Extracting temperature of plasma through fusion reactions within a transport approach

We have investigated the method of extracting the temperature from weighted proton-to-neutron yield ratio from fusion reactions as in the previous experiment~[W. Bang, {\it et al.}, Phys. Rev. Lett. \textbf{111}, 055002 (2013).] using the Texas Petawatt laser beam. The Coulomb explosion of deuterium clusters is simulated based on the particle-in-cell model in a box system with periodic boundary conditions, and fusion reactions are incorporated through the stochastic method. As long as the deuteron numbers in deuterium clusters follow a log-normal distribution, the low-energy part of the final deuteron spectrum can be fitted by a Maxwell-Boltzmann distribution, while there are more deuterons in the intermediate- and high-energy region compared to a thermal distribution, and dominate the weighted yield ratio. Therefore, the effective temperature extracted from the weighted yield ratio is generally higher than that from fitting the final deuteron spectrum. The local density fluctuation, which intrinsically exists due to the log-normal distribution of deuteron numbers, further enhances hot deuteron-deuteron collisions, and significantly affects the weighted yield ratio.