Experimental discrimination of ion stopping models near the Bragg peak in highly ionized matter

Abstract

The energy deposition of ions in dense plasmas is a key process in inertial confinement fusion that determines the α-particle heating expected to trigger a burn wave in the hydrogen pellet and resulting in high thermonuclear gain. However, measurements of ion stopping in plasmas are scarce and mostly restricted to high ion velocities where theory agrees with the data. Here, we report experimental data at low projectile velocities near the Bragg peak, where the stopping force reaches its maximum. This parameter range features the largest theoretical uncertainties and conclusive data are missing until today. The precision of our measurements, combined with a reliable knowledge of the plasma parameters, allows to disprove several standard models for the stopping power for beam velocities typically encountered in inertial fusion. On the other hand, our data support theories that include a detailed treatment of strong ion-electron collisions.

Figure 1. Experimental set-up.

(a) Schematics of the experimental set-up. Two high-energy laser beams are focused on a 100 μg cm⁻² thick carbon foil with a 1 mm diameter obtained by beam smoothing using RPP. The plasma electron density is measured with a laser interferometry diagnostic. The ion beam is collimated through a 0.5 mm diameter pinhole and degraded through a carbon foil before interacting with the plasma, and it is detected after a 462 mm TOF distance. (b) Time-integrated picture of one plasma shot registered with a digital camera. (c) Picture of the TOF detector displaying the ten diamond samples mounted on their printed circuit board.

Figure 2. Plasma characterization.

(a) Raw interferometry data for the plasma for t=11 ns (right), compared with the reference measurement (left). The target surface is located on the left side of each picture. (b) Comparison between measured and simulated free electron density (n_e) profiles along the ion axis (x) for t=7 and 11 ns respectively. Experimental error bars, of approximatively 20%, are not represented. (c) 2D map of the simulations of the free electron density (n_e) for t=7 ns, in units of cm⁻³ and represented in logarithmic scale. The solid arrow stands for the ion axis (1D plasma profile) used in the energy-loss calculations throughout the paper. The dotted arrows delimit the transversal region considered for the energy-loss calculation using a 2D plasma profile in Fig. 3c. (d) Energy-loss measurements for the same plasma versus the probing time, for argon projectile ions at a velocity ratio v_p/≈3.1. The data are compared with the predictions of the LP and the TM stopping-power models applying a Monte-Carlo description as well as the Gus'kov and the Kreussler models for the effective projectile charge state. The energy loss is normalized to its value in the solid target (100%), as well as the plasma areal density (ρR). The error bars correspond to one s.d. (1σ) of the uncertainty in the time shifts of the signals obtained from the TOF measurements.

Figure 3. Energy-loss results.

(a) Effective charge state of a nitrogen bunch in the plasma versus time according to a Monte-Carlo description as well as the Kreussler and the Gus'kov models, compared with the mean charge state in the solid target Z_sol=4.88. All values are averaged over the ion trajectory through the target. (b) Measured energy loss as a function of the bunch probing time and normalized to its value in the solid target (taken to be 100%) compared with the predictions of the LP and the TM stopping-power models applying the Monte-Carlo, Kreussler and Gus'kov projectile charge models respectively. The simulated target areal density (ρR), also normalized to its value in the solid target, represents the 3D plasma expansion dynamics. (c) Measured energy loss compared with the predictions of the LP, TM as well as BPS stopping-power models using the Gus'kov projectile charge model. The shaded areas show the differences between calculations considering the 1D (upper lines) or 2D plasma profile (bottom lines) respectively (cf. Fig. 2c). (d) Measured energy loss compared with calculations for the LP and TM stopping-power models using the Gus'kov projectile charge model, corresponding to the originally simulated density and temperature profiles (LP; TM), densities =n_e/2 and temperatures =T_e−40 eV (LP max; TM max) as well as densities =2 n_e and temperatures =T_e+40 eV (LP min; TM min), respectively. The shaded areas thus illustrate the maximum error in the energy-loss calculation due to uncertainties in the plasma parameters. Due to time averaging over the 5.5 ns bunch, the beam charge state in a as well as the energy loss in (b–d) for t=0 ns, are already larger than their respective values in the solid target. The error bars on the energy loss correspond to one s.d. (1σ) of the uncertainty in the time shifts in the detector signals.

Figure 4. Raw data from the TOF detector.

Sample of the detector signals through the solid target and through the plasma, for the shot corresponding to the data point at t=6.4 ns in Fig. 3b–d. The time shift (Δt) of each bunch is determined between the centre of mass of this bunch (plain vertical bar) and the associated reference time (dotted vertical bar). The shown reference times are extrapolated from the period of the undisturbed ion bunch signals few tens of μs after the plasma expansion. The average measured time shift in the solid foils (degrader, target and filter) is Δt=34.06±0.45 ns. Knowing the initial areal density of the target of 95±1 μg cm⁻² as well as of the filter, the projectile energy after the degrader is deduced to be E_p=0.576±0.005 MeV per nucleon, the time shift through the solid target Δt=2.39±0.45 ns and the energy loss in the solid target ΔE=0.82±0.14 MeV. Consequently, the first ion bunch that passed through the plasma features a time shift in the plasma target of Δt=3.43±0.47 ns (the global shift being Δt=35.10 ns). This corresponds to an energy loss in the plasma of ΔE=1.14±0.15 MeV, or 139.5±18.3% normalized to the value in the solid target.