Probing In-Solid Proton Energy Distributions in Laser-Driven Fusion via Nuclear Activation Diagnostics

The energy distribution of energetic protons inside a solid target is a key quantity governing nuclear reaction yields and energy deposition in high-intensity laser-driven fusion, including nonthermal proton--boron (p--B) schemes and proton fast ignition. Yet it has remained inaccessible to conventional particle diagnostics, which detect only ions escaping the target and are perturbed by intense plasma electromagnetic fields. Here we establish a quantitative diagnostic that uses nuclear activation reactions occurring within the target itself as an internal probe of the in-solid proton energy distribution. Applied to laser-driven p--B fusion experiments on the kJ-class laser, the method reconstructs an exponential-equivalent in-solid proton energy distribution from the absolute yields of $^{11}\mathrm{C}$ and $^{7}\mathrm{Be}$ produced via $\mathrm{^{11}B(p,n)^{11}C}$ and $\mathrm{^{10}B(p,\alpha)^{7}Be}$, and yields the absolute number of $\mathrm{^{11}B(p,2\alpha)^{4}He}$ reactions through a side-channel analysis with propagated cross-section uncertainties. This work opens a quantitative window onto the in-solid proton dynamics that drive nuclear reactions in laser-driven fusion experiments.