A semi-classical study of muon-enhanced proton-boron-11 fusion
A recent theoretical study has suggested that muons can enhance proton-boron-11 (p-$^{11}$B) reaction cross-section by several orders of magnitude in the low-energy regime. In this work, we investigate this reaction process using a semi-classical treatment, that is, a muon and a proton first form a muonic hydrogen atom p$μ$, which subsequently collides with a $^{11}$B nucleus. During the collision, the p$μ$ atom approaches the $^{11}$B nucleus and is then repelled by the Coulomb repulsive potential. At the distance of closest approach between the proton and the $^{11}$B nuclei in this classical scattering process -- namely, the classical turning point -- quantum tunneling through the Coulomb barrier can occur, allowing the proton to penetrate into the range of the nuclear force of the $^{11}$B and trigger the fusion reaction. We determine the turning point statistically by using the classical trajectory Monte Carlo method, where the initial phase-space distributions of the proton and muon are sampled from the ground-state microcanonical distribution. Our results show that, compared with the bare-nucleus case, the reaction cross-section is enhanced by several orders of magnitude in the low-energy region. A comparison with the static charge-shielding treatment reveals certain differences; however, both approaches demonstrate that the catalytic effect of the muon can significantly enhance the low-energy p-$^{11}$B reaction cross-section.