Self-consistency in the phonon space of the particle-phonon coupling model

In the paper the nonlinear generalization of the time blocking approximation (TBA) is presented. The TBA is one of the versions of the extended random-phase approximation (RPA) developed within the Green-function method and the particle-phonon coupling model. In the generalized version of the TBA the self-consistency principle is extended onto the phonon space of the model. The numerical examples show that this nonlinear version of the TBA leads to the convergence of results with respect to enlarging the phonon space of the model.

Figure 1

Dependence of the energy of $3_1^{-}$ state calculated in the CBA with ${\zeta }_{\text{min}}^2=0.5$ (red solid lines) and in the standard TBA (black dashed lines) on the value of the inverse cutoff parameter $1/{\kappa }_{\text{min}}$. The fine dashed horizontal line indicates the energy as found in a previous paper using standard TBA with optimized cutoff parameter $v_{\text{min}}=0.05$ [34]. The experimental values are given in Table 1.

Figure 2

Same as in Fig. 1 but for the reduced probabilities $B(E3;0_{\text{g.s.}}^{+}\to 3_1^{-})$ in units of their experimental values.

Figure 3

Same as in Fig. 1 but for the mean energies (Lorentzian parameter $E_0$) of the giant dipole resonance.

Figure 4

The giant dipole resonance (GDR) in ${}^{16}\mathrm{O}$ calculated in the CBA (red solid line), in standard TBA (black dashed line), and in RPA (blue dotted line). Experimental data from Ref. [37] are shown by the green circles.

Figure 5

Same as in Fig. 4 but for ${}^{40}\mathrm{Ca}$. Experimental data are taken from Ref. [38].

Figure 6

Same as in Fig. 4 but for ${}^{208}\mathrm{Pb}$. Experimental data are taken from Ref. [39].

Figure 7

Dependence of the number of the active TBA phonons in the CBA with ${\zeta }_{\text{min}}^2=0.5$ (red solid lines) and the number of the RPA phonons in the ordinary TBA (black dashed lines) on the value of the cutoff parameter ${\kappa }_{\text{min}}$.

Figure 8

Differences between the energies of $3_1^{-}$ state in the nucleus ${}^{16}\mathrm{O}$ as calculated in the CBA with different values of the cutoff ${\zeta }_{\text{min}}^2$, namely $\mathrm{\Delta }\omega \left(3_1^{-}\right)=\omega (3_1^{-};{\zeta }_{\text{min}}^2=0.6)-\omega (3_1^{-};{\zeta }_{\text{min}}^2=0.5)$ (red solid lines) and $\mathrm{\Delta }\omega \left(3_1^{-}\right)=\omega (3_1^{-};{\zeta }_{\text{min}}^2=0.7)-\omega (3_1^{-};{\zeta }_{\text{min}}^2=0.5)$ (black dashed lines).

Figure 9

Same as in Fig. 8 but for the nucleus ${}^{40}\mathrm{Ca}$.

Figure 10

Same as in Fig. 8 but for the nucleus ${}^{208}\mathrm{Pb}$.