  Odd-even double mass differences (DMD) of magic nuclei are found within the
approach starting from the free $NN$ interaction with account for
particle-phonon coupling (PC) effects. We consider three PC effects: the phonon
induced effective interaction, the renormalization of the "ends" due to the
$Z$-factor corresponding to the pole PC contribution to the nucleon mass
operator and the change of the single-particle energies. The perturbation
theory in $g^2_L$, where $g_L$ is the vertex of the $L$-phonon creation, is
used for PC calculations.
  PC corrections to single-particle energies are found self-consistently with
an approximate account for the tadpole diagram. Results for magic $^{40,48}$Ca,
$^{56,78}$Ni, $^{100,132}$Sn and $^{208}$Pb nuclei are presented. For lighter
part of this set of nuclei, from $^{40}$Ca till $^{56}$Ni, the cases divide
approximately in half between those where the PC corrections to DMD values make
agreement with the data better and the ones with the opposite result. In the
major part of the cases of worsening of description of DMD, a poor
applicability of the perturbation theory for the induced interaction is the
most probable reason of the phenomenon. For intermediate nuclei, $^{78}$Ni and
$^{100}$Sn, there is no sufficiently accurate data on masses of nuclei
necessary for finding DMD values. Finally, for heavier nuclei, $^{132}$Sn and
$^{208}$Pb, PC corrections always make agreement with the experiment better.

Nuclear Theory;Nuclear Experiment