Causes of fuel–ablator mix inferred from modeling of monochromatic time-gated radiography of OMEGA cryogenic implosions

In direct-drive inertial confinement fusion (ICF),¹ a shell is imploded via direct laser-light illumination. The shell is composed of an inner layer of fuel (typically equimolar deuterium and tritium, which maximizes yield) and a thinner outer “ablator” layer of non-fuel material (e.g., a C–H polymer), which is chosen for both its laser hydrodynamic efficiency (determined by the ratio A/Z of mass to atomic numbers) and its insusceptibility to laser−plasma instabilities. The implosion of a two-layer shell filled with gaseous fuel is used to briefly assemble a “hot spot” of high temperature (4 or more keV) and density. For a large-enough confinement time, the alpha particles generated by D–T fusion reactions will generate energy in excess of the heat conduction and radiation losses. This will lead to a runaway thermal instability referred to as “ignition,” in which some fraction of the fuel in the surrounding shell is consumed.

A determining factor in the performance of an ICF target is the hot-spot pressure, which reflects the energy density in the hot spot. The minimum energy needed for an imploding target to ignite is believed to scale with the hot-spot pressure at peak convergence, $pHS,$ as $Ek, min∼pHS−2$⁠; the smaller the hot-pot pressure, the greater is the shell kinetic energy needed to ignite. Similarly, the neutron yield, which is a function of hot-spot conditions and confinement time, is also estimated to depend on the hot-spot pressure, as $Y∼pHS7/5$ (Ref. 2). Hot-spot pressure in turn depends on the shell density: the lower the shell density as the shell is being decelerated, the greater is the pressure gradient decelerating the shell, and the larger is the radius at which the decelerating shell stagnates.² This greater radius leads to a greater hot-spot volume and, correspondingly, lower hot-spot pressure for a given hot-spot energy. For this reason, yield for a nonigniting implosion such as those conducted on the OMEGA laser is expected to scale as $Y1D ∼ vimp5.9ρRmax8/5$⁠, where (ρR)_max is the maximum shell areal density and v_imp is the peak shell speed. This scaling relation is only expected to hold if disruption of the shell due to hydrodynamic instability is minimal, and we have used the relationship $α∼vimp1/9ρRmax−2$ (Ref. 3) between the shell areal density and the adiabat α ≡ p/p_Fermi, the imploding-shell mass average of the ratio of the pressure to the Fermi pressure p_Fermi. A key implication is that greater shell areal density, assuming the shell remains integral, is expected to be needed for greater neutron yield, both for sub-ignition and igniting targets.

Various phenomena can reduce the shell areal density, such as heating of the shell by suprathermal electrons generated in the corona surrounding the imploding target; shock dynamics, which must be carefully engineered due to the increase in entropy generated by mistimed shocks; absorption of coronal radiation by the shell, which places an additional constraint on the choice of ablator material; and mixing of the ablator and shell fuel material due to hydrodynamic instabilities, particularly the Rayleigh–Taylor instability (RTI).

Mix of ablator material into the hot spot has been shown to be correlated with a decrease in performance (see discussion in Ref. 2). Epstein et al.⁴ considered the continuum hot-spot x-ray emission for sufficiently high x-ray photon energies that the assembled core is optically thin to free–free radiation. They demonstrate that in the absence of mix, the x-ray and neutron yields obey a simple scaling. When the expected filtered x-ray yield is compared to that measured for an ensemble of cryogenic implosions on the OMEGA Laser System⁵ spanning a range of calculated fuel adiabats, they find that below a threshold adiabat, there is an excess of core emission, consistent with a mix of carbon from the ablator into the hot spot. Hydrodynamic mix is detrimental to target performance not only due to shell breakup and reduction of areal density, but also because the presence of higher-Z material in the hot spot increases radiative losses during compression and burn.⁴ A mix of ablator material has also been observed at the National Ignition Facility (NIF)⁶ in laser-indirect-drive (LID) experiments, via measurements of hot-spot K-shell line emission,⁷ and in carbon continuum emission,⁸ consistent with hydrodynamic mix. Another possible signature of mix is early emission from the hot spot in OMEGA cryogenic implosions.⁹ Hu et al.¹⁰ showed by using radiation-hydrodynamic simulations that decompression of the shell related to laser imprint can have this effect.

Stoeckl et al.¹¹ showed that a cryogenic implosion backlit with a 20-ps short-pulse laser shows evidence of fuel–ablator mix. They document the experimental signatures of deviations from spherical symmetry of low harmonic order (i.e., “low modes”) including an index-2 Legendre mode pre-imposed on the target, and of the target stalk during the implosion. They also describe evidence of uniform (small-scale) mix of carbon from the ablator late in time throughout the shell, inferred by means of radiographic post-processing, and show that whereas the late-time radiograph of an implosion expected to be less unstable to Rayleigh–Taylor growth matches 1D simulations, a simulated radiograph from a more-unstable cryogenic implosion shows substantial agreement with experimental data only when the post-processed image is generated, assuming the presence of carbon mixed into the shell (or throughout the shell). Epstein et al.⁴ presented an in-depth analysis of the radiography of cryogenic targets, exploring the physics resulting from the mix of carbon into the fuel layer in the shell. They considered multiple backlit shots, including one of the shots discussed here (81 590), inferring levels of fuel–ablator mix at a very sensitive level (0.1% carbon).

The present paper builds on these findings through a systematic exploration of mix using 2D radiation-hydrodynamic simulations of two of the implosions described by Stoeckl et al.,¹¹ one of which is expected to experience minor perturbation growth due to the RTI, and the other, which has a lower shell adiabat and greater susceptibility to hydrodynamic instability. This includes several possible sources of mix as well as the effect on radiographs of discrepancies in the mass ablation rate and uncertainties in the initial ablator thickness. Our survey of these several possible causes of fuel–ablator mix shows that good agreement with experimental data particularly requires (a) modeling of laser imprint at an enhanced level (which we will argue is to be expected) and (b) sufficient unablated ablator mass at the time of the experimental radiograph, either due to an initial ablator thickness at the upper level of the range given by pre-shot target metrology or due to inaccuracy in the modeled ablation rate.

Section II of this paper describes the backlit cryogenic experiments. Section III describes the modeling, including various possible sources of mix. In Sec. IV, inferences of mix are discussed, followed in Sec. V by conclusions.

In the experiments considered here, a single short-pulse beam from OMEGA EP¹² is used to illuminate a Si-foil backlighter target near an imploding cryogenic (cryo) target after the end of the laser pulse but before stagnation or peak neutron production (see Ref. 11 for an in-depth description). The resulting radiation passes through the target to illuminate a quartz crystal imager. The crystal imager is cut with Bragg spacing tuned to the Si He_α line at 1.865 eV and is mounted on a curved substrate. This reflects the radiation from the Si backlighter to an x-ray framing camera that has an exposure time of 40 ps and a spatial resolution of 15 μm. The timing of the backlighter pulse was measured with a precision of 10 ps relative to the OMEGA laser using the neutron temporal diagnostic.¹³ The combination of a Si backlighter and a tuned monochromatic Bragg crystal imager is needed to ensure that the backlighter radiation is not overwhelmed by the core emission, which has a doubling time near peak emission of tens of picoseconds.

The image is subject to motion blurring due to the backlighter pulse duration, leading (for the implosion speeds considered here) to an effective resolution of ∼19 μm. The backlighter intensity, therefore, imposes two constraints on the experiment: First, it limits the maximum shell speed due to motion blurring because a less intense backlighter necessitates longer pulse duration. Second, it puts a limit on how close to peak core emission the image can be taken, given that the core emission must not overwhelm the backlighter emission.

The performance degradation of OMEGA cryogenic implosions, as measured by the reduction in areal density, has been found to be small or negligible (corresponding to a measured peak shell areal density of at least 85% of the 1D calculated value) for targets that satisfy IFAR < 20 (α/3)^1.1 (Fig. 1).² Here IFAR is the calculated in-flight aspect ratio, defined as the ratio of the shell radius and the shell thickness (given by the distance between the 1/e points with respect to the peak density). Both of these parameters are defined at the time when the shell radius is equal to 2/3 of the initial radius. This stability threshold represents the main hydrodynamic failure mode of an ICF target: if too thin, Rayleigh–Taylor growth at the ablation surface will feed through to the inner surface and grow as the shell decelerates due to pressure in the hot spot, or, if the growth is sufficient, the shell will be compromised. Additionally, for a given shell thickness (and corresponding IFAR), if the adiabat is too small, the ablative stabilization will be insufficient to prevent shell distortion and breakup.

Two backlit implosions are analyzed here: OMEGA shots 81590 and 82717 (Fig. 2), which by design lie on either side of this stability threshold line (Fig. 1). These shots are both triple-picket designs¹⁴ in which three short intensity spikes (pickets) at the start of the laser pulse are used to condition, using decaying shocks, the shell adiabat, such that if timed correctly the shocks generate a high adiabat near the ablation surface to reduce RTI during the pulse and a lower adiabat in the fuel, allowing for greater compressibility during stagnation.¹⁵ The properties of the designs are shown in Table I. Shot 81590 has a moderate adiabat and a lower IFAR. Shot 82717 has a lower adiabat and higher IFAR, making it the less stable of the two designs. The implosion speeds of the two designs (240 and 280 km/s, respectively) are comparable and significantly slower than typical cryo implosions, which generally have speeds v_imp of 300–450 km/s. These lower speeds reduce motion blurring of the radiographic image. As expected for the comparable implosion speeds, the neutron-averaged ion temperatures T_i,LILAC are comparable, with the higher implosion speed shot 82717 having a higher ion temperature and yield, as expected. The 1D ρR is expected³ to follow the dependence $ρR∼α−0.55vimp0.06E0.33,$ where E is the laser energy incident on the target, and the initial shell thickness is also comparable for the two shots. However, the higher adiabat shot 81590 has a smaller outer radius (444.4 vs 478 μm), so a lower mass (62.34 vs 71.37 μg) but a lower pulse energy (24.0 vs 25.2 kJ), and therefore a higher specific incident energy (386 vs 354 J/μg). This is compounded by the fact that the lower-adiabat shot 82717, given its larger initial radius, begins its coasting phase at 128 μm, compared to 102 μm for shot 81590. These factors combine to produce comparable peak (and neutron-weighted) 1D areal densities for the two shots.

The resulting calculated convergence ratio of these implosions is ∼24. Predictably, shot 81590 has a higher ratio of measured areal density to the calculated 1D value, about twice that of shot 82717. Similarly, the experimental yield divided by the 1D yield (the yield-over-clean, YOC) of shot 82717 is less than half that of shot 81590. Both implosions have a low YOC. The ion temperatures are comparable although the more-unstable implosion has a somewhat lower temperature. Both implosions have small offsets from target chamber center (under 10 μm) and low rms ice roughness (less than 1 μm), as determined by shadowgraphic measurement prior to the shot. The peak laser intensity of these designs was 300 TW/cm², ensuring that cross-beam energy transfer¹⁷ (CBET), while non-negligible, plays only a moderate role in reducing the drive by scattering incident laser light from the corona.

In this section, we describe the radiation-hydrodynamic modeling of these experiments in both 1D and 2D. A number of possible sources of mix are investigated. For the dominant sources, the simulations were post-processed in order to generate synthetic backlit radiographs for comparison with the experimental data. Because the synthetic radiograph calculated from a LILAC simulation of shot 81590 agrees well with the experimental radiographic lineout, the primary focus will be on shot 82717, for which this is not the case. The relative efficiency of each of these mechanisms is described, followed in Sec. IV by a discussion of signs of mix in these implosions.

The target designs presented here were simulated using the 2D radiation hydrocode DRACO.^18,19 DRACO is an arbitrary Lagrangian–Eulerian (ALE) code²⁰ employing a second-order accurate Winslow rezoning algorithm,²¹ which models two-temperature heat transport and multigroup radiative transport and performs material tracking using a second-order accurate volume-fraction contouring method. In the simulations described here, the first-principles equation of state²² (FPEOS) and opacity model²³ (FPOT) were used to model the CH and DT. These tables are calculated using density functional theory and have been shown to compare well to experimental data.

DRACO uses a 3D ray-based inverse bremsstrahlung energy deposition model for laser drive of the imploding target.^17,24 In laser-direct-drive implosions for these intensities and scale lengths, a significant fraction of the incident laser energy is scattered due to CBET, reducing the efficiency of the implosion. CBET is caused by seeded stimulated Brillouin scattering in which two beams interact by means of an intermediate ion-acoustic wave, increasing the scattered light and thereby reducing or preventing coronal absorption. The CBET model in DRACO uses an angular spectrum representation (ASR), which captures, for each computational zone, the accumulated intensity as a function of direction and frequency from all the rays from every beam traversing that cell, representing the field of pump rays. Probe rays crossing a cell interact with the pump field (represented computationally by the ASR), iteratively obtaining a self-consistent, energy-conserving solution. This model for CBET, combined with nonlocal electron heat transport, has been successful in matching a number of experimental observables in experiments on both the NIF and OMEGA.^24,25 DRACO uses the implicit Schurtz–Nicolaï–Busquet (iSNB) nonlocal electron heat-transport model (NLET).²⁶ The iSNB model computes, using an implicit algorithm, the nonlocal heat flux using multigroup diffusion by means of a multidimensional convolution integral, which has the effect of delocalizing the Spitzer–Härm heat flux. The iSNB model, based on the SNB model,²⁷ has demonstrated predictive capability for shock timing²⁶ and shell shape²⁸ in numerous experiments on OMEGA. Note that DRACO does not use a hydrodynamic mix model; all the mix observed here in simulations is due to directly modeled Rayleigh–Taylor growth or (as described below, for specifically indicated simulations) concentration gradient fluxing.

The 1D simulations were calculated with LILAC,²⁹ a Lagrangian radiation-hydrodynamic code that models all the same essential physical processes, including CBET and NLET, as DRACO, with the exception of concentration gradient fluxing.

Less-unstable shot 81590: The experimental backlit radiograph for the less-unstable implosion, 81590, shows substantial agreement with the synthetic radiograph of the post-shot LILAC simulation performed using the measured laser pulse, calculated with Spect3D³⁰ using the experimental filtering, backlighter spectrum, and point-spread function. Figure 3(a) shows the radiograph of the imploding target. In the radiograph, a large bright region, the image of the backlighter beam, is occluded by a dark ring caused by attenuation by the dense material of the shell. Inside this ring is a bright feature caused by core emission. A horizontal lineout of this radiograph is shown in Fig. 3(b) (black curve). The blue and red curves indicate lineouts from synthetic radiographs of LILAC and DRACO, respectively. The experimental lineout is corrected based on a model for the backlighter brightness profile.^4,11 As described by Epstein et al.⁴ and Stoeckl et al.,¹¹ because the brightness of the backlight is unknown, the simulated backlighter intensity was adjusted to reproduce the measured ratio of core self-emission to backlighter intensity at the time of the radiograph. The lineout shows the same central features of the 2D radiograph, particularly the central core emission and shadow cast by the shell. The shell used in shot 81590 was designed with a thick ablator (∼12 μm). This was done to ensure that some ablator remains after the end of the pulse in order to generate sufficient contrast to make the shell clearly distinguishable. DRACO simulations modeling the effect of laser imprint, including illumination intensity perturbation with Legendre modes ℓ = 2 through 150, show that the effect of imprint is minor; the fuel–ablator interface has a mix width of ∼10–20 μm, and no ablator material is conveyed due to mixing into the fuel. Correspondingly, the shell remains intact, and there is minimal impact on the yield and areal density due to imprint for this shot. The discrepancies between the experimental and simulated radiographs include notably two features that are prominent in the simulated lineouts but either reduced or absent in the experimental lineouts: the dips in the absorption at the inner and outer edges of the shell feature. The outer dip is due to limb darkening. The inner minimum is caused by higher density at the inner shell caused by both convergence and the outward-moving shock, and the absorption in this region is dominated by free–free absorption of the DT, which in turn depends on the square of the density.

More-unstable shot 82717: in contrast, a synthetic radiograph generated using LILAC output deviates markedly from the experimental backlighter transmission for the more-unstable implosion 82717. As mentioned above, according to LILAC, this shot is expected to have a lower adiabat and higher IFAR and also uses a thinner ablator (11.2 μm). Figure 4 shows the experimental radiograph for this shot as well as lineouts of synthetic radiographs from LILAC and DRACO, compared to the experimental lineout, corrected for the shape of the backlighter.

A small degree of pre-imposed asymmetry was introduced by using a shaped plastic shell with a 2-μm amplitude Legendre mode-1 variation in the shell thickness from one side to the other. As described by Stoeckl et al.,¹¹ the pre-imposed Legendre mode-1 perturbation in the experiment showed the expected mode-1 perturbation of the shell attenuation and allowed a direct measurement of the impact of this mode on distortion of the core emission. Since the focus of the present study is physical mechanisms capable of generating fuel–ablator mix, this mode, which accounts for an ∼7% asymmetry in attenuation, is not included in the DRACO simulations of this implosion, and the simulated radiograph lineout are compared to the experimental lineout, symmetrized about x = 0 by averaging for each |x| the values at –x and +x. The impact of long-wavelength non-engineered target and laser perturbations on mix is discussed in Sec. III B.

For this shot, the image was taken 50 ps closer to peak neutron production with correspondingly brighter core emission. (The bang times for shots 81590 and 82717 are 3.49 and 3.56 ns, respectively.) Notably, the lineout generated from a LILAC simulation (and from a DRACO simulation without perturbation seeds) has much lower shell absorption than the experimental image. When a nominal level of imprint is simulated, including again Legendre modes 2 through 150, with a resolution of 12 computational zones per wavelength for ℓ = 150, the agreement is better due to higher shell absorption, but a large discrepancy remains. As discussed in more detail below, we might expect the 3D instability growth to be somewhat larger than these 2D simulations. If the imprint amplitudes are increase by a factor of $2$⁠, the magnitude of the absorption matches that of the experiment. However, even with this amplitude increase, the radial extent of the absorption is less than that of the experimental lineout. A simulation with an expanded modal range of 2–200 has, for the $2$ case, an almost indistinguishable lineout.

As has been explored in detail in both Refs. 4 and 11, carbon is more opaque than hydrogen. The dominant absorption processes under compressed shell conditions are the bound-free absorption per ion of carbon, which is proportional to Z,⁴ and the free–free absorption of hydrogen (DT), which is proportional to n_eZ,² so that the absorption per ion of the carbon is greater than that of hydrogen by a factor of ∼1300. As they show, even a small amount of carbon mixed into the shell has a significant impact on the shell absorption. Figure 5 shows both the total mass density and the CH mass density for shot 82717 at the time of the radiograph for the nominal level of imprint as well as simulations where the imprint amplitudes were increased by constant factors 2 and $2$⁠. A linear scale is used in the upper row of contour plots in Fig. 5 to better show the distortion of the capsule, and the lower row uses a logarithmic scale to bring out the presence of plastic, given that even a small amount of carbon can have a significant impact on the optical depth. DRACO tracks multiple materials using second-order volume-fraction contouring, as mentioned above. The white line in the upper plots of Fig. 5 shows the 50% volume-fraction contour. For imprint simulations, this method, despite being second-order, generates some degree of material diffusion in which the initially sharp material boundary is spread over a finite range of radii. For the simulations shown in Fig. 5, the 5% and 50% volume-fraction contours are typically 2–5 μm apart in radius at the inner spikes, and in the corona, where the CH mass density is much less (and the CH absorption correspondingly less), the difference between the radii of the 5% and 50% contours is larger, as much as 20 μm. Note that this region of spreading is unrelated to the Fick's–law concentration diffusion discussed in Sec. III D. While even the nominal level of imprint generates an ∼150-μm mix region, enhanced levels of imprint transport CH much farther into the shell and closer to the core, even to the point where CH reaches the outward-moving shock front, although none of these simulations predicts that imprint seeds sufficient mix to introduce carbon into the hot spot. While imprint is not the only mechanism for transporting carbon deep into the shell, these simulations show it is efficient at doing so. The implication of the shortfall in absorption with the nominal imprint level is discussed below.

Figure 6 shows again the experimental radiograph for (a) shot 82717 compared with (b) 2D synthetic radiograph for the $2$-imprint simulation generated by using the noise spectrum extracted from the experimental radiograph and using a super-Gaussian model for the backlighter profile. Like that of shot 81590, the dark ring indicates absorption by the shell, which for this shot is off-center from the backlighter beam. As seen in the lineouts, the transmitted core emission is greater than the transmitted backlighter emission. The discrepancies seen in the lineout are shown here to be present as well: despite having comparable shell absorption near the inner edge of the core, the radial extent of the absorption well due to the shell is about half that in the experimental image, and the core emission region is larger. (Note that the shadow cast by the shell corresponds to the region with mass density over ∼2 g/cm³, and the mix region spans radii from the return shock well into the corona.) Figure 6 also shows [in (c) and (d)] a synthetic radiograph (c) and density plot (d) for a simulation which assumes 0.3 μm more initial CH ablator thickness, but the same total shell mass. This lies within the range of target metrology of the ablator thickness and has a significant impact on the synthetic radiograph. As will be shown in Sec. IV, this observation will provide the second key to agreement between modeling and experimental radiographic data.

The imprint spectrum used in DRACO is given by an analytical model describing the far-field laser intensity of super-Gaussian distributed phase plates (DPP's).³¹ In the simulations described here, a resolution of 12 zones per wavelength was used for the mode with the smallest initialized wavelength, ℓ = 150. A simulation with increased resolution was performed, as well as a simulation with a modal range extended to mode 200; in both cases, the differences were negligible. That a level of imprint beyond that predicted is needed to match the radiographic absorption of the shell has been observed³² and is unsurprising not least because in a 2D simulation the ablation-surface perturbations are modeled as axisymmetric rings rather than isolated spikes seeded by laser speckles, and rings have a different ratio of surface area to volume than isolated spikes. The discrepancy has been found to be minimal in at least one case, that of modeling imprint with DRACO in the context of an OMEGA plastic-shell implosion, which showed substantial agreement with experiment.¹⁰ Imprint is a complex process that goes beyond differences in the RTI growth in two and three dimensions since the initial stages involve not only the RTI but also the Richtmyer–Meshkov instability (RMI) and are affected by the dynamics of the multiple shocks and rarefaction waves that transit the shell prior to the acceleration phase.³³ Future simulations will model imprint in 3D in order to capture the necessary physical effects.

Another possible cause of mix is instability growth seeded by isolated shell-surface features that can include dust on the target surface, voids in the ice, or defects at the ice–ablator interface. For instance, radiation from tritium decay in permeation-filled targets can cause localized perturbations at the inner shell surface. Cryogenic targets may have as many as two to five clusters, ∼100 μm in extent, of dozens of these micrometer-sized features. Similar features at the ablation surface and in the shell have been shown to be capable of transporting ablator material into the core.³⁴ A necessary feature of such mix is that it must be isotropic to within the limits of the experimental resolution (i.e., spatially uniform in distribution) since the backlit radiographs show no evidence of individual jets or features (within the given spatial detector resolution). While these mechanisms are not studied here, they are consistent with the low performance of shot 82717, both in yield and areal density.

A series of DRACO simulations were also performed to assess the impact of other seeds, which are predominantly in low modes, on simulated backlit radiographs. Low modes have various sources, including feedthrough of outer-surface hydrodynamic growth due to laser imprint and surface roughness; beam mispointing; target offset, which primarily generates a Legendre mode-1 distortion; beam mistiming, which primarily generates a Legendre mode 2; ice roughness due to the cryogenic layering process; power imbalance between beams; and the static illumination pattern generated by the beam geometry and the ratio of the beam size to the target size. Simulations including the measured spot profile and OMEGA beam geometry were used to explore particularly beam mispointing, beam mistiming, and power imbalance. These simulations also included ice roughness using the measured inner-surface rms with a power-law power spectrum and a random phase for each mode. Note that the effect of an engineered low mode was explored by Stoeckl et al.;¹¹ as mentioned above, the target used in shot 82717 had a 2-μm amplitude ℓ = 1 Legendre mode on its ablator, with the peak thickness oriented to the left in the radiograph, 90° from the stalk. The engineered mode is not modeled here as it was fully explored in Ref. 11. Similarly, the impact of the stalk was explored in this same reference and elsewhere³⁵ and is not modeled here.

The impact of low modes caused by ice roughness and beam–port geometry, power imbalance, beam mistiming, and beam mispointing is shown in Fig. 7, which compares, at the time of the experimental radiograph for shot 82717, the mass density (color contours) and 50% DT-ice material-fraction contour (white lines) for implosions simulated with double the expected level of imprint. (Imprint was included to show the impact of low modes on an already perturbed target.) The impact of the low modes is indeed visible in the simulated radiographs, primarily in an ℓ = 2 distortion of the hot-spot emission. Simulations also show little mode coupling, as indicated by no increase in the degree of material mix into the inner regions of the shell. The distortions of the hot spot are also seen in simulations without laser imprint. What these simulations cannot show, being axisymmetric, is the impact of interactions between the phases in 3D of the different low-mode perturbations. While this is unlikely to affect transport of carbon into the shell, it may distort the hot spot and contribute to target performance.

One natural source of mix to consider is perturbation growth at the fuel–ablator interface seeded by perturbations on that surface. Unlike at the ablation surface, growth of modes here due to the RTI is not diminished by ablation, so no high-mode ablative cutoff exists. Beta decay from tritium decay in permeation-filled–target cases causes localized damage at the inner shell surface, potentially seeding perturbation growth. This may play a role in the known relationship between target age and performance.³⁶ In addition, the surface roughness of the polystyrene ablator shell may seed perturbation growth. The average power spectrum for surface perturbations has a power-law dependence of amplitude on Legendre mode number $L$ of ∼$L$^−3/4. Simulations of shot 82717 were performed with a piecewise power-law function approximating the measured polystyrene power spectrum. Figure 8 shows the mass density at 3.54 ns (the time of the radiograph) of (a) a simulation including modes 2–100 and (b) a second simulation with triple the expected amplitude. In this figure, the CH–DT interface is indicated by the outer white contour. For the nominal case, the effects of the perturbation are small, having a negligible impact on the shell mass density. For the enhanced simulation, the impact is more significant on the shell mass density, but there is still minimal material mix due to perturbation growth.

This interface, if unstable, is classically unstable, being a buried layer interface and not subject to the cutoff of the growth rate under ablation at a high perturbation mode number. For this reason, simulations were also performed of a mode ℓ = 500 Legendre-mode interfacial perturbation for a range of modal amplitudes. Similar to the simulations modeling modes 2–100, little effect is observed: the interface roughness between the CH and the DT layers has an rms of ∼1 μm, but the perturbation on the inner and outer edges of the shell is less than 0.1 μm. This simulation shows no significant mix. The reason can be found in the time history of the interface, shown in Fig. 9, which shows the Atwood number A of that interface as well as its acceleration, with the laser pulse shown for reference. The acceleration history has three sharp peaks due to the passage of the picket shocks, followed by the shell acceleration period. Prior to shell acceleration, the interface is subject to RMI growth due to the impulsive acceleration of the shocks and RTI growth due to the rarefaction waves that follow each picket, during which both the acceleration g and the Atwood number A are negative and Ag > 0; however, the magnitude of the rarefaction acceleration is small. During the majority of the drive pulse, when the acceleration is large and negative, A is negative, only changing sign halfway through the drive pulse, at around 2.6 ns, shortly before the interface is ablated. This is simply not enough time for significant perturbation growth and mix.

One further question arises regarding interfacial growth as a source of mix: For reasons given above, the shell speeds for these implosions are at least 100 km/s less than typical OMEGA cryo implosions, so that the interface acceleration is potentially less than is typical. In Fig. 10, the same quantities are shown for shot 90288, one of the highest-performing OMEGA cryo shots, with a yield of 1.6 × 10¹⁴ a YOC of 40%, a neutron-averaged ion temperature of 4.55 keV, an areal density of 160 mg/cm², and an inferred hot-spot pressure of 52.7 Gbar. The peak shell speed for this implosion was, according to 1D simulations, nearly 500 km/s, almost twice that of 82717, but the interfacial growth is not significant. This is simply because the ablator, by design, fully ablates by ∼1.2 ns, and although the acceleration of the interface becomes positive at ∼700 ps, the Atwood number of the interface is negative until shortly after 1 ns. Additionally, the pulse used for shot 90288 used only a single picket, not three, reducing the effects of RMI. For these reasons, it is considered unlikely that interfacial growth due to perturbations of the sort modeled is responsible for fuel–ablator mix in this more-characteristic cryo implosion. The possibility of isolated defect growth, for which the initial amplitude may be significantly larger, remains a possible source of mix, even for shot 90288.

Another mechanism investigated as a means of transporting carbon deep into the imploding shell is concentration diffusion. This is one of many kinetic effects that have received attention in recent years in both direct- and indirect-drive ICF.³⁷ Concentration diffusion is expected to occur wherever a concentration gradient exists in a multispecies plasma.³⁸ The flux in this case is given by Fick's law³⁹ and is proportional to the concentration gradient. A second-order accurate ion concentration diffusion model was implemented in DRACO to investigate this effect. The plasma diffusion coefficient is given by D = (Z_eff + 1)k_BT/(mν). The effective ionization Z_eff is given by the formula of Bastea⁴⁰ for a binary mixture, $Zeff2=⟨Z5/3⟩⟨Z⟩1/3$ (valid for small or moderate charge asymmetries, i.e., $Zi/Zj<10).$ T is the ion temperature, m is the average mass per ion, and ν is the collision frequency, given by the sum of the pairwise species collision frequencies,⁴¹ 

where n is the ion number density, e is the electron charge, ln Λ_ij is the Coulomb logarithm for collision between species i and j, and the Kronecker delta δ_ij, used to avoid double counting. Kinetic effects are expected to be significant when the Knudsen number $Kn=max{λij}/L$ is large, where λ_ij is the ion–ion mean-free path, and L is the hydrodynamic scale length. Equivalently, the scaling of the diffusion coefficient is D ∼ T^5/2n⁻¹, so that concentration diffusion is most efficient in regions of high temperature and low density, where the concentration gradient is small.

Simulations were performed of shot 82717, modeling concentration-gradient diffusion. These simulations were performed both with and without laser imprint. In all cases, the amount of material fluxed due to concentration diffusion was negligible, and no impact was found on the dynamics of the implosion, the density or temperature profiles in the shell or at the interface, or on the perturbations on the shell due to laser imprint. At the interface in the shell, the concentration gradient is indeed large, the gradient length scale being comparable to the mean-free path for the cold shell material, but due to the low temperatures and high densities, the concentration flux remains small throughout the implosion, increasing only upon complete ablation of the CH ablator, which occurs late in time. Once the CH is ablated, the temperature at the interface quickly rises and the density drops, causing the diffusion coefficient to rise, but the region where the concentration gradient is large has also ablated, so that concentration diffusion is able to flux material within the corona but not within the shell.

In Sec. III, mixing was considered due to low modes (from ice roughness as well as target offset, beam mistiming, beam mispointing, and power imbalance); mid- and high-order modes due to imprint and perturbations at the ice–ablator interface; and kinetic effects due to the concentration gradient at this interface. This is not an exhaustive list of possible causes of mix, as will be discussed in this section. Of those mix mechanisms considered, only imprint is found to efficiently transport carbon deep into the fuel layer, and, even so, it fails to match the radial extent of the shadow cast by the shell in the experimental radiograph.

A key to understanding this discrepancy is provided by comparing the 1D synthetic radiographs of shots 81590 and 82717. The former implosion is expected to have little or no perturbation growth yet has much greater attenuation due to the shell than shot 82717. As was mentioned above, this is because shot 81590's CH ablator is more than half a micrometer thicker than that of shot 82717, resulting in unablated CH at the time of the radiograph. Figure 11(a) shows the location of the fuel–ablator interface at the time of the radiograph for the LILAC simulations of both shots, showing that for shot 82717, the CH is much farther out and does not contribute significantly to the shadow cast by the shell. Figures 11(b) and 11(c) show the impact in 1D simulations of shot 82717 of increasing the initial ablator thickness by 0.25, 0.5, and 1 μm, while retaining the same shell mass. Because these are 1D simulations, the material interface radius does not match that of the 2D imprint simulations (Fig. 5). Figure 11(b) shows the mass density as a function of transverse distance through the target, as well as the ionization state Z, which serves as a proxy for the material location. The mass density shows the distended shell with a spike just inside 50 μm, where the outward-moving shock is moving through the shell. The location of the fuel–ablator interface is shown to move inward, as expected. The unablated CH creates in 1D a thin low-density (few-g/cm³) shell surrounding the fuel, extending to a minimum radius of ∼100–140 μm. Due to the high absorption of carbon, this has a strong effect on the shell shadow, so that for even 0.25 μm the attenuation is noticeably greater throughout the shell region—something that imprint by itself is unable to do in 2D simulations of shot 82717.

Figure 12 shows the horizontal radiographic lineout compared to the experimental lineout for the DRACO simulation with $2$-increased imprint but with an additional 0.3 μm of CH [and with the total mass held constant; see Figs. 6(c) and 6(d)]. In contrast to either the no-imprint, extra-CH case (Fig. 11) or the run with the nominal CH thickness and $2$ imprint, this combination matches both the width and depth of the shell shadow quite well, diverging from the experimental data only at the very bottom of the well, where the simulation has a sharper minimum. An additional 0.3 μm of CH is within the uncertainty of the initial ablator thickness, which is measured to within ±0.3 μm. (It is possible, but costly, to measure the initial shell thickness to a precision of ±0.1 μm.) The ablation velocity during the implosion is ∼4 μm/ns, and the entire CH ablator is ablated by the end of the laser pulse in either case, and by the time of the radiograph, in simulations of the implosion, the highly perturbed CH–DT interface ranges in radius from ∼50 to 200 μm. The effect of the initially thicker ablator is to delay final ablation of the CH and shift the CH–DT interface inward by ∼10 μm. The characteristic wavelength of the material interface perturbations in the simulation is smaller than the resolution of the image with the result that the additional shift effectively adds a uniform enhancement of carbon density outside the decelerating shell; the “cloud” of carbon is now shifted inward and, due to convergence, has a higher density. This cloud has the effect of attenuating the backlighter x rays and widening the shadow seen in the simulated radiograph.

In these simulations, we have explored the impact of uncertainty in the ablator thickness. It is also possible that the simulated ablation rate is greater than the actual ablation rate. In experiments performed by Michel et al.,⁴² coronal self-emission imaging was used to generate high-resolution tracking of the ablation front, allowing an experimental measurement of the ablation rate. In their experiment, the coronal emission from one side of the target is imaged on the other side of the target. These data agree well with LILAC prior to the complete ablation (burnthrough) of the ablator; however, the experiment indicates a delayed burnthrough, suggesting the experimental ablation rate is less than that in LILAC. The results from Ref. 42 are consistent with those for shot 82717.

Some possible causes for the discrepancy between modeled and observed ablation rates include (a) the equation of state (EOS), (b) the coefficient of thermal conduction, and (c) the heat-transport model itself.

Taking these in order:

• The more-compressible SESAME EOS has a higher hydrodynamic efficiency than FPEOS due to its higher ablation surface densities during the acceleration phase of the implosion; however, this effect is not large for shot 82717. In LILAC simulations where only the EOS is varied, the peak mass density in the shell during the laser pulse is a few percent greater than that with FPEOS, leading to an ∼10% greater ablation speed. In Fig. 13, profiles of the density and average ionization state Z from LILAC simulations of shot 82717 with FPEOS and SESAME EOS models are compared. For comparison, a simulation with an additional 0.3 μm of initial ablator thickness (but the same overall shell mass, to allow comparison at a given time) is also shown. As expected, SESAME has a greater peak density. However, the impact of the greater compressibility on the ablation is minimal; the location of the fuel–ablator interface differs by only 0.4 μm. In contrast, in the simulation with an additional 0.3 μm of CH, the material interface is smaller by 3 μm, which is enough to significantly impact the width of the shell shadow in the 2D synthetic radiograph.

• LILAC and DRACO use a “hybrid” Spitzer–Lee–More (SLM) conductivity^29,43 [see Ref. 43, Eq. (11)], multiplied by a function which models the effects of degeneracy. In addition to flux-limited thermal conduction, LILAC also models the scattered incident laser light due to CBET as well as NLET. To compare different thermal conduction models, a baseline LILAC simulation of shot 82717 was performed without CBET and NLET. Instead, a time-dependent flux limiter was tuned to replicate the shell trajectory and peak shell speed of the CBET and NLET simulation. This baseline was then compared to a calculation using the thermal conductivities of Hu et al.⁴³ which were computed using Kohn–Sham quantum molecular dynamics (QMD) calculations based on density functional theory, over the coupling parameter and degeneracy parameter range relevant to ICF. In simulations with these conductivities, the yield was found by Hu et al. to diverge from those using the Spitzer–Lee–More conductivities by as much as 20%. This is due especially to greater low-temperature, high-density conductivities, which leads to more-efficient conduction losses from the hot spot near peak neutron production. The higher conductivity has also been seen by Hu et al. to increase heat flow into the cold shell, reducing the temperature gradient in the conduction zone and reducing the ablation efficiency. In the context of the backlit shots, the latter effect is sufficient to move the material interface outward radially by ∼2.6 μm. This is comparable in magnitude to the effect of an additional 3 μm of initial ablator thickness. However, because the QMD conductivity is greater than the SLM conductivity, it moves the interface in the wrong direction (out rather than in) to explain the properties of the experimental radiograph.

• In the simulations shown here, as mentioned above, a fixed multiple of the CBET gain is used to reduce the laser energy deposited in the corona of the imploding shell. This combines with nonlocal electron heat transport to determine the modeled ablation rate. One avenue for investigation underway is the use of pump depletion in the CBET modeling. In the simulations reported here, conservation of energy is obtained through an iterative process governed by a proportional-integral-differential (PID) controller. In any given time step, the light scattered by CBET is drawn from the angular spectrum representation of a pump field in each computational zone. In a more recently developed pump-depletion model, the transferred energy is extracted from a particular beam rather than a pump field. This scheme is comparable to experimental observations but without recourse to a multiplier in the CBET gain. Simulations of shot 82717 have been performed using this improved model, with nearly identical results. An excellent test of the nonlocal electron heat transport model would be a comparison with a radiation-hydrodynamic code that uses a kinetic rather than fluid model of the plasma.

Two backlit cryogenic OMEGA implosions, expected to lie on either side of the observed areal-density degradation boundary (Fig. 2), were previously described and analyzed by Stoeckl et al.¹¹ They found that the less unstable of the two showed substantial agreement with a 1D radiation-hydrodynamic simulation, but the observed enhanced absorption in the more unstable one could be explained by only the presence throughout the shell of carbon from the ablator. In this work, we build on the work of Stoeckl et al.¹¹ and Epstein et al.⁴ by exploring possible causes of mix. Low modes, while possible causes of performance degradation, are found to be incapable of the required mixing. Similarly, kinetic effects at the fuel–ablator interface may be ruled out due to the small mean-free paths for the “cold” shell material. Mix due to classical instability growth at the material interface is also found to be insufficient because the period of instability is too small. In contrast to these mechanisms, as expected, laser imprint is found to be capable of the small-scale mix needed, but only if the imprint is greater than would be expected for a pure 2D simulation. Furthermore, we have found that imprint alone is incapable of reproducing the thickness of the observed shell shadow. This feature provides evidence for unablated CH at the time of the radiograph. This may be due to discrepancies between the calculated and experimental mass-ablation rates or errors in target metrology (which only measured the ablator thickness to ±0.3 μm) or both.

This latter effect was unexpected and is significant in that it points to the need for higher-precision metrology for high-performance cryogenic implosions. This is necessary for reproducibility in experimental campaigns with cryogenic targets and would allow future backlit experiments to further constrain errors in the simulated ablation rate. Since the experiments here were performed, the backlighter has been enhanced, improving the brightness by a factor of ∼5. This will allow both backlit experiments with shell speeds more typical of cryogenic implosions and radiography closer to peak neutron emission.

When the above physical effects are taken into account, close agreement is found between simulated and experimental radiographic lineouts. Even with these effects, the simulated yield is greater than the experimental value: $yield Exp /yield2−D∼10%$⁠. Of course, not all the known causes of performance degradation are included in these simulations, including notably 3D effects and mix due to small-scale features such as voids in the ice, particles on the target surface, and isolated features at the fuel–ablator interface. The results presented do not provide a full explanation of target performance; however, they do provide clear evidence of the role of imprint-driven fuel–ablator mix and motivate greater precision in measurement of target dimensions in order to better constrain the mass-ablation rate—a key parameter in ICF. This will allow exploration of the remaining causes of performance degradation—a necessary step if higher, ignition-scaled performance is to be achieved on OMEGA.

This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0003856, the University of Rochester, and the New York State Energy Research and Development Authority.

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

The authors have no conflicts to disclose.

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