Relativistic dynamical effects in proton emission: the Wentzel-Kramers-Brillouin method for 1+1 dimensional Dirac equation

Starting from the $1+1$ dimensional (one spatial and one temporal dimension) Dirac equation, we employ the Wentzel-Kramers-Brillouin (WKB) approximation to derive the corresponding relativistic penetration probability. The derivation shows that the semiclassical momentum is determined by the Schr\"odinger-equivalent potential $U_{\text{eff}}(r) = S(r) + \frac{E}{m}V(r) + \frac{S^{2}(r)-V^{2}(r)}{2m}$, instead of the simple sum of scalar and vector potentials $S(r)+V(r)$, which has been adopted widely in the studies of relativistic quantum tunneling. We then quantify the relativistic dynamical effects in proton emission by comparing the results obtained with $U_{\text{eff}}(r)$ and those obtained with $S(r)+V(r)$. Incorporating $U_{\text{eff}}(r)$ systematically reduces the penetration probability and the assault frequency, and consequently increases the predicted half-life. The relativistic dynamical effect becomes more pronounced with higher orbital angular momentum and can reach about $84\%$ in the half-life of $^{144}\mathrm{Tm}$.