Cone-Guided Fast Ignition with no Imposed Magnetic Fields

Simulations of ignition-scale fast ignition targets have been performed with the new integrated Zuma-Hydra PIC-hydrodynamic capability. We consider an idealized spherical DT fuel assembly with a carbon cone, and an artificially-collimated fast electron source. We study the role of E and B fields and the fast electron energy spectrum. For mono-energetic 1.5 MeV fast electrons, without E and B fields, the energy needed for ignition is E_f^{ig} = 30 kJ. This is about 3.5x the minimal deposited ignition energy of 8.7 kJ for our fuel density of 450 g/cm^3. Including E and B fields with the resistive Ohm's law E = \eta J_b gives E_f^{ig} = 20 kJ, while using the full Ohm's law gives E_f^{ig}>40 kJ. This is due to magnetic self-guiding in the former case, and \nabla n \times \nabla T magnetic fields in the latter. Using a realistic, quasi two-temperature energy spectrum derived from PIC laser-plasma simulations increases E_f^{ig} to (102, 81, 162) kJ for (no E/B, E = \eta J_b, full Ohm's law). This stems from the electrons being too energetic to fully stop in the optimal hot spot depth.


Integrated PIC-hydrodynamic modelling with Zuma-Hydra
We summarize Zuma and the Zuma-Hydra coupling here (see [3] for details). Zuma treats the fast electrons by standard relativistic-PIC methods, and injects them according to prescribed distributions. They undergo energy loss and angular scatter following [5] and [6]. The background plasma is treated as a collisional fluid with fixed ions. We eliminate physics on fast (Langmuir or light wave) time scales via reduced equations of motion. In particular, the background current is given by Ampère's law without displacement current: J b = -J f + µ 0 -1 ∇×B where (b, f) denotes (background, fast) current. The magnetic field is evolved by Faraday's law ∂B/∂t = -∇×E. a strozzi2@llnl.gov The electric field comes from Ohm's law, the massless force law for background electrons: (E C , E NC ) are (collisional, collisionless) terms. η and β are tensors due to nonzero Hall parameter ω ce τ e . We follow "notation II" of Ref. 7 and use their approximate values for η and β. We distinguish between the full Ohm's law Eq. (1) and the resistive Ohm's law E = ηJ b with scalar (unmagnetized) η. η is found following Lee and More [8] with Desjarlais' improvements [9], and the charge state from Ref. 9's modified Thomas-Fermi approach. At the start of a Zuma-Hydra coupling step, Hydra transfers plasma conditions to Zuma. Zuma advances for many of its own timesteps, accumulating energy and momentum deposition rates in each zone. Hydra then runs for many timesteps, using the deposition from Zuma. The cycle then repeats. Both codes were run on Eulerian cylindrical RZ grids. Fast electrons are injected into Zuma from a distribution factorized into an energy times an angle spectrum. The energy spectrum is either mono-energetic or the PIC-based form from Ref. 3: ε = E/T p where E is the kinetic energy and T p /m e c 2 =[1+a 0 2 ] 1/2 -1 is the ponderomotive temperature in the nominal laser intensity, with a 0 2 =I 0 λ 0 2 /1370 PW cm -2 µm 2 . The form Eq. (2) fits well the results of a 3D, 360 fs laser-plasma full-PIC simulation with the PSC code [10]. The average ε in this spectrum is <ε> = 1.02. We scale the spectrum ponderomotively as we vary intensity, though this choice has not yet been validated by PIC runs. The energy spectrum is a key aspect of fast ignition, and is not fully understood. Other work reports a cooler, more favourable scaling [11]. Our PIC data clearly shows two distinct temperatures, which may be important in interpreting experimental data. More details are in Ref. 3. We use λ 0 = 527 nm, and 52% laser to fast electron power conversion. The fraction of light absorbed by matter in the PSC run was higher, but not all was into fast electrons. Longer PSC runs show a third, high-temperature component with poor fuel coupling. The solid angle spectrum is dN/dΩ = exp[-(θ/Δθ) 4 ] with θ the velocity-space polar angle relative to the z axis. Δθ=10 o (average θ =6.9 o ) for our artificially collimated source; the PSC results were matched by the divergent Δθ=90 o (<θ> =52 o ).

Results with a mono-energetic source
We model an idealized DT fuel assembly with a carbon cone and initial 100 eV temperature, shown in Figure 1. Our peak density ρ=450 g/cm 3 and ρr=3.0 g/cm 2 give an ideal burn-up fraction of 1/3 and fusion yield of 64.4 MJ. Atzeni [12] has found from 2D hydrodynamic simulations with prescribed heating (not self-consistent fast electron dynamics) that the minimal heat deposited in the hot spot needed for ignition is 140 kJ / (ρ/100 g/cm 3 ) 1.85 , which for our ρ is 8.7 kJ. The fast electron source in Zuma is injected 20 µm to the left of the cone tip at z = 0.
We first find the ignition energy E f ig for an artificially collimated (Δθ=10 o ), 1.5 MeV monoenergetic source. The optimal DT hot-spot depth [12] is ρΔz = 1.2 g/cm 2 , which removes at most 1.3 MeV from a fast electron. 1.5 MeV electrons mostly stop in this depth. The time pulse is a 19 ps flattop, and intensity profile I(r) = I 0f exp[-(r/r spot ) 8 ] with r spot =18 µm. Neither of these is optimized for this case, but this r spot gave the smallest E f ig for Δθ=10 o , the PIC-based energy spectrum, and the full Ohm's law [3]. Figure 1 displays the yield vs. fast electron energy E f . For no E and B fields, ignition occurs for E f ig = 30.4 kJ. This is 3.5x Atzeni's minimum, due to including a cone, finite beam divergence and angular scattering, and un-optimized time pulse and spot shape. Including fields and the resistive Ohm's law E = ηJ b reduces E f ig to 20.3 kJ. This is due to magnetic selfguiding by the fast electrons, as shown in the current plots in Figure 2 and magnetic field plots in Figure 3. However, using the full Ohm's law increases E f ig to > 40 kJ (numerical problems occurred for larger E f ). Figure 2 shows less fast current reaching the fuel for this case than the other two. The effect of E and B fields is similar to Ref. 13: including just the resistive E produces selfguiding of collimated fast electrons, while the full Ohm's law reduces the coupling. In Ref. 13 the full Ohm's law gives better coupling than the no-field case, but we see the opposite. The relative ordering is thus not universal. Our scenario differs from Ref. 13 in the laser pulse, spot shape, and plasma profiles. We have not identified which aspect leads to the different ordering, or simulated their scenario with our codes. The reduced coupling is likely due to the ∇n eb ×∇T eb magnetic field arising from E∝∇p eb . A spherically-symmetric n eb and an azimuthally-symmetric T eb give ∂B/∂t = -(en eb ) -1 ∇n eb ×∇T eb → ∂B ϕ /∂t = -(en eb R) -1 (dn eb /dR)( ∂T eb /∂θ) .
φ is the azimuthal angle, and θ the polar angle with respect to positive z. Between the cone tip and the fuel, dn eb /dR < 0 while ∂T eb /∂θ > 0 once heating by fast electrons begins. We therefore generate a positive B ϕ , which exerts an outward radial force on fast electrons with v z >0. Such a B ϕ is seen in Figure 3 for z = 50-70 µm and r < 20 µm. We did not toggle specific non-resistive terms in Ohm's law, e.g. tensor vs. scalar transport coefficients due to Hall parameter ω ce τ e > 0. For the Full Ohm's law run with 1.5 MeV source spectrum and E f = 20 kJ at time 7 ps, ω ce τ e was >1 in most of the transport region and peaked at 13.4. This indicates magnetized transport coefficients should be used.

Results with PIC-based source energy spectrum
Zuma-Hydra runs were performed with the PIC-based energy spectrum Eq. (2), instead of a monoenergetic one (but the same Δθ=10 o ). Figure 1 reveals a several-fold increase in E f ig over the monoenergetic cases. The total fast-electron energy is E f = ATI 0f , where A and T are the source area and duration. For our time pulse and spot shape, regardless of energy spectrum, E f [kJ] = I 0f / 5.77×10 18 W/cm 2 . For our PIC-based spectrum, we have laser intensity I 0 = = I 0f /0. MeV spectrum to ignite. This is a substantial increase but less than the factor 26.3 2 /(2.78*30.4) = 8.2 our naïve stopping model requires to deposit the same 26.3 kJ. This may stem from increased stopping of the cold part of the PIC source (our estimate assumes all electrons have the average energy), or more angular scatter in the 1.5 MeV case.
Our work suggests 300 kJ of short-pulse laser is adequate for fast ignition -if the divergence problem is solved. This can be reduced by approaches to cool the source spectrum: LPI that produces an intrinsically cooler spectrum (e.g. shorter laser wavelength), targets with larger spot sizes or fuel dimensions (lower density), and longer laser pulses; or by enhanced electron stopping (micro-instabilities, orbit roll-up by magnetic fields).
This work was performed by LLNL under U.S. DoE Contract DE-AC52-07NA27344, and partly supported by LDRD project 11-SI-002 and the Office of Fusion Energy Sciences.