Abstract
The termination of the slow neutron capture process (s-process) in the Pb--Bi mass region plays a key role in shaping the abundances of the heaviest stable nuclei produced in stellar environments. In this work, we revisit the classical description of the s-process termination developed by Clayton and co-authors, incorporating modern experimental neutron capture cross sections and exploring the sensitivity of the results to variations in neutron irradiation parameters. The reaction network describing the Pb--Bi--Po termination cycle is reformulated as a system of Bateman equations and solved numerically using the burn-up matrix formalism combined with the Pad\'e approximation for the matrix exponential. We investigate the influence of updated $(n,\gamma)$ cross sections for Pb and Bi isotopes, as well as variations in the shape and scale parameters of the neutron exposure distribution, on the isotopic abundances, the timescale required to reach equilibrium, and the behavior at maximum neutron exposure. Our results confirm the robustness of the quasi-stationary equilibrium condition $\sigma_A N_A = \mathrm{const}$ under modern nuclear data, while demonstrating that updated cross sections lead to a noticeable redistribution of isotopic abundances and modify the timescale for reaching the equilibrium plateau. The contribution of the s-process termination to the total abundances of Pb and Bi isotopes is further examined using different neutron exposure distributions. We find that the resulting s-process abundances are largely insensitive to the neutron capture cross section of $^{209}\mathrm{Bi}$ over the investigated range, confirming that $^{208}\mathrm{Pb}$ effectively acts as the termination point of the s-process. Among the considered exposure models, the exponential neutron exposure distribution provides the best overall description of the final s-process abundances. These results are in good agreement with observational abundance constraints and demonstrate that the main qualitative features of the classical Clayton scheme remain valid, while quantitative predictions are significantly refined by modern nuclear data and irradiation models.