Unified Model of Heated Plasma Expansion

TL;DR

This work develops a collisionless two-fluid + Poisson model for heated plasma expansion into vacuum and introduces a three-parameter self-similar framework that remains valid under external electron heating. The analysis yields a unified description spanning five dynamical regimes, governed by the ion-acoustic correlation length $λ_s$, Debye length $λ_D$, and heated-domain length $L$, with key parameters $η=(ω_{pi0}/γ)^2$ and $|ζ_c|$ determining regime transitions. The model provides scaling laws for $λ_D$, $λ_s$, and $L$, resolves charge-separation effects near the expanding front, and offers insights into optimizing laser-plasma prepulse interactions for ion acceleration and surface modification. This framework informs experimental design by linking laser parameters and target properties to the intermediate-asymptotic plasma dynamics and energy partition among electrons, ions, and the electrostatic field.

Abstract

Motivated by the need to predict plasma density and temperature distributions created in the early stages of high-intensity laser-plasma interactions, we develop a fluid model of plasma expansion into vacuum that incorporates external heating. We propose a new three-parameter family of self-similar solutions for plasma expansion that models a wide range of spatiotemporal variations of the electron temperature. Depending on the relative scales of the heated plasma domain $L$, the Debye length $λ_D$ and an emergent ion-acoustic correlation length $λ_s$, characterized by the parameters $λ_s/λ_D$ and $L/λ_s$, a spectrum of dynamical behaviors for the expanding plasma are identified. The behavior is classified into five dynamical regimes, ranging from nearly quasineutral expansion to the formation of bare ion slabs susceptible to Coulomb explosion. The limiting self-similar solutions are analyzed, and the dynamics in the five asymptotic limits in the parameter space are detailed. Scaling relations for the length scales and energies of the expanding plasma are proposed. The self-similar framework is applied to laser-plasma interactions, specifically addressing the plasma dynamics at a target surface during prepulse-target interactions. The results offer insights into the expansion behavior based on the laser-plasma parameters, and scaling relations for optimizing laser-plasma schemes and guiding experimental designs in high-intensity laser experiments.

Figures (10)

• Figure 1: Parameter space of expansion phenomena characterized by $\frac{L}{\lambda_s}$ and $\frac{\lambda_s}{\lambda_D}$. The shaded regions represent the limits of Coulomb explosion precursor, quasineutral expansion, ablation, and expanding hot electron cloud
• Figure 2: (a) Ion density, ion velocity and electrostatic field profiles for $\eta \in \left\{10^{-2},10^{-1},1,10,10^{2}\right\}$ plotted in different colors, for $-\zeta_c \in \left\{0.13,0.35,1,2.82,7.94\right\}$ corresponding to each column from the left. The dashed gray lines in the $P_i$ plots represent $P_i=\zeta$ while the colored dashed lines in the $Q$ plots represent $\zeta=\zeta_f$ with $\zeta_f$ of the solutions for the corresponding colors. The $\zeta=0$ interface confining the initial plasma to the negative half-space is shown with the gray solid line in $P_i$ and $Q$ plots, and the black dot-dashed lines in the $N_i$ plots is the initial density profile. $(b)$ The parametric variation of $Q_f$ with respect to $\log_{10}(\eta) \in [-2,2]$ and $\log_{10}(-\zeta_c) \in [-1,1]$ with contours of fixed $Q_f$ shown with black and blue$($for $Q_f=1)$ lines. In $(b)$ and $(c)$, orange and purple contours are $|\zeta_c|=\zeta_{Df}$ and $|\zeta_c|=\Delta\zeta_i$. $(c)$ Schematic of the $\log_{10}(\eta)-\log_{10}(-\zeta_c)$ space, showing the $5$ asymptotic regimes. The points in the parameter space for which the profiles are plotted in $(a)$ are marked with green dots, and the sonic ion exit contour $P_f = 1$ is shown in blue
• Figure 3: Ion and electron density profiles for $\eta=10^{-2}$ and $\zeta_c=-100$ are plotted with colored lines. The shielded electrostatic field is shown in green. The ions vary on the $\zeta-$scale $\Delta\zeta_i = Q_f\approx 8.578\times 10^{-2}$ while electron density and the field vary on the scale $\zeta_{Df} \approx 16.49$
• Figure 4: Ion density profile (Purple) and electron density profile (Yellow) for $\eta = 10^{-2}$ and $\zeta_c = -0.126$. The black, dashed line is the analytic approximation, equation \ref{['Le_Lz_Approx_Ni']}, for the $N_i$ profile.
• Figure 5: Plasma profiles for $\eta=36$ and $\zeta_c=-10^{-3}$ : $(a)$ Ion density profile in purple solid line, and its analytical approximation near $\zeta_c$ and origin (Eq.\ref{['den_3,4_near0']}) in black dashed line; $N_i$ and $N_e$ in the ion density tail region are shown in the inset plot. $(b)$ Ion velocity(blue solid line) and electric field(green solid line) profiles