The particle-number-conserving (PNC) method in the framework of the deformed shell model is employed to study the properties of the newly discovered short-lived neutron-deficient nucleus ${}^{224}\mathrm{Np}$ and its $\alpha$ decay. The calculated energy of the $\alpha$ particle lies within 300 keV of the experimental data. This is the first application of the PNC method to the region of neutron-deficient nuclei. This work provides the first attempt to combine the microscopic PNC theory with empirical formulas to study the nuclear $\alpha$ decay. The configurations of ground states are assigned as $\pi 5/2^{-}\left[523\right]\otimes \nu 5/2^{+}\left[633\right]$ for ${}^{224}\mathrm{Np}$, $\pi 1/2^{-}\left[530\right]\otimes \nu 3/2^{+}\left[642\right]$ for ${}^{220}\mathrm{Pa}$, $\pi 3/2^{+}\left[651\right]\otimes \nu 1/2^{-}\left[501\right]$ for ${}^{216}\mathrm{Ac}$, and $\pi 7/2^{-}\left[514\right]\otimes \nu 3/2^{-}\left[501\right]$ for ${}^{212}\mathrm{Fr}$. The proton subshell at $Z=92$ disappeared in ${}^{224}\mathrm{Np}$. Low-lying excited states are predicted for nuclei along the $\alpha$-decay chain by the PNC method. Based on the PNC predicted $\alpha$-decay energy and the assigned configurations, the $\alpha$-decay half-lives are calculated by the empirical formulas, in which the angular momentum taken away by the $\alpha$ particle is taken into account. The angular momentum has an important effect on the $\alpha$-decay half-life. The errors of the empirical formula calculation are of two orders of magnitude with the experimental data.

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• Alpha decay
• Nuclear decay
• Nuclear many-body theory
• Nuclear spin & parity
• Nuclear structure & decays
• Shell model
• A ≥ 220