质子发射中的相对论动力学效应：1+1维狄拉克方程的温策尔-克拉默斯-布里渊（WKB）方法
Relativistic dynamical effects in proton emission: the Wentzel-Kramers-Brillouin method for 1+1 dimensional Dirac equation
AI总结：
该研究从1+1维狄拉克方程出发，用WKB方法推导相对论穿透概率，发现薛定谔等效势更准确，纳入该势会降低穿透概率、增加半衰期，且效应随轨道角动量升高更显著，在$^{144}\mathrm{Tm}$半衰期中达约84%。
AI中文摘要：
从1+1维（一个空间和一个时间维度）狄拉克方程出发，我们采用温策尔-克拉默斯-布里渊（WKB）近似推导对应的相对论穿透概率。推导表明，半经典动量由薛定谔等效势 $U_{\text{eff}}(r) = S(r) + \frac{E}{m}V(r) + \frac{S^{2}(r)-V^{2}(r)}{2m}$ 决定，而非相对论量子隧穿研究中广泛采用的标量势与矢量势的简单和 $S(r)+V(r)$。随后，我们通过比较采用 $U_{\text{eff}}(r)$ 和采用 $S(r)+V(r)$ 得到的结果，量化质子发射中的相对论动力学效应。纳入 $U_{\text{eff}}(r)$ 会系统性降低穿透概率和碰撞频率，进而增加预测的半衰期。相对论动力学效应随轨道角动量升高而更显著，在 $^{144}\mathrm{Tm}$ 的半衰期中可达到约84%。
英文摘要：
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 Schrödinger-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 $^{144}\mathrm{Tm}$.
