Upper Limit of Fusion Reactivity in Laser-Driven $p+{^{11}{\rm B}}$ Reaction

TL;DR

The paper addresses the problem of maximizing the fusion reactivity $\langle \sigma v \rangle$ for laser-driven $p+^{11}{\rm B}$ fusion in tabletop experiments. It adopts a 1D plasma-expansion (TNSA) framework to model the proton-energy distribution, employing the self-similar form $f_{p,ss}(E)$ and ponderomotive scaling for $T_e$ to compute $\langle \sigma v \rangle$ by convolving with the cross-section $\sigma(E_r)$; the analysis reveals how $T_e$ and $\omega_{pi} t_{acc}$ shape the resonance overlap. The key finding is a maximum $\langle \sigma v \rangle = 8.12 \times 10^{-16}\ \mathrm{cm^3\,s^{-1}}$ achieved at $k_B T_e = 10$ MeV and $\omega_{pi} t_{acc} = 0.503$, indicating an upper bound on table-top p+11B reactivity under this geometry. This work provides practical guidance for optimizing laser-driven fusion yields and clarifies that further improvements require incorporating beam evolution and electron screening effects via more rigorous kinetic modeling and experiments.

Abstract

We explore the averaged fusion reactivity of the $p+{^{11}{\rm B}}$ reaction in tabletop laser experiments using a plasma expansion model. We investigate the energy distribution of proton beams accelerated by lasers as a function of electron temperature $T_e$ and the dimensionless acceleration time $ω_{pi} t_{\rm acc}$, where $ω_{pi}$ is the ion plasma frequency. By combining these distributions with the fusion cross-section, we identify the optimal conditions that maximize the fusion reactivity, with $\left\langle σv \right\rangle = 8.12 \times 10^{-16}\,{\rm cm^3/s}$ at $k_B T_e = 10.0\,{\rm MeV}$ and $ω_{pi} t_{\rm acc} = 0.503$. These findings suggest that an upper limit exists for the fusion reactivity achievable in laser-driven $p+{^{11}{\rm B}}$ fusion experiments, even under optimized conditions.

Figures (6)

• Figure 1: Schematic diagram of the experimental setup for the $p+^{11}{\rm B}$ reaction in a pitcher-catcher type. Protons are accelerated from the foil target by a short-pulse laser, and the accelerated proton beam interacts with a $^{11}{\rm B}$ plasma ionized by the secondary plasma.
• Figure 2: Proton number as a function of $I\lambda^2$, compared with published experimental data. The prediction is obtained using $\tau = 500\,{\rm fs}$, $r_0 = 10\,{\rm \mu m}$, and $d_t = 10\,{\rm \mu m}$, and by integrating the self‑similar $dN/dE$ spectrum over a 1 MeV bin centered at 10 MeV. The circular data points are extracted from Refs: Osakaosaka, CUOSCUOS, RAL VULCANRal_Vulcan1Ral_Vulcan2, LULI fuchs_laser-driven_2006, RAL PW RAL_PW, and Nova PW PhysRevLett.85.2945.
• Figure 3: Fusion cross-section for the $p+^{11}{\rm B}$ reaction Nevins_2000 and distribution function as a function of $E_p$. The black solid line represents $\sigma(E_r)$ with $E_r \simeq (\mu/m_p) E_p$, while the blue dashed line corresponds to $f_p(E_p)$ obtained from the self-similar solution. For the $f_p(E_p)$ numerically obtained, the red dash-dotted, yellow dotted, and orange dash-dot-dot lines indicate $\omega_{pi} t_{\rm acc} = 1$, $1.5$, and $5$, respectively. For all $f(E_p)$ calculations, $k_B T_e = 2\,{\rm MeV}$ is adopted.
• Figure 4: This figure is the same as Fig. \ref{['fig:dndE_wt=1']}, but here, $f_p(E_p)$ is presented for different temperatures while keeping $\omega_{pi} t_{\rm acc} = 1$ fixed. The black solid line represents $\sigma(E_r)$ with $E_r \simeq (\mu/m_p) E_p$, while the blue dashed and blue dash-dotted lines correspond to $f_p(E_p)$ obtained from the self-similar solution for $k_B T_e = 2\,{\rm MeV}$ and $k_B T_e = 8\,{\rm MeV}$, respectively. For the numerically obtained $f_p(E_p)$, the red dotted and green dash-dot-dot lines indicate $k_B T_e = 2\,{\rm MeV}$ and $8\,{\rm MeV}$, respectively.
• Figure 5: Averaged fusion reactivity $\left\langle \sigma v \right\rangle$ of $p+{^{11}{\rm B}}$ reaction as a function of $\tau_{\rm laser}$ and $I$. For this calculation, we adopt $\lambda = 1\,{\rm \mu m}$, $r_0 = 10\,{\rm \mu m}$, and $d_t = 10\,{\rm \mu m}$.