Branching of High-Current Relativistic Electron Beam in Porous Materials

Propagation of high-current relativistic electron beam (REB) in plasma is relevant to many high-energy astrophysical phenomena as well as applications based on high-intensity lasers and charged-particle beams. Here, we report a new regime of beam-plasma interaction arising from REB propagation in medium with fine structures. In this regime, the REB cascades into thin branches with local density a hundred times the initial value and deposits its energy 2 orders of magnitude more efficiently than that in homogeneous plasma, where REB branching does not occur, of similar average density. Such beam branching can be attributed to successive weak scatterings of the beam electrons by the unevenly distributed magnetic fields induced by the local return currents in the skeletons of the porous medium. Results from a model for the excitation conditions and location of the first branching point with respect to the medium and beam parameters agree well with that from pore-resolved particle-in-cell simulations.

Figure 1

(a) Density (in units of $n_{b0}$, same below) of Si atoms in the pristine foam. Densities at $t=333\mathrm{fs}$ of the REB as it propagates in (b) foam and (c) a homogeneous target at the same average density of the foam. The inset in (c) shows an enlargement of the marked region ($49.5\mathrm{\mu }\mathrm{m}<x<50.5\mathrm{\mu }\mathrm{m}$, $-0.5\mathrm{\mu }\mathrm{m}<y<0.5\mathrm{\mu }\mathrm{m}$). The electron density at the caustics in (b) is up to $\sim 100n_{b0}$. (d) Evolution of the energy loss $\eta =1-⟨E_{k,b}⟩/⟨E_{k,b0}⟩$ of the REB. Here, $⟨E_{k,b}⟩$ is the average kinetic energy of all the beam electrons in the simulation box and $⟨E_{k,b0}⟩$ corresponds to its initial value. (e) Evolution of the target electron and ion kinetic energies ${\mathcal{E}}_{e^{-}}$ and ${\mathcal{E}}_{\mathrm{ions}}$ in the region $13.5\mathrm{\mu }\mathrm{m}<x<25.5\mathrm{\mu }\mathrm{m}$, $-5.5\mathrm{\mu }\mathrm{m}<y<5.5\mathrm{\mu }\mathrm{m}$. (f) Evolution of the azimuthal magnetic energy ${\mathcal{E}}_{B_z}$, longitudinal and lateral electric energy ${\mathcal{E}}_{E_x}$ and ${\mathcal{E}}_{E_y}$ in the region.

Figure 2

Distributions of (a) longitudinal current density $j_x$ (in units of $|j_{b0}|$), (b) azimuthal magnetic field $B_z$ (in tesla), (c) lateral, and (d) longitudinal electric field $E_y$ and $E_x$ in (V/m) at $t=333\mathrm{fs}$ in the selected region in (e). (e) The lateral force $F_y=e(c{B}_z-E_y)$ (in arbitrary units) acting on the beam electrons. The dashed lines mark the selected region ($13.5\mathrm{\mu }\mathrm{m}<x<25.5\mathrm{\mu }\mathrm{m}$, $-5.5\mathrm{\mu }\mathrm{m}<y<5.5\mathrm{\mu }\mathrm{m}$).

Figure 3

Phase space $(y,p_y)$ of the beam electrons at (a1) $x=0.1\mathrm{\mu }\mathrm{m}$, (a2) $4.3\mathrm{\mu }\mathrm{m}$, (a3) $24.3\mathrm{\mu }\mathrm{m}$, where the red dashed lines mark the singularity regions, (a4) $44.3\mathrm{\mu }\mathrm{m}$, and (a5) $80\mathrm{\mu }\mathrm{m}$. (b) Scintillation index $\mathrm{\Sigma }$ corresponding to the beam electron density (black dashed curve). The light-blue dashed curves are results from three other foam samples with the same statistic properties. The blue solid curve shows the ensemble average of the four samples.

Figure 4

The distance $d_0$ for different beam Lorentz factors $\gamma$ (in red, with $n_{b0}=1.72\times {10}^{25}{\mathrm{m}}^{-3}$ and $l_c=8\mathrm{\mu }\mathrm{m}$), initial beam density $n_{b0}$ (in green, with $\gamma =100$ and $l_c=8\mathrm{\mu }\mathrm{m}$), and average pore size $l_c$ (in blue, with $n_{b0}=1.72\times {10}^{25}{\mathrm{m}}^{-3}$ and $\gamma =100$). The error bars and central markers represent the standard deviation and mean value of $d_0$, obtained from simulations of four foam samples. The curves are from Eq. (4).

Figure 5

(a) Beam density (colored) at $t=120\mathrm{fs}$ from a 3D simulation for REB branching in porous foam. The background gray color represents the pristine foam structure. (b) Transverse cut of the beam density at $x\sim 8.7\mathrm{\mu }\mathrm{m}$ (marked by the red rectangle) showing the caustics. The electron density at the caustics exceeds $\sim 370n_{b0}$.