Data-driven assessment of magnetic charged particle confinement parameter scaling in Magnetized Liner Inertial Fusion experiments on Z William E. Lewis, Owen M. Mannion, D. E. Ruiz, Christopher A. Jennings, Patrick F. Knapp, Matthew R. Gomez, Adam J. Harvey-Thompson, Matthew R. Weis, Stephen A. Slutz, David J. Ampleford, and Kristian Beckwith Sandia National Laboratories, Albuquerque, New Mexico 87185 USA (Dated: 28 March 2023) In magneto-inertial fusion, the ratio of the characteristic fuel length perpendicular to the applied magnetic field R to the α-particle Larmor radius rL,α is a critical parameter setting the scale of electron thermal-conduction loss and charged burn-product confinement. Using a previously developed deep-learning-based Bayesian inference tool, we obtain the magnetic-field fuel-radius product BR ∝ R/rL,α from an ensemble of 16 Magnetized Liner Inertial Fusion (MagLIF) experiments. Observations of the trends in BR are consistent with relative tradeoffs between compression and flux loss as well as the impact of mix from 1D resistive radiation magneto-hydrodynamics simulations in all but two experiments, for which 3D effects are hypothesized to play a significant role. Finally, we explain the relationship between BR and the generalized Lawson parameter χ. Our results indicate the ability to improve performance in MagLIF through careful tuning of experimental inputs, while also highlighting key risks from mix and 3D effects that must be mitigated in scaling MagLIF to higher currents with a next-generation driver. Magneto inertial fusion (MIF) refers to inertial confinement fusion (ICF) schemes that rely on the application of an external magnetic field to reduce thermal-conduction losses and aid in charged fusion-product trapping. Appealing to the generalized Lawson parameter χ we can understand the tradeoffs made in MIF in attempting to achieve conditions necessary for self-heating (χ > 1) as compared to ICF and magnetic confinement fusion (MCF). The parameter χ is given by χ = (2/3) * (fD * fT / (1 + <Z>)^2) * (Eα * τ / VHS) ∫_V PHS * (<σv>DT / T^2) dV, (1) where Eα = 3.5 MeV is the energy of a DT fusion α-particle, fD(T) is the fraction of deuterium (tritium) in the fuel, <σv>DT is the (ion temperature dependent) DT fusion reactivity, PHS is the hotspot pressure, T is the fuel temperature, τ is the relevant confinement time (e.g. energy confinement time for MCF or burn duration for ICF/MIF), VHS is the hot spot fuel volume, and <Z> is the average ionization. Note that the criterion χ > 1 is a statement that energy deposited by the 3.5 MeV α-particles created by DT reactions exceeds the internal energy of the fuel. As the 3.5 MeV α's predominantly lose energy to the hot electrons, Te is the relevant fuel temperature, while the reactivity <σv>DT depends explicitly on the ion temperature. It is common practice, however, to make the approximation T = Te ≈ Ti in ICF/MIF as Ti can be readily obtained from neutron energy spectra. This approximation is supported by comparison of measurements of electron temperature from x-ray data with ion temperatures estimate from neutron time-of-flight (nToF) peak shape for the fuel plasma in the Magnetized Liner Inertial Fusion (MagLIF) MIF platform discussed in this manuscript. Furthermore, electron-ion collision time estimates are much smaller than the burn width (τei ≪ τburn) for both MagLIF and spherical ICF implosions at the National Ignition Facility (NIF). This indicates sufficient time for the two to equilibrate so that the assumption Te ≈ Ti may also be appropriate for ICF implosions. Table I shows estimates of key quantities in the generalized Lawson criterion for ICF capsule implosions on NIF the Hybrid-E (HyE) campaign, the MIF platform MagLIF at Sandia National Laboratories Z Pulsed Power Facility, the MCF Joint European Torus (JET) tokamak, and MCF Wendelstein 7-X (W7-X) stellarator at the Max-Planck Institute for Plasma Physics. TABLE I. Estimates of key quantities for generalized Lawson parameter for ICF, MIF and MCF platforms described in the text: - NIF HyE: τ = 1.2 × 10^-10 s, PHS = 3.1 × 10^11 bar, VHS = 3 × 10^-13 m^3, T = 5 keV - MagLIF: τ = 2 × 10^-9 s, PHS = 1 × 10^9 bar, VHS = 1 × 10^-10 m^3, T = 3 keV - JET: τ = 5.1 × 10^-1 s, PHS = 1 bar, VHS = 100 m^3, T = 20 keV - W7-X: τ = 2.2 × 10^-1 s, PHS = 0.5 bar, VHS = 25 m^3, T = 3.5 keV Critically, Eq. 1 is derived assuming local deposition of α-particles. In reality, fusion α's have finite range to slow down in the hot-spot. An important modification would therefore include the fraction ηα of α's that redeposit their energy into the fuel. The quantity ηα will, in general, depend in a complicated way on the fuel geometry, but will clearly be an increasing function of the ratio lfuel/lα, where lfuel is a characteristic length scale of the fuel and lα is the characteristic α-particle stopping length. In capsule implosions on the NIF and OMEGA, ρlα ≈ 0.025 T^5/4 / (1 + 0.0082 T^5/4) is the α-particle range (in g cm^-2) for T in keV. The condition lfuel/lα > 1 can be recast as a requirement on the temperature dependent ratio of fuel areal density to the α-particle range, (ρR)/(ρlα) > 1. At T = 10 keV, this corresponds to ρlfuel ∝ ρR ≳ 0.4 g cm^-2, at which point the hot-spot areal density exceeds the α-particle range so that a significant fraction of the α-particles energy is redeposited. For magnetized fuel configurations, on the other hand, there are two relevant length scales for the α-particles. First, there is the α-particle Larmor radius rL,α ∝ 1/B, where B is the strength of the magnetic field providing confinement of the α-particles. This should be compared to a length scale R transverse to the magnetic field, so that R/rL,α ∝ BR is a parameter characterizing how well confined the α's are in the transverse direction. In MagLIF, B ~ O(1 - 10 kT) with lfuel ~ R ≈ 50 µm, while for MCF, B ~ 1 - 10 T with lfuel ~ 1 m, showing that this MIF concept makes a tradeoff between field and fuel dimensions in determining the critical fast charged-particle confinement parameter BR. Magnetic flux compression and assessment of BR as a function of a variety of experimental inputs in MagLIF will be the topic of this work. Provided the α-particles are sufficiently confined along the transverse extent, there is a length scale Z characterizing the extent of fuel along the magnetic field direction which should be compared to the α-stopping length lα. Doing this, we see that for a sufficiently strong magnetic field, ρZ rather than ρR is the relevant areal density for setting the fraction of α-particles stopped in the fuel. In MagLIF ρZ ~ 0.1 - 1 g cm^-2 with Zfuel ~ 1 cm. For perfectly closed particle orbits in a MCF device, Z -> ∞. However, the fuel pressure is significantly lower so that the (energy) confinement time must increase to result in similar performance as shown by Tbl. I. In addition to aiding in confinement of charged fusion-products, the inclusion of a magnetic field additionally reduces electron thermal-conduction losses. This can therefore alter the power balance between α-heating and energy-loss mechanisms. This allows for the possibility of achieving self-heating across a much wider range of fuel areal-densities as shown by the Lindl-Widner diagram for a cylindrical DT plasma (neglecting energetic end-losses) in Fig. 1. As seen in Fig. 1, for BR ~ 0.35 - 0.5 MG·cm, the condition for large enough ρR required to achieve self-heating is significantly relaxed, eventually becoming nearly independent of ρR for high enough T and BR ≳ 0.6 MG·cm. In Fig. 2 we indicate the three key stages of pre-magnetization, laser preheat, and magnetically driven implosion in a MagLIF experiment that lead to fusion relevant conditions. MagLIF uses an approximately 0.75 - 1 cm tall beryllium cylindrical tube, or liner, with initial inner wall radius Rin,0 on the scale of 2 - 3 mm. The liner aspect ratio is defined by AR = Rout,0 / (Rout,0 - Rin,0), where Rout,0 is the initial radius of the outer wall of the cylinder. In this work, experiments with AR ∈ {6, 9, 11} will be assessed. The target is filled with gaseous deuterium (DD) fuel at O(1 mg cm^-3). External field coils are used to apply an O(10 - 20 T) axially oriented magnetic field, which serve to suppress radial thermal-conduction losses. A multi-kJ laser couples 0.5-2 kJ of preheat energy into the fuel, setting the fuel on a higher adiabat to reduce the compression needed to achieve fusion relevant conditions, while also ionizing the fuel, locking the magnetic field into the hot electrons allowing for effective compression of magnetic flux in the final stage. Finally, an O(20 MA) ~ 100 ns rise time current pulse is delivered by the Z Machine, which provides a magnetic pressure to the outside of the liner, causing the target to implode radially inward. When the thermal pressure of the fuel exceeds the magnetic pressure driving the implosion, the liner begins to decelerate and eventually stagnate, with the fuel achieving fusion relevant PHSτ (~ 1 GBar ns) and temperature (~ 3 keV) as indicated in Tbl. I. Flux compressed axial magnetic field strengths are on the order of Bz ~ O(1 - 10 kT), sufficient to trap charged fusion-products. For pure DD fuel, DD reactions are the primary interaction, while DT fusion can only occur as a secondary reaction whereby a 1.01 MeV triton interacts with a background deuteron in the fuel: Primary: D + D -> n (2.45 MeV) + 3He, or p + T (1.01 MeV) Secondary: D + T -> n (14.1 MeV) + α where energies are reported in the center of momentum frame. However, the birth Larmor radius for the 1.01 MeV tritons is nearly identical to that of 3.5 MeV α-particles, with rL,α ≈ 1.07 rL,t and stopping length within a factor of about two, lα ≈ 0.5 lt. As a result, the dynamics of magnetized 1.01 MeV tritons in DD fuel is expected to be a good surrogate for α-particles in a hydrodynamically equivalent DT fuel. In order to understand the effectiveness of flux compression and the associated magnetic confinement parameter BR from experiments, a constraining measurement must be made. The shape of the secondary DT neutron energy spectrum emitted along the axial and radial directions of the experiment, as well as the ratio of the secondary DT to primary DD neutron yields, are all sensitive to magnetization of the plasma and can be used to diagnose BR in an experiment. BR(t) = Φz(t) / (π R(t)) = (2 / R) ∫_0^{R(t)} r Bz(r, t) dr, (4) where Φz is the axial flux for the magnetic field profile Bz(r, t) of radius R defined to be the interface between the deuterium fuel and inner liner wall. When quoting simulation results for BR, we report the burn-averaged value given by: (BR)_f^sim = (1 / YDD) ∫_t (dYDD/dt) [ (2 / R) ∫_0^{R(t)} r Bz(r, t) dr ] dt, (5) The experiments analyzed in this manuscript include simultaneous changes to nine experiment input conditions. Seven of these inputs are indicated in Tbl. II including preheat energy deposited, fill density, target height, initial liner inner radius, liner aspect ratio, dielectric coating, and applied magnetic field strength. TABLE II. Experimental input conditions for experiments analyzed in the text: - z3040: AR = 6, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 0.775 ± 0.108 kJ, epreheat = 6.71 ± 0.94 kJ/mg, Rin,0 = 2.325 mm - z2977: AR = 6, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 0.868 ± 0.084 kJ, epreheat = 7.52 ± 0.73 kJ/mg, Rin,0 = 2.325 mm - z2839: AR = 6, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 1.069 ± 0.091 kJ, epreheat = 9.26 ± 0.79 kJ/mg, Rin,0 = 2.325 mm - z3143: AR = 6, Bz,0 = 10 T, fill density = 1.03 mg/cm3, Epreheat = 0.888 ± 0.108 kJ, epreheat = 5.08 ± 0.62 kJ/mg, Rin,0 = 2.325 mm - z2851: AR = 6, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 0.752 ± 0.080 kJ, epreheat = 6.51 ± 0.69 kJ/mg, Rin,0 = 2.325 mm - z2591: AR = 6, Bz,0 = 10 T, fill density = 0.65 mg/cm3, Epreheat = 0.731 ± 0.076 kJ, epreheat = 8.83 ± 0.91 kJ/mg, Rin,0 = 2.325 mm - z2966: AR = c6, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 1.006 ± 0.087 kJ, epreheat = 8.71 ± 0.75 kJ/mg, Rin,0 = 2.325 mm - z3180: AR = c9, Bz,0 = 10 T, fill density = 1.03 mg/cm3, Epreheat = 0.915 ± 0.108 kJ, epreheat = 5.23 ± 0.62 kJ/mg, Rin,0 = 2.325 mm - z3236: AR = c9, Bz,0 = 10 T, fill density = 1.03 mg/cm3, Epreheat = 1.350 ± 0.114 kJ, epreheat = 7.72 ± 0.65 kJ/mg, Rin,0 = 2.325 mm - z3019: AR = c9, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 0.933 ± 0.085 kJ, epreheat = 8.08 ± 0.74 kJ/mg, Rin,0 = 2.325 mm - z3075: AR = c9, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 0.935 ± 0.085 kJ, epreheat = 8.10 ± 0.74 kJ/mg, Rin,0 = 2.325 mm - z3135: AR = c9, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 0.999 ± 0.087 kJ, epreheat = 8.65 ± 0.75 kJ/mg, Rin,0 = 2.325 mm - z3018: AR = 9, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 0.945 ± 0.085 kJ, epreheat = 8.18 ± 0.74 kJ/mg, Rin,0 = 2.325 mm - z3240: AR = c9, Bz,0 = 15 T, fill density = 0.68 mg/cm3, Epreheat = 0.897 ± 0.084 kJ, epreheat = 7.54 ± 0.710 kJ/mg, Rin,0 = 2.325 mm - z3179: AR = c9, Bz,0 = 15 T, fill density = 0.68 mg/cm3, Epreheat = 0.946 ± 0.085 kJ, epreheat = 8.19 ± 0.740 kJ/mg, Rin,0 = 2.325 mm - z3076: AR = c11, Bz,0 = 10 T, fill density = 0.68 mg/cm3, Epreheat = 0.851 ± 0.108 kJ, epreheat = 4.01 ± 0.51 kJ/mg, Rin,0 = 3.155 mm TABLE III. Summary of stagnation parameters for the experiments considered in this manuscript: - z3040: CR = 31 (+5/-3), Tion = 2.18 ± 0.06 keV, PHS = 0.953 ± 0.054 Gbar, χ = 0.0193 ± 0.0013, BR = 0.233 (+0.016/-0.013) MG·cm - z2977: CR = 30 (+4/-4), Tion = 2.37 ± 0.09 keV, PHS = 0.704 ± 0.055 Gbar, χ = 0.0166 ± 0.0023, BR = 0.196 (+0.011/-0.010) MG·cm - z2839: CR = 27 (+4/-4), Tion = 2.01 ± 0.11 keV, PHS = 0.758 ± 0.067 Gbar, χ = 0.0123 ± 0.0019, BR = 0.215 (+0.013/-0.012) MG·cm - z3143: CR = 24 (+2/-1), Tion = 2.21 ± 0.09 keV, PHS = 0.617 ± 0.070 Gbar, χ = 0.0121 ± 0.0014, BR = 0.371 (+0.052/-0.044) MG·cm - z2851: CR = 31 (+6/-5), Tion = 1.44 ± 0.09 keV, PHS = 0.808 ± 0.095 Gbar, χ = 0.0029 ± 0.0005, BR = 0.460 (+0.060/-0.060) MG·cm - z2591: CR = N/A, Tion = N/A, PHS = N/A, χ = N/A, BR = 0.250 (+0.052/-0.020) MG·cm - z2966: CR = 36 (+1/-4), Tion = 1.82 ± 0.08 keV, PHS = 0.655 ± 0.061 Gbar, χ = 0.0074 ± 0.0010, BR = 0.495 (+0.030/-0.030) MG·cm - z3180: CR = 34 (+4/-6), Tion = 2.06 ± 0.09 keV, PHS = 1.694 ± 0.154 Gbar, χ = 0.0128 ± 0.0018, BR = 0.439 (+0.042/-0.040) MG·cm - z3236: CR = 28 (+9/-2), Tion = 2.89 ± 0.10 keV, PHS = 1.307 ± 0.070 Gbar, χ = 0.0624 ± 0.0058, BR = 0.311 (+0.041/-0.032) MG·cm - z3019: CR = 32 (+2/-5), Tion = 2.27 ± 0.09 keV, PHS = 1.071 ± 0.102 Gbar, χ = 0.0286 ± 0.0032, BR = 0.352 (+0.023/-0.025) MG·cm - z3075: CR = 25 (+3/-2), Tion = 2.10 ± 0.09 keV, PHS = 1.118 ± 0.121 Gbar, χ = 0.0189 ± 0.0025, BR = 0.420 (+0.038/-0.039) MG·cm - z3135: CR = 30 (+4/-5), Tion = 2.47 ± 0.09 keV, PHS = 0.846 ± 0.072 Gbar, χ = 0.0265 ± 0.0028, BR = 0.460 (+0.030/-0.031) MG·cm - z3018: CR = N/A, Tion = N/A, PHS = N/A, χ = N/A, BR = 0.490 (+0.046/-0.064) MG·cm - z3240: CR = 40 (+6/-3), Tion = 2.27 ± 0.09 keV, PHS = 0.789 ± 0.056 Gbar, χ = 0.0200 ± 0.0024, BR = 0.400 (+0.042/-0.037) MG·cm - z3179: CR = 34 (+6/-7), Tion = 3.31 ± 0.10 keV, PHS = 1.226 ± 0.072 Gbar, χ = 0.0843 ± 0.0100, BR = 0.470 (+0.040/-0.036) MG·cm - z3076: CR = 30 (+2/-2), Tion = 2.08 ± 0.11 keV, PHS = 0.475 ± 0.056 Gbar, χ = 0.0081 ± 0.0012, BR = 0.480 (+0.063/-0.048) MG·cm To understand the trends, we recast Eq. 4 as: BR(t) = CR(t) * Ψ(t) * (BR)0, (7) where CR(t) ≡ Rin,0 / Rin(t) is convergence ratio, Ψ(t) = Φ(t)/Φ0 is magnetic flux retention (0 ≤ Ψ ≤ 1), and (BR)0 is initial BR. Dimensionless parameters characterizing flux loss: Magnetic Reynolds number: Rm ≡ 0.01 * [BR(T cm)] * [R_dot(km/s)] / [T(keV)] Nernst parameter: Ne ≡ 200 * [T(keV)] / ([BR(T cm)] * [R_dot(km/s)]) * β_∧^uT(xe) Relative scaling predicted by simple 1D Kraken simulations appears to describe changes to BR between different experiment configurations quite well in most cases. For experiments exhibiting large positive ΔBR, mix appears to be a reasonable explanation for the observed enhancement in BR. As seen from the data, increasing preheat specific energy results in higher fuel adiabat and improved χ, but due to Nernst advection reducing flux retention, gains may saturate unless accompanied by increased magnetic field or higher aspect ratio targets. Key takeaways and future directions: 1. Comparison to 1D simulations: 1D simulations capture overall trends and relative scaling between target designs well, though 2D/3D effects like end-losses and MRTI feedthrough introduce discrepancies in high aspect ratio targets (e.g., cAR11 shot z3076) and misaligned preheat shots (z3240). 2. 1D inference model: Non-axisymmetric effects and 3D helical structures seen in MagLIF stagnation will be addressed in future work by post-processing 3D Kraken simulations. 3. Mix reduces thermal gradients and Nernst advection, leading to higher inferred BR but degraded Lawson parameter χ due to radiative cooling. 4. Scalings to next-generation drivers must account for the tradeoffs between preheat energy, Nernst flux loss, compression, and mix.