Phonon-particle coupling effects in odd-even double mass differences of semi-magic nuclei

A method is developed to consider the particle-phonon coupling (PC) effects in the calculation of the odd-even double mass differences (DMD) in semi-magic nuclei starting from the free $NN$ potential. The PC correction $\delta \Sigma^{\text{PC}}$ to the mass operator $\Sigma$ is found in $g_{\mathrm{L}^2}$-approximation, $g_{\mathrm{L}}$ being the vertex of creating the $L$-phonon. The tadpole term of the operator $\delta \Sigma^{\text{PC}}$ is taken into account. The method is based on a direct, without any use of the perturbation theory, solution of the Dyson equation with the mass operator $\Sigma(\varepsilon) = \Sigma_0 + \delta \Sigma^{\text{PC}}(\varepsilon)$ for finding the single-particle energies and $Z$-factors. In its turn, they are used as an input for finding different PC corrections to the DMD values. Results for a chain of even semi-magic nuclei $^{200-206}$Pb show that the inclusion of the PC corrections makes agreement with the experimental data significantly better.