The density-dependent cluster model combined with relativistic mean-field theory is used to explore the structure and $\alpha$ decay for the neutron-deficient nuclei with $89\le Z\le 94$, including two newly discovered nuclei ${}^{207}\mathrm{Th}$ [Phys. Rev. C 105, L051302 (2022)] and ${}^{214}\mathrm{U}$ [Phys. Rev. Lett. 126, 152502 (2021)]. The effective nucleon-nucleon interactions and matter density distributions from the relativistic mean field are employed to construct the $\alpha$-daughter potential with a double-folding model. The Pauli blocking effect is considered by normalizing the strength of the $\alpha$-daughter potential with the Bohr-Sommerfeld quantization condition. The $\alpha$-preformation factor is calculated with the cluster formation model. The calculated $\alpha$-decay half-lives for the 106 observed nuclei with $89\le Z\le 94$ are in excellent agreement with experimental data. Extending this model to the unknown nuclei ${}^{201–204}\mathrm{Ac}$, ${}^{205–206}\mathrm{Th}$, ${}^{209–210}\mathrm{Pa}$, ${}^{212–213,220}\mathrm{U}$, ${}^{215–218,221}\mathrm{Np}$, and ${}^{220–227}\mathrm{Pu}$, the evolution of the $N=126$ shell closure with neutron number is explored for the high-$Z$ isotopes. The available $\alpha$-decay energies, preformation factors, and $\alpha$-decay half-lives show a regular change with increasing neutron number. Especially, the robustness of the $N=126$ shell closure is shown up to the Pu isotopes.

• Received 29 August 2022
• Revised 8 November 2022
• Accepted 1 December 2022

DOI:https://doi.org/10.1103/PhysRevC.106.064310