Measurements of ion velocity separation and ionization in multi-species plasma shocks

Shocks provide a valuable testbed for fundamental studies of high-energy-density (HED) physics. Near a strong shock front, a plasma is not accurately treated using the hydrodynamic equations: instead, the details of microscopic physical processes must be evaluated to understand the shock structure.^1,2 These processes can include collisional evolution of the particle velocity distribution functions in the case of collisional shocks,^3,4 and additionally coupling to the electromagnetic fields in collisionless shocks.^5–8 The microscopic processes can influence macroscopic quantities, such as thermal and electrical conduction, which are important to astrophysical scenarios and applications such as inertial confinement fusion (ICF). However, in collisional shocks, few experimental studies of the structure inside a shock front have been performed.^9,10 Additionally, when multiple ion species are present, separation of the ion species within the shock introduces an additional structure in the shock front.^11,12

Theoretical investigations of strong plasma shock fronts have found that the shock-front structure depends on the strength of the shock. Zel'dovich and Raizer derive a simple scaling for the width of the electron preheat layer, the region in front of the density jump with increased electron temperature, as the square-root of the ratio of ion and electron masses times the ion-ion mean-free-path in the shocked region (⁠ $λiimi/me$⁠).¹³ Despite being an order-of-magnitude estimate, this expression accurately predicts the scale of the preheat layer in kinetic shock simulations.^14,15 These kinetic simulations, however, demonstrate significantly non-thermal behavior of the ions, notably the coexistence of a forward-streaming and a background ion distribution in the preheat region. This prediction has recently been confirmed by experimental evidence using Thomson scattering in a hydrogen plasma forming a strong collisional shock front.¹⁰ Other recent experiments have studied the structure of electric fields associated with strong plasma shocks.⁹ However, much work remains to be done on the general characterization of collisional shock-front structure in plasmas, including multiple-species effects in the shock front and the scaling of the shock width with shock strength.

This manuscript presents measurements of the collisional shock front structure in a two-species plasma. In experiments using the 60-beam OMEGA laser,¹⁶ strong shocks were driven into a gas volume containing two elements, hydrogen and neon in an atomic ratio of 98:2. The volume was interrogated using a 263.25 nm (4ω) probe beam impulse, and the Thomson-scattered light was recorded and used to infer temperature, density, and flow velocity as the shock passed through the probed region. These experiments demonstrate velocity separation between two populations of hydrogen and change in the neon ionization state within the shock front.

This paper is organized as follows: Section II describes the design and Sec. III the results of the experiments. Section IV interprets these results in comparison to theoretical models, discusses the neon ionization rate and the shock width, and compares with the results of kinetic ion Fokker-Planck simulations. Finally, Sec. V concludes.

The experimental layout is shown in Fig. 1. The target was a cylindrical polyimide tube, 6 mm long with a 2.1 mm outer diameter and a 50 μm thick wall, filled with 1 atm (0.111 mg/cc) gas. Ten beams containing a total of 2.1 kJ in a 0.6 ns square pulse were used to drive an 800 μm diameter circular region on a 2 μm-thick ablator foil composed of silicon dioxide that was mounted on one end of the gas tube, launching a strong shock into the gas. To probe the experiment, the 1 μm-thick polyimide foils covering laser-cut diagnostic windows in the gas tube were destroyed using an additional 75 J of laser energy per window in a 0.6 ns square pulse, timed 3.7 ns before the probe beam. This delay was sufficient for the electron density in the destroyed window plasma to reduce below the critical density for 4ω light, allowing the probe beam and scattered light to pass.

The Thomson scattering system collects scattered light with a scattering angle of ∼60° and records it using two spectrometers.¹⁷ When configured for $4ω$ operation, a narrow-band spectrometer records light with small wavelength shifts in the range 261.5–264.5 nm, while a broad-band spectrometer records light with large wavelength shifts in the range 205–240 nm. These ranges are sensitive to light scattered from ion acoustic waves (IAW) and electron plasma waves (EPW), respectively. For this experiment, the EPW data were generally compromised by a broad thermal background, so only the IAW data will be discussed here. The k-vector for scattered light was aligned with the direction of shock propagation, such that shocked plasma flowing away from the ablator produced a blue-shift in the scattered light spectrum. The scattered light was collected from an ellipsoidal region defined by the overlap of the probe beam and telescope optics; the size of the sampled region in the direction of shock propagation was approximately 100 μm.

In addition to Thomson scattering, an x-ray framing camera was used to record time-resolved x-ray emission images of the target.¹⁸ The framing camera was positioned nearly perpendicular to both the axis of shock propagation (inner angle = 75°) and the window axis (87°), and recorded four pinhole images on each of four gated strips separated in time by 1 ns steps. The field of view for each image was 3.25 mm, centered 3.2 mm along the tube axis from the ablator. Pinholes were 50 μm in diameter. The images record x-ray self-emission from the plasma inside the shock tube, filtered by the 50 μm wall of the polyimide tube and 200 μm of beryllium at the camera.

Thomson scattering data were recorded on several experiments, mapping out the passage of the shock front between 2.7 and 5.2 ns after the drive, as shown in Fig. 2. With this scattering geometry and plasma state, the dimensionless scattering parameter $α≡1/kλDebye>1$⁠, so the spectrum is dominated by collective scattering.¹⁹ In this regime, the ion acoustic wave feature in the Thomson scattered spectrum encodes information about the flow velocity, temperature, and ion composition of the probed plasma.²⁰ The IAW data demonstrate arrival of the shock front in the sampling volume beginning at approximately 3.2 ns. Prior to this time, the scattered light spectrum is symmetric around the initial probe wavelength of 263.25 nm, indicating no flow velocity. After 3.2 ns, the scattered light begins to show a blue-shifted feature consistent with some ions flowing forward in the direction of the shock. By 4.5 ns, the feature appears to have reached a new equilibrium with a systematically blue-shifted ion feature. (The red-shifted feature that appears at 4.9 ns is likely to be probe light scattered into the telescope from gradients near the windows; a self-consistent interpretation of all three peaks as originating from the same sample region could not be found.) We can conclude from these data that the shock front is actively passing through the sampling volume for a period of about $Δt=1.2$ ns. The shock front velocity is approximately the free-streaming velocity (distance from the ablator divided by the sampling time, $vFS≡$ (2750 $μ$m)/(⁠ $tsample)≈$ 720 μm/ns). Using these two values, the shock front width is estimated to be approximately $Δtvsh≈860$ μm. This width is 8× the size of the probed volume, confirming that the data record the ion properties within the shock front.

Analysis of the IAW data is performed by forward-fitting of a scattered spectrum model to the data. The form factor of the scattered light is calculated using the susceptibility χ_j calculated for each ion population and for a population of electrons, as¹⁹ 

where $ϵ=χe+∑jχj$⁠. It is important to note that the susceptibility terms are complex numbers, and adding multiple ion terms does not in general produce a simple superposition of ion features. The fitting model also takes into account the finite size of both the probe and the telescope optics.²¹ In this dataset, the electron plasma wave (EPW) spectral data were not of sufficient quality to directly infer the electron density and temperature in the sampled region. Instead, an electron density was assumed based on the initial electron density of the gas, and the electron temperature was initially assumed to be equal to the temperature of the ion populations. After the fit of ion properties was complete, the electron temperature was allowed to vary to find a best fit. Because the spectral power of the scattered light is not absolutely calibrated for the OMEGA diagnostic, the relative fraction of each ion population is fit rather than the absolute density.

A model assuming two thermal, flowing populations of ions within the sampled volume was used to interpret the data. This two velocity population ansatz was introduced by Mott-Smith for strong shocks in gasses²² and was applied to interpret experiments studying strong shocks in a hydrogen plasma.¹⁰ Each population is defined as a flowing Maxwellian, resulting in five free parameters for the fit: flow velocity and temperature for each population, and the relative density of the two populations. The two-population fit successfully reproduced the features observed in the data after the arrival of the shock front, as shown in Fig. 3. In particular, the data appear to demonstrate the coexistence of a broad, forward-streaming hydrogen population, and a narrow, non-flowing population: these will be referred to as the ‘hot’ and ‘cold’ populations, respectively. While the hot population is defined as pure hydrogen, the cold population includes the initial fraction (2%) of neon.

The results of these fits to the data in the range 3.2–4.0 ns are shown in Fig. 4. Note that, as time increases, the shock is passing through the sample region: thus, later time corresponds to a changing position in the shock front, from upstream to downstream. At 3.3 ns, the fit shows the cold ion population to be dominant (80% of the ions). In this region, the cold population is not moving, and the hot population is flowing forward at $450 μ$ m/ns. Between 3.4–3.7 ns, the cold population begins to increase in velocity and temperature, and the hot population reaches its maximum velocity of 540 μm/ns. From 3.7–3.9 ns, the fraction of ions in the hot population begins to increase, ultimately becoming the dominant feature at 3.9 ns. The cold population continues to decrease in fraction, increase in velocity, and increase in temperature through the end of the signal. After 3.9 ns, the data were found to be fit better using a model with doubly ionized neon in the cold population, indicating a change in the ionization state within the shock. Note that the jump in the best-fit values at 3.3 ns is likely produced by the inaccuracy of using the two-population ansatz to approximate the actual ion distribution function in the experiment early in time. When using an approximate model, continuous changes in the underlying distribution can result in a rapid jump in the best fitting values. This will be discussed more in relation to the fully kinetic simulations presented in Sec. IV D.

The Thomson scattering data recorded in the range 4.6–5.2 ns appear to show a steady-state flow after the shock front has passed, with a prominent bifurcated ion-feature. The most plausible interpretation of these Thomson scattering data is that the plasma consists of a combination of shocked H + Ne plasma with an admixture of 10%–20% silicon dioxide plasma from the ablator, as shown in Fig. 5. A hydrogen/neon combination alone would be consistent with this feature only if the neon were fully ionized (which is unlikely at this stage, as will be discussed below) and the neon fraction had increased to 9%, more than four times the initial fraction. The flow velocity of the plasma in this period is $450±20 μ$m/ns, similar to the velocity of the hot population within the shock front. The ion temperature is inferred to be 520 ± 100 eV.

Self-emission x-ray images recorded on each experiment show the progress of plasma flowing down the shock tube: images recorded between 1.2 and 4.2 ns on one shot (80060) are shown in Fig. 6. A calculation of the total x-ray attenuation by these filters indicates ×10⁻³ transmission for silicon Kα x-rays at 1.74 keV, with rapid exponential decay (e-folding constant < 0.09 keV) below that value.²³ As the silicon produces the most energetic spectroscopic line in the experiment, and the silicon dioxide plasma should have comparable electron temperatures and higher or comparable electron densities to the shocked gas, the x-ray images are thus interpreted as representing the location of the residual ablator plasma. A tungsten wire was attached to the outside of the tube within the field of view of the x-ray camera, and is clearly visible in the data. Reference to the apparent position of this fiducial feature was used to measure the absolute position of the ablator plasma over the course of the experiment. At 3.2 ns (when the Thomson probe fires in this experiment), the x-rays indicate that the silicon is freely flowing down the shock tube. By 4.2 ns, the plasma blowing down from the ablated windows is observed to pinch off the flow of the silicon plasma; this should not affect the Thomson scattering data.

Analyzed data for the position of the gas/ablator interface for all four shots are plotted in Fig. 7(a). Excellent reproducibility is shown in the expansion of the silicon dioxide plasma away from the initial position for three shots. These trajectories are also well matched by the trajectory taken from a 1D-HYADES simulation²⁴ of the experiment (purple dashed line), assuming an electron flux limiter of 0.07 and laser absorption of 25% in the ablator. The fourth shot (80060) suggests a similar velocity, but with an offset in position of approximately 300–400 μm. This offset is somewhat larger than the estimated uncertainty in the position of the fiducial wire (∼100 μm); however, it is possible that the fiducial was compromised during handling. These trajectories confirm the interpretation of the post-shock Thomson scattering data as containing silicon dioxide in the sample position (2.75 mm from the ablator) at 4.6–5.4 ns. The Thomson data indicate that the plasma remains predominantly hydrogen at this time; the observed mixture could arise due to ion diffusion²⁵ and/or shock-induced kinetic mix at the gas/ablator interface.²⁶ This suggests that the observed x-ray feature does not correspond to a sharp interface between the initial gas and initial ablator and instead documents some contour of silicon density in the mix layer at this boundary.

While the absolute positioning and timing of the diagnostic may be uncertain, the velocity may be calculated with high precision by comparing the relative position and relative timing between multiple images recorded on the same shot. The results of this analysis are shown in Fig. 7(b); error bars represent standard deviation in the three comparisons recorded on each shot that were averaged to generate each datapoint and are approximately 10%. This analysis indicates that the velocity of the gas/ablator interface follows a similar trend on all shots, slowing with time. A linear model provided a good fit to the velocity datapoints with a reduced chi-square value of 1.42 and is included in the figure. Using this model, the velocity of the gas/ablator interface is estimated to be 640 μm/ns at 3.2 ns, when the shock is first observed in the sample region, and 530 μm/ns at 4.4 ns, after the shock has fully passed. From the Rankine-Hugoniot relations, the relationship between the shock front velocity and the velocity of the piston driving the shock is expected to be $vshock=vpiston(γ+1)/2≈vp×4/3$ for an ideal fluid with $γ=5/3$⁠. The predicted shock velocity from the x-ray-inferred piston velocity is thus in the range 850–700 μm/ns, which is similar to the observed shock velocity (720 μm/ns). Note that the velocity of the hot ion population inferred from Thomson scattering (∼500–560 μm/ns) is consistent with the lower end of the observed piston velocity range. After the shock passes the sample region, the model velocity (480 μm/ns at 4.9 ns) is also similar to the piston velocity inferred from Thomson scattering (450 ± 20). The slightly excessive velocity of the x-ray feature relative to the inference from Thomson scattering could be interpreted as a diffusion velocity for silicon relative to the background flow, as the mix layer at the material boundary grows with time.

The characteristics of a shock are determined primarily by the dimensionless Mach number, defined as the ratio of the shock front velocity to the upstream sound speed (⁠ $M≡vsh/c0$⁠). Weak shocks [ $(M−1)≪1$] do not generate large mean-free-paths relative to the system scale size and are well characterized by hydrodynamic treatments.² In contrast, moderate [ $(M−1)∼1$] and strong shocks (⁠ $M≫1$⁠) generate large ion mean-free-paths, driving ion species, and temperature separation and non-thermal ion distributions near the shock front that are not accurately captured by hydrodynamic theory. The Mach number of the shocks produced in these experiments may be determined by comparing the measured shock velocity with the inferred sound speed in the unshocked plasma, which is calculated as $c0=γ(Z+1)T0/A$⁠, where (A, Z) are the average ion mass and charge, respectively. The Thomson scattering data prior to the incidence of the shock front measure an ion temperature in the range 50–60 eV. Assuming the initial ion fraction $fNe=2%$⁠, singly ionized neon, and an ideal equation of state (⁠ $γ=5/3$⁠), this temperature implies a sound speed of 110–120 μm/ns. Compared with the inferred shock velocity of 720 μm/ns, the Mach number is estimated to be M = 6.0–6.7 in the present experiments: well into the strong shock regime.

The Rankine-Hugoniot equations fully define the relationship between upstream and downstream properties of a steady-state shock given three variables, often the upstream pressure (P₀) and density (ρ₀) and the Mach number.²⁷ These equations may be used to derive the shocked temperature T_sh in terms of the upstream temperature T₀ and the shock velocity

The second term provides a constant offset approximately equal to the pre-shock temperature and may be neglected when $Tsh≫T0$⁠.²⁸ For a strong shock (⁠ $M≫1$⁠), the shocked temperature grows as the square of the shock velocity. In these experiments with $vsh≈720 μ$ m/ns, $T0=50$–60 eV, and doubly ionized neon, Eq. (2) predicts a post-shock temperature of $Tsh≈730$ eV. This value is somewhat higher than Thomson scattering measurements in the fully shocked plasma, which are best fit by a temperature of 520 ± 100 eV. The reduced temperature may be partially explained by radiation, which is ignored in the analytical calculation.

An additional energy sink present in the experiment is the ionization of the neon. Prior to the shock, the plasma is relatively cold (50–60 eV, heated only by x-ray preheat from the laser-driven ablator) and low-density, and the Thomson scattering data are consistent with $ZNe≈1$⁠. However after the shock, the temperature and density are sufficiently high to expect full ionization in the steady state. The total binding energy of a neon atom is $EB=3.512$ keV per ion.²⁹ As such, the energy absorbed by full ionization of the 2% neon in the shocked plasma would amount to a reduction in the average particle thermal energy of $E¯ion=EBfNe/(1+⟨Z0⟩)=35$ eV, approximately 5% of the predicted total thermal energy. This is also accompanied by an increase in the number of electrons sharing the thermal energy. If the shocked plasma were a closed system, the ultimate temperature might be estimated as the predicted temperature averaged over the increased number of particles: $Tfinal=(Tsh−E¯ion)(1+⟨Z0⟩)/(1+⟨Zfinal⟩)$⁠. Including both of these effects, neon ionization might be expected to reduce the final temperature of the shocked plasma from 730 to ∼650 eV (12%). This is slightly hotter than the Thomson scattering measurement reported above, but the difference is potentially observable. Of course, the shocked plasma is not a closed system in this experiment, and in particular, we expect losses due to electron thermal conduction and radiation to be significant; nevertheless, this calculation shows that the ionization of neon is expected to perturb the energy balance in these experiments.

Fits to the ion acoustic wave data near the onset of the shock front feature (∼3.2 ns) were consistent with singly ionized neon; however 0.7 ns later, the data became most consistent with doubly ionized neon. Considering that the ionization of a neon ion removes energy from the local plasma and that different ionization states constitute different ion species for diffusive processes, the structure of the shock ought to depend in some way on the ionization rate of the neon. Smith and Hughes present ionization rates for neon as a function of plasma temperature and electron density:³⁰ the timescale for neon to achieve one e-folding toward ionization equilibrium in the post-shock plasma conditions (assuming $ne≈2×1020$⁠) is approximately $τNe≈1.0$ ns. This is similar to the shock passing time in this experiment. Within the precursor region, the neon ionization has a very minimal impact: liberating the second electron only requires an additional 41 eV per ion, accounting for less than a percent of the thermal energy in the plasma at this time. Indeed, most of the total binding energy (73%) is contained in the last two electrons; taking the ionization rate calculation at face value, these would become the dominant transition after about 1.4 ns, after the shock front has passed. As such, the effect of neon ionization on the region of the shock front probed here is expected to be small.

The ion-ion mean-free-path is calculated using the temperature, density, and composition of the shocked plasma.³¹ Assuming a shocked electron and ion temperature of 450 eV, an ion density equal to $3.7×$ the initial density (as predicted by the Rankine-Hugoniot equations with $M=6,γ=5/3$⁠), and doubly ionized neon, the mean-free-path of hydrogen ions in the shocked plasma is estimated to be 30 μm. From the theoretical scaling that the electron preheat layer (and the region of non-thermal ion distributions) extends $λiimi/me$ ahead of the ion density jump, this calculated mean-free-path would thus predict a preheat layer that is 1.3 mm wide, somewhat larger than ∼0.86 mm observed in the experiment (Sec. III A).

The importance of the neon ionization rate in the structure of the shock front becomes evident here: if the neon were instead assumed to be fully stripped, the hydrogen mean-free-path would be reduced to 10 μm and the predicted shock-width to 0.4 mm, smaller than the observed values. Thus, one explanation for the discrepancy between the observed and calculated shock width could be that the neon is more ionized than expected within the density jump: an average charge state of $⟨ZNe⟩=$ 4–5 would reduce the hydrogen mean-free-path sufficiently to match theory to the observed shock width. Given the calculated e-folding constant for neon ionization in the shocked plasma conditions, ionization to this range should take 0.3–0.5 ns. During this time, the ionizing neon is flowing away from the density jump at the difference between the shock-front and the piston velocity, $vsh−vp≈200$ μm/ns in this experiment.

To estimate whether this is a plausible explanation, a simple model was developed for the ionization state of the neon relative to the location of the density jump in a steady-state shock:

for initial charge state Z₀ and ionization rate τ_Ne. Using values from the present experiment, this model estimates that the neon charge state increases to ∼3 within one hydrogen mean-free-path of the density jump. This scale of the ionization gradient is neither obviously perturbative nor obviously negligible to the dynamics in the preheat region. On the one hand, the size of the electron preheat layer is determined by electron heat conduction ahead of the density jump. The relevant scale length should thus be the electron mean-free-path, which is roughly half the hydrogen mean-free-path and therefore implies a shorter boundary layer that is more isolated from the neon ionization gradient. On the other hand, the ions streaming forward into the preheat layer and forming the ‘hot’ population should be weighted toward more energetic ions due to their longer mean-free-paths (⁠ $λ∝ϵ2$⁠); this would imply a larger boundary layer, and an increased effect due to the ionization gradient. This may explain the relative coldness of the observed ‘hot’ population, which the Thomson scattering measurements indicate are only 200–300 eV in the precursor, much less than 450 eV measured after the shock has passed. In summary, the neon charge gradient should be expected to curtail hydrogen mobility near the density jump, but not the scale of the electron preheat region. More detailed measurements of the ionization gradient will be necessary to test this hypothesis.

A likely explanation for the narrowness of the shock front relative to theoretical predictions is that the shock is not yet fully formed. Vidal, et al. presented a Fokker-Planck simulation of a M = 5 shock forming after a hydrogen flow strikes a perfectly reflecting boundary.¹⁴ The width of the shock is shown to asymptotically approach its final value from below, with the steady-state result appearing after the shock had separated by approximately one shock width from the piston. This example was used as a simple model for evaluating the extent of shock formation in gas-jet experiments on OMEGA,¹⁰ with the time to achieve steady state estimated as $tS≈8Δx/vsh$⁠. Applying that model, the shock in these experiments is sampled in the range 30%–50% of the steady-state time. This shock is thus more fully developed than the shock in the previous work (⁠ $t/tS≈$ 17%), and the width is correspondingly larger relative to the predicted scaling: $Δx/λiimi/me≈2/3$ in the present work, compared to $∼3/8$ previously. These two datapoints are both consistent with an asymptotic width increase in the form $[1−exp (−t/tSτ)]$⁠, with the e-folding constant $τ≈0.3$⁠. However, more research would be necessary to confirm this hypothesis. Development of a detailed model for shock formation and improved measurements of the shock formation dynamics are areas for future work.

To support the understanding of the shocks in these experiments, 1D-planar simulations were performed using the Fokker-Planck code iFP.³² In these simulations, a steady-state shock with M = 6 was simulated using 98% H plus 2% singly ionized neon. The ion distribution functions were simulated to capture non-thermal behavior in the preheat layer; electrons were treated as a fluid. In these simulations, the shock reproduces the expected values for flow velocity (520 μm/ns) and temperature (730 eV) in the shocked region. The hydrogen ion distribution is plotted in Fig. 8(a), with position relative to the shock front (note the shock is propagating toward – x), and the bi-modal nature of the ion distribution in the preheat region (⁠ $x∈[−3.1,0]$⁠) is clearly visible. The hydrogen mean-free-path in the shocked plasma is calculated to be $λii=74.5 μ$m: thus, the width of the shock preheat layer in the steady-state simulation is 41.1 $×λii$⁠, a factor of $0.96×$ the theoretical prediction. The neon did not participate in the hydrogen shock in these simulations, due to its much longer thermal mean-free-path: $λi,j∝(mi+mj)/mjnjZi2Zj2$⁠, so in the case with trace singly ionized neon, $λNe/λH≈10$⁠. Instead, the neon formed a trailing shock front 5 mm (7.5λ_Ne) behind the hydrogen shock. Ionization of the neon, which was not included in this model, would reduce the neon mean-free-path and shorten the neon density jump.

A two-Maxwellian distribution was fit to the hydrogen population, and a single-Maxwellian distribution was fit to the neon, for comparison with the Thomson scattering data; the results are shown in Figs. 8(b)–8(d). This approximation of the full kinetic distribution functions using the two-population model does not capture in detail the ion distribution functions evolving within the strong plasma shock, and this imperfection results in ambiguity in the two-population fits. In particular, the hot proton population was not strictly Maxwellian in the preheat region, and thus, various solutions were possible depending on how the fit was performed. Primarily, the fit temperature was sensitive to the fitting method: emphasizing the tails of the distribution resulted in a lower temperature, whereas emphasizing the bulk resulted in a higher temperature (and slightly slower velocity). The plotted fit results transition from emphasizing the tail to emphasizing the bulk at $x=−1.4$ mm; some neighboring datapoints from the alternate method are included for comparison. This rapid change in the best-fit solution is similar to that observed in the best-fits to the data at 3.3 ns (Fig. 3), supporting the inference that this jump is due to the imperfection of the two-population ansatz early in time. In principle, a more detailed model could provide a better comparison between the data and the full ion-kinetic physics captured by the simulation. However in practice, the restricted parameter space of the two-population model is required for robust interpretation of the data in the presence of noise and other imperfections. Comparison between the two-population models thus provides a baseline for comparison between the data and simulations.

The general trends observed in the data are visible in the Fokker-Planck simulation: the cold population is accelerated and heated; the hot population becomes dominant at the density jump. After the density jump passes, a residual cold population drags and thermalizes with the neon, increasing the neon velocity and temperature; the hot population reaches its final temperature (equilibrating with electrons) and velocity. However, there are some notable differences between these simulations and the data. First, the data show that the hot population composes roughly 20% of the ions in the shock preheat region, whereas the simulations predict that the hot population does not reach this fraction until just before the density jump. This may be explained by the incomplete formation of the shock. Very early in the process of shock formation, the population of forward-streaming ions may be somewhat larger than in the steady state case.¹⁴ This observation could alternatively be explained if the hydrogen in the gas tube was not completely ionized as the shock began to pass through. The Thomson scattering ion feature consists of light scattered from electrons as they collectively shield ions; unionized atoms would not be observed, increasing the apparent fraction of the hot, forward-streaming population. Full ionization of the hydrogen gas requires only 13 eV per atom and thus does not significantly impact the shock energetics, given the plasma temperatures in this experiment. Second, the simulated hot population temperature in the preheat region is comparable to the post-shock values (700 eV), whereas in the data the hot population was substantially colder (200–300 eV). The simulations also predict that the hot population velocity should exceed the piston velocity (and indeed the shock velocity), whereas in the data the hot population peak velocity was 540 μm/ns at 3.6 ns, slower than the piston velocity at that time (∼600 μm/ns). We have hypothesized that these differences are due to a damping effect of the neon ionization gradient within the shocked plasma, which reduces the fluence of energetic protons into the preheat layer while less strongly affecting the electron conduction. Future development of the simulation to include a self-consistent ionization model for the neon will allow this hypothesis to be tested.

Simulations were also performed in which a pure hydrogen plasma with constant initial density and temperature was flowed into a reflecting boundary to generate an M = 6 shock. Assuming that the growth of the shock-width follows an asymptotic single-exponential trend, an e-folding constant of 1.7 ns was found to match the simulation. Note that this is somewhat faster than is expected from the Vidal heuristic discussed above: for this experiment, the Vidal steady-state timescale $tSτ≈2.9$ ns. Sampling this simulation at 3.3 ns, the width of the preheat region is 0.72× of the theoretical value $λiimi/me$⁠, less than the steady state value in that simulation (0.87) but similar to the value observed in the experiments (∼0.66). The exact dynamics of shock front formation, in particular as a function of Mach number, remains an area for future research.

In conclusion, the ion velocity structure of strong (M = 6.0–6.7) collisional plasma shocks was probed in a hydrogen-neon mixture. Thomson scattering of a $4ω$ probe beam demonstrated that two distributions of ions exist within the preheat region: a still, cold background population of hydrogen and singly ionized neon, and a hot population of hydrogen flowing ahead of the shock front near the piston velocity. X-ray data were used to measure the location and velocity of the interface between the (H,Ne) plasma and the (Si,O) pusher, confirming the velocity measurements from Thomson scattering. The Thomson scattering data also measure the transition of the neon from singly to doubly ionized, which occurs approximately 0.7 ns after the onset of the shock. A calculation of the neon ionization gradient in the shocked plasma suggests this gradient is too slow to affect the size of the electron preheat region, but may inhibit hydrogen transport into the hot population and thus change the shock's velocity structure. Neon ionization can also absorb enough energy to observably reduce the temperature in the post-shock region. Fokker-Planck simulations of an M = 6 shock with conditions matched to the experiment demonstrate a similar morphology to the shock front as observed in the experimental data. Discrepancies include a higher predicted temperature and velocity for the forward-streaming ion population in the shock preheat region, supporting the hypothesis that the neon ionization reduces ion flow into this region.

Future experiments will continue to probe the dynamics and structure of strong shocks in plasmas. Areas that merit further research include the scaling of shock width as a function of Mach number, the effect of the ionization gradient on the shock structure, and the separation of ion species at the shock front. Connecting the Thomson scattering results with maps of shock-generated electric fields from proton radiographs⁹ will further constrain shock dynamics. Creating and studying fully formed shocks and ionization processes may require increased drive energies and experimental timescales of tens of nanoseconds. With the implementation of an Optical Thomson Scattering system³³ and a proton radiography capability,³⁴ such an experiment is now possible at the National Ignition Facility. Progress in this area will further improve the understanding of collisional processes in plasmas and provide a valuable dataset for improving the performance of high-fidelity physics simulations.

This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. DE-AC52–07NA27344 and was also supported by the Los Alamos National Laboratory LDRD program and the LANL Institutional Computing Program under Contract No. DE-AC52–06NA25396.