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Identification of electrostatic two-stream instabilities associated with a laser-driven collisionless shock in a multicomponent plasma

Youichi Sakawa, Yutaka Ohira, Rajesh Kumar, Alessio Morace, Leonard N. K. Döhl, and Nigel Woolsey
Phys. Rev. E 104, 055202 – Published 4 November 2021
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  • INTRODUCTION
  • PARTICLE-IN-CELL SIMULATION IN A…
  • LINEAR ANALYSIS OF INSTABILITIES
  • DISCUSSION
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Electrostatic two-stream instabilities play essential roles in an electrostatic collisionless shock formation. They are a key dissipation mechanism and result in ion heating and acceleration. Since the number and energy of the shock-accelerated ions depend on the instabilities, precise identification of the active instabilities is important. Two-dimensional particle-in-cell simulations in a multicomponent plasma reveal ion reflection and acceleration at the shock front, excitation of a longitudinally propagating electrostatic instability due to a nonoscillating component of the electrostatic field in the upstream region of the shock, and generation of up- and down-shifted velocity components within the expanding-ion components. A linear analysis of the instabilities for a C2H3Cl plasma using the one-dimensional electrostatic plasma dispersion function, which includes electron and ion temperature effects, shows that the most unstable mode is the electrostatic ion-beam two-stream instability (IBTI), which is weakly dependent on the existence of electrons. The IBTI is excited by velocity differences between the expanding protons and carbon-ion populations. There is an electrostatic electron-ion two-stream instability with a much smaller growth rate associated with a population of protons reflecting at the shock. The excitation of the fast-growing IBTI associated with laser-driven collisionless shock increases the brightness of a quasimonoenergetic ion beam.

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    • Received 28 April 2021
    • Revised 24 August 2021
    • Accepted 13 October 2021

    DOI:https://doi.org/10.1103/PhysRevE.104.055202

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Laser driven ion accelerationPlasma instabilities
    Plasma Physics

    Authors & Affiliations

    Youichi Sakawa1,*, Yutaka Ohira2, Rajesh Kumar3, Alessio Morace1, Leonard N. K. Döhl4,†, and Nigel Woolsey4

    • 1Institute of Laser Engineering, Osaka University, Suita, Osaka 565-0871, Japan
    • 2Department of Earth and Planetary Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
    • 3Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
    • 4York Plasma Institute, Department of Physics, University of York, Heslington, York YO10-5DD, United Kingdom

    • *sakawa-y@ile.osaka-u.ac.jp
    • †Present address: Glen Eastman Energy b.v., Van Nelleweg 1, Expeditiegebouw, 3044 BC Rotterdam, The Netherlands.

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    Issue

    Vol. 104, Iss. 5 — November 2021

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    • Figure 1
      Figure 1

      The temporal evolution of (a)–(g) proton phase space and (h)–(n) their corresponding velocity spectrum taken at Δx = 3 μm in the upstream region (shown by vertical dotted lines in phase space) in a C2H3Cl plasma. The color scale shows the number of ions in a log scale.

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    • Figure 2
      Figure 2

      The velocity spectrum (open circles) of the upstream expanding protons taken at Δx = 3 μm of the upstream region in a C2H3Cl plasma at (a) t = 2.0, (b) 3.0, and (c) 4.0 ps, which is replotted from Fig. 1. Note that the y axis is in the linear scale. A sum (black line) of the three 1D shifted-Maxwellian distributions [low (green line), medium (red line), and high (blue line) velocity components] are used to fit the upstream expanding protons.

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    • Figure 3
      Figure 3

      Temporal evolution of the upstream expanding protons taken at Δx = 3 μm of the upstream region in a C2H3Cl plasma. (a) Peak number density dN/dv, (b) velocities v/c at the peak number density, and (c) proton temperature TP of the low (L), medium (M), and high (H) velocity components of the fitted 1D shifted-Maxwellian distributions.

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    • Figure 4
      Figure 4

      Results of the linear analysis of the electrostatic two-stream instability for a C2H3Cl plasma when ne=0 and Ti=2×10−5 MeV. [(a) and (c)] The imaginary part (Im [ω]/ωpe) and [(b) and (d)] real part (Re [ω]/ωpe) of normalized frequencies versus the normalized wave number in the x direction (kv0/ωpe, where v0=vP−ref−vCl). Panels (a) and (b) [(c) and (d)] are plotted for kv0/ωpe<5[kv0/ωpe<0.35]. The thin red, blue, green, and orange lines in (b) and (d) are the fast and slow Ref-P, Exp-P, Exp-C, and Cl modes, respectively.

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    • Figure 5
      Figure 5

      Ion species effect of the electrostatic two-stream instability for a C2H3Cl plasma when ne=0 and Ti=2×10−5 MeV. [(a), (c), (e), (g), (i), and (k)] The imaginary part (Im [ω]/ωpe) and [(b), (d), (f), (h), (j), and (l)] real part (Re [ω]/ωpe) of normalized frequencies versus the normalized wave number in the x direction (kv0/ωpe). The thin red, blue, green, and orange lines in (b) and (d) are the fast and slow Ref-P, Exp-P, Exp-C, and Cl modes, respectively. Panels (a), (b), (e), (f), (i), and (j) [(c), (d), (g), (h), (k), and (l)] are plotted for kv0/ωpe<5[kv0/ωpe<0.35]. [(a), (b), (c), and (d)] nP−ref=0; [(e), (f), (g), and (h)] nP−exp=0; and [(i), (j), (k), and (l)] nC=0.

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    • Figure 6
      Figure 6

      Results of the linear analysis of the electrostatic two-stream instability for a C2H3Cl plasma when ne=7.0×1020cm−3 and Ti=2×10−5 MeV. [(a) and (c)] The imaginary part (Im [ω]/ωpe) and [(b) and (d)] real part (Re [ω]/ωpe) of the normalized frequencies versus normalized wave number in the x direction (kv0/ωpe. Panels (a) and (b) [(c) and (d)] are plotted for kv0/ωpe<5[kv0/ωpe<0.35]. The thin red, blue, green, and orange lines in (b) and (d) are the fast and slow Ref-P, Exp-P, Exp-C, and Cl modes, respectively.

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    • Figure 7
      Figure 7

      Results of the linear analysis of the electrostatic two-stream instability for a C2H3Cl plasma when ne=0 and Ti=0.02 MeV. [(a) and (c)] The imaginary part (Im [ω]/ωpe) and (b), (d) real part (Re [ω]/ωpe) of normalized frequency versus the normalized wave number in the x direction (kv0/ωpe). Panels (a) and (b) [(c) and (d)] are plotted for kv0/ωpe<3[kv0/ωpe<0.35]. The thin red, blue, green, and orange lines in (b) and (d) are the fast and slow Ref-P, Exp-P, Exp-C, and Cl modes, respectively.

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    • Figure 8
      Figure 8

      Results of the linear analysis of the electrostatic two-stream instability for a C2H3Cl plasma when ne=7.0×1020cm−3 and Ti=0.02 MeV. [(a) and (c)] The imaginary part (Im [ω]/ωpe) and [(b) and (d)] real part (Re [ω]/ωpe) of normalized frequency versus the normalized wave number in the x direction (kv0/ωpe). Panels (a) and (b) [(c) and (d)] are plotted for kv0/ωpe<3[kv0/ωpe<0.35]. The thin red, blue, green, and orange lines in (b) and (d) are the fast and slow Ref-P, Exp-P, Exp-C, and Cl modes, respectively.

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    • Figure 9
      Figure 9

      Results of the linear analysis of the electrostatic two-stream instability for a C2H3Cl plasma including expanding C6+ ions, Cl15+ ions, protons, and reflected protons. The normalized growth rate (Im [ω]/ωpe) versus the wave number in the x direction (kv0/ωpe). Ion temperature effect of the unstable modes for (a) ne=0 and (b) ne=7.0×1020cm−3 for Ti=2×10−5 MeV (red marks), Ti=0.01 MeV (blue marks), Ti=0.02 MeV (green marks), and Ti=0.4 MeV (black marks). Ref-P mode (small-k roots, filled circles), Exp-P mode (medium-k roots, filled triangles), and Exp-C mode (large-k roots, filled diamonds) are shown. Arrows are guide to eyes.

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    • Figure 10
      Figure 10

      The temporal evolution of (a) the peak amplitude of the electrostatic fluctuation Ex,k and (b) the dominant kx in the PIC from the power spectrum of Ex taken at the width of Δx = 10 μm in a few μm upstream region of a shock in a C2H3Cl plasma. Solid and dotted lines in (a) represent growth rates of Ex, γ=1.0×1012s−1 and 2.4×1011s−1, derived from the exponential fits to the data at t=2.25–2.75 ps and 2.75–4.0 ps, respectively.

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    • Figure 11
      Figure 11

      Proton temperature TP dependence of (a) the normalized maximum growth rate (γmExp−P/ωpe) and (b) the normalized wave number in the x direction (kmExp−Pv0/ωpe) of Exp-P mode obtained from the linear analysis for ne=7.0×1020cm−3 at 4.0 ps. The drift velocity of the expanding protons vP−exp used in the linear analysis is 0.075c (filled circles) and 0.060c (open circles). The normalized growth rate (γPIC/ωpe) and the wave number in the x direction (kxPICv0/ωpe) obtained from the PIC simulation at 4.0 ps are 5.6×10−4 [horizontal solid line in (a)] and 0.37, respectively.

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    • Figure 12
      Figure 12

      (a) The electron phase space γv/c versus x, in a C2H3Cl plasma at t = 4.0 ps. The color scale shows the number of electrons in a log scale. (b) The electron energy spectrum taken at Δx = 20 μm in the upstream region of the shock front in a C2H3Cl (circles) plasma at t = 4.0 ps. The electron energy spectrum is fitted with a 2D-relativistic Maxwellian f(E)=aEexp(−E/Te) (solid line), where a is constant, E is the energy of electrons, Te gives the electron temperature in the upstream region. (c) The temporal evolution of Te (open triangles) calculated by fitting a 2D-relativistic Maxwellian at each time. The Te is very well fitted with a sigmoid function, S(t)=1/(1+e−at) (blue solid line), and the derivative (dTe/dt) of S(t) (dot-dashed line), which peaks at t = 1.75 ps. Here a is a fitting constant. Normalized temporal evolution of the laser intensity (black solid line), which peaks at t = 1.5 ps, is also shown as a reference.

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    • Figure 13
      Figure 13

      (a) The temporal evolution of ETNSA measured from the PIC (open circles, red line is guide to eyes) and derived from the vCl (blue lines) shown in (b). (b) The velocity of Cl15+ ions vCl (open triangles) derived from the peak of velocity spectrum dN/dvCl taken at Δx = 3 μm in the upstream region. The vCl is fitted to the 2nd order polynomial from t = 1.0 to 2.75 ps (green line), and a logarithmic curve from t = 2.25 to 4.0 ps (blue line). In (a), ETNSA derived from the vCl follows a ∝t from t = 1.0 to 2.75 ps (blue dotted line) and a 1/t dependences from t = 2.25 to 4.0 ps (blue solid line). Note that the x axis is in the log scale.

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