We examine the potential that imposed magnetic fields of tens of Tesla that increase to greater than 10 kT (100 MGauss) under implosion compression may relax the conditions required for ignition and propagating burn in indirect-drive inertial confinement fusion (ICF) targets. This may allow the attainment of ignition, or at least significant fusion energy yields, in presently performing ICF targets on the National Ignition Facility (NIF) that today are sub-marginal for thermonuclear burn through adverse hydrodynamic conditions at stagnation [Doeppner et al., Phys. Rev. Lett. 115, 055001 (2015)]. Results of detailed two-dimensional radiation-hydrodynamic-burn simulations applied to NIF capsule implosions with low-mode shape perturbations and residual kinetic energy loss indicate that such compressed fields may increase the probability for ignition through range reduction of fusion alpha particles, suppression of electron heat conduction, and potential stabilization of higher-mode Rayleigh-Taylor instabilities. Optimum initial applied fields are found to be around 50 T. Given that the full plasma structure at capsule stagnation may be governed by three-dimensional resistive magneto-hydrodynamics, the formation of closed magnetic field lines might further augment ignition prospects. Experiments are now required to further assess the potential of applied magnetic fields to ICF ignition and burn on NIF.

In inertial confinement fusion (ICF), around a milligram of deuterium-tritium (DT) fuel is rapidly compressed to high temperatures sufficient for thermonuclear fusion to commence. The complete burn of a 50:50 mix of DT through the fusion reaction 2H + 3H → n + 4He + 17.6 MeV releases a specific energy of 3.4 × 1011J/g. The National Ignition Facility (NIF) is seeking to demonstrate laser-driven ICF ignition and thermonuclear burn in the laboratory for the first time by indirect-drive.1 Here, laser energy is first converted to x-rays in a hohlraum enclosing the spherical shell fuel capsule. The radiation absorbed in an ablator surrounding the fuel implodes the capsule at high velocity until spherical convergence and back pressure decelerates this hydrodynamic “piston,” compressing its fuel, and converting its kinetic energy into thermal PdV work on a central gas “hotspot.” Under the right conditions, this could initiate fusion ignition, i.e., a thermonuclear burn wave propagating out from the hotspot into the colder compressed fuel heated by the deposition of the 3.5 MeV DT alpha particles resulting in fusion energy yields of ≳1 MJ.

The criteria for ICF ignition and propagating burn2 are determined by the time-dependent balance between hotspot energy gains (PdV work and fusion alpha particle deposition) and losses (hydrodynamic expansion, electron heat conduction, and radiation) Achievement of ignition requires a well-formed hotspot with a central ion temperature of Ti ∼ 12 keV, a hotspot areal density of ρRHS ∼ 0.4 g/cm2, and pressure of several-hundred-GBar.

The NIF ignition campaign has made significant progress, returning data of unprecedented quality and diagnostic precision with over 80 cryogenic DT target implosions to date, driven by “low-foot” and “high-foot” laser pulse shapes for capsules comprising either plastic, diamond, or beryllium ablators and fuel of solid cryo-layered-DT or DT-wetted CH foams.3–7 However, the hotspot pressures remain around 50% of that required for ignition so that the resulting fusion yields are around a factor of ten too low to initiate ignition via bootstrap alpha heating.

In this paper, we report on an update of preliminary work8 that suggested that an imposed magnetic field that compresses to high values under NIF implosion conditions might relax the minimum hotspot conditions required for thermonuclear burn. In this extended study, we incorporate detrimental target perturbations paralleling those observed in recent NIF experiments and post-shot simulations—i.e., low mode shape distortions, hotspot compression losses due to residual kinetic energy (RKE), and higher mode Rayleigh-Taylor instabilities. In this way, we seek to show that the relatively simple expediency of applying an imposed field of tens of Teslas may recover ignition and high fusion yields in today's typical NIF capsule implosions that are otherwise sub-marginal for propagating burn because of unfavorable stagnation hydrodynamics.

Detailed two-dimensional (2D) and three-dimensional (3D) simulations of current NIF experiments show reasonable agreement with the data and suggest that a reliable predictive capability is developing to understand ignition processes.9,10 Modeling of the better performing “high-foot” implosion design11—where capsules are driven with a higher radiation temperature during the early phase of the implosion—indicates that these are more stable to higher-mode ablation front instabilities and that stagnation conditions now appear to be degraded mainly by long wavelength, low-mode hohlraum radiation flux asymmetries, although defects seeded by the capsule support tent and capsule fill-tube for the DT fuel also play a role.9,10,12 Associated high resolution 2D and 3D studies have also proposed that low-mode deviations from spherical hohlraum drive symmetry may lead to coherent vortex structures and it is the instability of these structures that in turn leads to turbulence and energy loss in the hotspot.13 

Figure 1(a) illustrates the starting point for our studies, where we adopt a standard NIF ignition-scale capsule with a 62 μm-thick cryogenic solid-DT fuel layer. The ablator is 76 μm-thick high-density-carbon (HDC), i.e., diamond, doped with 0.25 at. % tungsten to absorb non-Planckian L-band and M-Band radiation from the uranium-lined hohlraum.5 Diamond has high density (3.5 g/cm2) compared with the other candidate ablator materials—i.e., plastic CH (1.08 g/cm2) and Be (1.8 g/cm2)—and consequently can be driven with a shorter laser pulse length due to shorter transit times of the ablation-driven shocks across the capsule; shorter pulse lengths result in less hohlraum plasma filling, better laser beam propagation, and the potential for better capsule drive symmetry.5 The general conclusions of this paper should apply to NIF capsules with either diamond or CH ablators and with fuel layers of either cryo solid-DT4,5 or cryo DT-wetted-foams.7 But Be ablators6 may be too conductive to permit initial field penetration in time before the capsule is imploded.

We drive the hohlraum with a three-shock, laser pulse [Fig. 1(b)] that is adiabat-shaped14 with initial 40 TW picket and 442 TW main drive. The 11.7-ns pulse results in a laser energy of 1.86 MJ—close to the maximum obtainable and demonstrated in the best performing NIF high-foot shots.15 Simulations in this paper were performed in 2D with the LASNEX radiation-hydrodynamics code.16,17 We first ran an integrated hohlraum-capsule calculation to model relevant physics in the laser propagation and x-ray conversion processes for a uranium hohlraum of 0.672-cm diameter and 0.6 mg/cm3 He gas fill. We then performed subsequent 2D capsule-only simulations where only the imploding capsule was modeled with the surrounding hohlraum replaced by an x-ray drive mapped from the frequency-dependent source from the integrated calculation. (Of course, this results in a radiation flux that is uniform versus polar angle; we will modify this condition below.) The resulting x-ray radiation drive temperature is also shown in Fig. 1(b).

In a first set of simulations, we ran the capsule in Fig. 1 with just standard higher-mode “roughness” perturbations applied to the outer ablator surface, the interior doped ablator interfaces, and to the inner DT ice surface, where the perturbation mode spectra are obtained from characteristic measurements from target fabrication. However, under such conditions, the target ignites and burns with full fusion yields of ∼20 MJ. This result is unsurprising given that NIF capsules were originally designed to ignite and burn under such “X1” standard higher-mode roughnesses. And once ignition and propagating burn occur, the fusion yield is then limited only by the inertial tamping of the compressed fuel's areal density which, for these well-formed implosions, was ρR ∼1.4 g/cm2.

Our next step was to make this capsule sub-marginal for ignition by applying additional perturbations to approximate stagnation conditions modelled in Refs. 9, 10, 12, and 18 that compared NIF experiments with simulations, and then examine conditions under which an imposed magnetic field might recover ignition. Specifically, we sought to encompass the recent experimental observations that:

  1. The cold fuel “piston” is not round at stagnation and has non-zero Legendre P2-P4 mode structure with components of up to tens-of-percent that result in less ideal spherical compression and a larger surface area for hotspot heat conduction loss;

  2. The areal density of the piston can have variations of up to a factor of two as a function of equatorial angle which results in less efficient hotspot tamping with less momentum/energy transfer to the hotspot thermal content;

  3. The residual kinetic energy remaining in the cold fuel structure at stagnation can be in excess of 50% of the peak inflight value, resulting in inefficient conversion of shell kinetic energy to hotspot thermal energy.

The capsule support tent and fill-tube for the DT fuel can also impose higher-mode perturbations that can penetrate the hotspot and this will be considered below.

Accordingly, we degraded the capsule performance by: (a) applying a low-mode polar-angle-dependent radiation flux perturbation to the frequency-dependent radiation source driving the capsule of the form ϕ(θ) = ϕ0[1 + a Cos(P(θ + 180°/P)], where θ is the polar angle, and with P =4 and a =0.1, reasonably representative of shape distortions observed, and (b) increasing the residual kinetic energy (RKE) remaining at stagnation. RKE features are likely 3D in form9,10 and moreover, it is unclear how they are appearing at stagnation as current imaging techniques are insufficient to diagnose. Accordingly, we define an “effective-RKE” remaining at stagnation, RKEeff, by reducing the net inflight kinetic energy EK in the DT-fuel through lowering the drive temperature such that RKEeff= EK, peak − EK, where EK, peak = 13.5 kJ is the peak kinetic energy at the full drive conditions shown in Fig. 1. Thus, less energy is available for compression and results in lower hotspot thermal energy. The intention is not to precisely benchmark a specific experimental shot but rather decrease capsule performance to around that observed in better performing but still degraded NIF shots to date.

Figure 2 shows the resulting fusion yields from LASNEX simulations under the above observed conditions of imposed drive shape perturbation and increasing effective-RKE and no imposed magnetic field. The resulting density contours at stagnation in the r-z plane are also indicated at three representative points A, B, and C; a 90° capsule segment is shown where the capsule has rotational symmetry around the z axis. Note the increasing RMS variation in shell areal density, ρR, with equatorial angle as the effective-RKE is increased. At low effective-RKE and with only modest shape distortions, the capsule ignites at on-axis ion temperatures of Ti ∼ 12 keV and full yields of ∼20 MJ are obtained. As the effective-RKE and shape distortions increase, the fusion yield falls over three orders of magnitude to low values where energy deposition from fusion alpha particles is negligible so that the hotspot hydrodynamic parameters are determined only by the energy imparted by the perturbed stagnating shell.

The dashed magenta line in Fig. 2 indicates fusion yield obtained from an artificially imposed “burn-off” condition where fusion alphas in the simulations are allowed to escape with their full 3.5 MeV birth energy without deposition; in this case, fusion yield results only through modest hotspot heating (maximum Ti ∼ 4 keV) from shell PdV work alone. Comparison of the region where the two curves separate indicates that minimum fusion yields of approximately ∼20 kJ (i.e., ∼4 kJ of deposited alpha energy) are required for detection of alpha particle heating and the onset of the ignition “cliff”. Figure 2 also indicates the fusion energy yield (26.2 kJ or 9.3 × 1015 14 MeV neutrons) from NIF shot N140304—the highest performing ICF experiment to date.15 Other similarly performing shots were N140120 (26.1 kJ) and N140520 (25.3 kJ). It is important to note that, in terms of alpha deposition and resulting fusion yields, such shots are on the foot of the alpha-heated ignition cliff; as illustrated below, this is the regime where magnetic fields may offer the most benefit to enhancing ignition performance.

In compressing an ICF capsule in an imposed magnetic field, the conducting plasma is characterized by large values of the Magnetic Reynolds Number—i.e., ratio of the magnetic advection to magnetic diffusion. Given that NIF capsule implosion times are less than one-hundredth of characteristic magnetic resistive diffusion times, the flux would be effectively frozen in to the converging plasma shell and the field would compress to high values depending on the capsule convergence ratio. That applied magnetic fields might improve the prospects for ICF which was proposed over 30-years ago.19–23 They have been adopted in recent concepts for magnetically compressed liner implosions,24–26 in our preliminary studies of ICF implosions8 and have been reported in a related paper.27 We have performed an associated study of (essentially uncompressed) imposed magnetic fields in NIF hohlraums and indicated that they may have the potential to reduce laser-plasma instabilities, improve laser beam transport, and modify the transport of hot electron preheat to the capsule.28 

The perpendicular electron heat conduction in a magnetized plasma would reduce as kTe where k = k(1 + (ωceτei)2) where k is the regular, unmagnetized thermal conductivity, ωce (∼B) is the electron gyrofrequency in the magnetic field B, and τei is the electron-ion collision time.29 Suppression of k is achieved for ωceτei> 1 requiring compressed fields ≳103 T (10 MGauss). At higher compressed fields of ≳104 T (100 MGauss), the Larmor gyro orbits of Eα = 3.5 MeV DT alpha particles, gα = (2mEα)1/2/qB, are localized such that their deposition range becomes less than a characteristic hotspot radius. Accordingly, reducing electron heat conduction and enhancing alpha energy deposition density result in an associated reduction in the minimum hotspot conditions (with no field) required for ignition and propagating burn2 of (ρR)HST ≳ 2.5 g/cm2 keV or  ≳ 10 Atm s, where (ρR)HS, T, P, and τ are the hotspot areal density, temperature, pressure, and confinement time, respectively. The compressed fields may also aid in suppressing shorter-wavelength Rayleigh-Taylor (RT) modes, thus slowing instability growth during late time capsule deceleration and reducing hotspot mix; this will also be examined below.

Laser-driven magnetic field compression in ICF targets has been performed on the OMEGA laser facility where seed fields of ∼8 T were compressed to ∼8000 T (80 MGauss). Measureable increases in ion temperatures and neutron yields were detected,30 resulting from reduced electron heat conduction (fusion burn-product charged particles were not relevant to these low yield experiments).

We envisage that an axial magnetic seed field could be provided for NIF targets with a thin, single-layered solenoid wound on the outside of the hohlraum driven by a co-located pulsed power supply. The coil would be pulsed and during the time that the field is maintained before the coil reaches melt temperature (∼few-μs), the laser would fire and the capsule imploded (∼10 ns). It is beyond the scope of this paper to discuss engineering details of such systems but we have constructed such a coil driven by a 4 μF, 40 kV power supply on a test stand (no laser or ICF targets). The 9-turn, center-fed coil was wound of 24-AWG wire on a 10 μm-wall-thickness uranium cylinder designed as a proxy for a NIF 0.672-cm-diamter uranium hohlraum and of the same dimensions. We obtained fields of up to 58 T over ∼2 μs pulse length before coil disassembly. Eddy currents induced in the uranium wall had no significant effect on the wall motion or the internal field rise time during the pulse; gold hohlraums might be more problematic as their electrical conductivity is a factor of ∼20 higher.

The colored curves in Fig. 3 show fusion yield enhancement predicted by LASNEX for initial imposed axial fields of 20 through 70 T for the same inflight conditions—i.e., same shape and RKE perturbations—as for the black (no-field) curve reproduced from Fig. 2. These simulations accommodated a full set of magneto-hydrodynamic phenomena including J × B forces (magnetic pressure), Braginskii cross-field transport model for degenerate matter, magnetic diffusion, the Nernst term, and orbit-following-deposition of fusion burn product charged particles along the field lines. The result of the imposed field is to translate the ignition cliff to the right so that implosions can achieve ignition with enhanced fusion yields at inflight conditions that would otherwise result in only low yields. At low perturbations, the capsule is seen to attain around the same full yield of ∼20 MJ irrespective of the imposed field because here, with a robust igniting hotspot, the fusion yield is essentially determined only by the tamping areal density of the surrounding cold fuel shell. By contrast, at the other end of the curves, there is little benefit of the field because the high capsule perturbations always result in deleterious hotspot conditions.

The most marked benefit of the applied field is seen for capsule implosions that, without field, are on the foot of the ignition cliff and would ordinarily obtain only around twice the fusion yield relative to burn-off conditions. That is, for capsules where alpha heating is only approximately doubling the fusion yield over that attainable from shell PdV work heating alone. This is the regime where our best performing NIF experimental shots lie today—e.g., shots N140304 (26.2 kJ), N140120 (26.1 kJ) and N140520 (25.3 kJ)15—and thus they may be appropriate candidates for fusion yield enhancement and perhaps ignition under imposed fields.

Our preliminary study of applying imposed fields to earlier NIF ignition capsules reported in Ref. 8 used a simple, arbitrary multiplier on capsule roughness to initially degrade fusion yield performance. The essential difference in this study is that we sought to incorporate detrimental perturbations—i.e., low-mode shape and residual kinetic energy—paralleling those observed in Refs. 9, 10, 12, and 18 that compared post-shot simulations with recent NIF experiments, and accordingly, to obtain reduced fusion yield performances in the range that have been experimentally observed.15 The results in Fig. 3 look qualitatively similar to the results obtained in our earlier study because any increasing perturbation source applied to an otherwise robustly igniting hotspot will eventually result in a steeply falling ignition cliff. This is due to the strong bootstrapping effect of fusion alpha particle deposition around marginal ignition conditions but this does not imply that the physics of one perturbation source is directly equivalent to another other than they both can cause failing hotpots under high enough perturbation conditions.

In Fig. 4, we take a lineout from points B (B0 = 0) to BB (B0 = 50 T) in Fig. 3 for capsules with performance that ordinarily—i.e., with no applied field—would be on around the foot of the ignition cliff, and plot the fusion yields versus the imposed field B0. An optimum in the enhanced yield performance is seen for imposed fields around 50 T. Lower initial fields result in larger alpha particle gyro orbits with less localization within the hotspot. By contrast, fields higher than 50 T result in increasing shell shape distortion at stagnation—and, thus, further departure from ideal spherical hotspot symmetry—due to compressed field back pressure and increasing asymmetry in both electron heat conduction and alpha transport. This is discussed further below.

To examine the physics in more detail, we take the two points B (B0 = 0) and BB (B0 = 50 T) in Fig. 3 and, in Fig. 5, compare their density and temperature contours at stagnation in the r-z plane. Note these are for the same capsule with the same perturbations (shape and RKE) and same inflight characteristics, and differ only by the application of a 50 T field in one case. The burn-averaged (Brysk) ion temperatures are 3.8 keV (no B0) and 17.9 keV (B0 = 50 T) with corresponding fusion yields of 0.036 MJ and 9.8 MJ, respectively. Thus, the magnetic field has lifted a capsule at the bottom of the ignition cliff into the ignition regime and enhanced the fusion yield by a factor of over 200.

For the 50 T case BB, the compressed field lines are shown at increasing increments of magnetic flux; the average field over the hotspot is 5.3 × 104 T (530 MGauss), a compression ratio of >1000. Given this is 2D compression of an axial field B0 with minimal flux diffusion, this is consistent with the field scaling with capsule convergence ratio, CR, as Bz ∼ B0 CR2, where CR ∼ 32. The electron magnetization Hall parameter ωce τei in the hotspot is >20; thus, perpendicular electron heat conduction is fully suppressed, whereas electrons parallel to the field conduct normally.

Regarding the reduction of alpha deposition ranges, the Larmor orbit radius, gα, of 3.5 MeV alphas is around one third of the conventional hotspot radius as shown for case BB in Fig. 5. Note that this is the initial birth radius because the alpha particle drags down in energy, spiraling inwards on itself. Thus, alphas are localized within the hotspot at these compressed fields. By contrast, the conventional hotpot is essentially demarcated by the regular (linear) alpha range of ρRHSρRα ∼ 0.4 gm/cm2 as shown in the no-field case B. However, because of the perpendicular truncation of the alpha range, the alpha particles leave the hot spot predominantly axially along the field lines, causing non-uniformities in the heat deposition profile. As described above in connection with Fig. 4, the contribution of such non-uniformities to shape asymmetries at stagnation increases with increasing field resulting in optimum imposed fields of around 50 T.

From the table in Fig. 5, a comparison of the burn-off (i.e., no alpha deposition) simulation equivalents shows only a modest 60% increase in fusion yields due to the imposed field because here fusion alphas are irrelevant to the hydrodynamics and only suppression of electron heat conduction is contributing. We note that if fusion alpha particles were irrelevant in a stagnated implosion and only electron heat transport contributing, then maximum initial fields of only ∼10 T would be required to fully suppress transverse electron heat flux and maximize performance because electrons are fully magnetized at this field (i.e., have Hall parameter, ωceτei≫ 1). We have observed this behavior in separate simulations of imposed fields applied to DT gas capsules with diamond shells that act as pusher/ablators—i.e., capsules similar to the NIF “Symcaps” or “2D-ConA” targets that are used for initial tuning experiments3,4 but of higher convergence. Here, the contribution of fusion alpha particles to thermonuclear burn is negligible and fusion only results from PdV work heating of the shell on the gas. In these cases, we saw only at most a factor of ∼1.8 increase in the fusion yield at optimum fields of ∼5–10 T due only to the higher electron (and associated ion) temperature. This is consistent with results from the OMEGA laser in Ref. 30 discussed above, where similar enhanced yields were observed with 8 T fields applied to gas capsules.

Accordingly, the major contribution of imposed magnetic fields of around 50 T for enhancing fusion yields and ignition in presently performing NIF capsules would accrue from their potential to confine alpha particles potentially resulting in igniting hotspots for otherwise the same implosion dynamics.

In addition to the radiation-flux-driven asymmetries leading to low-mode shape and RKE perturbations that have been proposed as the dominant mechanism for yield degradations in the present NIF experiments, perturbing effects also been attributed to the capsule support tent and fill-tube.10,12 These are higher-mode Rayleigh-Taylor (RT) instabilities—for example, the fill-tube perturbation can present as a thin jet of higher-z material piercing the hotspot—so they were not included in the low-mode simulations above. However, in a magnetized implosion, the field lines are frozen into the shell. So, for shorter wavelength RT modes to grow during late time capsule deceleration and penetrate the hotspot boundary, they would have to cause a sharp curvature in their entrained flux. This requires increased field line bending energy and, therefore, may be stabilizing to such interchange-like modes.

To examine this potential, in Fig. 6 we show the effect of applying two single-mode (cosine) perturbations of mode number 2π/λ = 12 and 24 but of large initial 5 μm amplitude to the inner DT-ice surface of the capsule in Fig. 1 for imposed fields of B0 = 0, 20, and 70 T. These simulations have no other applied perturbations (i.e., no shape or RKE perturbations). The plots show shell density contours at stagnation. The frozen-in field lines are evident for the magnetized cases as is the evident damping of the non-linear RT growth. These are relatively low-mode numbers so we might expect increased stabilization for higher modes due to increased bending energy required. Accordingly, in future magnetized implosions it might be effective if capsule supports and fill-tubes are positioned so that their perturbations are radially directed perpendicular to the field.

Of course, these 2D simulations require shell perturbations to be axisymmetric. Thus, 3D simulations are needed to fully quantify the extent to which compressed fields might suppress higher-mode RT growth.

In summary, our simulations suggest that imposed seed magnetic fields of tens of Tesla—with optimum values around 50 T—that compress to greater than 104 T (100 MGauss) under implosion may reduce hotspot conditions required for ignition and propagating burn in indirect-drive ICF. This accrues through range reduction of fusion alpha particles, suppression of electron heat conduction, and potential stabilization of higher-mode hydrodynamic instabilities. This may permit recovery of ignition, or at least increased fusion yields in presently performing indirect-drive NIF targets that today are sub-marginal for thermonuclear burn through adverse hydrodynamic conditions at stagnation. In particular, we see the most potential for capsule implosions that ordinarily, without field, are on the foot of the ignition cliff and where alpha particle heating is only approximately doubling the fusion yield over that attainable from shell PdV work heating alone. This is the regime where our best performing NIF experimental shots lie today and thus, they may be appropriate candidates for fusion yield enhancement and perhaps ignition under imposed fields.

A caveat is that our findings are contingent on a 2D interpretation of the physics. Thus, an interesting question is whether the 3D hydrodynamics of the RKE plasma structure around stagnation might form orthogonal toroidal field components. That is, would the intense field gradients resulting from tangled, wound-up field lines promote reconnection of otherwise frozen-in axial flux to form closed field lines in the hotpot?31 If so this might further augment ignition prospects because electron and alpha charged particle transport is then suppressed in all directions. But this further enhanced ignition probability may come at the expense of high fusion yield because alpha particles might then be inhibited from full burn propagation into the cold compressed fuel. Such rich physics involves complex 3D resistive MHD and is likely beyond the scope of any present radiation-hydrodynamics code to accommodate with confidence. Accordingly, proof-of-principle experiments are now essential to examine the potential of imposed magnetic fields to indirect-drive ICF ignition and burn.

We are pleased to acknowledge informative discussions with J. H. Hammer, H. F. Robey, J. N. Nuckolls, and J. D. Moody of Lawrence Livermore National Laboratory. This work was performed under the auspices of U.S. DOE by Lawrence Livermore National Laboratory (LLNL) under Contract No. DE-AC52-07NA27344 and supported under LLNL Laboratory Directed Research and Development 14-ERD-028.

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