Role of hot electrons in shock ignition constrained by experiment at the National Ignition Facility

Shock ignition (SI)^1–7 is a potential high gain approach to direct drive inertial confinement fusion (ICF).^8–10 Gain is the ratio of released fusion energy to input laser energy and is a key indicator of whether a method is suitable for energy generation. Direct drive ICF uses a laser ablated material to drive a deuterium–tritium ice shell toward its gas filled core. For conventional hotspot ignition, the inertia of the capsule wall must provide both the energy required for fusion and the confinement of the hotspot for the tens of picoseconds necessary to achieve thermonuclear burn. SI separates the assembly of the hotpot from the ignition energy requirement. The implosion is carried out below the self-ignition velocity due to a less energetic assembly pulse, and a late-stage shock introduces the additional energy required to ignite the hotspot. The ignition shock collides with an outward traveling shock created by the reflection of the first shocks at the center. The shock collision, if timed accurately, provides the increase in hotspot density and temperature required for ignition. The benefit of this method is that it exhibits the potential for high gain at lower laser energies,⁴ but it also opens up the ignition design space to lower velocity implosions that have been demonstrated to be more stable to hydrodynamic instabilities.¹¹ This is significant since hydrodynamic instabilities have been one of the major limiting factors on the success of ICF. For simulated high gain, laser driven implosions rely on either large target mass (and large laser energies) or near adiabatic compression (measured by shell adiabat¹⁰). The failure of the low adiabat (⁠ $α < 2$⁠) implosions to perform as simulated¹² has led the laser driven ICF community to develop higher adiabat (⁠ $α > 3$⁠) implosions. High adiabat implosions have led to better agreement with simulation enabling the incremental improvements for both indirect drive^13,14 and direct drive¹⁵ approaches to the point where hydrodynamically scaled results are close to ignition (at $≃ 1.8 MJ$⁠). While SI may lead to modest improvements in implosion performance at $α > 2$ seen in Trela et al.¹⁶ and Anderson et al.,¹⁷ it also introduces the significant unknown of a high intensity laser pulse. For third harmonic Nd:glass lasers at $351 nm$ (this wavelength is used throughout the rest of the paper), the laser intensity exceeds the $∼ 10 15 W / cm 2$ limit where laser–plasma instabilities (LPI) are thought to play an increasingly dominant role in laser absorption.¹⁸ LPI have two significant impacts on the hydrodynamics of an implosion: reduced drive efficiency due to scattered light and the generation of hot electrons.

At intensities, $< 10 15 W / cm 2$⁠, implosion dynamics from experiments have been matched in simulations, using semi-empirical multipliers to account for LPI scattered light and hot electron preheat.¹⁹ For the targets investigated in Christopherson et al.,¹⁹ it was found that hot electron preheat led to an increase in adiabat of $10 % – 20 %$ and a reduction in stagnation areal density of $10 % – 40 %$⁠.

Simulation of hot electrons at SI relevant intensities (⁠ $> 10 15 W / cm 2$⁠) has shown conflicting results.^20,21 Initially, for SI, it was predicted that hot electrons have a negligible impact and in fact may improve drive efficiency due to their penetration depth.^7,22 Experiments appear to support this hypothesis, demonstrating the generation of strong shocks in the presence of LPI generated hot electron populations.^5,23,24 Due to the hot electron drive efficiency, the idea of electron shock ignition was also theorized²¹ where the aim of the final laser pulse is to maximize hot electron generation from LPI. However, kinetic modeling of hot electrons using a simplified LPI model embedded in a hydrodynamics simulation indicated that the production of hot electrons was found to be harmful to the implosion performance.^20,25,26 The deleterious effect of preheat²⁷ outweighs the benefit of drive support for hot electrons with a population temperature higher than 30 keV or mono-energetic hot electron populations with an energy exceeding $∼ 50$ keV (target dependent). Conversely, there is evidence that kinetically modeled hot electrons can provide benefit to implosion performance across a broad range of population temperatures^28–30 (up to 70 keV). Within these simulations, the LPI scattered light energy fraction is not explicitly related to the LPI hot electron energy fraction, which may explain the difference in impact of hot electrons compared to those that use simplified LPI models.

There are several hydrodynamic factors that may change the impact of hot electrons. Christopherson et al.¹⁹ show that preheat is worse for lower adiabat implosions, and Colaïtis et al.²⁰ demonstrate that the inclusion of an ablator can help to reduce the penetration (thus, preheat) of the hot electrons. Trela et al.¹⁶ and Bel'kov et al.³¹ demonstrate that the timing of the high-intensity spike can also change the amount of hot electron preheat. The target and laser pulse shape design might be key to determine whether hot electrons have a harmful or beneficial effect to SI.

Even if a specific target and laser pulse shape is considered, there is still significant uncertainty over the LPI and the hot electron population generated. Fully kinetic simulations (such as particle-in-cell codes) are too expensive to run for the durations required to understand the LPI over a nanosecond pulse. Linear LPI theory does not include the non-linear interaction of many laser–plasma waves with non-thermal particle distributions at a range of densities. Despite these complexities, there is evidence, from experiment^27,32 supported by PIC simulation,³³ that stimulated Raman scattering (SRS) off plasma at $10 % – 25 %$ of the critical density is the dominant LPI and source of hot electrons in the $> 3 × 10 15 W / cm 2$ intensity regime.

In addition to difficulties with theory and simulation at SI relevant conditions, there is not a consensus in observation from experiment on hot electron characteristics. It is possible to use planar or conical geometry targets on lower energy facilities to increase laser intensity and make plasma conditions more closely map to those expected during SI implosions.^34–40 This is done to investigate the LPI and hot electrons that are generated at $> 10 15 W / cm 2$ intensities, long density scale lengths $≃ 500 μ m$⁠, and electron temperatures $≃ 3 keV$ predicted to occur in SI plasmas.^18,27 Recent experiments conducted in planar or conical geometry at the OMEGA Facility^41,42 have attempted to infer the fraction of laser energy converted to hot electrons (η) and the temperature of the population (T_h).^32,43,44 Zhang et al.⁴³ use a planar target and predict $η < 0.01$ in $T h ≃ 30 keV$⁠. Tentori et al.⁴⁴ also use a planar target but predict $η ≃ 0.10$ in $T h ≃ 30 keV$⁠. Scott et al.³² use a conical target and predict $η ≃ 0.025$ in $T h ≃ 45 keV$⁠. The discrepancies are significant as the population is at the edge of what will cause acceptable preheat for SI.⁴⁵ There are many possible explanations for the differences; nonetheless, it limits our ability to make predictions for SI implosions.

There are two additional aspects, which are not well characterized by experiments or simulation; these are hot electron refluxing and emission angle (Ω_h). Hot electron refluxing occurs when the accumulated charge on a target prevents the escape of the electrons. An analysis of implosion capsule charging was presented in Sinenian et al.⁴⁶ and Volpe et al.⁴⁷ with MeV scale charge. Reflux information extracted from planar and conical targets might not be relevant to implosions due to the difference in target geometry, but the planar target experiment in Pisarczyk et al.⁴⁸ demonstrates similar accumulation of charge with electrons refluxed up to MeV energy scales. The refluxing hot electrons are expected to lead to additional coronal heating.⁴⁹ The most significant experimental constraint on hot electron emission angle can be seen in Yaakobi et al.⁵⁰ where the population is inferred to have a solid angle of $Ω h > 2 π sr$⁠. PIC simulation and LPI theory give a much narrower angle, $Ω h ≃ 1.8 sr$⁠,^20,33 which is also in closer agreement with the preheat values inferred by Christopherson et al.¹⁹ The narrower emission angle will be used throughout the rest of this paper. The larger the emission angle, the more important refluxing is as a larger fraction of hot electrons will miss the target.

This paper presents results from a semi-empirical kinetic 3D Monte Carlo transport model for hot electrons, which is run inline with a radiation-hydrodynamics code (Odin). The hot electron distribution was constrained by a polar direct-drive (PDD)^51–54 solid target experiment on the National Ignition Facility⁵⁵ (NIF). This experiment achieved $I ≃ 3 × 10 15 W / cm 2$ intensities, density scale lengths $L n ≃ 600 nm$⁠, and electron temperatures $T e ≃ 2.5 keV$ close to the conditions predicted in SI (see Table II). The experiment was carried out with a spherical target and PDD beam geometry as expected for SI (on the NIF) removing the geometric limitations of previous experiments.^32,43,44 It was found in simulation of the solid target direct drive NIF experiment that hot electrons change the shock timing due to deeper target penetration (than the laser) and lead to a modified density profile featuring a larger radius ablation front and a single shock front (in comparison a simulation without hot electrons predicts two separate shocks). When applied to the SI implosion, the hot electron population is simulated to have a negative impact on performance, reducing peak areal density by $∼ 35 %$⁠. In addition, features similar to those seen in simulations of the solid target experiment were observed, such as a modified shock timing and smaller in-flight-aspect-ratio (IFAR). These features contributed to the degradation in performance; however, the hot electron preheat was the most significant limiting factor. In Sec. II, the model will be described; then, in Sec. III, the free parameters and how they were constrained will be detailed. In Sec. IV, the model was then applied to a SI simulation, and the impacts are compared to a simulation without the hot electron population.

Figure 1(a) gives a schematic representation of how hot electrons are transported in this model. The Odin code is a 2D radiation-hydrodynamic, arbitrary Lagrangian–Eulerian code, which is run in polar coordinates. The laser drive is modeled via purely radial rays, which deposit energy via inverse bremsstrahlung until they reach the quarter critical surface (n_c/4) at which point if the intensity is above a threshold, they transfer energy into hot electrons and lose energy to SRS scattering. Any laser energy reaching the ray-turning point is deposited as thermal energy. A drive reduction multiplier is applied to the experimental laser powers before the rays are initialized.

The hot electron particle tracking is modeled in 3D by rotating the 2D Odin grid. Hot electrons are emitted in a solid angle (Ω_h) centered parallel to ray propagation. Hot electrons are sampled from a thermal distribution. The emission angle of the electrons is randomly and uniformly distributed within the solid angle. Each electron follows a path based on the stopping and scattering formula from Berger et al.,⁵⁶ Davies,⁵⁷ and Robinson et al.⁵⁸ When the hot electrons are generated, half the energy taken from the ray is put in the hot electron population (⁠ $η n c / 4$⁠) and the other is discarded from the simulation to account for the energy loss due to scattered SRS light. Figure 1(b) shows a density plot of a solid target (without color scale) with the hot electron paths overlaid. In the simulation, both the emission solid angle (⁠ $Ω h ≃ 1.8 sr$⁠) and the size of the target viewed from the quarter critical surface $Ω ≃ 1.2 sr$ are similar; in addition, the target is “thick,” meaning that the majority of hot electrons are stopped if incident upon it. In these simulations, we disregard the reflux population.

The above-described method includes free parameters for both the hydrodynamic and hot electron models. The aim of the NIF solid target experiment (N190204-002) is to constrain these free parameters in an environment as similar to SI as possible (calibrate the simulations) but to have the benefit of a solid target for more accurate shock tracking. The experiment consists of a $1100 μ m$ radius solid target being illuminated by 184 beams in a PDD configuration. The central $1000 μ m$ is deuterated plastic (CD), and the outer $100 μ m$ is plastic (CH). A zinc backlighter is illuminated by 2 quads to give radiographic information of shock timing,⁵⁹ used when constraining the hydrodynamic simulations. The filter-fluorescer diagnostic system (FFLEX) diagnostic⁶⁰ is used to measure hot electron induced bremsstrahlung x rays, from which the hot electron population's temperature $T h = 55 ± 2 keV$ and total energy $E h = 35 ± 7 kJ$ are inferred.⁶¹ (The full-aperture backscatter system (FABS) diagnostic⁶² indicates that SRS is the dominant cause of the hot electrons.²⁷) The FFLEX diagnostic is shielded from the zinc backlighter using a gold disk. The experimental laser pulse shape can be seen in Fig. 2(b) as the dashed black line. It reaches a peak power of 350TW and a peak intensity (calculated at initial target radius) of $3.0 × 10 15 Wcm − 2$⁠. The FFLEX diagnostic was calibrated for hot electrons using a 3D Maxwellian⁶¹ so that distribution is used throughout the simulations to best reflect the energies observed by the diagnostic.

The experiment features a low intensity ramp used to create the ablation-plasma density scale length and temperatures anticipated in SI followed by a high intensity peak. LPI are expected to occur most during the high intensity peak $I > 10 15 W / cm 2$ so when the experiment is simulated, hot electrons and the SRS multiplier are only applied during this part of the laser pulse (supported by observations from the FFLEX diagnostic, and the FABS diagnostic⁶²). The solid black line in Fig. 2(b) corresponds to the simulation laser energy absorbed, once the loss multipliers are applied. (Multipliers are the only loss mechanism since all the laser energy remaining in the ray is deposited into thermal electrons at critical density.) The drive reduction multiplier is 0.6 applied throughout the pulse (40% of laser energy removed before the ray is launched).

Figure 2(a) shows the experimental shock timing and the close match achieved by a simulation using the parameters listed in Table I including the effect of hot electrons. The laser drive multiplier and flux limiter are chosen to match the shock timing shown in Fig. 2(a). The hot electron emission solid angle (Ω_h) is constrained by PIC simulation.³³ The hot electron conversion fraction at the quarter critical density (⁠ $η n c / 4$⁠) is selected to deposit the $35 kJ$ observed by FFLEX, and the hot electron population's temperature (T_h) is also constrained by the FFLEX diagnostic.

Displayed in Fig. 3 are snapshots at $4.0 ns$ from the simulations of the NIF solid target experiment (N190204–002). The snapshots show the comparison of two simulations: one with hot electrons shown by the solid lines and one without hot electrons shown as the dashed lines, other than this the simulations' setup and multipliers are identical. The energy allocated to hot electrons is modified using the SRS multiplier. The SRS multiplier applied to a ray at the quarter critical density is 0.6 for simulations with hot electrons (20% discarded as scattered light and 20% put into the hot electron population, listed in Table I as the fraction $η n c / 4 = 0.2$⁠) and 0.8 for the simulation without hot electrons (20% of the ray energy removed from the simulation as scattered light). The solid red line in Fig. 2(b) gives the simulated power deposited by hot electrons (equivalent to an energy of $35 kJ$⁠, which matches the experimental observations of the FFLEX diagnostic).

In Fig. 3, two separate shock fronts are seen only in the simulation without hot electrons. The cause is the change from low intensity ramp to a steeper ramp at $3.0 ns$ in the laser pulse [Fig. 2(b)]. The second shock travels faster and the two will coalesce (at $5.0 ns$⁠). In the simulation with hot electrons, the hot electrons penetrate deeper into the target than the laser and deposit their energy beyond the ablation front. The hot electrons are able to support the shock front created by the low intensity ramp leading to only one shock front.

At $4.0 ns$⁠, shown in Fig. 3, the difference in location of peak pressure between the two simulations is $≃ 30 μ m$⁠. The shocks travel at $≃ 300 μ m / ns$ (relative to a laboratory rest frame) giving a shock timing difference at $4.0 ns$ of $≃ 0.1 ns$⁠. This gap will extend to $≃ 0.3 ns$ (⁠ $≃ 90 μ m$⁠) by $5.0 ns$ when the two shocks (seen only in the simulation without hot electrons) coalesce. After the shocks coalesce, the difference in shock timing between the two simulations remains constant at $≃ 0.3 ns$⁠. This is because the shocks' velocity is similar in each simulation from $5.0 ns$ onwards. It can be inferred that it is the penetration depth of the hot electrons, which has changed the timing of the first shock. Once the two shocks, in the simulation without hot electrons, have coalesced (at $5.0 ns$⁠), both simulations have the same shock pressure and shock velocity indicating a similar overall drive efficiency. In Fig. 3, preheat is visible as an increase in pressure at $R < 790 μ m$⁠; however, it has less of an impact on the shock pressure (the absolute difference in pressure ahead and behind the shock) than the effect of the hot electron energy deposited behind the shock front at $790 < R < 900 μ m$⁠. For this reason, it is stated that hot electron penetration, not preheat or drive efficiency, has changed the shock timing.

Figure 3 also demonstrates that the volumetric heating of the target by the hot electrons leads to a reduced ablator density and increased ablation front density scale length. In an implosion, this will lead to a lower in-flight-aspect-ratio (IFAR). IFAR is the ratio of initial target radius (R_i) to the minimum in-flight width of the shell wall (Δ), $IFAR = R i / Δ$⁠, and is a key metric for implosion efficiency.⁹ This result could not be obtained with kinetic simulation of mono-energetic hot electrons.

The NIF solid target experiment (N190204–002) has been simulated and observations matched, constraining the parameters shown in Table I. The calibrated parameters can now be applied to a SI simulation.

The SI capsule design is shown in Fig. 4 in addition to two of the laser pulse shapes simulated. The design is based on Atzeni et al.⁶³ However, a larger implosion velocity $V i = 330 km / s$ and a larger adiabat were used (minimum adiabat at peak velocity $α min = 2.5$ and a fuel averaged adiabat at peak velocity $α ave = 3.5$⁠) due to evidence of experimental limitations of implosions with low adiabats.¹² The use of higher adiabats results in a much lower predicted yield than most SI investigations;^4,7 for this implosion, the direct comparison is Atzeni et al.⁶³ which used an adiabat of $α ave ≃ 1.6$ and achieved a gain of G > 50. For the simulations presented in this paper, the implosions fall short of the ignition threshold having a similar areal density and hotspot temperature (the metric of performance often used sub-ignition) to scaled versions of the current best direct drive implosions.^15,64 Using a larger adiabat means that the results should be more experimentally reproducible, and there is evidence, suggesting that higher adiabat implosions are less susceptible to preheat.¹⁹ The laser pulse shape requires $< 1 MJ$ of laser energy and power $< 500 TW$⁠, meaning that this design is accessible by the current generation of laser facility.

The parameter calibration is only accurate if the plasma conditions are similar between the NIF solid target experiment and the SI simulation. Table II gives details of the plasma conditions simulated in the case of the experiment and SI implosion for comparison. The laser pulse shape, target size, and resultant plasma conditions are similar so the inferred impact of hot electrons on SI represents an acceptable extrapolation. A notable difference between the simulations is target convergence. The SI capsule is half of its initial radius during the high intensity pulse while the NIF solid target is approximately the same radius. The convergence would likely have a negative impact on the laser coupling (absorbed over emitted laser energy, $E Labs / E L$⁠) reducing it further. This means that $E L = 542 kJ$ is an underestimate for the laser energy required to drive such an implosion, and $542 < E L < 1000 kJ$ is likely.

Target convergence and a less energetic high intensity pulse (⁠ $> 1.0 × 10 15 Wcm − 2$⁠), compared to the NIF experiment (N190204-002), are the main reasons for the difference in total absorbed hot electron energy (E_h) between the two simulations shown in Table II. The NIF solid target experiment, $E h = 35 kJ$⁠, shows almost triple the energy deposited by hot electrons compared to the SI simulation $E h = 13 kJ$⁠. Despite the lower total energy absorbed, the SI implosion features more preheat (⁠ $E h 1$ in Table II) due to lower areal density between the ablation front and the shock front. Preheat was defined as heating in the unshocked material for the NIF solid target experiment but in the case of the SI implosion all material is shocked (due to the picket pulses), so it is defined as heating ahead of the ignitor shock. The difference in definition also contributes to the difference in $E h 1$ seen in Table II.

For shock ignition, the target cannot be assumed to be thick, stopping the majority of hot electrons incident (the NIF solid target experiment simulations demonstrated this property). Hot electron reflux is not simulated. The fraction of hot electron energy, which reaches the edge of the corona [ $E reflux / ( E h + E reflux )$], is larger for SI than for the solid target due to the solid angle the target makes when viewed from the quarter critical surface [which is $Ω ≃ 0.67$ for SI and $Ω ≃ 1.2$ for the NIF experiment (N190204-002)]. Due to the time of flight, it is likely that the reflux population (E_reflux) does not play a role in the implosion except as wasted drive energy. The distance to the coronal edge, where refluxing occurs, during the high intensity pulse, is $≃ 10 mm$⁠, at 10% the speed of light, a return journey for the fastest hot electron along the most direct path would take $≃ 700 ps$⁠; at which point, the shock collision would have already occurred, with a bang-time $≃ 300 ps$ later.

Figure 5 shows the impact of hot electrons on SI implosions. The baseline performance of a conventional central hotpot implosion is demonstrated by the green dash-dotted line. In the other three simulations, performance is improved by adding a late-time SI spike (as shown in Fig. 4). By putting extra laser energy into the SI spike, rather than the pulse's plateau, one can increase the hotspot temperature without significantly modifying the shell's IFAR or peak velocity. This demonstrates the ability of SI to increase the design space of hydrodynamically stable implosions.

The best performing implosion in Fig. 5 is shock ignition simulated without hot electrons (orange-solid line) in which both hotspot ion temperature and peak areal density are approximately double that seen in the baseline case (green dash-dotted line). When a hot electron population is added to the simulations, implosion performance is reduced by 35% (as measured by peak areal density). The 35% can be compared to the $10 % – 40 %$ degradation seen in direct drive conventional hotspot implosions due to hot electrons.¹⁹ The amount of degradation is similar despite the larger SI hot electron population cumulative energy because the population is generated in the late stages of the implosion. At late times, the shell has a larger areal density, and the shell has a higher adiabat; in addition, more hot electrons miss the target due to convergence. It is also likely that the two plasmon decay (TPD) hot electrons seen at lower intensities (used for conventional hotspot implosions) have a higher temperature and are, thus, more likely to cause preheat.²⁷ It is clear that hot electrons pose a threat to implosion performance for both types of direct drive fusion, so mitigation strategies are key. Delaying the SI laser spike by $0.1 ns$ (hence, the ignitor shock timing) mitigated the negative impact of the hot electrons from 35% to 25% reduction in peak areal density. The optimization in performance can be seen in Fig. 5 as the difference between the red-dotted and the purple-solid line, although much of the benefit (improved compression) is in the cold fuel not the hotspot, which is plotted.

The laser pulse shape that was optimized with hot electrons is compared to the simulation without hot electrons in Fig. 6. The maximum pressure of the shock is similar despite $≃ 20 %$ of the laser spike going into a hot electron population. This result is similar to that seen in the NIF solid target simulation (in Fig. 2). The minimum pressure ahead of the shock is increased from $≃ 0.3$ to $≃ 0.6 Gbar$ due to hot electron preheat. The preheat, not the loss of drive, leads to the majority of the degradation in performance. The doubling of pressure is visible throughout the hotspot and shell, which is in line with the doubling of adiabat observed at peak velocity from $α min = 2.5$ to $α min = 6$⁠. Figure 6 also shows the expansion of the outer edge of the target (ablation front) due to the hot electrons depositing energy within the shell. This decreases the IFAR and reduces the efficiency of energy transfer to the hotspot. A similar expansion of the target's ablation front is visible in the NIF solid target simulation (in Fig. 2). An additional effect is a lower shock pressure gradient when simulated with hot electrons due to the volumetric energy deposition of the thermal distribution of hot electrons around the location that the shock forms. This shallower shock pressure gradient will result in a weaker shock collision and energy dissipation as the shock travels, which contributes to the reduction in performance and the increased adiabat seen when hot electrons are included in the simulation.

In the NIF solid target experiment, there is uncertainty in the measurement of the hot electron population. By using FFLEX and FABS diagnostic in combination, the experiment represents the state of the art in LPI diagnosis. The experiment was carried out at the National Ignition Facility, which is one of few facilities capable of creating SI conditions in a near spherical geometry (not planar or conic target), limited by polar configuration of the ports. It is thought that beam geometry plays a large role in many LPI. Despite this, the results reflect a single experiment and are, thus, vulnerable to the variability observed in ICF experiments. Furthermore, while the conditions in the NIF solid target experiment are close to SI, the experimental laser intensity is $3 × 10 15 W / cm 2$⁠, which is lower than that used in our simulations (⁠ $1 × 10 16 W / cm 2$⁠, see Table II). At the experimental intensities, TPD is expected to play a larger role and may increase the observed hot electron temperature and reduce the hot electron energy deposited. The quarter critical surface is the region where absolute backscattered SRS occurs, and so, it is used as the launch site for the hot electrons in our simulations; however, there is evidence that SRS occurs at a range of densities below quarter critical as well.^27,32,36,40 Given the experimental evidence and PIC simulations,³³ the most probable values for the hot electron population were used in each case, but these are scaled from the NIF experiment and different laser intensities may generate a different hot electron distribution.

For a thermal distribution of hot electrons, we find higher temperatures more disruptive to the implosion and lower temperatures behave more similarly to simulations that ignore the impact of hot electrons. It is worth noting that even without the impact of hot electrons, there is the impact of lost drive energy of the LPI scattered light. If the hot electrons are emitted further from the target at densities less than quarter critical, hot electron reflux plays a larger role. It is likely that more of the drive energy is wasted in the corona due to a larger fraction of hot electrons missing the target, but there will also be less target preheat, so the implosion is likely to perform slightly better.

This paper presents an investigation of the impacts of hot electrons on a SI implosion and a design that may be possible on current facilities. The SI simulations were constrained by an easier to diagnose solid target experiment undertaken on the NIF where the plasma conditions and laser pulse shape were similar to that of SI. The 25% reduction in peak areal density is an indication that hot electrons play a key role in SI. Similar performance reductions are observed in conventional hotspot ignition,¹⁹ meaning that SI still offers a useful method to expand the design space for direct drive implosions. When designing future direct drive or PDD campaigns, this work gives evidence that the baseline performance of a conventional central hotpot ignition implosion can be improved without significantly impacting the implosion velocity, a key marker of hydrodynamic stability; however, hot electrons will preheat the cold fuel and reduce the IFAR unless fully mitigated.

Significantly, the adiabat used in this investigation was higher than in most SI ignition simulations. A higher adiabat reduces the possible performance of an implosion but adheres to the empirical observations in several experimental campaigns on the NIF and the OMEGA facility.^12–15 The larger adiabat makes the design more robust not only to hydrodynamic instabilities but also to laser–plasma instabilities as the resulting hot electrons are predicted to have a lesser impact on high adiabat implosions.¹⁹ This latter point may be under represented in the literature investigating the benefits of high adiabat implosions due to the difficulties in simulating LPI and hot electrons.

This paper demonstrates that shock retiming can be used to reduce the negative impact of hot electrons, but additional strategies, such as thicker targets,²⁰ doping the ablator,⁶⁵ shorter wavelength, and broadband lasers,⁶⁶ will be key for attempts at direct drive ignition.

The subject of hot electron generation and source characteristics is led by experiment. At present, there are limited data and significant measurement uncertainty for SI conditions. The simulations in this work uses new data to characterize the hot electron population and to classify its impacts on SI relevant target designs. It is clear that there is need for further experiments to reduce the uncertainty as the potential impact on implosion performance is significant. Proposed facilities with a higher experimental repetition rate will improve the statistical significance of future LPI and hot electron measurements. In addition, experimental diagnosis of the fraction of hot electron energy that is deposited on the inner surface of the shell (which can be achieved using a doped target) will be essential, as it is this preheat which has the most detrimental effect on the implosion. The measurement will give key insight to experimentally constrain the hot electron emission angle and reflux behavior.

This work was carried out within the framework of the UK Research and Innovation (UKRI), Engineering and Physical Sciences Research Council (EPSRC) under Grant Nos. EP/P026796/1, EP/P023460/1, EP/P026486/1, and EP/L01663X/1. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2019–2021 and 2021–2024 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. The involved teams have operated within the framework of the Enabling Research Project No. ENR-IFE.01.CEA “Advancing shock ignition for direct-drive inertial fusion.” Part of this work was performed at the Laboratory for Laser Energetics for the U.S. Department of Energy National Nuclear Security Administration under Grant No. DE-NA 0003856.

The authors have no conflicts to disclose.

Duncan Barlow: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review and editing (equal). Matthias Hohenberger: Formal analysis (equal); Methodology (equal). Nigel C. Woolsey: Funding acquisition (equal); Project administration (equal); Resources (equal); Writing – original draft (equal); Writing – review and editing (equal). Philip Bradford: Writing – original draft (equal); Writing – review and editing (equal). Matthew Khan: Writing – original draft (equal). Tony D. Arber: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Writing – original draft (equal); Writing – review and editing (equal). Tom Goffrey: Software (equal); Supervision (equal); Writing – original draft (supporting); Writing – review and editing (supporting). Keith Bennett: Software (equal). Robbie H. H. Scott: Conceptualization (supporting); Formal analysis (supporting); Methodology (supporting); Project administration (supporting); Resources (supporting); Supervision (supporting); Writing – original draft (supporting); Writing – review and editing (supporting). Kevin Glize: Formal analysis (equal); Methodology (equal). Wolfgang Theobald: Formal analysis (equal); Methodology (equal). Kenneth S. Anderson: Formal analysis (equal); Methodology (equal). Andrey A. Solodov: Formal analysis (equal); Methodology (equal). Michael Jonathan Rosenberg: Formal analysis (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.