Ion acceleration at two collisionless shocks in a multicomponent plasma

Intense laser-plasma interactions are an essential tool for the laboratory study of ion acceleration at a collisionless shock. With two-dimensional particle-in-cell calculations of a multicomponent plasma we observe two electrostatic collisionless shocks at two distinct longitudinal positions when driven with a linearly polarized laser at normalized laser vector potential $a_0$ that exceeds 10. Moreover, these shocks, associated with protons and carbon ions, show a power-law dependence on $a_0$ and accelerate ions to different velocities in an expanding upstream with higher flux than in a single-component hydrogen or carbon plasma. This results from an electrostatic ion two-stream instability caused by differences in the charge-to-mass ratio of different ions. Particle acceleration in collisionless shocks in multicomponent plasma are ubiquitous in space and astrophysics, and these calculations identify the possibility for studying these complex processes in the laboratory.

Figure 1

The normalized initial electron density profile used in PIC simulations for $a_0=3.35$. The laser is from the left-hand side of the simulation box. The density profile consists of an exponentially increasing $5\mu \mathrm{m}$ scale-length laser-irradiated front region, followed by $5\mu \mathrm{m}$ uniform central region, and an exponentially decreasing rear-side profile with $30\mu \mathrm{m}$ scale length. To avoid boundary effects, the simulations use 40 and $100\mu \mathrm{m}$ vacuum regions at the front and rear of the target, respectively.

Figure 2

Phase-space of (a) protons and (b) ${\mathrm{C}}^{6+}$ ions for ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$ plasma at $a_0=3.35$ and at $t=4.0$ ps. The horizontal lines represent the lower threshold $v_L^i$ for ion reflection and the shock velocity in proton density $V_{sh}^{\mathrm{P}}$. The color scale shows the number of ions in a log scale.

Figure 3

(a) The electrostatic field $E_x$ (left axis, blue line) and potential $\varphi$ (right axis, red line) at $t=3.0\mathrm{ps}$. The normalized proton $n_H/n_{cr}$ (left axis, blue line) and carbon $n_{\mathrm{C}}/n_{cr}$ (right axis, red line) densities at (b) $t=3.0\mathrm{ps}$ and (c) $t=4.0\mathrm{ps}$ in ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$ plasma for $a_0=10$. Panel (a) is shown across a narrow longitudinal range compared with panels (b) and (c).

Figure 4

The first and second columns indicate the ion phase-space and the velocity spectra respectively for $a_0=10$ and at $t=4.0\mathrm{ps}$. The velocity spectra are taken in the upstream region immediately in front of the shock across $\mathrm{\Delta }x=3\mu \mathrm{m}$. (a), (b) Results for protons from a single-component H plasma. (c), (d) Results for protons from a ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$ plasma. (e), (f) Results for ${\mathrm{C}}^{6+}$ ions from a ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$ plasma. (g), (h) Results for ${\mathrm{C}}^{6+}$ ions in single-component C plasma. The vertical lines in phase- space in panels (a), (c), (e), and (g) identify the position of the shock front. In panels (b), (d), (f), and (h), moving left to right, the dotted lines indicate the positions of the lower threshold velocity $v_L^i$, shock velocity $V_{sh}^i$, and the maximum velocity of the reflected ions, $2V_{sh}^i-v_L^i$. The color scale shows the number of ions on a log scale.

Figure 5

The phase space for (a) protons and (b) ${\mathrm{C}}^{6+}$ ions from a $\mathrm{CH}$ plasma for $a_0=10$ and at $t=4.0\mathrm{ps}$. The vertical lines identify the positions of the shock front associated with the protons (solid line) and the ${\mathrm{C}}^{6+}$ ions (dashed line). The horizontal lines indicate the positions of the lower threshold velocity $v_L^{\mathrm{P}}$ and shock velocity $V_{sh}^{\mathrm{P}}$ of protons, and the maximum velocity of the reflected protons, $2V_{sh}^{\mathrm{P}}-v_L^{\mathrm{P}}$. The color scale shows the number of ions in a log scale.

Figure 6

(a) The electron energy distribution taken at $t=4.0\mathrm{ps}$ in the upstream region of the shock front for different laser intensities corresponding to $a_0=3.35$ ($\times$), 10 ($+$), 20 (-), and 33 ($|$) for ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$. A sum of two (bulk and tail) 2D relativistic Maxwellian is used to fit to the electron energy distribution shown by the solid lines for $a_0=10$, 20, and 33. The bulk (dotted line) and tail (dashed line) components for $a_0=33$ are shown. (b) The electron temperatures as a function of $a_0$.

Figure 7

The $a_0$ dependence of (a) shock velocities $V_{sh}^i$, (b) mean velocities of the expanding ions $v_m^i$, (c) ion-acoustic velocities $c_s^i$, (d) difference between the shock velocity and mean velocity of the expanding ions $v_{df}^i=V_{sh}^i-v_m^i$, and (e) the corresponding Mach number $M^i=v_{df}^i/c_s^i$ at $t=4.0$ ps for protons in a single-component H plasma ($\circ$), ${\mathrm{C}}^{6+}$ ions in a single-component C plasma ($▵$), and protons () and ${\mathrm{C}}^{6+}$ ions () in a multicomponent ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$ plasma.

Figure 8

(a) The temporal evolution of shock positions $X_{sh}$ and shock velocities $V_{sh}$ for protons in a single-component H plasma (shown in red) and a multicomponent ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$ plasma (shown in blue) for $a_0=3.35$. $X_{sh}$ data are shown as open circles (a single-component H plasma: ) and open triangles (a multicomponent ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{C}\mathrm{l}$ plasma: ). The derivative of $X_{sh}$ with respect to time gives $V_{sh}$. $X_{sh}$ (dotted lines) and $V_{sh}$ (solid lines) rise exponentially with time. (b) The temporal evolution of $X_{sh}$ and $V_{sh}$ for protons (shown in red) and ${\mathrm{C}}^{6+}$ ions (shown in blue) in a multicomponent ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$ plasma for $a_0=33$. $X_{sh}$ data are shown as open circles (protons: ) and open triangles (${\mathrm{C}}^{6+}$ ions: ). The time dependencies of $X_{sh}$ and $V_{sh}$ are best represented by a third-order (dotted lines) and second-order (solid lines) polynomials, respectively.

Figure 9

The $a_0$ dependence of (a) energy $E$ per nucleon and (b) the number $dN/dE$ at $E$ of reflected protons and ${\mathrm{C}}^{6+}$ ions at the peak of the energy distribution at $t=4.0$ ps for protons in single-component H ($◯$), ${\mathrm{C}}^{6+}$ ions in single-component C ($▵$), and protons () and ${\mathrm{C}}^{6+}$ ions () in ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$ plasmas.

Figure 10

The spatial profile of electrostatic potentials in (a) a multicomponent ${\mathrm{C}}_2{\mathrm{H}}_3\mathrm{Cl}$ plasma and (b) a single-component hydrogen plasma at $t=2.5$ (blue curves) and 4.0 ps (red curves) for $a_0=3.35$. The vertical lines indicate the position of the shock fronts.