Renormalizing random-phase approximation by using exact pairing

A fully self-consistent renormalized random-phase approximation is constructed based on the self-consistent Hartree-Fock mean field plus exact pairing solutions (EP). This approach exactly conserves the particle number and restores the energy-weighted sum rule, which is violated in the conventional renormalized particle-hole random-phase approximation for a given multipolarity. The numerical calculations are carried out for several light-, medium-, and heavy-mass nuclei such as ${}^{22}\mathrm{O}$, ${}^{60}\mathrm{Ni}$, and ${}^{90}\mathrm{Zr}$ by using the effective MSk3 interaction. To study the pygmy dipole resonance (PDR), the calculations are also performed for the two light and neutron-rich ${}^{24,28}\mathrm{O}$ isotopes, whose PDRs are known to be dominant. The results obtained show that the inclusion of ground-state correlations beyond the random-phase approximation (RPA) by means of the occupation numbers obtained from the EP affects the RPA solutions within the whole mass range, although this effect decreases with increasing the mass number. At the same time, the antipairing effect is observed via a significant reduction of pairing in neutron-rich nuclei. The enhancement of PDR is found in most neutron-rich nuclei under consideration within our method.

Figure 1

The IS dipole transition probabilities of ${}^{22}\mathrm{O}$ obtained within the SC-HFEPRPA with $\eta =1$ and $\eta \ne 1$.

Figure 2

The IS (a) and IV (b) dipole transition probabilities of ${}^{22}\mathrm{O}$ obtained within the SC-HFEPRPA by using different cutoff energies $E_c$.

Figure 3

Same as Fig. 2 but for ${}^{90}\mathrm{Zr}$.

Figure 4

The IS (a) and IV (b) dipole transition probabilities of ${}^{22}\mathrm{O}$ obtained within the SC-HFEPRPA by removing one and two hole levels.

Figure 5

The IS (a) and IV (b) dipole transition probabilities of ${}^{22}\mathrm{O}$ obtained within the SC-HFEPRPA by removing and adding two particle levels.

Figure 6

The IS (a) and IV (b) dipole transition probabilities of ${}^{22}\mathrm{O}$ obtained within the SC-HFEPRPA by using different GSC factors $D_{\text{ph}}$.

Figure 7

The IS (a) and IV (b) dipole transition probabilities of ${}^{22}\mathrm{O}$ obtained within different approaches. The sticks and lines represent the probabilities $B(E1,0\to 1^{-})$ and strength functions $S\left(E\right)$ (the smoothing parameter $\varepsilon =0.4\mathrm{MeV}$), respectively.

Figure 8

Same as Fig. 7 but for ${}^{60}\mathrm{Ni}$.

Figure 9

Same as Fig. 7 but for ${}^{90}\mathrm{Zr}$.

Figure 10

GDR cross-sections for ${}^{22}\mathrm{O}$, ${}^{60}\mathrm{Ni}$, and ${}^{90}\mathrm{Zr}$ obtained within different methods. Results in (a), (b), and (c) are obtained by using the MSk3 interaction, whereas those in (d) are obtained by using the ${\mathrm{SkM}}^{*}$ interaction. The experimental data for ${}^{22}\mathrm{O}$, ${}^{60}\mathrm{Ni}$, and ${}^{90}\mathrm{Zr}$ are taken from Refs. [16, 36, 37], respectively. The dotted line in (a) is predicted by the SC-HFEPRPA by using $E_c=130\mathrm{MeV}$.