2. HYDRA MHD simulation method

We use the radiation-hydrodynamics code HYDRA to simulate the NIF hohlraum experiment N120321. This shot used a 4-shock, ‘low-foot’ laser pulse (shown in figure 2), a plastic ablator ( $\text{C}_{0.42}\text{H}_{0.57}$ plus small amounts of O impurity, and Si dopant to control X-ray preheat) with a DT ice layer, and a depleted uranium (DU) hohlraum with a thin $0.7~{\rm\mu}\text{m}$ inner gold coating. The hohlraum fill gas was $0.96~\text{mg}~\text{cm}^{-3}$ of He. The methodology is the standard one in use for hohlraum simulations at LLNL (Jones et al. Reference Jones, Cerjan, Marinak, Milovich, Robey, Springer, Benedetti, Bleuel, Bond and Bradley2012), entailing the ‘high-flux model’ with detailed configuration accounting (DCA) for non-LTE material properties, and an electron thermal flux limit of 0.15 times the free-streaming value (Rosen et al. Reference Rosen, Scott, Hinkel, Williams, Callahan, Town, Divol, Michel, Kruer and Suter2011). The runs use a 3-D mesh with one zone and periodic boundary conditions in azimuth, and are effectively cylindrical 2-D $(r-z)$ . The mesh is managed with an arbitrary Lagrangian–Eulerian (ALE) approach, designed to keep the mesh as Lagrangian as possible. We use the full, incident laser energy of 1.52 MJ, and neither remove measured backscatter nor degrade the laser power as is needed to match X-ray drive data in high gas fill targets (Jones et al. Reference Jones, Cerjan, Marinak, Milovich, Robey, Springer, Benedetti, Bleuel, Bond and Bradley2012). Our HYDRA simulations are therefore not proper post-shots, but address the role of an imposed field in ignition-scale designs, and provide relevant plasma conditions for hot electron studies.

A distinct aspect of the present work is the inclusion of an initial axial magnetic field $B_{z0}$ . HYDRA’s MHD package (Koning et al. Reference Koning, Kerbel and Marinak2006) was used to model $B_{z}$ and the resulting $B_{r}$ required by $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{B}=0$ . No azimuthal field is produced for our axisymmetric geometry and simple Ohm’s law. The MHD package uses a 3-D finite element method, with appropriate boundary conditions to be effectively axisymmetric. We used the Ohm’s law

where ${\it\eta}$ , $\boldsymbol{J}$ and $\boldsymbol{v}$ are scalar resistivity, net current and centre-of-mass velocity, respectively. The $B$ field is evolved via $\partial _{t}\boldsymbol{B}=-\boldsymbol{{\rm\nabla}}\times \boldsymbol{E}$ . The MHD package as used here affects the matter in three ways: (i) the $\boldsymbol{J}\times \boldsymbol{B}$ force, (ii) ${\it\eta}J^{2}$ (Ohmic) and other heating terms, and (iii) a tensor electron thermal conductivity:

The Righi–Leduc effect, with a separate conductivity along $\boldsymbol{B}\times \boldsymbol{{\rm\nabla}}T_{e}$ , is currently neglected ( $T_{e}$ is electron temperature). An artificial flux limit $f$ is imposed, as is typical in hohlraum simulations. Specifically, the component of the heat flux $\boldsymbol{q}_{e}$ along each logical index (not physical) coordinate $\hat{i}$ is limited: $\boldsymbol{q}_{e}\boldsymbol{\cdot }\hat{i} =\min [\hat{i} \boldsymbol{\cdot }{\it\kappa}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}T_{e},fq_{FS}]$ where $q_{FS}\equiv n_{e}T_{e}^{3/2}/m_{e}^{1/2}$ is the free-streaming heat flux.

The anisotropic heat conduction has the largest effect in our simulations. We expect $B\gtrsim 1~\text{T}$ to significantly reduce ${\it\kappa}_{\bot }$ below its unmagnetized value ${\it\kappa}_{\Vert }$ . The Hall parameter $H\equiv {\it\omega}_{ce}{\it\tau}_{ei}$ for thermal electrons is

For each ion species, $Z_{i}$ is the ionic (not nuclear) charge, $n_{i}=f_{i}n_{I}$ and $n_{I}=\sum _{i}n_{i}$ is the total ion number density. In practical units, $B_{0}\;(\text{T})=4.73(n_{e}/n_{cr})Z_{eff}/T_{e}~(\text{keV})^{3/2}$ with $n_{cr}=9.05\times 10^{21}~\text{cm}^{-3}$ the critical density for light of wavelength 351 nm. For $Z_{i}=2$ He at $n_{e}=0.1n_{cr}$ and $T_{e}=3~\text{keV}$ , typical of the underdense hohlraum fill, we find $\ln {\it\Lambda}_{ei}=7.9$ and $B_{0}=1.43~\text{T}$ . Given $B_{z0}=70~\text{T}$ , most of the underdense plasma fill should be strongly magnetized. ${\it\kappa}_{\bot }$ decreases with $H$ according to

with the $Z_{i}$ -dependent fitting coefficients $p_{j}$ given in Epperlein & Haines (Reference Epperlein and Haines1986). For $Z_{i}=2$ He, ${\it\kappa}_{\bot }/{\it\kappa}_{\Vert }=0.1$ for $H=1.6$ . ${\it\kappa}_{\Vert }$ is found either from the Lee & More (Reference Lee and More1984) formulation to include dense-plasma effects, or interpolation from an advanced table. The Epperlein and Haines results are used to include electron self-collisions ( $Z_{i}<\infty$ ) and dependence on $H$ .

3. MHD simulation results

The HYDRA runs of NIF shot N120321 with and without MHD are qualitatively similar. The principal difference is the MHD run has higher electron temperature in some regions and a wider channel of He fill gas at the equator. This leads to better inner beam propagation to the wall (less inverse-bremsstrahlung absorption in the low- $Z$ fill), and results in a less oblate capsule. Figure 3 shows the material regions and electron density for the two runs at 18 ns, during the rise to peak power. We use this time for ZUMA simulations of SRS hot electrons. A more detailed density plot is shown below in figure 7. Electron temperature $T_{e}$ with and without MHD at several times during peak power is displayed in figure 4. The field increases $T_{e}$ mainly in the laser-heated gold, such as the outer-beam ‘bubble’ $(r,z)\approx (0.2,-0.3)~\text{cm}$ . The He gas fill is also hotter with the field, but less so than the gold. The low density plasma outside the laser entrance hole (LEH) is cooler with the field, though the total energy in this region is small.

Figure 3.

Plasma conditions from HYDRA simulations of NIF shot N120321 at 18 ns, used in SRSPEAK ZUMA run series. The left $(z<0)$ half of the hohlraum is plotted, but the simulation included both halves. Top half ( $r>0$ ): without MHD, bottom half ( $r<0$ ): with MHD and initial axial magnetic field $B_{z0}=70~\text{T}$ . (a) Material region: DT, He, CH (two green regions), Au and DU indicate the materials deuterium–tritium, helium, plastic, gold and depleted uranium. (b) Free electron density in units of critical density for 351 nm light.

Figure 4.

(a,c,e) Electron temperature in keV from HYDRA MHD simulations shown in figure 3, at several times. (b,d,f) Temperature difference with MHD minus without MHD. Light blue contour marks region boundary for DT, CH and Au. Dashed contours are temperature differences of $\pm 0.5~\text{keV}$ .

Figure 5.

$|\boldsymbol{B}|$ at 18 ns from HYDRA MHD simulation shown in figure 3. The dark red colour in the capsule is the uncompressed field of $70~\text{T}$ . White curves are stream lines (integral curves) of the vector field $(B_{z},B_{r})$ . Light blue boxes and text indicate ZUMA hot electron sources for runs with and without MHD.

Figure 6.

Capsule shell density at 21.5 ns from HYDRA simulations shown in figure 3 without MHD (top half, $r>0$ ), and with MHD and $B_{z0}=70~\text{T}$ (bottom half, $r<0$ ). The shell is oblate without MHD, while it is close to round with MHD. This reflects the improved inner beam propagation with the field.

The $B$ field for the MHD run is plotted in figure 5. It roughly follows the MHD frozen-in law, and advects with the radial motion of the ablator and high- $Z$ wall. The compressed field approaches 300 T and continues to grow with time. The white streamline that just touches the capsule outer radius at $(z,r)=(0,0.1)$ roughly separates field lines that are still connected to the capsule ( $r<0.1~\text{cm}$ at $z=0$ ), from those that have advected with the ablated blowoff and no longer connect to the capsule.

The capsule density at the end of peak power is plotted in figure 6. The dense fuel is oblate without the field, but becomes close to round with it.

4. ZUMA hot electron simulation method

We propagate hot electrons through fixed plasma conditions from HYDRA using the hybrid-PIC code ZUMA in a ‘Monte-Carlo’ mode. We do not include forces from $E$ and $B$ fields, except when we include a specified (static) $B$ field. The background plasma properties are not updated. In other work, HYDRA and ZUMA have been coupled to run in tandem, and applied to fast ignition designs (Strozzi et al. Reference Strozzi, Tabak, Larson, Divol, Kemp, Bellei, Marinak and Key2012). The hot electrons undergo collisional energy loss off background electrons, and angular scattering off background electrons and ions. We neglect collisions among hot electrons, since their density is much less than the background species. We use the formulas in Robinson et al. (Reference Robinson, Strozzi, Davies, Gremillet, Honrubia, Johzaki, Kingham, Sherlock and Solodov2014) for a fast electron with $v\gg v_{Te}$ . The energy loss rate (stopping power) is given by

$C_{e}\equiv e^{4}/4{\rm\pi}{\it\epsilon}_{0}^{2}$ , the fast electron kinetic energy $E=m_{e}c^{2}{\it\epsilon}$ , ${\it\gamma}={\it\epsilon}+1$ is the Lorentz factor and $p={\it\gamma}m_{e}v$ . $n_{e}$ is the total (free plus bound) background electron density. $L_{d}$ given above is valid for energy loss off free electrons, or bound electrons for sufficiently high $E$ or $n_{e}$ (the ‘density effect’). This assumption may not be valid for all electrons. The angular scattering rate is $n_{e,f}$ is the free-electron density, $\langle Z^{2}\rangle =\sum _{i}f_{i}Z_{i}^{2}$ and we use the same ion species notation as after (2.3) except $Z_{i}$ is the nuclear (not ionic) charge. $l_{s}$ is a screening length, which we take to be the free-electron Debye length. For neutral atoms, it should be replaced by the atomic radius. In any event, we impose a minimum of 1 on $L_{d}$ , $L_{si}$ , and $L_{se}$ .

We run ZUMA in 2-D cylindrical geometry. ZUMA currently operates with constant (but different) grid spacing $\text{d}r$ and $\text{d}z$ . The HYDRA plasma conditions are interpolated onto a uniform mesh with $\text{d}r=\text{d}z=3~{\rm\mu}\text{m}$ using the OVERLINK package (Grandy Reference Grandy1999). This small spacing is needed to resolve small features, such as the gold wall and DT layer. The ZUMA time step is 1 fs, which is chosen to adequately resolve the dependence of $\text{d}E/\text{d}t$ on $E$ for small $E$ and high $n_{e}$ . ZUMA stops following electrons when $E<5.11~\text{keV}$ and locally deposits their kinetic energy. In this paper, ZUMA injects hot electrons from a distribution that is a product of an energy spectrum $\text{d}N/\text{d}E$ times a polar angle spectrum $\text{d}N/\text{d}{\it\Omega}$ . For a thermal spectrum with a ‘temperature’ $T_{h}$ , we use a relativistic Maxwell–Jüttner distribution:

$C$ is a normalization constant, and the two bracketed factors are absent for a non-relativistic Maxwellian.

5. Mono-energetic electron propagation through capsule at peak power: CAPTEST series

This section considers the propagation of electrons directly incident on the capsule during peak laser power, as a function of electron energy. We call this the CAPTEST series of ZUMA runs, and stress this source is not realistic for LPI-generated hot electrons. Rather, our purpose is to understand where electrons that reach the capsule deposit their energy, and which energies pose the greatest preheat risk. We use plasma conditions from our HYDRA simulation of NIF shot N120321 with no MHD at time 18 ns. The same conditions are used in the SRS-relevant SRSPEAK series discussed below in § 7. The time 18 ns is during the rise to peak power (see figure 2) and has significant inner-beam SRS. An analogous time in shock-timing (‘keyhole’) shots has been identified as possibly having a large hot-electron preheat effect (Robey et al. Reference Robey, Celliers, Moody, Sater, Parham, Kozioziemski, Dylla-Spears, Ross, LePape and Ralph2014).

Figure 7.

Electron density (free plus bound) at 18 ns in HYDRA simulation of NIF shot N120321 with no MHD (discussed in § 3), used for CAPTEST series of ZUMA runs. (b) Zooms on capsule. Material regions are separated by solid lines and labelled with text as in figure 3. The ZUMA hot electron source is indicated.

Figure 8.

Energy deposited by 175 keV mono-energetic hot electron source, without (a) and with (b) angular scattering, for CAPTEST ZUMA run series.

Figure 7 shows the total (free plus atomically bound) electron density in the HYDRA simulation. Mono-energetic hot electrons are injected in a cylinder of radius $500~{\rm\mu}\text{m}$ at $z=-0.1~\text{cm}$ , with an initial velocity in the $z$ direction. The hot electrons experience energy loss and (in some runs) angular scatter, but no forces from $E$ or $B$ fields. The resulting energy deposited per volume, zoomed on the capsule, is plotted in figure 8. The case with angular scattering shows large spreading of the hot electrons in the dense CH (plastic) ablator. Since the absolute number of hot electrons introduced is arbitrary, we express the deposition as energy density per injected hot electron energy.

Figure 9.

Fraction of injected hot electron energy deposited in different regions, or escaped to boundaries, for CAPTEST ZUMA run series. Plot on (a,b) is (without, with) angular scattering.

Figure 10.

Solid curves: ${\it\phi}_{DT}=$ fraction of injected hot-electron energy deposited in DT, with (black) and without (red) angular scatter, for CAPTEST ZUMA run series. Closed circles are ZUMA simulation points. Dashed curves: power-law fits to $E\geqslant 200~\text{keV}$ points. Thin blue curve is approximate form explained in text.

We plot the fraction of injected energy that is deposited in various regions or escapes to the boundaries in figure 9. The deposition in DT (essentially the dense fuel layer, not the less-dense proto-hotspot) is shown in figure 10. All electrons with $E<125~\text{keV}$ stop in the CH ablator. This sets a minimum energy hot electrons must have when they reach the capsule (at the time 18 ns) to reach the DT layer. Above this energy, the fraction deposited in DT increases, until the hot electrons have enough energy to not stop in the DT. With no angular scatter (figure 9 a), hot electrons eventually cross the capsule, and exit the problem through the LEH. Angular scatter lowers and spreads out the peak in coupling to DT. It also causes some hot electrons to reach the Au/U hohlraum wall. In both cases, the deposition in the He hohlraum fill gas is negligible.

A simple model illustrates the basic features of figure 10, especially for no angular scatter. Imagine a hot electron starting at position $z_{0}$ in the CH ablator, with initial energy $E_{0}$ and $v_{z}>0$ . We use a 1-D slab geometry with CH from $z_{0}$ to $z_{1}$ , DT from $z_{1}$ to $z_{2}$ and CH for $z>z_{2}$ . We seek the fraction of initial energy deposited in DT, ${\it\phi}_{DT}=E_{DT}/E_{0}$ , where $E_{DT}$ is the energy deposited in DT. The hot electron loses energy as it moves to increasing $z$ according to $\text{d}E/\text{d}z=-f/2E$ , where $f$ is a constant and we include only the leading-order dependence of stopping power on energy, for $E\ll m_{e}c^{2}$ . Integrating from $z_{a}$ to $z_{b}$ gives $E_{b}^{2}=E_{a}^{2}-f(z_{b}-z_{a})$ . An electron with $E_{0}<E_{01}$ fully stops in the CH with $z<z_{1}$ , one with $E_{0}>E_{02}$ crosses the DT, i.e. stops at $z>z_{2}$ and one with $E_{01}<E_{0}<E_{02}$ stops in the DT layer, i.e.  $z_{1}<z<z_{2}$ . A straightforward calculation gives

$E_{01}^{2}=f_{CH}(z_{1}-z_{0})$ and $E_{02}^{2}=E_{01}^{2}+f_{DT}(z_{2}-z_{1})$ . For $E_{0}=E_{01}+{\it\delta}E$ with ${\it\delta}E$ small, ${\it\phi}_{DT}\approx (2{\it\delta}E/E_{01})^{1/2}$ , and for $E_{0}\gg E_{02}$ , ${\it\phi}_{DT}\approx f_{DT}(z_{2}-z_{1})/2E_{0}^{2}$ . The simple model for ${\it\phi}_{DT}$ , with $E_{01}=135~\text{keV}$ and $E_{02}=155~\text{keV}$ , is plotted as the solid blue curve in figure 10. The model is close to the red, no-scattering result, though the capsule curvature smears the peak compared to the simple model. Figure 10 also includes least-square power-law fits to the $E_{0}\geqslant 200~\text{keV}$ results: ${\it\phi}_{DT}=(E_{0}/52.3~\text{keV})^{-1.79}$ without angular scatter, and ${\it\phi}_{DT}=(E_{0}/62.7~\text{keV})^{-1.75}$ with scatter. These are both close to the $E_{0}^{-2}$ scaling of our simple model for $E_{0}\gg E_{02}$ .

Figure 11.

Coupling to DT of a Maxwell–Jüttner hot electron source. Blue: $E\ast \text{d}N/\text{d}E$ for Jüttner source with $T_{h}=50~\text{keV}$ . Black: ${\it\phi}_{DT}=$ energy fraction deposited in DT, with angular scatter (solid black curve from figure 10). Red: product of black and blue curves.

Figure 12.

Black (left $y$ -axis): injected hot electron energy fraction coupled to DT, for Maxwell–Jüttner source with temperature $T_{hot}$ . Red (right $y$ -axis): hot electron energy with peak coupling to DT (i.e. peak of red curve from figure 11).

Figure 13.

Material region (a) and laser intensity summed over all beams (b) at 1 ns from HYDRA simulation with no MHD of NIF shot N120321, used for PICKET series of ZUMA runs. Hot-electron source is indicated as white box and is placed only in the $z<0$ half of the two-sided ( $z<0$ and $z>0$ ) ZUMA domain.

We apply our mono-energetic results to a thermal spectrum in figures 11 and 12, and find DT preheat comes mainly from hot electrons with energies ${\gtrsim}160~\text{keV}$ , for $T_{h}>20~\text{keV}$ . Figure 11 shows the coupling to DT of a thermal, Maxwell–Jüttner spectrum with $T_{h}=50~\text{keV}$ . The black curve is the DT coupling fraction from figure 10 (also given there by a black curve), and the blue curve is the thermal energy spectrum $E\ast \text{d}N/\text{d}E$ for $T_{h}=50~\text{keV}$ . The red curve is their product, namely the energy coupled to DT by electrons of a given energy, in a thermal spectrum. The red curve exhibits behaviour akin to the ‘Gamow peak’ in fusion reactions, with a location determined essentially by the steeper black curve. Figure 12 shows the overall ${\it\phi}_{DT}$ integrated over the thermal spectrum versus $T_{h}$ . This peaks slightly above 5 % near $T_{h}=90~\text{keV}$ . The red curve is, as a function of $T_{h}$ , the hot electron energy of maximum $E_{DT}$ , i.e. the energy of the peak in the red curve in figure 11. This increases slowly with $T_{h}$ , and is at ${>}160~\text{keV}$ for all $T_{h}$ of interest. It is thus important to correctly model these hot electrons to calculate DT preheat, even for $T_{h}\ll 160~\text{keV}$ .

6. Hot electron propagation in early-time picket: PICKET series

This section studies hot-electron dynamics during the initial laser ‘picket’, and the effects of an axial $B$ field. The principal way hot electrons are produced during the picket is LPI in the LEH. This can be TPD for $n_{e}\approx n_{cr}/4$ , Raman scattering or a multi-beam variant of it (Michel et al. Reference Michel, Divol, Dewald, Milovich, Hohenberger, Jones, Hopkins, Berger, Kruer and Moody2015). NIF experiments have shown the picket hot electrons can be reduced by shaping the picket pulse, for instance by turning the inner beams on before the outers to blow down the window at low power. Experiments at the Omega laser studied hot electrons from TPD during window burn down (Regan et al. Reference Regan, Meezan, Suter, Strozzi, Kruer, Meeker, Glenzer, Seka, Stoeckl and Glebov2010).

The DT fuel is particularly sensitive to hot electrons produced during the early time picket pulse: ${\rm\Delta}\text{entropy}=E_{DT}/\text{temperature}$ , so a small $E_{DT}$ added when the fuel is cold produces a large entropy increase. In addition, melting the cryogenic DT layer before the first shock arrives causes the inside surface to expand, which can degrade the ability to perform shock timing (C. A. Thomas, 2015, private communication). For indirect-drive ignition designs, this occurs for $E_{DT}\sim 0.1~\text{J}$ . NIF ignition-relevant hohlraum experiments show total hot-electron energies $E_{h}\sim 1~\text{J}$ with $T_{h}\sim 80~\text{keV}$ . Calculations typically show $E_{DT}/E_{h}\sim (2{-}5)\times 10^{-3}$ , giving preheat $E_{DT}\sim (2{-}5)\times 10^{-3}~\text{J}$ well below melt.

We use HYDRA plasma conditions at 1 ns, shortly after the outer-beam power has peaked, for ZUMA calculations. We call this the PICKET run series. Figure 13 shows the material regions and laser intensity. We source the hot electrons in a $500~{\rm\mu}\text{m}$ radius circle at the left-side LEH ( $z=-0.45~\text{cm}$ ), which is roughly the extent of high laser intensity. Since TPD does not generally produce collimated hot electrons, we use an isotropic source with velocity-space $\text{d}N/\text{d}{\it\Omega}$ constant ( ${\it\Omega}$ is solid angle in velocity) for polar angles between 0 and $90^{\circ }$ , and zero otherwise (i.e. uniform in the forward-going half-space). The energy spectrum is a Maxwell–Jüttner with $T_{h}=80~\text{keV}$ , which is consistent with hard X-ray data on NIF (discussed below).

We expect a 70 T axial field to strongly magnetize the hot electrons in the low-density hohlraum gas fill, and guide them to the capsule. Recall that we inject a divergent hot-electron source, so the question is whether the field confines them in space. It will not collimate them, i.e. reduce their velocity-space divergence. The electron Larmor radius $r_{Le}\equiv p_{\bot }/eB$ , which for $E=100~\text{keV}$ and $B=70~\text{T}$ is $r_{Le}=16.0~{\rm\mu}\text{m}\cdot \sin {\it\alpha}$ ( ${\it\alpha}$ is the angle between $\boldsymbol{B}$ and $\boldsymbol{p}$ ). This is much less than the relevant plasma scale lengths. Also, the cyclotron period ${\it\tau}_{ce}=2{\rm\pi}{\it\gamma}m_{e}/eB$ is $0.510~\text{ps}\cdot {\it\gamma}$ for $B=70~\text{T}$ , which is much shorter than the propagation time through the hohlraum. Figure 14 plots ${\it\tau}_{ce}$ and the time for $90^{\circ }$ root-mean-square angular scatter, ${\it\tau}_{as}$ : $\langle {\it\theta}^{2}\rangle =(90^{\circ })^{2}$ for $\text{d}t={\it\tau}_{as}$ in (4.2). We consider two fully-ionized cases: one representative of the hohlraum fill: $0.96~\text{mg}~\text{cm}^{-3}$ of He at $T_{e}=1~\text{keV}$ , and one of the ablator: $1~\text{g}~\text{cm}^{-3}$ of $C_{1}H_{1}$ at 200 eV. The plot shows all hot electrons are magnetized in the He, while those with $E>300~\text{keV}$ in the CH are. Even if ${\it\tau}_{ce}$ exceeds ${\it\tau}_{as}$ , $r_{Le}$ is much smaller than typical capsule dimensions ${\sim}$ 100’s  ${\rm\mu}\text{m}$ .

Figure 14.

Time for $90^{\circ }$ angular scatter ${\it\tau}_{as}$ for: blue curve: $0.96~\text{mg}~\text{cm}^{-3}$ of He at $T_{e}=1~\text{keV}$ (typical for hohlraum gas fill in NIF shot N120321), green curve: $1~\text{g}~\text{cm}^{-3}$ of $C_{1}H_{1}$ at 200 eV (typical for the ablator). Red dashed curves are cyclotron period ${\it\tau}_{ce}$ for $B_{z}=1$ and $70~\text{T}$ . Most hot electrons are (strongly, weakly) magnetized in the (He gas, CH ablator).

The energy deposition is shown in figure 15 for the ZUMA runs with no $B$ field (top half), and with a uniform $B_{z}=70~\text{T}$ field (bottom half). Table 1 lists the fraction of injected hot-electron energy deposited in different regions. With no $B$ field, the hot electrons propagate essentially freely in the He gas fill. They mostly deposit in the hohlraum wall, and a small fraction deposits in the ablator. This is expected based on the solid angle subtended by these regions. With a uniform $B_{z}=70~\text{T}$ , the hot electrons are strongly magnetized in the He gas and guided to the capsule. They mostly deposit in the ablator, out to a radius comparable to that of the source. This asymmetric preheat, occurring mostly in the poles, may drive capsule asymmetries. The energy deposited in the DT layer is ${\sim}12\times$ higher with the 70 T field. Whether this is a preheat concern depends on the spectrum and total energy of hot electrons produced.

The lack of deposition in high $Z$ with the field means the same hot electron source produces many fewer hard X-rays. This is a diagnostics concern, since hard X-rays are generally used to deduce hot electrons on NIF. One such principal diagnostic is the FFLEX hard X-ray ( ${>}10~\text{keV}$ ) detector (Dewald et al. Reference Dewald, Thomas, Hunter, Divol, Meezan, Glenzer, Suter, Bond, Kline and Celeste2010; Hohenberger et al. Reference Hohenberger, Albert, Palmer, Lee, Döppner, Divol, Dewald, Bachmann, MacPhee and LaCaille2014), with 10 channels filtered for different energy ranges. Energetic electrons lose energy by collisions with background electrons and by bremsstrahlung radiation. Radiation loss ${\sim}EZ\times$ collisional loss, with the two equal in gold for $E=10$ MeV. Only electrons that deposit energy in high- $Z$ material, such as the hohlraum wall, produce enough hard X-rays for FFLEX to detect. Hot electrons striking the capsule during the picket have been measured on ‘re-emit’ experiments, where the capsule is replaced by a high- $Z$ (e.g. bismuth) ball.

Figure 15.

Hot electron energy deposition for PICKET series of ZUMA runs with no magnetic field (top) and a uniform $B_{z}=70~\text{T}$ field (bottom).

Table 1.

Fraction of injected hot-electron energy deposited in different regions, for PICKET (1 ns) and SRSPEAK (18 ns) series of ZUMA runs. $(x)$ is ratio of with $B$  field/MHD to without. Fractions do not sum to unity because some hot electrons escape from problem boundaries.

7. Hot-electron propagation during peak power: SRSPEAK series

We now consider ZUMA simulations of propagation of a realistic hot-electron source produced by Raman scattering on the inner beams during peak power. We use the same plasma conditions at 18 ns that were used in the CAPTEST series (i.e. a simulation of NIF shot N120321 with the full, incident laser power on each cone), along with conditions from a HYDRA run with an initial $B_{z0}=70~\text{T}$ axial field and the MHD package active. The hot-electron source has a Maxwell–Jüttner energy spectrum with $T_{h}=30~\text{keV}$ . This temperature is from the FFLEX data at 18 ns (rise to peak power) on NIF shot N130517, which is analogous to N120321 (Robey et al. Reference Robey, Celliers, Moody, Sater, Parham, Kozioziemski, Dylla-Spears, Ross, LePape and Ralph2014). Once peak power is reached, $T_{h}=18~\text{keV}$ is consistent with FFLEX data. The injected angle spectrum is $\text{d}N/\text{d}{\it\Omega}=\exp [-(({\it\theta}-27^{\circ })/10^{\circ })^{4}]$ , which is directed along roughly the bisector of the two NIF inner beams at ${\it\theta}=23.5^{\circ }$ and $30^{\circ }$ .

We find strong sensitivity to what field lines the hot electrons start on – namely, whether or not the field lines connect to the capsule. Hot electrons are injected in the three locations indicated as sources 1, 2 and 3 in figure 5: from $r=0$ to 0.06 cm at $z=-0.4$ cm, and from $r=0.12$ to 0.18 cm, at $z=-0.25$ and $-$ 0.2 cm. The energy deposition versus space is plotted in figure 16, and the total into various materials is given in table 1. With no MHD, the fraction of hot-electron energy deposited to DT varies from $(1.2{-}3.6)\times 10^{-4}$ over the three sources. The field strongly magnetizes the hot electrons in the He fill gas for all three sources, as in the PICKET series of § 6. Also like the PICKET series electrons from source 1, in the LEH, are guided to the capsule. The deposition in DT and CH is greatly increased compared to the no-MHD case. For source 2, located deeper in the hohlraum and off axis, electrons are injected on field lines that do not connect to the capsule. The resulting deposition in He gas is significantly increased compared to the no-MHD case, while that into the CH ablator and especially the DT layer are reduced. The situation reverses for source 3, which is slightly closer to the capsule in $z$ than source 2. Some electrons now start on field lines that connect to the capsule, which results in much higher DT deposition. It is not presently known where in the hohlraum SRS hot electrons are produced, so we cannot say whether the field increases or decreases DT deposition. As with the PICKET series, the fraction of hot-electron energy deposited in high- $Z$ material is lower with the field, especially for source 1. Hard X-ray diagnostics may thus not be reliable indicators of hot electron preheat.

Figure 16.

Hot electron energy deposition for SRSPEAK series of ZUMA runs without MHD (top $r>0$ ) and with MHD (bottom $r<0$ ). (a–c) plots are for source locations 1, 2, 3 indicated in figure 5.