Effect of the Multipole Pairing and Particle-Hole Fields in the Particle-Vibration Coupling ofPb209. I.

The particle-vibration-coupling formalism for nuclei around closed shells is studied and a microscopic description is given. We consider as collective modes of excitation of the closed-shell-system ($N_0$) ground state both particle-hole excitations and excitations that change the number of particles by 2 (pairing mode). To describe the pairing modes, multipole pairing residual interactions are used. The odd particle is allowed to couple to both types of vibrations. The coupling Hamiltonian is linear in the collective phonons and the particle variables. The effective Hamiltonian is constructed by a perturbative method. Explicit expressions are given for the spectroscopic amplitudes and spectroscopic factors associated with the ($N_0-1\mathrm{}$) ($t,p$) ($N_0+1\mathrm{}$) and ($N_0+2\mathrm{}$) ($p,d$) ($N_0+1\mathrm{}$) reactions, respectively. The concept of an effective transfer operator is used. Special attention is paid to the ${}_{}{}^{207}{}_{}{}^{}\mathrm{Pb}\mathrm{}(t,p)\mathrm{}$ and ${}_{}{}^{210}{}_{}{}^{}\mathrm{Pb}\mathrm{}(p,d)\mathrm{}$ reactions.