The Non-thermal Energy Window for Laser-Driven Nuclear Reactions

Abstract

Astrophysical nuclear reaction rates in stellar environments are governed by the Gamow window, where Maxwell-Boltzmann distributions and quantum tunneling probabilities combine to produce effective reactivity. However, this conventional formulation is inadequate for the non-thermal ion distributions generated in ultra-intense laser-plasma interactions. Here, we introduce an analytical framework, based on a Target Normal Sheath Acceleration (TNSA) mechanism, to evaluate nuclear reaction rates under these non-equilibrium conditions. We identify a new effective energy window and analytical expression of the fusion reactivity distinct from the conventional Gamow window, providing a predictive tool for laboratory astrophysics experiments designed to replicate astrophysical nuclear processes using laser-driven nuclear reactions.

Figures (4)

• Figure 1: Schematic of the pitcher-catcher experimental setup. A high-intensity laser pulse (pitcher) irradiates a thin foil target, accelerating a beam of ions, which is then incident on a secondary target (catcher). Nuclear reactions occur as the laser-accelerated ions interact with the secondary target.
• Figure 2: Energy spectra of protons (red solid line) and deuterons (blue dashed line) calculated from a self-similar solution, compared with the experimental proton-beam data (black dots) Fuchs_PRL. We adopt a laser intensity of $I = 3 \times 10^{19}\,\mathrm{W/cm^2}$ and a wavelength of $\lambda = 1.057\,\mu\mathrm{m}$, consistent with the 100-TW laser experiment at LULI Fuchs_PRL. Under these parameters, corresponding to $k_B T_e = 2.068\,\mathrm{MeV}$, the acceleration time satisfies $\omega_{pi}t_{acc} = 11.08$ for protons and $\omega_{pi}t_{acc} = 7.839$ for deuterons.
• Figure 3: $I(E_i)$ (black solid line), $(\sqrt{E_i}/E)f_{i, ss}(E_i)$ (red dashed line), and $P(E)$ (blue dotted line) as a function of $E_i$ for the D+D reaction. For this calculation, the electron temperature is fixed as $k_B T_e = 2.068\,\mathrm{MeV}$, consistent with Fig. \ref{['fig:compare_exp']}. The peak energy is found to be $E_i =E_0 = 0.305\,\mathrm{MeV}$. The shaded region at $E_i = E_{0,G}=0.494^{+0.046}_{-0.048}\,{\rm MeV}$ denotes the conventional Gamow peak energy in a thermal plasma, calculated using $k_BT_{\rm eff,D} = 0.7 \pm 0.1 \text{ MeV}$, where $k_BT_{\rm eff,D} =0.7\,{\rm MeV}$ is the best-fit value reported in Ref. Fuchs_PRL.
• Figure 4: Ratio of the reactivity to the astrophysical $S$-factor at the peak energy, $\langle \sigma v \rangle / S(E_0)$, for the D+D reaction. The maximum occurs at $k_B T_e = 10.85\,\mathrm{MeV}$, corresponding to $I \lambda_\mu^{2} = 6.752 \times 10^{20}$, where $\langle \sigma v \rangle / S(E_0) = 3.356 \times 10^{8}\,\mathrm{cm\,s^{-1}\,MeV^{-1}}$. Each marker denotes the reactivity obtained from numerical integration using the corresponding $\omega_{pi} t_{\rm acc}$ values for different experimental conditions: XG-III at LFRC (blue circles), LFEX at ILE (green squares), 150-TW Ti:Sa laser at RRCAT (red triangles), ELFIE laser at LULI (cyan diamonds), and Pico2000 at LULI (magenta inverted triangles).