A note on thermal effects in non-linear models for plasma-based acceleration

TL;DR

This work extends the foundational cold-plasma bubble theory of Lu et al. to warm electron plasmas in the blowout regime of plasma wakefield acceleration. By introducing a thermal momentum spread via a Maxwell–Boltzmann sampling and deriving a temperature-dependent, stochastic blowout-radius equation, the authors obtain ensemble-averaged bubble dynamics $\langle r_b(\xi)\rangle$ and fluctuations that scale with the thermal spread $\mu_i$. Comparisons with PIC simulations show that warm plasmas contract both the longitudinal and transverse bubble sizes and broaden the electron sheath, and that a temperature-dependent reparametrization of the source term is essential for accurate reproduction. The results provide a more realistic description of PWFA wakes under finite-temperature conditions and suggest pathways to incorporate thermal effects into more sophisticated wakefield models and closures.

Abstract

We investigate the impact of a non-negligible background temperature on relativistic plasma wakefields generated when a beam of charged particles passes through a neutral plasma at rest. Our analysis focuses on the blowout regime, where the plasma response is highly non-linear: plasma electrons are radially blown out and expelled away from the propagation axis of the beam particles, creating a region (bubble) of ions without electrons. Our study builds upon earlier investigations for non-linear models of plasma wakefields developed neglecting plasma temperature. In the presence of a non-zero background temperature, we characterize the bubble in terms of its transversal and longitudinal sizes as a function of the temperature. Model predictions and parametrizations are studied in combination with PIC simulations, and correctly reproduce the temperature induced contraction of both the longitudinal and transverse bubble sizes.

Figures (8)

• Figure 1: Representative sketch coming from PIC simulations of the first electron depletion bubble generated by a relativistic electron bunch moving at the speed of light, in two different configurations of initial background plasma temperature. The observable represented in the figure is the number density, normalized w.r.t. the initial unperturbed value $n_0$, that consists of two contributions. The first is from the bi-Gaussian driving bunch $n_b$, characterized by normalized rms-sizes $\sigma_r=0.1$ and $\sigma_z=\sqrt{2}$, normalized peak amplitude $\alpha=84$, and propagating from right to left at the speed of light. The second contribution comes from the background plasma density $n_e$, initially set at uniform density $n_0=10^{16} cm^{-3}$ and uniform temperature $k_b T_i = 0 \; \rm{keV}$ (top half of the figure) and $k_b T_i = 1 \; \rm{keV}$ (bottom half of the figure). Also shown in gold is the curve $r_b(\xi)$, which marks the interface between the blowout region -- where all plasma electrons have been expelled -- and the outer region. Quantities $\ell_\bot$ and $\ell_\parallel$ measure respectively the transversal and longitudinal extents of this interface.
• Figure 2: Numerical solutions of \ref{['eq:blowout-radius-equation']}, obtained using a fixed sheath thickness parameterization $\Delta=0.1r_b+1.0$ as in lu-2006lu-2006-b. Panel (a) shows a representative solution with an initial background plasma temperature of $k_b T_i = 1\; \rm{keV}$. In the panel is shown an ensemble of electron trajectories (gray) obtained from 500 stochastic realizations of the initial conditions detailed in \ref{['eq:stochastic-sampling', 'eq:stochastic-initial-velocity']}, and in orange the resulting average $\langle r_b \rangle$. Panel (b) compares this average trajectory (dashed orange line) against a cold-plasma realization of the model. While the mean trajectory remains close to the cold prediction, the trajectory bundle widens significantly, suggesting that thermal fluctuations contribute notably to the system's behavior. This effect is quantified in panel (c), where the proposed fluctuation measure $\Delta r_b(\xi)$, evaluated at $\xi=11.0$ (highlighted in panel (a) by the black segment), is plotted as a function of the initial plasma temperature. The plot is in logarithmic scale.
• Figure 3: Analysis of the first derivative $r_b'$. Panel (a) shows an ensemble of derivatives' trajectories (gray) obtained from $500$ stochastic a realization of the initial conditions, for an initial plasma temperature of $k_b T_i = 1 \; \rm{keV}$. The orange curve denotes the ensemble average $\langle r_b' \rangle$. Panel (b) compares this average trajectory (dashed orange line) with the cold-plasma prediction, showing little to no difference between the two cases. Panel (c) shows $\Delta r_b'(\xi)$, defined analogously to \ref{['eq:thermal-fluctuations']}.
• Figure 4: Radial cross section of the source term $S=-\rho-J_z$ for different temperature realizations, at a fixed value of the co-moving coordinate $\xi_0 = 7.0$, corresponding to the longitudinal center of the Gaussian bunch. As the background temperature increases, the electron accumulation region at the $r_b$ interface broadens and its peak amplitude decreases. This behavior highlights the need for a temperature dependent parametrization of the source term $S$ through a suitably defined $\Delta(\mu_i)$.
• Figure 5: Parametric scan of the electron sheath width parameters $(\epsilon,\Delta_l)$, shown through color maps of the relative error metric $C(\epsilon,\Delta_l)$, as defined in \ref{['eq:cost-function']}. Each panel corresponds to a different background temperature: $k_b T_i = 0,1$ and $2\; \rm{keV}$ (panels from top to bottom). The metric quantifies the agreement between the bubble radius $r_b^{(sim)}$ extracted from PIC simulations and the theoretical prediction $r_b^{(th)}(\epsilon,\Delta_l)$. Minima in the plots indicate the most suitable parametrization in order to reproduce the simulation results at each temperature.