Kinetic simulations of fusion ignition with hot-spot ablator mix

Inertial confinement fusion fuel suffers increased x-ray radiation losses when carbon from the capsule ablator mixes into the hot-spot. Here, we present one- and two-dimensional ion Vlasov-Fokker-Planck simulations that resolve hot-spot self-heating in the presence of a localized spike of carbon mix, totalling $1.9\%$ of the hot-spot mass. The mix region cools and contracts over tens of picoseconds, increasing its $\alpha$ particle stopping power and radiative losses. This makes a localized mix region more severe than an equal amount of uniformly distributed mix. There is also a purely kinetic effect that reduces fusion reactivity by several percent, since faster ions in the tail of the distribution are absorbed by the mix region. Radiative cooling and contraction of the spike induces fluid motion, causing neutron spectrum broadening. This artificially increases the inferred experimental ion temperatures and gives line of sight variations.

Figure 1

Results of two one-dimensional kinetic simulations, the first with carbon ablator mix in a Gaussian profile of waist $5\mu \mathrm{m}$ in the center of the simulation domain and the second with the same carbon mass spread uniformly across the hot-spot. The (i) panels in the top row show the number density for the combined deuterium-tritium ion species at three separate times (a) $0\mathrm{ps}$, (b) $20\mathrm{ps}$ and (c) $40\mathrm{ps}$. The bottom row (ii) panels show the deuterium-tritium temperature. Data for other species are not shown. The carbon ions, assumed fully ionized, constituted $1.9\%$ of the hot-spot mass and had peak number density $8\times {10}^{29}\mathrm{m}{}^{-3}$ in the localized case and ${10}^{29}\mathrm{m}{}^{-3}$ in the uniform case.

Figure 2

Distribution function $v^2{f}_0(x,v)$ of the supra-thermal $\alpha$ particles in the localized mix case, shown at times (a) $6\mathrm{ps}$ and (b) $20\mathrm{ps}$. Fusion $\alpha$ particles are created with speed $1.3\times {10}^7\mathrm{ms}{}^{-1}$. The thermalized $\alpha$ particles in the red stripe (exceeding the color scale) at the bottom of each plot have a thermal velocity $v_{\text{th}}\simeq \sqrt{k_{\text{B}}{T}_{\text{e}}/m_{\alpha }}=3\times {10}^5\mathrm{ms}{}^{-1}$ and constitute the bulk of the $\alpha$ particle number density. The central mix region is more effective at slowing the $\alpha$ particles. Equation (2) shows that the integral of these distribution functions along the $v$ axis gives the $\alpha$ particle number density, shown in (c) for both time-steps.

Figure 3

Fusion reactivity of the deuterium-tritium plasma from the one-dimensional simulation at $40\mathrm{ps}$, calculated from a numerical integral of Eq. (16) using the simulated distribution function with localized carbon mix. The reactivity is normalized to the reactivity of a Maxwellian distribution with identical fluid moments. The normalized profile of the carbon mix is also overlaid with the dashed line.

Figure 4

Results of the two-dimensional kinetic simulation after $40\mathrm{ps}$, with carbon ablator mix localized in a Gaussian spike of waist $5\mu \mathrm{m}$ in the $x$ direction, waist $20\mu \mathrm{m}$ in the $y$ direction and centered at $y=-25\mu \mathrm{m}$. The peak carbon number density was $8\times {10}^{29}\mathrm{m}{}^{-3}$ and all other parameters are equivalent to Fig. 1. (a) The temperature of the deuterium tritium species, showing the radiative cooling in the mix region. (b) The ratio of the deuterium-tritium fluid kinetic energy to the total energy, given by ${mn\left|\mathbf{u}\right|}^2/\left(2U\right)$. (c) The volumetric fusion reaction rate. (d) The ratio of the fusion reaction rate to that of a Maxwellian with equal density, temperature and fluid velocity.

Figure 5

Synthetic normalized neutron spectra generated from the $40\mathrm{ps}$ time-step of the two-dimensional simulation shown in Fig. 4. The solid line shows the case where the broadening from fluid motion is neglected, the dashed line shows the calculation with the broadening due to fluid motion for a line of sight in the $x$ direction, and the dotted line shows the same for the $y$ direction. The fitted temperatures are $4026\mathrm{eV}$, $4464\mathrm{eV}$, and $4526\mathrm{eV}$, respectively. Neutron scattering was neglected.