Phonon–particle coupling effects in the single-particle energies of semi-magic nuclei

A method is presented to evaluate the particle–phonon coupling (PC) corrections to the single-particle energies in semi-magic nuclei. In such nuclei, always there is a collective low-lying $2^+$ phonon, and a strong mixture of single-particle and particle–phonon states often occurs. As in magic nuclei the so-called $g_{\mathrm{L}^2}$ approximation, where $g_{\mathrm{L}}$ is the vertex of the $L$-phonon creation, can be used for finding the PC correction $\delta\Sigma^{\mathrm{PC}}(\varepsilon)$ to the initial mass operator $\Sigma_0$. In addition to the usual pole diagram, the phonon “tadpole” diagram is also taken into account. In semi-magic nuclei, the perturbation theory in $\delta\Sigma^{\mathrm{PC}}(\varepsilon)$ with respect to $\Sigma_0$ is often invalid for finding the PC-corrected single-particle energies. Instead, the Dyson equation with the mass operator $\Sigma(\varepsilon)=\Sigma_0+\delta\Sigma^{\mathrm{PC}}(\varepsilon)$ is solved directly, without any use of the perturbation theory. Results for a chain of semi-magic Pb isotopes are presented.