  Phonon coupling (PC) corrections to magnetic moments of odd neighbors of
magic and semi-magic nuclei are analyzed within the self-consistent Theory of
Finite Fermi Systems (TFFS) based on the Energy Density Functional by Fayans et
al. The perturbation theory in g_L^2 is used where g_L is the phonon-particle
coupling vertex. A model is developed with separating non-regular PC
contributions, the rest is supposed to be regular and included into the
standard TFFS parameters. An ansatz is proposed to take into account the
so-called tadpole term which ensures the total angular momentum conservation
with g_L^2 accuracy. An approximate method is suggested to take into account
higher order terms in g_L^2. Calculations are carried out for four odd-proton
chains, the odd Tl, Bi, In and Sb ones. Different PC corrections strongly
cancel each other. In the result, the total PC correction to the magnetic
moment in magic nuclei is, as a rule, negligible. In non-magic nuclei
considered it is noticeable and, with only one exception, negative. On average
it is of the order of -(0.1 - 0.5) \mu_N and improves the agreement of the
theory with the data. Simultaneously we calculated the gyromagnetic ratio
g_L^{ph} of all low-lying phonons in 208Pb. For the 3^-_1 state it is rather
close to the Bohr-Mottelson model prediction whereas for other L-phonons, two
5^- and six positive parity states, the difference from the Bohr-Mottelson
values is significant.
