A near one-dimensional indirectly driven implosion at convergence ratio 30

In indirectly driven inertial confinement fusion (ICF) implosions, a spherical capsule is illuminated with X-rays produced when laser energy is absorbed in the walls of a cavity or hohlraum containing the capsule.¹ Inside the capsule shell is an equi-molar mixture of deuterium and tritium (DT) isotopes, most of which is contained in an ice layer formed on the shell inner radius. Ablation pressure generated by the X-ray absorption in the capsule shocks and spherically compresses the remaining shell and the ice layer, and the shocks and compressive work heat the DT vapor to several kilo-electron volts (keV), forming the “hot spot.”² At these temperatures and densities, fusion reactions occur in the gas and if the areal density is large enough, alpha particles deposit their energy, heating the gas and leading to more fusion (“self-heating”). To date, ICF implosions at the National Ignition Facility (NIF)³ have achieved self-heating conditions such that the alpha particle heating deposits roughly an equivalent amount of energy as the work from compression.⁴ 

Early results from ICF implosions on the NIF indicated that reduced fusion performance was correlated with a mix of ablator material into the hot spot.⁵ A subsequent campaign of improved implosions significantly reduced the ablator-hot spot mix, but fusion performance remained significantly below 1D predictions.⁶ A dynamic model that assumes incomplete confinement of the hot spot has been used to obtain reasonable agreement with the measured yield of the improved implosions.⁴ Simulations of and data from NIF ICF implosions indicate that the thin plastic membrane, or “tent,” used to support the capsule, is a significant seed for an ablation front instability, which, if allowed to grow, results in a major disruption to the hot spot formation.^7,8 There is also experimental indication that the capsule fill tube may seed an ablation front instability which is suspected to significantly impact performance.⁹ Finally, a combination of symmetry measurements and simulations indicates that small fluctuations in the low-mode symmetry of the hohlraum drive result in a significant asymmetry at stagnation that robs the implosion of kinetic energy for compression.¹⁰ Taken together, ablation front growth from engineering features and time-dependent low-mode drive asymmetries may account for the majority of the deficit between measured performance and performance expected from a symmetric implosion.⁷ 

One is therefore inclined to hypothesize that if these degradations to confinement are removed, agreement with the prediction of 1D radiation-hydrodynamics simulations can be recovered. However, some have speculated that even in the absence of ablation-front degradations to confinement, there may still be a significant reduction in 1D predicted yield due to instability growth of the cryogenic DT layer into the hot spot.¹¹ Both incomplete confinement and reduction in hot spot volume due to instability growth represent deviations from a 1D fluid model of the transfer of kinetic energy from the imploding shell to the hot spot. Others have hypothesized that separation of deuterium and tritium ion species, particularly during convergence of the main shock,¹² or ion-ion mean free path effects near the hot spot/cold-fuel boundary¹³ can result in a deviation from predictions of stagnation from fluid hydrodynamic codes that do not include these kinetic effects. A cryo-layered implosion platform that demonstrates 1D performance would be very useful in sorting through these mechanisms.

Analytic scaling, including the effect of self-heating, can be applied to ICF implosions to estimate the increase in work done on the implosion and therefore the laser energy necessary to achieve significant capsule gain (∼MJ's) starting from an implosion with only tens of kJ's of self-heating.¹⁴ However, this scaling assumes a spherically symmetric assembly of fuel in order to estimate the effects of both alpha deposition into and conduction and radiation losses from the DT fuel. Without spherical symmetry, detailed two- or three-dimensional simulations are necessary to include the effects of implosion non-uniformities on capsule performance; if the asymmetry is not highly reproducible, predictions of the performance improvement one might expect from a given increase in laser energy become suspect. Therefore, it is highly desirable to demonstrate spherically symmetric layered DT implosions on the NIF to underwrite confidence in the models of hot spot power balance if one wishes to estimate the laser energy necessary to achieve high gain from an indirectly driven ICF implosion.

In this paper, Sec. II provides the details of the indirect-drive platform designed to produce a 1-D implosion. Section III summarizes the data from the series of experiments conducted, including shots used to check the trajectory and symmetry of the implosion, as well as the cryo-layered experiment itself. Here the data are compared to 1D simulations of the capsule implosion. Section IV leverages the agreement between simulation and data to examine a constraint on the hot spot power balance. A planned subsequent experimental series is also described, in which an attempt will be made to reproduce this level of agreement with a bigger, faster implosion that will potentially add significant energy to the hot spot via alpha deposition.

The design of the NIF 2-Shock implosion platform¹⁵ deliberately attempts to avoid the effects from asymmetry and ablation front instability described above. “High-foot” implosions on the NIF have demonstrated a reduction in the ablative Rayleigh-Taylor instability growth as a result of increasing the drive temperature during the early part of the pulse over the previous “low-foot” implosions.¹⁶ However, for thinner high-foot capsules, measurements¹⁷ and simulations⁷ indicate that the tent still has a significant effect on performance. The 2-Shock platform doubles down on this approach to ablative stabilization, further increasing the temperature of the drive “foot” to ∼140 eV in hopes of eliminating altogether the effects of the tent and fill tube.

The strategy for low-mode symmetry control has evolved significantly in the eight years since ICF implosion experiments began on the NIF. Initially, He gas fill at ∼1 mg/cm³ was used as a means to hold back the hohlraum walls as the laser power was converted into X-rays, minimizing the motion of the wall material, the emission from which drives the capsule.¹⁸ This density of He gas resulted in ∼15% of the incident laser energy being reflected back out of the hohlraum due to stimulated Raman scattering and stimulated Brillouin scattering.¹⁹ During this time, a second type of laser scatter was employed to manipulate drive symmetry through the transfer of power from outer- to inner-cone beams. This cross-beam energy transfer (CBET) was controlled by applying a small wavelength difference between the cones;²⁰ however, the effect was not well predicted by simulations and the amount of power transferred was predicted to vary as a function of time in an un-controlled fashion.

The 2-Shock platform of Ref. 15 attempts to minimize laser-plasma interactions in order to maintain positive control over symmetry by reducing the He density to ∼0.03 mg/cc, or the minimum that is necessary to conduct heat from the cryogenic capsule.²¹ Because the hohlraum wall plasma is now much more free to expand inward and interfere with laser beam propagation, the ratio of hohlraum to capsule radii is increased from 2.8 to 4.25. The result is a platform that can reproducibly generate an implosion that remains spherically symmetric through stagnation.¹⁵ 

Because the non-cryo-layered capsule described in Ref. 15 was conservatively designed to maximally protect the gas-ablator interface from the hydrodynamic growth at the ablation front, it has a peak shell velocity of ∼250 km/s, too slow to be interesting to study cryo-layered ICF performance. Therefore, while the capsule outer radius remains fixed at 680 μm, the CH shell thickness was reduced from 175 to 120 μm, and the DT ice layer was designed to be only 40 μm thick. As a result, the cryo-layered 2-Shock capsule is predicted to have a fuel velocity of 315 km/s and a convergence ratio (CR) of 30, as defined by the initial capsule inner radius over the radius of the hot spot emission at stagnation, versus a CR of 12 for the Ref. 15 capsule. The final target design is shown in Fig. 1. With this reduction in shell thickness from that of the capsule in Ref. 15, the predicted mass remaining at stagnation is reduced from 24% to 17.5% (including ice layer mass). However, the layered 2-Shock design reported here still produces an implosion that exhibits a high degree of stability at the fuel-ablator interface. Figure 2(a) plots the Atwood Number, defined as $(ρice−ρshell)/(ρice+ρshell)$ between peak acceleration and the arrival of the stagnation shock just following peak velocity. The normalized density profile at peak velocity is shown in Fig. 2(b).

This capsule was contained in a 5.75 mm diameter, 9.4 mm long gold hohlraum with the aforementioned 0.03 mg/cc He fill. It was suspended in the hohlraum between two 45 nm sheets of polyamide, the “tent” as described above. The inner and outer surface roughness spectra of the plastic ablator were characterized at or below the roughness specifications for the National Ignition Campaign.²² The DT was supplied to the capsule via a 10 μm diameter fill tube and formed a layer of ice that also conformed to the National Ignition Campaign specifications. Finally, the capsule was driven with the laser pulse shown in Fig. 3(a), producing the temperature history at the capsule shown in Fig. 3(b).

Prior to imploding the cryo-layered capsule, a series of three “tuning” experiments were conducted to evaluate the effectiveness of the re-designed 2-Shock platform. Shock speed and shock merger depth were checked with VISAR measurements of shock velocity at the capsule pole and equator in the “keyhole” geometry.²³ In-flight symmetry was checked using the 2D Convergent Ablator platform²⁴ with a D₂ gas-filled capsule. Backlit radiographic images are taken of the capsule starting at 300 μm radius to generate a time history of the in-flight symmetry. The measured shell trajectory and P2 asymmetry as a function of time are shown in Fig. 4(a), where the “Limb Min” measurements are taken from the inner surface limb of the image, and the “Max Slope” values are from the outer edge. Apparent “jumps” in the trajectory occur between imaging strips of the X-ray framing camera. An image of the capsule at 7.25 ns is shown in Fig. 4(b), when the measured limb-minimum P2 is $3.8±1.7μ$m. Note the absence of any horizontal features (similar to those seen in Ref. 8) associated with ablator instability growth seeded by the capsule support tent. Composite analysis of a series of 7 images spanning from 7.125 to 7.375 ns is used to calculate a mass remaining at 7.25 ns of 16 ± 2%, in good agreement with the calculated value of 17.5%.

Another preparatory experiment, shot N160721, was performed as a symmetry capsule or “symcap”²⁵ without a DT ice layer, only DT gas fill, in order to evaluate the time-dependent symmetry of the hot spot at stagnation. As a mass surrogate for 40 μm of DT ice, the capsule shell was 10 μm thicker than the cryo-layered design shell, for a total of 130 μm. The gas fill density for this implosion was reduced from the Ref. 15 fill of 8.5 mg/cc to 5 mg/cc. Even at this increased convergence, the hot spot X-ray emission revealed excellent symmetry control including an absence of any symmetry swings. Figures 5(a) and 5(b) show time-resolved X-ray emission images of the implosion at the time of peak emission, from equatorial and polar views, respectively. Also shown in the figure are time histories of the mode-2 symmetry from both views, computed from a series of images near peak emission taken ∼25 ps apart. The emission history is shown as the dashed line on the plots.

Having measured the symmetry of the shell and hot spot based on X-ray emission, it remains to measure the symmetry of a high-convergence cryo-layered implosion. Just as for the symcap implosion, X-ray self-emission from equator and pole can be evaluated; however, the cryo-layered implosion employed time-resolved penumbral imaging²⁶ at the equator to capture the self-emission detail at high-resolution (∼5 μm). Additionally, for the layered experiment, two diagnostics were employed to evaluate the symmetry of the nuclear performance. The Neutron Imaging System imaged neutrons emitted from the implosion along an equatorial axis²⁷ and the Neutron Activation Diagnostics (NADs) were arrayed around the NIF target chamber to evaluate the time integrated symmetry of neutron production.²⁸ 

The primary neutron image of the layered implosion, NIF shot N161004, is shown in Fig. 6(a). It has a size (P0) of 20 μm and a P2 moment of +2%. The higher resolution time-integrated penumbral reconstruction of the X-ray emission is shown in Fig. 6(b). The size of the X-ray emission from that image (P0) was 18.7 μm with a P2 moment of +4.5%, which together with an initial capsule inner radius of 581 μm gives a convergence ratio of 30. The NAD data are shown in Fig. 7 compared with the data from N140520, the highest performing high-foot implosion.²⁹ Immediately apparent are the size of the error bars for N161004 as compared with a much higher yield shot. The low signal precludes a determination as to the significance of the observed fluence anisotropy.

From the imaging data, we conclude that N161004, the cryo-layered implosion, retained the good symmetry control of its predecessors down to high convergence. Next, we consider the neutron time-of-flight (NTOF) data, collected along 5 lines-of-sight as shown in Fig. 8. Temperatures for the DD and DT reactions can be extracted from the DD and DT peaks of the NTOF spectra centered near 2.4 and 14 MeV, respectively. The quantity reported in the data is actually derived from the second moment of a Gaussian fit to these peaks that includes a scattering contribution;³⁰ the two leading contributions to the second moment are the ion thermal temperature and the variance of the fluid velocity along the line of sight of the detector.³¹ Therefore, we refer to these quantities as “apparent” temperatures, as the fluid velocity variance component can vary considerably depending upon the line-of-sight.

The DT apparent temperatures ranged from 2.68 to 2.89 keV, and the DD apparent temperatures from 2.60 to 2.73 keV depending on the line-of-sight. However, the error bars on the individual measurements themselves are $±∼250$ eV, dominated by systematic uncertainty,³⁰ so the spread in temperatures is not as much an indication of asymmetry as of uncertainty in the measurements. The ∼4% difference between DT and DD apparent temperatures is consistent with the difference in the temperature dependence of the reaction rates³² combined with a 25% difference in the fluid velocity variance contribution due to the mass difference.³³ The similarity of the two apparent temperatures, therefore, is a strong indication that the two ion species, D and T, experienced the same conditions of temperature and density averaged over the burn, in contrast to the pictures presented in Refs. 12 or 13. Finally, the DT neutron yield obtained from the NTOF data was $1.64×1014±1.6%$ neutrons in the 13–15 MeV range, and the DD neutron yield was $7.29×1011±8.7%$ neutrons in the 2.2–2.7 MeV range. The ratio of roughly ∼220 between the yields is consistent with the ratio of the reactivities at a temperature of 2.75 keV.³² 

To evaluate the degree to which the N161004 implosion can be considered one-dimensional, and to attempt to constrain methods and approximations in our models of stagnation, we made detailed comparisons of the data to 1D capsule-only simulations of the implosion. A frequency-dependent drive source was created for the 1D simulations from a single two-dimensional integrated laser ray-trace radiation-hydrodynamic simulation of the full target. Adjustments were made to the input laser power for the integrated simulations of the tuning experiments to match observed shock breakout times and shock velocities as well as peak X-ray emission times at stagnation. These adjustments were then applied to the laser power history recorded for N161004 and used in an integrated simulation of that experiment, with the peak X-ray emission time checked for consistency. The frequency-dependent X-ray flux just outside the capsule surface in this 2D simulation of N161004 was recorded and spatially averaged for application in 1D simulations.

Table I provides a comparison of key observed and inferred quantities with 1D simulations run in LASNEX³⁴ and HYDRA.³⁵ For the simulated values of DT neutron yield, the values in parentheses are from calculations that do not include heating due to alpha particle deposition. Down-scattered ratio (DSR) is the ratio of 10–12 MeV to 13–15 MeV neutrons as extracted from the neutron time-of-flight (NTOF) detector signal. The apparent temperatures (T_app) listed for the data are the ranges reported from the various lines-of-sight as discussed above. The apparent DT and DD temperatures for the 1D simulations are calculated in an analogous manner as for the data; a fourth order Gauss-Hermite function is fit to the 14 and 2.4 MeV peaks and the second moment is extracted from the fit to evaluate the apparent temperature.

The variation in the inferred pressure in the Data column of Table I comes from the variation in the fusion reaction rate which is a function of the spread in apparent DT temperature measurements. In each case, the simulations were run without any mechanism to mix the ablator, DT ice, or DT gas regions. The comparison between simulated and measured performance for a high-convergence cryo-layered implosion is remarkable when compared with all other such implosions on the NIF. Figure 9 plots the ratio of the measured 13–15 MeV neutron yield to the yield from 1D simulation as a function of convergence ratio. The point at the upper left is the non-layered, low-convergence indirect drive exploding pusher (IDEP).³⁶ N161004 clearly stands above the historical trend of NIF cryo-layered implosions.

Because the 1D models are in as good agreement with the stagnation conditions as has been demonstrated above, we can apply constraints defined by the data in order to evaluate stagnation parameters in the model that are not directly observed. If we assume that stagnation is the time period of minimum volume, such that the hydrodynamic work is essentially zero, the two loss terms in the hot spot power balance are from Bremsstrahlung radiation and electron heat conduction.² Radiation-hydrodynamics codes such as LASNEX and HYDRA will calculate the frequency dependent radiation field explicitly, but typically do not include kinetic models of electron thermal transport. Instead, they use a non-flux-limited Spitzer-Harm model adjusted for dense plasmas³⁷ for the electron heat conduction; therefore, it is relevant to try and constrain the thermal conduction coefficient if we wish to have a more predictive model of the stagnation.

Like the inferred hot spot stagnation pressure, the hot spot density at stagnation is a quantity that takes into account all of the stagnation observables: neutron yield $(Yn)$⁠, temperature $(Ths)$⁠, DSR, hot spot volume $(Vhs)$⁠, and burn width $(τburn)$⁠. We calculate the hot spot density from these observables

where $⟨σv⟩(Ths)$ is the D-T reaction rate per unit volume as a function of temperature,^32,$A¯$ is the average atomic mass of the reactants and N_A is the Avogadro's number. If we again take the range of apparent temperatures from N161004, we find hot spot densities in the range of 47.5 to 55.9 g/cc. Calculating the hot spot density from the simulated set of the same five observables using Eq. (1) provides a convenient method to check if any given simulated stagnation is consistent with the data. Taking the LASNEX 1D model, we apply a multiplier to the Spitzer-Harm thermal conductivity and vary the multiplier until the simulated hot spot stagnation density is no longer consistent with data. Doing so, we find that this constrains the electron thermal conductivity in the model to within ±25% of the nominal value. This is a factor of two improvement over the previously reported constraint on a model for electron heat conduction in a near-1D NIF implosion.³⁶ 

Of course the remaining term in the hot spot power balance for an implosion with sufficient number of fusion reactions is alpha heating, or the deposition of energy from the fusion burn products. An experimental demonstration of a 1D implosion that includes this term represents the next step in the process of validating models of stagnation. A campaign is now underway to implode a larger 2-Shock capsule at higher velocity in order to reach the self-heating regime. The target is compared to that from N161004 in Fig. 10(a); a 990 μm outer radius capsule replaces the original 680 μm outer radius capsule. The hohlraum diameter and length are also increased, but the case-to-capsule ratio is reduced significantly from 4.25 to 3.15. In order to mitigate the impact of wall blowoff on inner beam propagation in the tighter geometry, the hohlraum is filled with 0.3 mg/cc He. Finally, the temperature drive pulse is shown in Fig. 10(b), compared with the original pulse. At 303 eV, the peak temperature is only slightly lower than the N161004 pulse, but the increase in capsule absorbed energy will significantly increase the implosion velocity.

In an attempt to isolate the effects of alpha heating on the stagnation, a pair of cryo-layered implosions is planned for the larger 2-Shock target. The first implosion will contain “dudded” fuel, with an isotopic ratio of 70%/20%/10% for tritium, hydrogen, and deuterium, respectively. This so-called “THD” implosion is designed as a hydrodynamic surrogate for the 50/50 DT-fueled self-heating implosion.³⁸ With a predicted yield of 8.7 × 10¹⁴ neutrons, this THD experiment will produce roughly 10 times fewer neutrons than the 50/50 DT shot, allowing the larger, faster two-shock cryo-layered capsule to implode and stagnate without generating enough DT fusion reactions to contribute a source to the hot spot power balance. This yield should be sufficient, however, to significantly reduce the error bars in the flange NADs data from Fig. 7 and permit an evaluation of ρR asymmetry. The data from this implosion can then be checked for consistency with 1D simulation results prior to performing the DT implosion. If consistency with 1D simulation can again be achieved for the THD implosion in the same way it has been for N161004, then the 50/50 DT-fueled implosion results can be used to place a constraint on alpha deposition, as the other terms in the hot spot power balance will have been fixed by the THD implosion.

To illustrate the importance of constraining the model of alpha heating, Fig. 11 plots how one of the best-performing implosions on the NIF might scale in total yield as the laser energy is increased. The solid curve represents a hydro-scaling of NIF shot N170601,³⁹ a high-density carbon (HDC) capsule cryo-layered implosion. Under hydro-scaling assumptions,⁴⁰ the adiabat and velocity of the implosion are fixed, and the improvement in implosion performance comes only from the increase in the physical size of the capsule. The horizontal axis is then consistent with the assumption that the case-to-capsule ratio remains fixed, and the laser energy increases with the hohlraum wall area to maintain a constant temperature drive on the capsule in accordance with the hohlraum wall albedo.¹ This is considered to be a quite conservative approach to scaling prediction, as no improvements to the implosion quality that would otherwise increase the velocity, adiabat, or fuel areal density are allowed. The dashed lines on either side of the solid hydro-scaling trajectory represent a variation of ±20% in the fraction of alpha energy deposited in the hot spot. The lower dashed grey curve represents the performance of N170601 under hydroscaling in the absence of alpha heating. Taking, for example, a laser energy of ∼2.8 MJ, we find that under this variation, predictions of fusion yield range from 200 kJ to 1 MJ; this range only widens as the scale increases.

We have described the first 2-Shock cryo-layered experiments on the NIF, designed specifically to minimize low-mode asymmetries and ablation front instabilities. With the series of experiments detailed herein, the platform illustrates that once these effects, which lead to significant thinning and perforation in the cold fuel assembly, are removed, stagnation performance described by 1D models can be recovered. We compare the observables with those generated from two such models and find good agreement in both cases for two independent ICF codes. We show for the LASNEX code how this agreement allows us to establish a more restrictive constraint on our model of the electron thermal conduction losses from the hot spot.

In matching clean 1D predictions, the 2-Shock experiments have demonstrated a first step towards validation of predictions of scaling to increased facility laser energy. The subsequent 2-Shock campaign is directed at achieving a more general validation of stagnation models to include heating due to alpha deposition. The behavior of the dashed lines in Fig. 11 that bound a 20% uncertainty in alpha heating implies that it should become easier to constrain models of alpha deposition with high-yield implosions as the self-heating begins to dominate the hot spot power balance and the sensitivity of the yield increases. If, however, we wish to predict how much laser energy will be required to reach a certain stagnation condition based on lower-yield implosions, we will require carefully controlled stagnation conditions to validate the models used for these predictions. This effort will potentially provide a critical underwriting of our confidence in predictions of scaling at higher yield.

The authors would like to acknowledge the efforts of the NIF Operations, Laser Performance, Target Diagnostics, and Target Fabrication Teams. This work was performed under the auspices of the Lawrence Livermore National Security, LLC (LLNS), under Contract No. DE-AC52–07NA27344 and by General Atomics under Contract No. DE-NA0001808.