System of equations of the quasiparticle-phonon nuclear model with allowance for phonon scattering at finite temperature

The formalism of the quasiparticle-phonon nuclear model, which describes interaction with $(2p-2h)$ configurations, is developed for the case of a finite temperature. With exact allowance made for the population numbers of the single-phonon levels, a system of coupled equations is obtained for the Green's functions with the propagators of transition from one to two phonons and vice versa, for the phonon correlations in the ground state, and for phonon scattering. A one-to-one correspondence is established between this system and the system of equations for the coefficients of the wave function of an excited state in the quasiparticle-phonon model at finite temperature. The effect of phonon scattering is estimated numerically, and the diagrams summed in the quasiparticle-phonon model in this case are compared with the diagrams of other approaches.