Design and modeling of indirectly driven magnetized implosions on the NIF Open Access

Magnetized inertial confinement fusion (ICF) is an old idea^1–3 that has received significant interest in recent years. The basic concept is to reduce the loss of energy (carried by electrons and fusion-generated charged particles) from the implosion hotspot and thereby relax the requirements for ignition and energy gain. This thermal insulation and resulting improved fusion performance have been demonstrated in the three major approaches to ICF: laser direct drive, magnetic direct drive, and indirect (x-ray) drive. Laser-driven cylindrical and spherical implosions at the OMEGA laser facility have compressed externally imposed B fields to large values^4–6 and demonstrated increased fusion yield and ion temperature $Ti$⁠. The magnetically driven MagLIF concept uses the pulsed-power Z Machine at Sandia National Laboratories to implode a cylindrical metal liner that contains fusion fuel with an imposed axial magnetic field and preheated by a laser.^7,8 The imposed field is needed to confine DT fusion alpha particles and has been shown experimentally to improve performance. An imposed magnetic field can also improve the fast ignition concept⁹ by guiding relativistic electrons to the hotspot and mitigating their angular divergence. This was proposed in a modeling study¹⁰ and has recently been explored in experiments with kilotesla laser-driven fields.¹¹ 

This paper focuses on magnetized indirect-drive ICF, which has been researched at Lawrence Livermore National Lab (LLNL) for over a decade.^12,13 Recent experiments at the National Ignition Facility (NIF)¹⁴ have studied hohlraum-driven implosions of gas-filled capsules, with imposed initial axial magnetic field up to 28 T.^15,16 These have demonstrated increased fusion yield and $Ti$⁠, in line with radiation-magneto-hydrodynamic (rad-MHD) modeling. We continue to build on this work, with the goal of using magnetic fields to enhance high-yield designs in the ignition regime.

The role of magnetic fields in fusion devices can be understood by considering two dimensionless measures: the electron Hall parameter $He≡ωceτei$ (where $ωce≡eB/me$ is the electron cyclotron frequency and $τei$ is the electron–ion (e–i) collision time) and $β≡2μ0p/B2$ (where p is the thermal pressure). For all systems currently under study, the minimum requirement to be magnetized is $He>1$⁠. This means the electron thermal conductivity $κ$ perpendicular to $B$ is significantly reduced. Within this class, systems can be divided based on $β$⁠. For $β≫1$⁠, magnetic pressure has a small direct effect, and we call this regime “magnetically insulated” ICF. Magnetized ICF schemes for NIF and other laser facilities fall in this domain. $β∼1$ is the “magneto-inertial fusion” regime, where magnetic and thermal pressures are comparable.¹⁷ This includes the magnetic compression stage of pulsed-power driven MagLIF, though the imploded fuel has $β≫1$ (so the laser-driven “mini-MagLIF” scheme at OMEGA¹⁸ is magnetically insulated ICF). Other interesting MIF approaches include the sheared-flow stabilized Z-pinch¹⁹ and the dense plasma focus.²⁰  $β≪1$⁠, where magnetic pressure dominates the confinement, is the regime of classic magnetic-fusion systems such as tokamaks and stellarators.

We now estimate the magnetic field requirements for magnetizing a typical NIF hohlraum-driven hotspot. High-performing capsules with a cryogenic DT ice layer have a measured convergence ratio $CR=$ (initial capsule radius)/(stagnated hotspot radius) $∼30$⁠. We consider an initial axial B field of a uniform 40 T in the capsule and make the optimistic assumption that magnetic flux is conserved in the implosion according to the frozen-in law of ideal MHD. The field on the equator is then enhanced by $CR2∼900$⁠, giving $B∼$ 36 kT. A marginally igniting hotspot has a density $∼$100 g/cm³ and temperature $∼$6 keV, which gives the Hall parameter $He∼9.5$ and $β∼90$⁠. To estimate the effect on $κ$⁠, for a purely axial field the effective $κ/κ||∼1/3+(2/3)κ⊥/κ||$⁠, with $κ||$ being the unmagnetized, axial value. $κ$ is reduced in the two perpendicular directions. For $Z=1$ and $He=9.5$⁠, the Epperlein and Haines²¹ fits for magnetized $κ$ give almost full suppression: $κ⊥/κ||=0.012$⁠. Even $He=1$ gives significant reduction: $κ⊥/κ||=0.32$⁠. These parameters thus give substantial margin for processes that reduce the benefits of magnetization. These include flux loss due to effects such as resistive diffusion and the Nernst effect, or non-axial imploded field geometries. From this simple estimate and detailed rad-MHD modeling, the LLNL program aims to impose axial fields up to 100 T on layered cryogenic capsules driven by NIF hohlraums.

This paper is organized as follows: Section II describes the constraints on, design of, and results from magnetized NIF hohlraum-driven gas-filled capsule experiments. Rad-MHD modeling of these shots with the Lasnex code and Lasnex Hohlraum Template (LHT) common model is presented in Sec. III. Section IV contains a modeling study of MHD and imposed-field effects on one NIF shot from the BigFoot campaign,^28,29 upon which the magnetized platform is based. Conclusions and future work toward magnetized ignition on NIF are given in Sec. V. Two appendixes give details on the Lasnex LHT model and Lasnex's MHD package.

This section presents the rationale behind and results of NIF experiments with hohlraum-driven gas-filled capsules, with and without an imposed magnetic field. This work has demonstrated increased hotspot ion temperature and yield with a field and has been partially reported elsewhere.^15,16 The new aspects here are a detailed discussion of the “WarmMag” (WM) platform's design rationale and early results from recent experiments with increased hotspot temperature (with and without field).

Our “WarmMag” platform is designed to demonstrate magnetic insulation of an indirect-drive ICF hotspot, subject to various practical constraints. The platform has three designs: the initial low-T design with a prolate implosion shape, the final low-T design with a round shape when magnetized, and the high-T design with increased capsule hotspot temperature. We magnetize a NIF hohlraum by wrapping it with an external coil and running current through it via a pulsed-power system. The most salient present constraint is that only room-temperature NIF experiments (293 K) can be magnetized. A room temperature pulsed power system consisting of a 4- $μ$F capacitor and a spark gap trigger system has been in use for a variety of NIF target experiments since 2016. Pre-magnetizing cryogenic DT-layered implosion targets requires integration of a pulsed power system with a cryogenic target positioner. NIF has designed this system, but the completion date for construction is currently under evaluation. Room-temperature hohlraum implosions are fielded somewhat routinely on NIF.²² They are limited to gas-filled capsules or “symcaps,” with no ice layers and no keyhole shock-timing experiments (these require a reentrant gold cone filled with liquid D₂ inserted into the capsule).

There are limits on the room-temperature pressures and compositions of capsule and hohlraum gas fills. Our diamond or high density carbon (HDC) capsule can hold very high pressures of H and He isotopes without permeation, so this does not affect our gas choice. The hohlraum gas fill is limited by not bursting the laser entrance hole (LEH) windows, for which the probability gradually increases with pressure. To minimize the likelihood of bursting, we typically use $∼1 μ$m thick windows vs $∼1/2 μ$m for cryo experiments. We also follow the common NIF practice of using neopentane (C₅H₁₂) for room-temperature hohlraum fills, with the same fully ionized electron density as the He in the analogous cryo design.²² This corresponds to a much lower initial fill pressure.

Another constraint is having the field permeate the hohlraum and capsule before the laser fires, without significantly perturbing either. The field must soak through the high-Z hohlraum with acceptably small $J×B$ force, Joule heating, and temporal delay due to resistive diffusion.²³ All these effects increase with hohlraum conductivity. The most limiting for our current gas-filled experiments is the $J×B$ force. We require the resulting radial wall motion to be less than $50 μ$m, which is comparable to NIF laser beam pointing error. Pure gold hohlraums are too conductive to meet this, though pure uranium hohlraums might be acceptable. Future experiments with ignition-quality cryogenic DT ice layers have more stringent constraints, which require lower conductivity. For instance, the hohlraum and coil must allow adequate thermal control and spatial uniformity, and radiation from a Joule-heated hohlraum must not disturb the ice layer. Uranium is probably not adequate for layered targets with 60–70 T imposed field, so instead we use a novel, resistive hohlraum alloy AuTa₄.^24,25 MHD modeling indicates the field soak-thru is rapid enough to provide a sufficiently uniform initial B field in the capsule. The spatial variation is actually dominated by the coil structure, and essentially reaches its steady-state profile before the lasers fire. For a $Bz0=40$ T field and a 1-mm capsule radius, $∂zBz[r=0,z]≈0.6$ T/mm. Our HDC capsule is non-conductive so field soak-through poses no challenges, though it may be for a conductive capsule made of, e.g., beryllium.

The final constraint due to magnetization is the risk of laser optics damage due to stimulated Brillouin scattering (SBS).²⁶ Magnetic fields introduce risk like any new target, and laser energy and power must be increased in safe increments. The new risk with fields is that the pulsed-power system could prefire before the laser. The current melts the coils and induces $J×B$ motion, all of which can leave the target in an unknown state (with unknown SBS risk) when the laser fires. Pulser prefire has not been observed on any NIF shot to date, but its risk is unacceptably high. Every magnetized NIF shot must, therefore, pass a stringent SBS risk rule, which can only be relaxed when the prefire risk is made small enough. The SBS rule is a limit on the “equivalent fluence” of laser power vs time, which is a complicated functional somewhat between total energy and peak power.²⁷ This usually limits hohlraum pulses to $∼$ 5 kJ/beam (or 1 MJ full NIF). A low-power 1 TW/beam “caboose” at the end of the pulse for a standard hohlraum is considered to not add risk. The planned cryo pulser will include prefire mitigation and remove the SBS limit.

The room-temperature magnetized hohlraum project aims to understand magnetization with $≲$ 5 shots per year, which each use $≲$ 1 MJ of laser energy. We, therefore, start from an existing NIF platform that has both high performance and robustness, meaning shots are likely to be reproducible and not depart much from expectations. The BigFoot platform stands out in this regard.^28,29 It was deliberately designed for robustness, with a high implosion adiabat and dense HDC capsule. Both of these lead to short laser pulses, which require a relatively low density hohlraum gas fill of 0.3 mg/cm³ of He to adequately limit inward hohlraum wall bubble motion. This has been demonstrated to produce much less backscatter than platforms with longer pulses and higher hohlraum fill densities like the earlier LowFoot and HighFoot.⁷⁸ BigFoot achieved round implosions without using wavelength shifts $Δλ$ between the NIF lasers. This is done to control cross-beam energy transfer (CBET),^30,31 a nonlinear laser–plasma interaction that transfers power to the longer-wavelength laser in the plasma flow frame. $Δλ=0$ for all BigFoot and WarmMag shots. BigFoot has also given the highest fusion yield (29.9 kJ) of any NIF platform with $≤$ 1.3 MJ of laser energy, on NIF shot N190721-2.³² The record yield NIF shots use the Hybrid E platform³³ and have given much more $∼170×$ yield than BigFoot, but this platform uses at least 1.8 MJ of laser energy, which our SBS rule precludes. In addition to that, new ideas are more affordably tested with less laser energy and power, since optic damage increases rapidly with both. N190721-2 is an appealing BigFoot layered target for a baseline magnetized design. This was used in the alpha-heating campaign³² and gave an inferred yield amplification due to alpha heating of $∼2.5×$⁠. Adding an initial B field $Bz0≥20$ T (quoted at the capsule center) should measurably enhance the yield due to both reduced electron thermal loss and enhanced alpha heating.

N190721-2 unfortunately used 1.3 MJ of laser energy and is beyond our SBS risk limit. We base our initial design on an earlier BigFoot series with slightly less laser energy, specifically gas-filled capsule shot N161204-3. Table I and Figs. 1–3 present details on the shots discussed here. The first two WarmMag shots, N201228-1 (⁠ $Bz0=0$⁠) and N210301-1 (⁠ $Bz0=26$ T), stayed as close to N161204-3 as our constraints allowed (except for the reduced capsule fill density). The differences of this initial low-T platform vs N161204-3 are as follows:

• Reduced late-time laser power to meet SBS risk limit (see Fig. 1).

• Changed capsule gas fill from 6.74 mg/cm³ of D₃-³He₇ to 5.10 mg/cm³ of D₃-⁴He₇ to give more convergence and B-field compression.

• Changed hohlraum gas fill from 0.3 mg/cm³ of He to 0.258 mg/cm³ of C₅H₁₂ to prevent LEH window bursting.

• Changed hohlraum wall from 30 to 50  $μ$m thick Au to 15  $μ$m thick AuTa₄ to reduce electrical conductivity and field soak-thru issues.

The goals of the WarmMag NIF shots were to demonstrate that (1) hohlraum x-ray conversion was similar with or without a B field, and with an AuTa₄ instead of pure Au hohlraum; and (2) a repeatable comparison of nuclear performance in shots with and without B, with a roughly round hotspot. Our starting point was to hew as closely as possible to BigFoot shot N161204-3 given our constraints, and tuning implosion shape came later. The first shot N201228-1 had no imposed field, and no capsule fill due to a fielding issue. It was essentially a hohlraum coupling test. The AuTa₄ hohlraum and modified laser pulse gave x-ray drive as measured by DANTE⁷⁷ comparable to N161204-3 until the time of the caboose.¹⁶ The backscatter was very low on this and all WarmMag shots.

The next shot N210301-1 had imposed field $Bz0=26$ T and capsule fill. It gave similar x-ray drive and backscatter to the prior shot and produced a very prolate implosion: $P2/P0=63%$⁠. This is not a drawback of the platform or magnetization—as stated above, we did not design the shot to be round but to be as close as possible to BigFoot shot N161204-3. That shot had a moderately prolate $P2/P0=18.7%$⁠. Of the differences between BigFoot and WarmMag, the increased cone fraction (or ratio of inner-beam to total laser power) of $∼1/3$ in the low-power caboose likely contributed to the increased $P2$⁠. This was chosen to maximize the total power in the caboose. Other changes, like the use of an AuTa₄ instead of an Au hohlraum, or C₅H₁₂ instead of He hohlraum fill, could also affect implosion shape but do not suggest an obvious direction like the increased cone fraction. We do not think the imposed B field leads to increased $P2$⁠: a hohlraum run identical to the tuned one for N210301-1 in Table I but with no imposed B gave a slightly higher $P2/P0=70.4%$⁠, a 49% lower yield of 4.95 × 10¹¹, and a 28% lower burn-averaged $Ti$ of 2.69 keV.

Our next goal was to make the magnetized implosion shape round and finalize the low-T platform. We did this on the third shot N210607-2 by reducing the laser cone fraction. We only reduced the inner beam power: the outer beams were already at their SBS limit, so we could not raise them to maintain total power. We also changed the capsule fill to 3.99 mg/cm³ of pure D, to increase yield via increased D density and reduced hotspot radiative loss. This design was modeled in Lasnex to give a round implosion, which was basically achieved: $P2/P0=5.7%$⁠. Our next goal was an unmagnetized shot of this design. Several attempts had fielding issues, with the best one being N210912-1 (which still dropped 2 of the 48 NIF laser quads and thus had lower laser energy). We subsequently shot an intermediate $Bz0=12$ T on N220912-1, which had almost the same yield and $Ti$ as $Bz0=26$ T.

These results validate the expected $Ti$ scaling from Ref. 34 for gas-filled targets where reduced electron thermal conduction is the main B-field effect. The effective thermal conductivity for a strong, purely axial field and a spherical implosion is $κeff=1/3$⁠, which gives a $Ti$ increase of $κeff−2/7=1.37×$⁠. The WM low-T shot pair N210912-1 (⁠ $B=0$⁠) and N210607-2 (⁠ $B=26.1$ T) has a measured $Ti$ increase of $1.40×$⁠, and the WM high-T pair N230612-1 (⁠ $B=0$⁠) and N230212-2 (⁠ $B=28$ T) had $1.28×$⁠. Both of these are close to the ideal scaling. The measured yield increases are larger than the estimate from Ref. 34. This is due to that model being for layered targets where cold fuel ablates into the hotspot, and the confounding difference in delivered laser energy and hotspot $P2$⁠.

In 2023, our goal was to increase the capsule hotspot temperature while still respecting our constraints. This WarmMag “high-T” platform entailed four changes vs the low-T one:

• Extended caboose to increase laser energy and x-ray drive (see Fig. 1).

• Reduced LEH diameter from 3.45 to 3.13 mm, to reduce hohlraum x-ray loss.

• Reduced capsule fill density from 3.99 to 3.00 mg/cm³ to increase capsule convergence for a given x-ray drive.

• Increased laser cone fraction to maintain a round implosion shape.

These changes increased the hotspot temperature both with and without a B field, as demonstrated in our most recent shots N230212-2 and N230612-1.

Figures 2 and 3 provide a summary of the WarmMag NIF shots to date, with $Bz0$ as the abscissa in all plots. The delivered laser energy and cone fraction slightly increase with $Bz0$⁠, since both $Bz0=0$ shots N210912-1 and N230612-1 had two dropped laser quads. This accounts for some of the decrease in bangtime with $Bz0$ (besides any field effects). It also somewhat complicates drawing clear conclusions from the increase in $P2$⁠, yield, and $Ti$ with $Bz0$⁠. A compelling demonstration of the benefits of magnetization is the higher yield and $Ti$ in the low-T, $Bz0=12$ T shot vs the unmagnetized high-T shot: the field over-compensates for the higher laser energy, x-ray drive, and capsule convergence.

To address the confounding effects of delivered laser energy and hotspot $P2$ on magnetization, we performed unmagnetized hohlraum runs of magnetized shots N210607-2 and N230212-2 but with no imposed field and with the x-ray deposition symmetrized in the capsule (which leads to essentially round hotspots). The magnetized runs had DD neutron yields of 1.65  $×$ and 1.51  $×$ these unmagnetized runs, respectively, which we can take as the minimum “true” magnetic enhancement (minimum since varying the implosion shape could increase the magnetized yield). These are smaller enhancements compared to the runs for the unmagnetized analogous shots N210912-1 (2.67) and N230612-1 (2.37). Simply treating the enhancement as a product of magnetic and confounding effects, the latter has values of 1.62 and 1.57. Thus the yield increase from confounding effects (laser energy and hotspot shape) in the magnetized shots is roughly comparable to that due to magnetization.

We now describe our Lasnex rad-MHD modeling of the WarmMag NIF shots.  Appendix A presents details on Lasnex and LHT hohlraum modeling, and  Appendix B presents the MHD model in detail. Imposed field is done through an initial condition on the vector potential $Aϕ[r,z]$⁠, which produces $Brz$ but no $Bϕ$⁠, $(B→=B→rz+Bϕϕ̂)$ with $B→rz$ in the rz plane). We use the analytic solution for a solenoid of finite axial length and no radial thickness.³⁵ This idealization compares well to detailed calculations of the real NIF coils. Hohlraum modeling at LLNL has traditionally not included MHD effects, though work is under way to address this deficiency. Self-generated Biermann-battery fields are known to be present in the hohlraum and capsule, and experiments are ongoing to understand their role. Earlier hohlraum MHD modeling with the Hydra rad-MHD code³⁶ includes studies of self-generated fields³⁷ and imposed fields.³⁸ Compared to the earlier imposed-field work, the present work uses a more complete MHD model with current best practices. The prior work used the very simple Ohm's law $E=−u×B+(α||/(ene)2)J$ (in the notation of  Appendix B) with no azimuthal, self-generated B. The $J×B$ bulk force was included, along with a subset of terms in the electron energy equation. Anisotropic electron thermal conduction was included but no Righi–Leduc effect, i.e., $κ||e$ and $κ⊥e$⁠, but no $κ∧e$⁠. We believe the addition of the Nernst effect in the present work is probably the most physically important difference. The present work considers targets with low hohlraum gas fill, HDC ablator, and short laser pulses, while the prior work considered a “LowFoot” design with plastic ablator, high hohlraum gas fill, and long laser pulses.

Our modeling goals are to design the WarmMag NIF shots and to understand how well we can model MHD effects. We, therefore, follow the standard practice outlined in  Appendix A of tuning laser power multipliers and the CBET saturation clamp to match data, namely, shock velocity, bangtime, and hotspot $P2$ shape. This provides what we consider to be a reasonably realistic x-ray drive for the capsule. Subsequent capsule performance, such as yield and ion temperature, are untuned and thus an indication of simulation quality.

Shock velocity data are obtained from keyhole experiments, which can only be done cryogenically. We, therefore, use the BigFoot keyhole shot N161115-2 to develop laser power multipliers. This shot directly applies to the two BigFoot gas-filled shots we model, but not to the WarmMag platform (with different, e.g., wall material and hohlraum fill). We use the Automated NIF Tuning Suite (ANTS)³⁹ to develop our tuned parameters. Figure 4 plots the delivered laser power and multipliers, and Fig. 5 shows the resulting shock velocities. The power multipliers clearly improve the agreement between simulated shocks and data, but the agreement is not perfect. We deem this to be adequate for the present study of gas-filled capsules, which should be less sensitive to shock dynamics than DT ice-layered implosions. The power multipliers for N161115-2 are applied to all other shots, at equivalent times found by ANTS. Figure 4 illustrates this for N161204-3. A single peak power multiplier is chosen for each shot other than N161115-2 to match the implosion bangtime. For N161204-3, this starts around 4 ns. The CBET clamp $δnsat/ne$ is 0.005 until 3.8 ns (near the end of the rise to peak power), which is a typical value used for similar shots. After 3.8 ns, a new clamp is chosen to match the hotspot $P2$⁠. For all shots, the simulated bangtime is within 30 ps of data, and $P2/P0$ is within 2.9%.

We return to Fig. 3 to discuss the modeling results.  Appendix A provides more details on the peak power multipliers and CBET saturation clamp. The peak power multipliers vary from 0.85 to 0.92, without a clear trend with $Bz0$⁠: the multiplier increases toward unity for the low-T shots but decreases for the high-T ones. All WarmMag shots have multipliers equal to, or closer to unity than, BigFoot. These values and their variations are comparable to other NIF campaigns with low hohlraum fill density, such as Hybrid E. Physics deficiencies in the hohlraum opacity model likely explain at least part of the need for multipliers.⁷⁰ The peak CBET clamp $δnsat/ne$ increases substantially with $Bz0$⁠, from 5 × 10⁻⁵–5 × 10⁻⁴ for unmagnetized shots to 1.5 × 10⁻³–3.5 × 10⁻³ for magnetized ones (recall a larger clamp means less artificial modification to the physical CBET model). This results in significantly more CBET to the inner beams with increasing $Bz0$ (see Table II). Physical ion–wave nonlinearities can limit CBET for amplitudes $δnsat/ne≳0.01$⁠: values of that order may indicate real nonlinearity, while smaller values are likely “standing in” for incorrect plasma conditions relevant to CBET. We conclude that the tuning needed for the WarmMag platform is comparable to subscale BigFoot and that the only clear trend with imposed B is an increased CBET clamp. We do not understand this last trend, and note that it leads to clamps closer to the regime of physical nonlinearities. Our modeling of magnetized hohlraums seems at least as accurate as for unmagnetized ones.

With hohlraum tuning parameters chosen, the capsule yield and $Ti$ are untuned and reflect simulation quality. The $Ti$ simulated/experiment ratio is closer to unity (at most 1.066) for all but one WarmMag shot than the average of the two BigFoot shots (0.904). The exception is N210301-1 (the blue point: 0.888) which was extremely prolate and could have $Ti$ compromised by substantial plasma flow. The relative yield increase due to magnetization (see Table I) is $2.19−2.90×$ in the data and is similar but slightly lower on average in the modeling $(2.08−2.67×)$⁠. For $Ti$⁠, the relative increase due to magnetization is $1.28−1.4×$ in the data and slightly higher $(1.35−1.46×)$ in modeling. The yield and $Ti$ increases are roughly in line with modeling, especially given measurement error bars.

The major shortcoming of the modeling is that the absolute simulated yield is $1.95−2.72×$ the data in the WarmMag platform, while it agrees well for BigFoot $(0.83−0.94×$⁠). The discrepancy is slightly less for the magnetized shots, which indicates the field is at least not introducing new mysteries. This suggests the field may reduce capsule mix, but this is speculative. There are many differences between the BigFoot and WarmMag platforms, which could account for the difference in yield modeling. We are exploring these now and will report results in the future. In particular, modeling of capsule mix due to hydro instabilities, the fill tube, etc., remains to be done.

This section studies the role of MHD and imposed B fields in the BigFoot shot N161204-3. This is closer to a typical cryogenic NIF implosion than our WarmMag shots: it uses a pure Au hohlraum, He hohlraum fill, and no caboose on its laser pulse. We use our Lasnex modeling methodology and only vary the MHD physics included. We consider four integrated hohlraum modeling runs: (1) a base run with no MHD; (2) run 1 with self-generated azimuthal field and the full Nernst effect [Nernst multiplier $fN=1$⁠, see Eq. (B20)]; (3) run 2 but with our standard $fN=0.1$ (except $fN=1$ inside the capsule); and (4) run 3 but with a finite-solenoid imposed $Brz$ field and $Bz0=30$ T. All runs use the same laser power multipliers and CBET clamp as given in Table I for this shot, which corresponds to run 3.

Figure 6 shows $|B|$ and $Te$ in the hohlraum at several times. Self-generated field is produced mostly by the Biermann battery effect (misaligned $∇ne$ and $∇Te$⁠). This leads to highly localized field, such as at the LEH window boundary (2 ns) and where the laser is absorbed near the hohlraum wall. The imposed $Bz0=30$ T field roughly follows the frozen-in law: the field increases where plasma compresses normal to $B$ (like the hohlraum fill) and decreases where it expands (like the ablated hohlraum wall and capsule). The imposed $Brz$ field roughly “adds on” to the self-generated field. $Te$ is hotter for the imposed-field run in the LEH at 2 ns, as expected due to reduced electron thermal conduction. This difference is reduced at 4 ns, and the “gold bubble” where the outer beams hit the wall is hotter in the self-generated run. At 6 ns, the hohlraum fill is generally hotter in the self-generated run. This goes against the expectation that the main effect of the imposed field is to reduce thermal conduction. It also seems to contradict $Te$ measurements in hohlraums at the OMEGA Laser Facility.⁴⁰ Those hohlraums were much smaller in scale than our NIF experiments: they were driven by $∼1/40$ the laser energy, had equivalently shorter spatial dimensions and laser pulses, and had no capsule. In absolute terms, those experiments resemble the LEH early in time of the NIF shots, which are hotter with the imposed field. Analysis of the later times when the imposed-field run is cooler is ongoing. One possibility is the higher fill pressure, from both higher $Te$ and magnetic pressure, reduces compression and pdV heating.

The hohlraum x-ray drive as seen by the capsule is presented in Fig. 7. There is very little difference in total x-ray flux or the M-band fraction (photon energy $>1.8$ keV) between the four runs. MHD and imposed field thus have little effect on overall hohlraum energetics. Figure 8 shows the $P2$ Legendre moment of the x-ray flux. We follow the standard sign convention with $P2>0$ for a pole-hot x-ray drive, which produces an oblate (⁠ $P2<0$⁠) hotspot shape. The four runs are similar, but not identical: the no-MHD run has the most pole-hot drive, the self-generated with $fN=0.1$ the least, and the other two runs are in between. The peak-power CBET is very small in all four runs since we used the very low clamp of 1 × 10⁻⁴ from our tune for this shot. A better CBET model without the need for a clamp may reveal differences between the runs.

We now discuss the capsule physics of the four hohlraum runs, as seen in Table II and Fig. 9. The most salient effect of self-generated fields, as seen in comparing runs 2 and 3 vs 1, is the less pole-hot x-ray flux $P2$⁠. This is consistent with the hotspot $P2$ for runs with self-generated fields being prolate vs the round no-MHD run. The very low capsule Hall parameter <0.01 in Fig. 9 suggests that self-generated capsule fields have very little effect. This is as expected: high-mode asymmetries are required to generate significant magnetic flux in capsules⁴¹ and are not included here. The bangtimes are all close (within 60 ps), consistent with the close x-ray fluxes. The lower yield and $Ti$ vs no-MHD are likely due to the prolate shape. The only difference between the two self-generated runs with different Nernst multiplier is in hotspot $P2$⁠, again consistent with the x-ray flux $P2$⁠.

The main effect of the imposed $Bz0=30$ T field in run 4 is to increase yield by 56% and $Ti$ by 34% vs run 3. The x-ray flux $P2$ is slightly more pole hot, which is consistent with the slightly lower hotspot $P2$⁠. This is in contrast to a naïve expectation that an axial field leads to a more prolate hotspot, since electron thermal conduction is unmagnetized along z. To separate the effects of the imposed field on the hohlraum x-ray drive and capsule physics, we performed runs 5 and 6. These are capsule-only runs both driven by the x-ray flux from hohlraum run 3. This is done via a frequency-dependent source (FDS) imposed outside the capsule vs time and photon energy, but symmetric in space. Any asymmetry is the result of capsule physics. Run 5 with no imposed field has $P2/P0=−0.2%$⁠, which is very small and indicates residual numerical asymmetry in either the simulation or post-processing. The imposed-field run is prolate (⁠ $P2/P0=9.7%$⁠), as expected. In the imposed-field hohlraum run, the more pole-hot x-ray flux over-compensates for this effect.

This paper reviewed the magnetized NIF hohlraum experiments to date with room-temperature, gas-filled capsules, which we call the WarmMag platform. The emphasis was on the design considerations and modeling with the Lasnex rad-MHD code and the LHT common model. The WarmMag platform is based on BigFoot gas-filled capsules driven by $1.05−1.1$ MJ of laser energy. Changes were made to satisfy the constraints of allowing the field to soak thru the hohlraum wall without excessive $J×B$ motion (novel AuTa₄ hohlraum instead of standard Au or DU), room-temperature fielding (hohlraum fill gas of C₅H₁₂ instead of He), and a stringent laser energy limit $≲1$ MJ to avoid SBS optics damage risk. The two most recent shots in the “high-T” campaign on the WarmMag platform, N230212-2 and N230612-1, were presented for the first time. These increased the capsule compression and hotspot temperature with or without an imposed field and showed an imposed field gives a hotter hotspot. A compelling demonstration of the benefits of magnetization is that the lowest imposed field of 11.8 T in the “low-T” shot N220912-1 gave a higher neutron yield and ion temperature $Ti$ than the best-performing unmagnetized shot N230612-1 in the high-T campaign.

Lasnex modeling captures most of the NIF observations, with the notable exception of the absolute neutron yield being $2.0−2.7×$ higher than the data. Hohlraum modeling required similar laser power multipliers to match the implosion bangtime for WarmMag shots with and without the imposed field. The needed clamp $δnsat/ne$ on cross-beam energy transfer (CBET) to match hotspot $P2$ increased greatly with $Bz0$⁠, from 5 × 10⁻⁵–5 × 10⁻⁴ for unmagnetized shots to 1.5 × 10⁻³–3.5 × 10⁻³ for magnetized ones. This is approaching the level of 0.01 when physical ion-wave nonlinearities may play a role. With hohlraum tuning parameters (power multiplier and CBET clamp) chosen, the capsule outputs of yield and $Ti$ are untuned and indicate simulation quality. The modeling is close to the measured $Ti$ and relative yield and $Ti$ increase due to magnetization, but the modeled yields are too high. We will study yield degradation due to mix, hydro instabilities, and the fill tube in the future, all of which our current modeling neglects.

A rad-MHD study of effects from self-generated (mostly Biermann battery) and imposed fields (⁠ $Bz0=30$ T) on BigFoot shot N161204-3 showed both cause little change in the total x-ray drive and its M-band fraction. There is some effect on the x-ray drive's $P2$ moment, with the imposed field making it slightly more pole-hot. This leads to a slightly less prolate hotspot vs the self-generated run, though capsule-only modeling with the same, symmetric x-ray drive shows the imposed field makes the hotspot more prolate. The $P2$ variations due to imposed field in both the x-ray drive and hotspot shape are minor and should be tunable through standard techniques like laser cone fraction and wavelength shift (to control CBET). The optimal shape of a magnetized implosion is an open question, but it is likely not too far from round. Non-spherical implosion comes at a substantial cost in reduced compression: the limit of a cylindrical implosion gives a density increase $∼CR2$⁠, not $∼CR3$ as for a spherical one (CR is convergence ratio).

To close, we discuss ongoing work at LLNL toward magnetized indirect-drive ignition on NIF. Current experiments are limited by being room-temperature and having $≲1$ MJ of laser energy, so magnetized ignition work is mostly theory and modeling. This is broadly geared toward finding the best uses of imposed field for ignition designs. The main thrust is to increase the maximum achievable yield and gain on NIF. We start by imposing fields with $Bz0≤100$ T on the best performing NIF implosions, in the Hybrid E campaign. Early modeling results show up to tripling the yield with $Bz0∼60−70$ T. We are also investigating magnetized target designs for use on NIF “Enhanced Yield Capability” (EYC) which is a proposal for upgrading NIF to $2.6−3$ MJ of laser energy. As mentioned above, a cryogenic pulsed-power system, which mitigates the SBS optics risk due to prefire, has been designed, but its construction timeline is uncertain. Ongoing research on advanced hohlraum coils and pulser architecture is showing promise in achieving higher magnetic fields and easier target fabrication.

Future work will consider variants on these designs optimized to take advantage of the field, and other designs entirely. For instance, we speculate that a field may give the most benefit for designs that do not have a hot enough hotspot to self-heat and ignite without a field. We have also studied non-axial fields, such as magnetic mirrors or even closed field lines, which can give better performance. A major challenge is how to impose such field of reasonable magnitude without seriously disrupting the implosion (say by running a wire through the capsule). In addition to increasing the maximum yield, another goal is to achieve the same yield but with relaxed requirements on laser energy or target quality. This is appealing for “users” of ignition, who desire a $1−10$ MJ yield to conduct other high-energy-density (HED) experiments.

An important topic we are starting to address is the role of magnetic field in capsule degradation (“mix”) due to hydro instabilities or engineering features like the fill tube or capsule support tent. Some experimental⁴² and simulation⁴³ work exists, but this topic is little studied in ignition-relevant implosions. We are conducting more gas-filled capsule implosions on the NIF WarmMag platform to study how imposed field alters mix. These use the dual crystal backlighter imager (CBI) platform⁴⁴ to image high-Z tungsten x-ray emission from the implosion hotspot. Experiments at the OMEGA laser are under way to study magnetized mix, as well as magnetized CBET. Recent 3D MHD modeling of the Rayleigh–Taylor (RT) instability⁴⁵ in idealized settings shows reduced growth for k-vectors $k || B$⁠, but not for $k⊥B$⁠. In this case, growth can be enhanced by reduced thermal conductivity, which reduces ablative stabilization.⁴⁶ We plan to extend this work to integrated implosion modeling. A very desirable goal is to develop reduced mix models like those available in Lasnex (discussed in  Appendix A and Refs. 47–49) to include magnetization.

We look forward to this and related work culminating in magnetized ignition experiments on the NIF.

It is a pleasure to thank C. A. Thomas and K. L. Baker for fruitful discussions about the BigFoot platform. We are grateful to the NIF and OMEGA operations teams for fielding our nonstandard experiments, and to the target fabrication team for providing AuTa₄ hohlraums. 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. Partially supported by LLNL LDRD projects 20-SI-002 and 23-ERD-025.

The authors have no conflicts to disclose.

D. J. Strozzi: Conceptualization (equal); Data curation (equal); Investigation (lead); Methodology (lead); Writing – original draft (lead). H. Sio: Conceptualization (equal); Data curation (equal); Investigation (supporting); Methodology (supporting); Writing – review & editing (equal). G. B. Zimmerman: Methodology (supporting); Software (equal). J. D. Moody: Conceptualization (equal); Investigation (supporting); Writing – review & editing (equal). C. R. Weber: Methodology (supporting); Software (supporting). B. Z. Djordjevic: Investigation (supporting); Methodology (supporting); Writing – review & editing (equal). C. A. Walsh: Investigation (supporting); Methodology (supporting). B. A. Hammel: Conceptualization (supporting). B. B. Pollock: Investigation (supporting); Methodology (supporting). A. Povilus: Investigation (supporting); Methodology (supporting). J. P. Chittenden: Methodology (supporting). S. O'Neill: Methodology (supporting); Writing – review & editing (equal).

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

The rad-MHD modeling presented here uses the Lasnex simulation code,^50,51 which is a 2D axisymmetric (rz), arbitrary-Lagrangian–Eulerian (ALE) code. We use the Lasnex Hohlraum Template (LHT), a “best-effort,” version-controlled common model used by the LLNL ICF program. An earlier version of this model is described in Ref. 52. We describe the current recommended practice for “design-quality” modeling of NIF hohlraums. A typical LHT 2D hohlraum run takes 12–48 wall-clock hours on 36 Intel CPUs. The LHT also supports 1D and 2D capsule-only runs, driven either directly by a laser source, an x-ray frequency-dependent source with no angular spatial variation, or an x-ray “tallied flux” source with such variation. The external laser entrance hole (LEH) hardware is included, namely, the retainer ring and washer that secure the main window and thin secondary “storm window” that prevents NIF chamber gas from condensing. The LEH hardware can affect LEH plasma conditions and thus cross-beam energy transfer (CBET), which transfers energy among the NIF lasers.⁵³ The storm window itself is not currently included, due to mesh management challenges.

Laser model: The LHT treats the lasers with 3D ray tracing with an effectively infinite speed of light: at each time step, rays propagate through the target until the remaining power is <1 × 10⁻⁴ of incident. This includes refraction and inverse bremsstrahlung absorption as reduced by the Langdon effect.⁵⁴ The incident ray positions, k-vectors, and relative powers are chosen to accurately represent the near-field square lens aperture, far-field intensity profile due to phase plates, and pointing of each of the 192 NIF beams (or 96 in the more common one-sided runs shown here). The NIF beams are grouped into “quads” of four beams. The LHT follows this in that rays for the four beams in a quad are treated as a single laser with one power history, and function as a unit for CBET. The inline CBET model is used,⁵⁵ which currently treats the lasers as unpolarized quads and finds their 3D intensity on an auxiliary 3D mesh (the plasma conditions are still axisymmetric). The quad treatment involves $∼16×$ fewer couplings than the beam treatment and is thus much faster. The inline CBET model currently does not handle polarization, which an accurate beam model needs. CBET between two lasers transfers power to the one with the longer wavelength in the plasma-flow frame, which depends on both flow-induced Doppler shifts and differences in the incident (lab-frame) wavelengths. All four NIF cones (located at polar angles of 23°, 30°, 44°, and 50°) can have separate wavelengths. All shots discussed in this paper have one wavelength for all cones. Inline stimulated Brillouin (SBS) and Raman (SRS) backscatter models are available,⁵⁵ but not used in this paper: backscatter for the relevant shots is generally low.

Atomic and plasma models: Material zones are treated with local thermodynamic equilibrium (LTE) models for properties like x-ray opacity and equation of state, until they exceed a critical electron temperature $Tcr$ of 0.3 keV. We raise this to 1 keV “inside the capsule,” meaning all capsule regions other than the outermost one: the latter ablates into the hohlraum, is heated by the laser, and becomes part of the hohlraum fill. The higher $Tcr$ reduces computer cost and agrees with current LLNL practice for high-resolution capsule instability modeling. Above $Tcr$⁠, all properties come from LLNL's detailed configuration accounting (DCA) non-LTE atomic models.^52,56 DCA is not a single model but a framework that accommodates models of widely varying complexity and cost. The results shown here use the “November 2020” DCA models with details considered acceptable for evolving atomic populations “inline” in current ICF rad-hydro simulations. We also use a tabulated, steady-state non-LTE approach based on the Linear Response Method in the high-Z hohlraum wall.⁵⁷ These tables are generated with substantially more detailed models than the inline ones and greatly reduce the non-LTE computational cost.

The multi-ion species hydrodynamics package is used,⁵³ run here in an intermediate mode where each ion species has its own density and flow velocity but a single ion temperature (to reduce computer time and numerical issues). The package supports per ion species temperatures and heat fluxes. The plasma anisotropic stress tensor is also used. Electrons are not treated as an independent species, instead the plasma is assumed to be quasi-neutral: $ne=∑iZini$⁠. Separate electron and ion temperatures are used, and the net current in the MHD package is equivalent to a separate electron flow velocity. Ion–ion transport and coupling coefficients come from Ref. 58, while electron–ion (e–i) coefficients come from either the analytic dense-plasma model of Lee and More,⁵⁹ specifically the GMS-6 model from Table I of Ref. 60, or a more advanced LTE table if available.

Mix models: Reduced models of hydrodynamic mix due to Richtmyer–Meshkov (RM) and Rayleigh–Taylor (RT) instabilities are available. These are designed to capture mix physics without spatially resolving it and involve some parameters that must be tuned. The methodology is detailed in Ref. 49, which demonstrates good agreement between Lasnex mix modeling and experimental data for layered implosions with plastic capsules and separated nuclear reactants. RT growth is calculated via a buoyancy-drag model,⁴⁸ which creates zones of mixed material at user-specified interfaces. These mixed zones entail three subzones: one for each pure material, and a third that is atomically mixed. We use an RM model that limits the growth to keep the perturbation “finger tips” between the shocked material interface and the transmitted shock.⁴⁷ Mix models are not used in the present work, but will be to understand the over-predicted yields on WarmMag gas-filled capsules, and for modeling magnetized ignition designs.

Drive deficit: laser power multipliers: LHT modeling differs in important ways from NIF ICF data. We discuss three discrepancies and physics modifications to reduce them. The x-ray drive deficit refers to the fact that the total hohlraum x-ray drive is higher in modeling than data. This is seen in x-ray measurements with the DANTE detector, shock velocity measurements in “keyhole” shots with the VISAR detector, and in the time of peak capsule emission or “bangtime.”⁵² In this paper, we modify the laser power by a time-dependent factor (the same on all beams) to bring modeled shock velocity and bangtimes in line with measurements. Recent NIF experiments indicate improved hohlraum wall opacity models explain at least part of the drive deficit.⁷⁰ 

We also use an electron heat flux limiter f, which limits the magnitude of the electron heat flux q to some fraction of the free-streaming value: $q=min(fneTevTe,κ∇Te)$⁠. This is an old technique in HED modeling,⁶¹ intended to mimic physical flux inhibition by things like nonlocal transport, microturbulence, or magnetic fields. We follow the “high flux model”⁶² and use $f=0.15$ outside the capsule (as defined above), which generally has little effect on ICF hohlraums: similar results obtain with a larger value or the nonlocal Schurtz model.⁶³ Inside the capsule, we use $f=0.1$⁠, which has little effect and is done to agree with high-resolution capsule modeling practice. Jones et al.⁵² revisited a low $f≈0.03$ to explain the drive deficit by allowing more unabsorbed inner-beam laser light to escape the opposite LEH as “glinted light.”⁶⁴ Dedicated experiments to examine this show much less glint than $f=0.03$ modeling and generally support a high $f≈0.15$⁠.^65,66

Implosion hotspot shape: CBET clamp: A second discrepancy with NIF data is with the compressed hotspot shape, as measured by x-ray or neutron images near peak emission. Specifically, the second Legendre moment or $P2$ is a key quantity. A major factor in $P2$ is CBET between the inner and outer laser cones. This relies on accurate plasma conditions, especially in the LEH. This region has many overlapping laser beams and weakly collisional plasma. Accurate modeling likely needs better accounting for the lasers and kinetic or other effects beyond standard rad-hydro. LHT modeling generally produces too much CBET to be consistent with implosion shape data—assuming the non-CBET parts of the model that affect shape are correct. LLNL has for over 10 years used a time-dependent saturation clamp $δnsat/ne$ on the amplitude of the CBET-driven ion acoustic waves. This was introduced in Ref. 67 and is also discussed in Ref. 55. In the inline CBET model, the change in amplitude of light wave 0 due to CBET with light wave 1 is $∂za0∝a1δne$ with ion-wave amplitude $δne=min(δnsat,βa0a1*)$ and $β$ a function of plasma and light wave parameters other than amplitudes. In terms of intensity $Ii=aiai*$⁠, $∂zI0∝min(δnsat(I0I1)1/2,βI0I1)$⁠. When the clamp applies, CBET still increases with intensity but with a slower square-root rather than linear scaling. Physical ion-wave nonlinearities can set in for $δnsat/ne∼0.01$⁠. Values between 0.001 and 0.01 are typically needed to match shape data in hohlraums with low gas fill density $≤$0.6 mg/cm³. We stress that the clamp at least partly stands in for incorrect plasma conditions instead of ion-wave nonlinearity.

Outer-beam bubble conditions: Nernst multiplier: The final discrepancy we discuss is with NIF measurements of flows and $Te$ in the “gold bubble” plasma that expands from the hohlraum wall heated by the outer beams.⁶⁸ LHT modeling with a low flux limit $f=0.03$ is consistent with the data, while $f=0.15$ is not. This is at odds with the inner-beam glint results. It suggests a model with $f=0.15$ where the inners hit the wall but $f=0.03$ where the outers hit. Rosen found that the MHD model with self-generated fields does this, if a Nernst multiplier $fN=0.1$ is used to greatly reduce Nernst advection.⁶⁹ More recent work⁷⁰ suggests that hohlraum wall opacity models that better agree with x-ray drive measurements remove the need for $fN$⁠; we will examine this approach in the future. The Nernst effect advects B field from hot to cold electrons. In the hohlraum wall, this advects self-generated “Biermann battery” fields farther out in radius, to colder and denser plasma where the field is too small to magnetize the electron heat flow. The LHT adopts $fN=0.1$ outside the capsule but $fN=1$ (no modification) inside the capsule.

We describe the MHD package in detail since it is a key part of this work. Lasnex has had a “toroidal,” or azimuthal, field package for four decades.⁷¹ This includes a relatively complete implementation of Braginskii's classical fluid model.⁷² The “poloidal,” or rz, MHD model is a more recent addition and does not yet implement all of Braginskii.^73,74 Lasnex is an rz code in that all quantities only vary in r and z, but azimuthal $ϕ$ vector components are of course allowed. We write a vector $v=vrz+vϕϕ̂$ with $vrz=vrr̂+vzẑ$ the 2D vector in the rz plane. The relevant fields are the magnetic field $B$⁠, vector potential $A$⁠, electric field $E$⁠, net current $J$⁠, and center-of-mass or bulk flow velocity $u$⁠.

The Lasnex MHD model is typical of those used in fluid HED modeling. Electron inertia $∂tue$ and displacement current $∂tE$ are dropped. This eliminates Langmuir and light waves from the system, which allows an enormous increase in the time step stability limit (for explicit methods). We also do not consider any nonlocal kinetic effects, even though they could matter in hohlraum modeling with or without MHD.⁷⁶ “Anomalous” effects such as resistivity enhanced by micro-turbulence are also neglected. We roughly expect our model's accuracy to increase with density and collisionality. This should make it more applicable in the compressed capsule, and less applicable in the underdense hohlraum plasma and especially outside the LEH. These comments apply to hohlraums with and without an imposed field. The comparable accuracy of our LHT modeling of the WM NIF shots with and without a field suggests that the field does not introduce significant inaccuracy.