response is recorded after the pump has traveled the full domain (≈0.025 ns). The key findings are:

1. High‑density regime (n₀ ≥ 2 × 10¹⁵ cm⁻³): The electric field of the pump pulls ions modestly toward the axis, but the ion column remains weak. The electron wake quickly damps, the focusing field W_r = E_r − cB_φ collapses, and ΔE would be close to zero, consistent with experimental observations of rapid recovery.

2. Intermediate/low‑density regime (n₀ ≈ 10¹⁴ cm⁻³): The pump’s field and the ponderomotive force of the excited plasma wave jointly drive a strong radial ion pinch. PIC results show a pronounced on‑axis ion density spike that persists for tens of picoseconds, creating a localized increase in plasma frequency and sustaining the wake’s electric field. This leads to continued deceleration of the probe bunch, reproducing the large negative ΔE measured experimentally. The fluid model captures the overall density depletion but fails to reproduce the sharp ion spike and the associated phase‑mixing and wave‑breaking phenomena.

3. Very low densities (n₀ < 10¹⁴ cm⁻³): The ion pinch weakens again because the pump charge density exceeds the background plasma density, leading to a more linear plasma response. Both PIC and fluid simulations show a rapid decay of the wake, and ΔE approaches zero.

The comparative analysis demonstrates that while fluid models are adequate for describing average field evolution and are computationally inexpensive, they miss critical kinetic effects such as ion‑induced wave breaking, localized phase mixing, and the exact timing of ion column formation. PIC simulations, though more demanding, provide a faithful representation of the non‑monotonic recovery behavior and confirm that the interplay of ion pinching and the plasma‑wave ponderomotive force is the root cause of the observed density‑dependent recovery times.

In conclusion, the paper validates the experimental hypothesis that ion dynamics, specifically rapid radial pinching enhanced by the plasma‑wave ponderomotive force, govern the sub‑nanosecond recovery of wakefield accelerating structures. It also establishes a benchmark for future modeling: kinetic PIC simulations are indispensable for capturing early‑time ion‑electron coupling, whereas fluid models remain useful for longer‑time, large‑scale studies. The authors suggest that hybrid approaches—using fluid models for background evolution and embedding kinetic PIC patches where ion pinching is critical—could provide an optimal balance of accuracy and computational cost for the design of high‑repetition‑rate plasma accelerators.