Hydrodynamic instabilities seeded by the X-ray shadow of ICF capsule fill-tubes

Understanding and mitigating against the impact of hydrodynamic instabilities on indirect drive inertial confinement fusion (ICF^1–6) implosions has been an important goal for the capsule physics campaign^7–27 at the National Ignition Facility (NIF^28–31) since its inception. In this scheme, a cryogenic deuterium-tritium (DT) fuel layer is driven by the ablative acceleration of the outer shell irradiated by x-rays generated in an enclosing hohlraum. The hohlraum is driven with up to 1.9 MJ of laser energy up to 500 TW peak power. The shell accelerates to ∼400 km/s to achieve the ∼3000× solid DT fuel density required for ignition while maintaining a low fuel adiabat throughout the drive. Imperfections on the outer surface evolve during shock transit through the shell due to the ablative Richtmyer-Meshkov instability.^32–37 Further amplification follows during the acceleration phase of the implosion by the Rayleigh-Taylor (RT) instability.^38,39 These modulations feed through the shell to the DT ice layer and seed a subsequent phase of RT growth during the deceleration phase of the implosion towards peak compression. These imperfections can lead to a distorted, asymmetric hotspot which can significantly limit the temperature, pressure, and neutron yield attainable. In extreme cases, ablator material can penetrate through the shell into the hotspot to mix with the DT fuel prior to peak compression, further reducing the temperature by radiative cooling resulting in even greater yield degradation.^40–43 

The DT ice layer on the inside surface of indirect drive ICF capsules is typically ∼60 μm thick. For a ∼1 mm radius capsule, this is a considerably thicker layer than that can be produced from a diffusion filled capsule volume of DT gas at a reasonable temperature and pressure. Instead, the DT ice is continuously layered cryogenically at ∼20 K and ∼10 atm through a glass tube up to ∼30 μm outer diameter glued into the wall of the capsule. We previously reported on radiography experiments at the NIF characterizing the perturbation produced by the fuel fill-tube for indirect-drive ICF implosions.^44,45 These experiments revealed an unexpected mechanism for generating hydrodynamic instability seeds that we characterized as x-ray shadow imprint.⁴⁴ We hypothesized that the X-ray spots formed on the hohlraum wall by the outer beams early (<1 ns) in the laser drive cast x-ray shadows of the glass fill-tube onto the capsule surface until fill-tube expansion, and its concomitant opacity drop and increasing hohlraum reemission reduces the shadow contrast. We tested this hypothesis by repeating the experiment with a reduced diameter fill-tube and comparing to the analytic estimates of how the shadow imprint would scale with fill-tube diameter. The reduction in perturbation observed agreed with our estimate, confirming our hypothesis, and at the same time significantly mitigated against the effect of the shadow imprint from the outer beams.⁴⁵ In this paper, we summarize the experiments on the shadow imprint to date and report on a new mitigation scheme whereby the fill-tube is destroyed before the principal shadow forming outer-beams switch on, using hohlraum radiation produced by a 3× elevated and ∼1 ns extended pre-pulse on the inner-beams (We refer to this pre-pulse on the laser “foot” as a “toe”).

Figure 1 illustrates the Hydrodynamic Growth Radiography (HGR) experimental platform used for hydrodynamic instability growth measurements on the NIF.⁷ The cone attached to the side of the capsule provides an unobstructed view along the radiography axis through one side of the capsule only, without significantly impacting the dynamics of the implosion up to a convergence ratio (CR) of r₀/r ∼ 4, beyond which the argon self-backlighting platform is used as described elsewhere.⁴⁶ The first HGR experiments used a space resolving slit to measure the growth of 2D sinusoidal perturbations machined into the capsule surface with mode number near the peak of the simulated growth spectrum in the range of 30–90, corresponding to modulation wavelengths 200 μm–70 μm. Growth was measured in this way to test the hydrodynamic model under a range of drive conditions for plastic (CH) ablators⁹ and work is underway to repeat these measurements for beryllium and polycrystalline diamond (high-density carbon, HDC) ablator materials. Subsequent experiments using pinhole imaging extended this capability to measure 3D instabilities due to the native surface roughness of the capsule,¹⁵ the capsule support tent,⁴⁷ and more recently the DT fuel fill-tube.^44,45

In HGR experiments with the fill-tube aligned along the diagnostic axis, shadow imprint occurs predominantly due to five of the outer NIF laser quads as illustrated in Fig. 2, where a quad is a group of four NIF laser beams sharing common amplifier settings and target chamber ports. The inset in Fig. 2 shows the distribution of NIF beams on the hohlraum wall. For both the upper and lower beams, the outer quads arrive in two rings of 8 quads each at 44.5° and 50° to the hohlraum axis, offset vertically ∼2.5 mm and 3 mm from the target equator, respectively. The inner quads arrive in two rings of 4 quads each at 23.5° and 30° to the hohlraum wall, distributed closer to the equator. The beam pattern is clocked π/16 (π/8) radians between the upper and lower outer (inner) beams. The average azimuthal and polar radii of the inner and outer beam spots at the 50% contour are individually (0.82 × 1.18) mm and (0.64 × 0.52) mm, respectively. When overlaid in quads for optimum hohlraum drive, the corresponding spots are (1 × 1.65) mm and (0.8 × 0.67) mm. Two outer quads are removed from the hohlraum to drive the backlighter, leaving a gap in the ring of outer quads directly above and below the radiography axis, splitting the upper and lower shadows into the regions shown by raytracing⁴⁸ in the figure. The remaining outer beam spots on the hohlraum wall are below the horizon viewed from the capsule surface at the base of the fill-tube and hence cast no shadows. The inner quads at a more grazing angle produce larger spots with ∼7× lower intensity and result in considerably less contrast at the edges of the shadow.

Referring to the plot of the laser drive shown in Fig. 3, shadow imprint starts at ∼0.3 ns once the outer beams switch on. Shadow imprint stops at ∼0.7 ns, once the fill-tube becomes transparent due to expansion, or the hohlraum wall albedo increases sufficiently to dilute the shadow (as discussed in Sec. IV). By the end of the shadow imprint phase, an azimuthal mass areal density variation δρr is formed on the capsule surface near the fill-tube base due to modulation of the ablation rate as a function of angle around the fill-tube axis. Reduced ablation in the umbrae (set by the angular width of the laser spots subtended at the fill-tube, ∼1.3 mm at ∼3.3 mm ≈0.4 rad) leads to a pattern of radial ridges of excess ablator material standing ∼50–150 nm above the surrounding capsule surface in initial ablator density units. As the capsule converges, the ρR perturbation along each edge of the ridge grows, resulting in the modulations observed in the data.

The first measurements of shadow imprinted hydrodynamic instabilities were reported for a 10 μm diameter fill-tube on an HDC ablator.⁴⁴ The fill-tube was made from borosilicate glass with density 2.33 g/cc, 5 μm outer radius, and 2.5 μm inner radius. Figure 4 shows a 1.8 mm diameter capsule assembled with a 10 μm outer diameter fill-tube and an x-ray radiograph of the detail at the glue joint. The capsule ablator was 3.5 g/cc HDC with outer radius r₀ = 908 μm and thickness 64 μm. The inner 25 μm of the ablator was doped with 0.25 at. % tungsten, which for a layered fuel capsule acts as a pre-heat shield to control the ablator density profile near the DT fuel (for a detailed discussion on target surrogacy, see Ref. 49). The hohlraum was 10.13 mm long by 5.74 mm diameter with 3.38 mm diameter laser entrance holes (LEH), driven with a three-shock drive specific to HDC implosions^50,51 and shown in Fig. 3. Time-gated radiographs of the capsule transmission were recorded along the fill-tube axis at a convergence ratio r₀/r of 2.1 ± 0.1 using an array of thirty 16 μm diameter imaging pinholes (18 μm for the subsequent 5 μm diameter fill-tube shot) and an 80 ps duration gate. The source to pinhole distance was 80 mm and the magnification onto the gated microchannel plate was 7.8×. The x-ray backlighter comprised principally of vanadium He-α emission at ∼5.2 keV, isolated from the higher energy bremsstrahlung radiation using a vanadium filter and by limiting irradiation onto the vanadium foil to ∼8 × 10¹⁴ W cm⁻².

Figure 5 shows the fill-tube shadow expected from the relevant x-ray spots for the 10 μm fill-tube case obtained by raytracing,⁴⁸ and illustrates the impact of removing quads 26B and 31T from the hohlraum to drive the backlighter foil. In the regions where the corresponding 26B and 31T shadows would have fallen, the irradiance is increased relative to the adjoining regions where shadowing does occur, and hence more material is ablated forming troughs. A point on the capsule equator normally sees up to four outer quads from each hemisphere. Hence, removing one quad from each hemisphere with a view of the fill-tube reduces the irradiance and increases shadow contrast by approximately 25%. For DT layered implosions, the shadow imprinted perturbation will have an hourglass shape rather than a cross, although the solid angle of the envelope of the perturbation will be similar.

Figure 6 shows the radiograph of the shadow imprinted perturbation recorded for an HDC ablator with a 10 μm fill-tube at convergence ratio ∼2. The dashed lines correspond to the axes of the umbrae formed by raytracing the region comprising the capsule and fill-tube, using extended sources corresponding to the laser spots on the hohlraum wall.

To test the hypothesis that the modulations observed in the x-ray radiograph were indeed perturbations produced by shadow imprint of the fill-tube, we subsequently performed a similar measurement using a smaller fill-tube,⁴⁵ reducing the outer diameter from 10 μm to 5 μm, the inner diameter from 5 μm to 3 μm, and the wall thickness from 2.5 μm to 1 μm. The expectation was that a narrower, thinner wall fill-tube would project a shorter, narrower umbra that would become x-ray transparent sooner, resulting in reduced differential ablated mass across the shadow boundaries.

X-ray radiographs of the HDC implosions measured along the axes of the 5 μm and 10 μm fill-tubes are shown in the right two panels of Fig. 7, in units of areal density modulation ΔρR/ρR = ΔOD/OD taken at t = 6.7 ns, where $ΔOD=−Δln(I/I0)=Δ∫κρdR$⁠, $κ$ is the opacity at the x-ray backlighter energy, and $ρdR$ is the projected areal density. I and I₀ are the transmitted and incident backlighter intensities, respectively, and OD is the average optical depth of the shell. We cannot measure the effective shell OD directly as we do not have an in-situ measurement of the unattenuated backlighter intensity. We can estimate it however by taking the line integral along the capsule radius up to the ablation front from the numerical simulation. In the HDC ablator case, OD ∼ 0.7.⁵² This is justified as at radii greater than the ablation front, attenuation due to the ablated material is essentially isotropic as the material spreads into 2π from each point. This estimate is consistent with an analytic estimate of ∼0.6 based on the rocket model^2,53 with measured peak hohlraum temperature ∼293 eV and shell velocity ∼260 km/s and with ∼1/3 of the mass ablated during the shock compression phase.

We determine the convergence of the capsule by measuring the separation of a pair of divots laser-machined into the surface of the capsule separated 60° about the fill-tube on the equator. The reduction in separation from the initial value gives the convergence at the time of the radiograph. To improve the accuracy of centroiding the divot features in the radiograph, the later 5 μm fill-tube shot used shallower divots with reduced diameter: 50 μm vs 60 μm diameter and 0.35 μm vs 0.5 μm depth. The divot features in the 5 μm fill-tube radiograph therefore appear smaller than the 10 μm case. The location of the fill-tube relative to the divots is consistent with pre-shot target metrology, where the fill-tube hole was determined to be 40 μm to the left of and 37 μm above the nominal center axis in the unconverged frame. This is also consistent with radiographic data taken ∼375 ps later at CR ∼ 3 where a low-density (bubble) feature developed at the expected location of the fill-tube.⁴⁵ 

In the case of the 10 μm fill-tube, after correcting for the optical transfer function (OTF) of the imaging system Fig. 8, the peak to valley optical depth modulation corresponds to a ∼32 μm tall column of ablator material in initial ablator density units. The initial seed perturbation was estimated by calculating the difference in ablation rate between four and five quad illumination. From the plot of hohlraum temperature vs. time measured using the broadband time-resolved soft x-ray spectrometer Dante^55,56 (right hand axis in Fig. 3), during the first 0.3 ns of foot drive at an average T_r = 120 eV, the mass ablation rate for the HDC ablator ∼T_r³ is 2.0 μm/ns in the pre-shocked reference frame.^20,44,57 This gives a differential mass ablated across the shadow boundary of up to 1/5 × 2 μm/ns × 0.3 ns ≈ 120 nm. The time averaged brightness of the x-ray sources due to the inner beam spots will tend to dilute the shadow contrast, giving a lower bound on the differential ablation of ≈ 100 nm. The measured ∼32 μm thick perturbation at CR ∼ 2 therefore corresponds to an optical depth growth factor of ∼320× compared to the ∼100 nm seed depth.

For the 5 μm diameter fill-tube, the average depth of the ρR perturbation was reduced at least 5× per unit length of shadow compared to the 10 μm fill-tube case, consistent with the analytic scaling described in Ref. 45, where we postulate that the fill-tube material becomes transparent to the x-ray radiation principally due to expansion and hence a drop in areal density. The opacity of the fill-tube material is assumed to also drop concurrently⁴⁴ and for a thinner wall, the material will blow down more rapidly so this puts an upper limit on the final radius R_f at the onset of transparency. The areal density at which transparency occurs is a constant regardless of initial fill-tube radius and since the expanding fill-tube mass/unit length is conserved, $Rf scales as Ri2$⁠. For fixed expansion speed in the limit R_f ≫ R_i, the fill-tube radius can be approximated as increasing linearly with time such that $tf also scales as Ri2.$ Hence for similar ratio of wall thickness to diameter, the time to transparency $tf$ scales as the square of the initial outer radius, so a 2× reduction in $Ri$ gives a ∼4× reduction in differential ablated mass per unit length of shadow. As the length of the shadow is also proportional to $Rf$⁠, a 2× reduction in $Ri$ will also reduce the imprinted feature length by 4×. We estimated that a 10 μm diameter fill-tube goes transparent at 2R_f ∼ 100 μm (Ref. 44) based on the geometry and length of the imprinted features, so a 5 μm fill-tube should go transparent at 2R_f = 25 μm diameter, and indeed, Fig. 7 shows the evidence of reduced umbra length for the smaller fill-tube. On this basis, we estimate that the perturbed mass imprint for the 5 μm fill-tube was reduced at least ∼16× compared to the 10 μm fill-tube case, 4× per unit length due to the reduced ablation time, and an additional 4× due to the shorter average length of the umbrae. Also, since the feature size is reduced, the effective mode number of the perturbation is increased. Instability growth occurs at the penumbrae as shown schematically in Fig. 9, with transverse mass motion from trough to spike. Eventually, depending on the ratio of the umbrae to penumbrae widths, the two spikes of the shadow imprinted perturbation will merge into a single feature. This is a highly simplified picture; in reality, there is a range of modes growing as the mode number will decrease as one moves away from the base of the fill-tube. For the 10 μm fill-tube case with a ∼42 μm wavelength imprinted feature and capsule radius ∼900 μm, the mode number is of the order ∼64. This places the perturbation near the peak of the simulated growth factor spectrum at ∼140 as shown in Fig. 10(a).^58,59 For the 5 μm fill-tube case, the imprinted feature size will be (10/5)² = 4× less, ∼25 μm resulting in significantly reduced growth at ∼4× higher average mode number. The perturbation for the 5 μm fill-tube case will therefore be at least ∼16× less multiplied by the difference in RT growth factor, resulting in a final seed mass up to ∼200× less for the 5 μm fill-tube case depending on the lower order modal composition of the imprinted seed. The R_i² dependence in spoke length will also reduce the solid angle subtended by the spoke pattern by 16× from ∼300 msr (220 msr measured) for the 10 μm fill-tube to ∼19 msr for the 5 μm fill-tube.

We have also measured the growth of shadow imprinted instabilities for a plastic ablator (low pressure glow discharge polymer, “CH”) in a gold hohlraum.⁶⁰ In this case, the outer radius of the capsule was 904 μm, 167 μm thick with a 40 μm thick silicon doped layer starting 16 μm from the inner surface, with 5 μm at 2.4 Si at.%, 27 μm at 4.8 at. % and 8 μm at 2.4 at. %. For CH, the hohlraum was gold, 8.26 mm long × 4.69 mm diameter with 3.1 mm diameter laser entrance holes and 0.96 mg/cc ⁴He gas fill. The left panel of Fig. 7 shows the radiograph recorded for CH at a convergence ratio of ∼3. Shadow imprint switches off when either (i) the fill-tube expands sufficiently (50 μm in radius) to become transparent to the x-ray radiation produced at the laser spots on the hohlraum wall, or (ii) the hohlraum albedo at $≲$ keV rises sufficiently to dilute the shadow. We can estimate the cutoff times for each effect for the two ablators analytically: As was shown previously for HDC,⁴⁴ the fill tube expands with an average velocity of the order of the sound speed √(ZT_e/m_i) ∼ 70 μm/ns, assuming T_e ≈ 0.8 T_r and Z/A ≈ ½ and average T_r = 120 eV over the first 0.7 ns from the Dante measurement in Fig. 3. For CH, the foot power is ∼4× lower for the shadow forming outer beams as shown in Fig. 11. T_r in this case is ∼60 eV (Ref. 61) so we can expect the fill tube expansion time to increase from ∼1.7× to ∼1.2 ns (including fact that Z will be ≈40% lower). This is a lower limit, however, as the SiO₂ fill tube ablation rate ∼ T_r^2.5 will be 6× smaller and hence full transparency delayed by 6×. Specifically, scaling from HDC, we expect a SiO₂ ablation rate of ≈2T_r^2.5μm/ns = 0.6 μm/ns at T_r = 0.6 heV, hence would not burn through much of the 2.5 μm fill tube wall in 1.2 ns. Moreover, as the hohlraum albedo rises as 1 − 0.4/T_r^0.7t^0.4 with T_r in heV (100 eV units),² we can expect the albedo to reach 0.5 at ∼1.4 ns for the CH case. Hence, dilution of the shadow by rising albedo will occur prior to fill tube transparency, whereas for higher foot T_r HDC designs, the converse is true.

Scaling the mass ablation rate for CH from Ref. 57 as T_r³, at 60 eV, we get ∼0.7 μm/ns. So in the shadow, the differential ablation rate is of the order ∼0.14 μm/ns, giving an initial seed height equivalent to ∼190 nm of CH at initial density after 1.4 ns. The measured peak to valley optical depth for the shadow imprinted perturbations for CH in Fig. 7 is ΔOD ∼ 0.15⁶¹ at a lineout radius of 30 μm. After correcting for system OTF ∼ 0.27 at a wavelength λ ∼ 24 μm as set by twice the penumbra width corresponds to a ∼74 μm thick column of 3.6 at. % silicon doped GDP in initial ablator density units. The 3.6 at. % Si is the avg. concentration layer sensed by the perturbation at the time of the radiography. Based on a calculated 10 g/cc average shell density from measurements of the remaining shell areal density and thickness on previous plastic ablator shots,⁶² the amplitude of the modulation at CR ∼ 3 is ∼4 μm. This is 1/6th of the wavelength so just approaching the nonlinear growth phase at λ/2π. Hence, we are justified in comparing the inferred linear growth factor of 74 μm/0.19 μm ≈ 400 to that expected from the simulated spectrum shown in Fig. 10(b),^58,59 showing reasonable agreement at the relevant mode number. Further from the fill tube base, the penumbra width and hence wavelength increases, mode number decreases, and growth factor is expected to increase. However, that will be countered by adjacent penumbrae overlapping resulting in a reduced seed height. In effect, at very low foot T_r, the slower fill-tube expansion results in reduced imprint area and at higher less damaging mode numbers before the albedo rises, suggesting an alternate mitigation technique, although not as effective as reduced fill-tube diameter or enhanced inner-beam pre-pulse as described in Sec. V.

A larger fill-tube has significant advantages from a capsule manufacturing and fielding perspective. Assembling capsules with the smaller fill tubes is more difficult which results in a ∼50% successful production rate. Fielding the larger fill-tube is less subject to cryogenic fuel blockages causing interruptions to the fuel layering process which can take several days. One alternative approach to mitigating against shadow imprint due to the larger fill-tube is to use a pre-pulse on the inner laser beams to blow down the fill-tube prior to switching on the shadow forming outer beams.⁴⁴ To test this hypothesis, we repeated the 10 μm diameter fill-tube HGR experiment using a laser drive profile where the inner beam “toe” was extended from 0.3 ns to 1.2 ns and elevated in intensity by 3× as shown in Fig. 12. The results in Fig. 13 illustrate three main points. First, the absence of outer beam shadow imprint suggests that the pre-pulse did indeed successfully blow down the fill-tube prior to switching on the outer beams. Second, there is now evidence of shadow imprint associated with the 3× intensity inner beams, where two enhanced ΔρR/ρR spokes at ten and two o'clock correlate with raytracing for the expected shadow location for quads 26T and 36T shown in Fig. 6. Third, the central fill-tube perturbation has inverted from an RT spike to a nominally less disruptive RT bubble.

Together these observations suggest there is value in pursuing some form of pre-pulse to mitigate shadow imprint as an alternative to or in conjunction with reducing the fill-tube diameter. As the outer beam shadows are no longer visible, there may be a capacity to reduce the inner beam imprint by reducing the pre-pulse intensity while still blowing down the fill-tube before the outer beams switch on. However, the central fill-tube perturbation is still the dominant feature after shadow imprint is eliminated, so effort is being directed towards reducing the fill-tube diameter further or eliminating it entirely through the diffusion fill.

We have demonstrated that x-ray spots formed on the hohlraum wall by the drive beams for indirect drive ICF cast shadows of the fuel fill-tube on the capsule surface. Differential ablation at the shadow boundaries during the first few hundred picoseconds of the laser drive seeds perturbations which are hydrodynamically unstable under subsequent acceleration and can grow to impact capsule performance for both HDC and CH ablators.

The optical depth of the shadow imprinted modulations is comparable to the central fill-tube feature, but extends over a spherical cap with considerably larger solid angle as shown in Fig. 7. This likely explains why the jet of ablator material ascribed to the fill-tube and observed experimentally in self-emission on DT fuel implosions is larger than that predicted by simulations which omit the shadow imprint and may therefore contribute significantly to performance degradation.^4,63 We have characterized this shadow imprint mechanism and demonstrated two techniques to mitigate against it for high density carbon ablators, using (i) a reduced diameter fuel fill-tube, and (ii) a pre-pulse to blow down the fill-tube before the shadow forming x-ray spots develop. A factor of 2× reduction in fill-tube diameter reduced the seed perturbation by a factor of ∼4× per unit shadow length, and ∼16× in total seed mass and solid angle. A pair of layered fuel implosions was performed to make a direct A-B comparison of the impact of fuel fill-tube diameter on HDC capsule performance. The primary neutron yield for the 5 μm fill-tube case was increased ∼40% compared to the 10 μm fill-tube case. A subsequent A-B comparison for a pair of targets where both the hohlraum and capsule were scaled up in size by ∼10% showed a ∼70% increase in primary neutron yield for the 5 μm diameter fill tube case.⁶⁴ The smaller features shift the instability growth to a higher mode number and highlight the sensitivity to exact fill-tube dimensions. The 0.9 ns extended pre-pulse at 3× intensity on the inner beams successfully blew down the fill tube prior to the onset of significant shadow imprint. In the absence of diffusion fill, a combination of reduced fill-tube diameter and more optimized pre-pulse may be the most effective way to mitigate the impact of the fill-tube on capsule performance.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. The target fabrication was supported at General Atomics under DOE Contract No. DE-NA0001808.