Self-similar multishock implosions for ultrahigh compression of matter

TL;DR

The paper addresses achieving ultrahigh compression of uniform-density solid targets by synchronizing multiple converging shocks. It develops a class of self-similar solutions that generalize the Guderley problem to $N$ stacked shocks and derives a density-scaling law $\rho_r/\rho_0 \propto \hat{P}^{\beta(N-1)}$ with a $\beta(\gamma)$ exponent determined from perturbation theory. The results, verified by 1D hydrodynamic simulations, show robust agreement across perturbative and strongly nonlinear regimes up to $\hat{P} \sim 70$, with a saturation set by the asymptotic strong-shock limit; the approach inherently suppresses Rayleigh–Taylor instabilities due to its volumetric geometry. This analytic framework provides a principled route to instability-free, high-density compression designs in inertial confinement fusion and may inform advanced fusion concepts such as proton–boron where ultrahigh densities are desirable.

Abstract

We present a class of self-similar solutions describing ultrahigh compression of a uniform-density target by spherically converging, stacked shock waves. Extending the classical Guderley model, we derive a scaling law for the final density of the form $ρ_{r}/ρ_{0} \propto \hat{P}^{β(N-1)}$, where $N$ is the number of shocks, $\hat{P}$ the stage pressure ratio, and $β$ a numerical exponent determined by the adiabatic index $γ$. One-dimensional hydrodynamic simulations confirm the validity of this scaling across a broad parameter range. Notably, the relation remains accurate even in the strongly nonlinear regime up to $\hat{P} \sim 70$, well beyond the perturbative limit, highlighting the robustness and practical relevance of the model. Owing to its volumetric geometry, this compression scheme inherently avoids the Rayleigh--Taylor instability, which typically compromises shell-based implosions, and thereby establishes a theoretical benchmark for instability-free compression in inertial confinement fusion.

Figures (4)

• Figure 1: (a) Lagrangian trajectories (blue) showing the implosion of a solid DT sphere under two-stage pressure loading with $\hat{P} = 8$. Dashed lines: self-similar theoretical trajectories (from panel (b)) over-plotted for direct comparison with the simulations. Shocks are synchronized to collide simultaneously at the center at $t=t_c$. (b) Close-up view near $t = t_c$, where both shocks obey the scaling $r \propto |t - t_c|^\alpha$.
• Figure 2: Pressure profiles at three times (A, B, C) during convergence, corresponding to markers in Fig. 1(b). The peak pressure rises and the shock sharpens, preserving the two-step structure. Inset: Log–log scaling collapse confirms self-similar evolution. Minor deviations at large radii reflect the pre-asymptotic flow, which shrinks as the collapse proceeds.
• Figure 3: (a) Integrated trajectories in the U--Z plane for $\gamma = 5/3$, showing the first-shock (blue, $\alpha^\ast = 0.68838$) and second-shock (red, $\alpha_1 = 0.69345$) solutions. Thin solid curves indicate the sonic lines $Z = (U - \alpha)^2$. (b)--(d) Scaled radial profiles of density, velocity, and pressure during convergence. Solid lines: hydrodynamic simulations; dashed lines: self-similar solutions, showing excellent agreement.
• Figure 4: Final compression $\rho_r/\rho_0$ versus stage pressure ratio $\hat{P}$ for $N=1$–5 stacked shocks. Curves 1–5 denote $N=1$–5 (1 = single-shock/Guderley). $\rho_r$ is the reflected-shock crest density immediately after on-axis coalescence and is subsequently advected outward with the diverging shock. Dashed lines indicate the asymptotic limits.