Design of an inertial fusion experiment exceeding the Lawson criterion for ignition

We present the design of the first igniting fusion plasma in the laboratory by Lawson's criterion that produced 1.37 MJ of fusion energy, Hybrid-E experiment N210808 (August 8, 2021) [Phys. Rev. Lett. 129, 075001 (2022)]. This design uses the indirect drive inertial confinement fusion approach to heat and compress a central “hot spot” of deuterium-tritium (DT) fuel using a surrounding dense DT fuel piston. Ignition occurs when the heating from absorption of $\alpha$ particles created in the fusion process overcomes the loss mechanisms in the system for a duration of time. This letter describes key design changes which enabled a $\sim 3\text{–}6\times$ increase in an ignition figure of merit (generalized Lawson criterion) [Phys. Plasmas 28, 022704 (2021), Phys. Plasmas 25, 122704 (2018)]) and an eightfold increase in fusion energy output compared to predecessor experiments. We present simulations of the hot-spot conditions for experiment N210808 that show fundamentally different behavior compared to predecessor experiments and simulated metrics that are consistent with N210808 reaching for the first time in the laboratory “ignition.”

Figure 1

(a) Top: Calculated positions of the wall and capsule materials [gold-lined depleted uranium hohlraum (orange), HDC ablator (light gray), and DT ice layer (blue)] at peak power (6 ns) from radiation-hydrodynamic HYDRA simulations with simulated laser rays colored by relative power overlaid. Implosion symmetry control is accomplished via cross-beam energy transfer (CBET) in low-gas-filled ($\sim 0.3$ ${\mathrm{g}/\mathrm{cm}}^3$) hohlraums [6] from the “outer” laser beams ($44.5^{\circ }$ and ${50}^{\circ }$) to the “inner” beams (${23}^{\circ }$ and ${30}^{\circ }$) by detuning the wavelengths relative to each other ($\mathrm{\Delta }\lambda$). (a) Bottom: Hybrid-E target geometry at $t=0$ with laser beam pointing (laser cones are labeled by their respective polar angle) for the new (N210808, left) and previous (N210307, right) configurations. (b) The total (thick curves), “outer” cone (thin middle curves), and “inner” cone (thin lower curves) laser powers vs time (i.e., pulse shape) for experiments N210808 (red) and N210307 (blue). The balancing of power between the “outer” and “inner” cones together with CBET is tailored to give precise control of the time-dependent radiation drive symmetry. (c) Simulated hohlraum internal radiation temperature, $T_{\text{rad}}$, histories for N210808 (red) and N210307 (blue). The extended $T_{\mathrm{rad}}$, for N210808 compared to N210307 results in increased late-time ablation pressure and reduced “coast time.” The near vertical rise of the $T_{\text{rad}}$ at $t\sim 9.2\mathrm{ns}$ seen in the red curve is reheating of the hohlraum from the capsule fusion output. “Bang times” or time of peak neutron production and times of peak implosion velocity are also denoted. (d) Calculated change in cone powers as a result of CBET moving power from the “outer” beams to the “inner” beams, incident in black and post-CBET in red. This calculation uses multipliers on the incident powers to match experimental tuning data [7]. (e) Ratio of the “inner” cone power to total laser power for the incident pulse (black) and post-CBET pulse (red) from the calculation in (d). See the text for more information.

Figure 2

(a) Neutron yield as a function of laser energy for diamond (pink), plastic (green), and beryllium (gray) ablator designs. Hybrid-E experiments (e.g., N210207, N210307, and N210808) show the improvement in performance at fixed available laser energy through design changes to increase the scale of the implosion while maintaining high hot-spot pressures. The curves are direct hydrodynamic scalings from best performing smaller scale designs which require more laser energy to increase implosion scale. The Hybrid-E experiments are notably higher than a direct scaling of previous smaller scale diamond ablator designs. (b) Fusion energy as a function of scaled DT ice mass showing that, in addition to being able to field a larger scale implosion at fixed laser energy, the quality of the implosion compared to a hydrodynamic scaling of Bigfoot experiment N180128 [22] was improved on N210808. The upper points are the measured yields, which include the impact of $\alpha$ heating ($\alpha$-on), and the lower curve uses calculations to remove the impact of $\alpha$ heating ($\alpha$-off), verified in a THD experiment at 1.0 scale (dark point) and calculated otherwise (light gray points).

Figure 3

(a) Simulated Legendre decomposition of the radiation flux asymmetry (Pn = P1, P2, and P4) for N210808 plotted together with the scaled laser power as a function of time, and cone fraction (ratio of “inner” laser cones to total power) as a function of time. Also denoted is the range of $\pm 1\%$ radiation flux asymmetry. The calculated radiation flux symmetry is within specification during most of the pulse for P1 and P4. The P2 decomposition is within the specification during the “foot,” until the rise to peak power, but is allowed to swing during the peak of the pulse (see the text). These calculations do not adjust the drive of the “inner” cones and overestimate the P2 swing in the rise to peak. (b) Observed increase in hot spot P2 (more prolate) for a given amount of wavelength separation ($\mathrm{\Delta }\lambda$) over the calculated P2 at $\mathrm{\Delta }\lambda =0$ Å using the Callahan et al. semiempirical model [38].

Figure 4

Schematic of the N210808 Hybrid-E hohlraum and capsule configuration showing the nominal target dimensions for the hohlraum (left) and pie chart for the central DT-fuel filled capsules (right). Also shown are pie charts for N210307 and N210207. The diamond ablator consists of a $\sim 6$ $\mu \mathrm{m}$ inner undoped HDC layer, followed by a tungsten (W)-doped HDC layer at larger radii and an outer undoped HDC layer. The crystal structure for the diamond ablators used in N210307 and N210808 was nanocrystalline diamond (NCD) compared to microcrystalline diamond (MCD) used in experiment N210207. The DT ice and gas core radii are also listed.

Figure 5

Calculated DT fuel-HDC ablator Atwood number as a function of fuel-ablator radius for Hybrid-E designs using lower W-dopant diamond ablators (N210207 with 0.3% W dopant, blue) compared to higher W-dopant ablators in the longer “coast-time” configuration (N210307 with 0.44% W dopant, red). The improvement in Atwood number for the higher W-dopant ablators in the lower-coast design of N210808 is shown in black (KC789 with 0.42% W dopant). When the ablator density $>$ the DT fuel density (negative values on the $y$ axis) the implosion is more stable at the DT fuel-ablator interface.

Figure 6

(a) Simulated internal radiation temperature for N210808 (red) and N210307 (blue) where the inset is expanded on the first shock. (b) Radial pressure gradients of the three shocks launched by distinct changes in radiation temperature plotted as a function of initial radius vs time. The darker contours show the shock front moving through the various capsule layers (ablator, DT ice, and DT gas), which are denoted on the right-hand side. The ice-gas interface is also noted. The plot on the left-hand side is for experiment N210307, the middle is a preshot calculation for the adjusted N210808 pulse (see the text), and the right-hand plot is for the postshot delivered N210808 pulse. (c) Calculated leading shock front pressure in the diamond (red) is higher than the melting pressure for HDC along the shock Hugoniot ($\sim 10\text{–}12\mathrm{Mbar}$) [51]. The black curves denote where the temperature is $>$ 3 $\text{eV}$ behind the shock front.

Figure 7

Calculations of ablator mass remaining as a function of implosion velocity (a) and implosion velocity as a function of peak laser power (b) for preshot simulations of the N210808 hohlraum and LEH configuration for high (blue) and low (red) W-doped capsule batches (see Fig. 4). As the peak laser power is varied the total laser energy is conserved. The spread in velocities for a given peak laser power is a result of using a range to estimate uncertainty in the drive benefit of the smaller LEH (3.1 vs 3.64 mm) for the full 1.9 MJ laser energy, which was first fielded on this experiment N210808. The design point for N210808 was $\sim 5\%$ ablator mass remaining and $\sim 390\mathrm{km}/\mathrm{s}$, which led to the 430 TW requested laser power (with 440 TW actually delivered on N210808).

Figure 8

Relative impacts: Simulations of the total neutron yield for the changes made between the burning plasma regime (N210307) and this paper (N210808) including the lower-coast radiation drive, DT fill-tube size change, stability improvement, and capsule quality change. Postshot simulations including all degradations for that experiment, (a) for N210307 and (g) for N210808, are compared to experimental data (slashed bars). See the text for a description of the relative changes between the bars. The insets are simulated images of the density (left) and ion temperature (right); see also the text.

Figure 9

(a) “Hot-spot” ion temperature and $\rho \mathrm{R}$ evolution as a function of time for $\alpha$-heating experiment N170601 compared to burning plasma experiment N210307 and ignition by Lawson criterion experiment N210808. The direction of increasing time is shown with the arrows on the curves, and the overlaid red curves denote “bang time” (peak neutron production) and $\pm 50\mathrm{ps}$. The dashed curve is a theoretical model for conditions where the $\alpha$ heating balances the loss terms. (b) Calculated hot-spot energy partition as a function of time: PdV work on the hot spot [black (dot-dashed)], energy deposited by $\alpha$ heating (green), radiative (red), and conduction [magenta (dotted)] energy loss, and hot-spot internal energy [blue (dashed)]. Note that the negative PdV work is a loss term that is comparable to the radiative loss and exceeds the conduction loss by the time of peak neutron production. Inset: Simulated density (left, $\mathrm{g}/{\mathrm{cm}}^3)$ and ion temperature $T_{\mathrm{ion}}$ (right, keV) at the time of peak neutron production. Inset spatial scale in microns.