In this work we consider neutron particle-hole excitations across the $N=126$ shell gap and perform calculations for the $N=126$ isotones, ${}^{210}\mathrm{Po},{}^{212}\mathrm{Rn},{}^{214}\mathrm{Ra}$, and ${}^{216}\mathrm{Th}$, using the nucleon-pair approximation of the shell model with particle-hole excitations. The Hamiltonian is constructed starting from the realistic low-momentum interaction, to which simple and important modifications are further made. The calculated energy levels, magnetic moments, and $B\left(E2\right)$'s are in a good agreement with available experimental data, for both the low-lying states characterized by the neutron $N=126$ closed-core state and those by neutron particle-hole excitations, and on this basis, the $3_1^{-}$ states manifest themselves with a nature of the strong mixing between the two types of configurations. The available experimental datum of $B(E3;3_1^{-}\to 0_{\mathrm{g}.\mathrm{s}.}^{+})$ is reproduced, and the neutron particle-hole excitations are found to play a decisive role here. In addition, the experimental data for the $8_1^{+}$ states, in particular the $B(E3;{11}_1^{-}\to 8_1^{+})$'s, are well reproduced and interpreted.

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• Electromagnetic transitions
• Nuclear structure & decays
• Nucleon-nucleon interactions
• 190 ≤ A ≤ 219
• Shell model