Deep underground laboratory measurement of $^{13}$C($\alpha$,$n$)$^{16}$O in the Gamow windows of the $s$- and $i$-processes

The $^{13}$C($\alpha$,$n$)$^{16}$O reaction is the main neutron source for the slow-neutron-capture (s-) process in Asymptotic Giant Branch stars and for the intermediate (i-) process. Direct measurements at astrophysical energies in above-ground laboratories are hindered by the extremely small cross sections and vast cosmic-ray induced background. We performed the first consistent direct measurement in the range of $E_{\rm c.m.}=$0.24 MeV to 1.9 MeV using the accelerators at the China Jinping Underground Laboratory (CJPL) and Sichuan University. Our measurement covers almost the entire i-process Gamow window in which the large uncertainty of the previous experiments has been reduced from 60\% down to 15\%, eliminates the large systematic uncertainty in the extrapolation arising from the inconsistency of existing data sets, and provides a more reliable reaction rate for the studies of the s- and i-processes along with the first direct determination of the alpha strength for the near-threshold state.