Spatiotemporal dynamics of ultrarelativistic beam-plasma instabilities

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Spatiotemporal dynamics of ultrarelativistic beam-plasma instabilities

P. San Miguel Claveria et al.

Phys. Rev. Research 4, 023085 – Published 2 May 2022

Abstract

An electron or electron-positron beam streaming through a plasma is notoriously prone to microinstabilities. For a dilute ultrarelativistic infinite beam, the dominant instability is a mixed mode between longitudinal two-stream and transverse filamentation modes, with a phase velocity oblique to the beam velocity. A spatiotemporal theory describing the linear growth of this oblique mixed instability is proposed which predicts that spatiotemporal effects generally prevail for finite-length beams, leading to a significantly slower instability evolution than in the usually assumed purely temporal regime. These results are accurately supported by particle-in-cell (PIC) simulations. Furthermore, we show that the self-focusing dynamics caused by the plasma wakefields driven by finite-width beams can compete with the oblique instability. Analyzed through PIC simulations, the interplay of these two processes in realistic systems bears important implications for upcoming accelerator experiments on ultrarelativistic beam-plasma interactions.

Received 27 June 2021
Accepted 28 February 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.023085

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Advanced accelerator test facilities Laboratory studies of space & astrophysical plasmas Plasma instabilities Plasma-beam interactions

Particle-in-cell methods

Plasma PhysicsAccelerators & Beams

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Article Text

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Issue

Vol. 4, Iss. 2 — May - July 2022

Subject Areas

Plasma Physics

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Images

Figure 1
Simulated instability dynamics for ultrarelativistic ( $γ_{b} = 2 \times 10^{4}$ ), dilute ( $α = 0.03$ ) electron beams of various normalized lengths ( $k_{p} σ_{x}$ ). (a) Snapshots of the beam density profile in the comoving coordinates $(ξ, τ) = (v_{b} t - x, t)$ for different beam lengths. (b) 2D Fourier spectrum of the $E_{y}$ field fluctuations at $c τ = 3338 k_{p}^{- 1}$ for $k_{p} σ_{x} = 10$ . (c) Transverse electric field $E_{y, rms} = {〈 E_{y}^{2} 〉}^{1 / 2}$ (solid line) and magnetic field $B_{z, rms} = {〈 B_{z}^{2} 〉}^{1 / 2}$ (dashed line) averaged over $ξ \in [ξ_{peak} - σ_{x} / 2, ξ_{peak} + σ_{x} / 2]$ ( $ξ_{peak}$ the position of the beam center in the comoving coordinates) as a function of the beam propagation distance in the plasma $(c τ)$ and the beam length. The dotted line plots the theoretical growth of the OTSI, Eq. (1), in the infinite-beam case. The evaluation of the dominant $k_{⊥}$ in Eq. (1) is carried out using the electrostatic result $〈 E_{y}^{2} 〉 / 〈 E_{x}^{2} 〉 ≃ {(k_{⊥} / k_{p})}^{2}$ . (d) Effective growth rate ( $Γ / Γ_{OTSI}$ ) vs $k_{p} σ_{x}$ within the central slice of the beam (see text for details).
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Figure 2
2D PIC simulations of the OTSI induced by a steplike $e^{-} e^{+}$ pair beam and comparison with linear theory in the spatiotemporal regime for $γ_{b} = 2 \times 10^{4}$ (a)–(c) and the temporal regime for $γ_{b} = 20$ (d). (a), (d) Spectral amplitude $| {\tilde{E}}_{y} (k_{x}, k_{⊥}) |$ of the dominant oblique mode ( $k_{x} = k_{p}$ , $k_{⊥} ≃ 3 k_{p}$ ) as a function of $ξ$ for different propagation distances $c τ$ . (b) Same quantity but as a function of $c τ$ for different beam slices $ξ$ . In (a) and (b), the simulation data (solid lines) is fitted to the theoretical law $A exp [(3 / 2^{2 / 3}) Γ_{OTSI} {(ξ / c)}^{1 / 3} τ^{2 / 3}]$ for $ξ \leq ξ_{sat}$ (dashed lines). (c) Saturation position $ξ_{sat}$ [also shown in (a) as circles] vs $c τ$ (filled circles), compared with the theoretical expectation $ξ_{sat} \propto τ^{- 2}$ (red dashed line). Dashed lines in (d) plot the theoretical temporal growth of $| {\tilde{E}}_{y} (k_{p}, 3 k_{p}) |$ at different times $c τ \geq 17 k_{p}^{- 1}$ .
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Figure 3
Simulated electron-beam-density maps at different propagation distances in a uniform plasma of density $n_{p} = 1 \times 10^{19} {cm}^{- 3}$ (a)–(f), $2.5 \times 10^{19} {cm}^{- 3}$ (g)–(l), and $5 \times 10^{19} {cm}^{- 3}$ (m)–(r). The transverse beam profile is taken to be either finite with $σ_{r} = 10 μ m$ rms width [(a)–(c), (g)–(i), and (m)–(o)] or infinite, i.e., with transverse periodic boundary conditions [(d)–(f), (j)–(l), and (p)–(r)]. In all cases, the beam has a 10-GeV energy ( $γ_{b} = 2 \times 10^{4}$ ), a Gaussian longitudinal profile with $σ_{x} = 5 μ m$ rms length, a transverse normalized emittance $ε_{n} = 3 mm mrad$ , and a peak density $n_{b} ≃ 1.5 \times 10^{18} {cm}^{- 3}$ [i.e., $α ≃ 0.15$ for (a)–(f), $α ≃ 0.06$ for (g)–(l), and $α ≃ 0.03$ for (m)–(r)], which would correspond to a total beam charge of $2 nC$ in 3D. The insets show the transverse beam phase space along the slices indicated by the dashed blue lines.
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