Production probability of super-heavy nuclei in fusion

The synthesis of super-heavy nuclei (SHN) through fusion reactions is a critical area of nuclear physics, offering insights into nuclear stability and the limits of the periodic table. However, theoretical predictions of evaporation residue cross sections $\sigma_{\rm {ER} }$ remain challenging due to large uncertainties arising from complex reaction mechanisms and sensitive model parameters. In this work, a new and analytical formula is proposed for systematically describing the production probabilities of SHN with atomic number $Z\ge 110$, based on barrier tunneling concept. Together with the empirical barrier distribution method for describing capture, an improved model, EBD3, reproduces 64 measured $\sigma_{\rm {ER} }$ within one order of magnitude, with a root-mean-square deviation of 0.351. The model successfully captures key quantities in fission-like process, including fission barrier height, mass asymmetry, depth of capture pocket and the effective fusion barrier height. Predictions for the synthesis of element 119 are presented, identifying promising projectile-target combinations such as $^{45}$Sc + $^{249}$Cf with a maximum cross section of $107.5^{+120}_{-56.7}$ fb. The maximum cross section falls to $3.2^{+3.6}_{-1.7}$ fb for $^{54}$Cr + $^{243}$Am at the optimal incident energy of 244 MeV.