Polarization corrections to single-particle energies studied within the energy-density-functional and quasiparticle random-phase approximation approaches

Background: Models based on using perturbative polarization corrections and mean-field blocking approximation give conflicting results for masses of odd nuclei.

Purpose: We systematically investigate the polarization and mean-field models, implemented within self-consistent approaches that use identical interactions and model spaces, to find reasons for the conflicts between them.

Methods: For density-dependent interactions and with pairing correlations included, we derive and study links between the mean-field and polarization results obtained for energies of odd nuclei. We also identify and discuss differences between the polarization-correction and full particle-vibration-coupling (PVC) models. Numerical calculations are performed for the mean-field ground-state properties of deformed odd nuclei and then compared to the polarization corrections determined using the approach that conserves spherical symmetry.

Results: We have identified and numerically evaluated self-interaction (SI) energies that are at the origin of different results obtained within the mean-field and polarization-correction approaches.

Conclusions: Mean-field energies of odd nuclei are polluted by the SI energies, and this makes them different from those obtained using polarization-correction methods. A comparison of both approaches allows for the identification and determination of the SI terms, which then can be calculated and removed from the mean-field results, giving the self-interaction-free energies. The simplest deformed mean-field approach that does not break parity symmetry is unable to reproduce full PVC effects.

Figure 1

Comparison of polarization corrections of selected orbitals in ${}^{100}$Sn, determined using the HF and RPA methods and Skyrme EDF SV [48]; see text. Lines connect the values obtained for different projections of the angular momentum $|m_{\lambda }|=\frac{1}{2},...,j_{\lambda }$ (from left to right).

Figure 2

Polarization corrections of $|m_{\lambda }|=j_{\lambda }$ orbitals in ${}^{100}$Sn, determined by not including (left bars) and including (right bars) the orbital-dependent terms in the RPA matrices; see text. The order of orbitals is the same as shown in Fig. 1. Contributions coming from four RPA channels $J^{\pi }=0^{+}$, 1${}^{+}$, 2${}^{+}$, and 3${}^{+}$ are shown separately (note very different scales).

Figure 3

Contributions to polarization corrections of $|m_{\lambda }|=j_{\lambda }$ orbitals in ${}^{100}$Sn, coming from different $J^{\pi }$ RPA channels, determined for the Skyrme EDF SV [48]. The order of orbitals is the same as shown in Fig. 1. Contributions are ordered according to the value of $J$, with the $0^{+}$ channels shown nearest the abscissa.

Figure 4

Same as in Fig. 3, but for the $|m_{\lambda }|=\frac{1}{2}$ orbitals.

Figure 5

Same as in Fig. 1, but for the Skyrme EDF SLy5 [49]. The RPA results correspond to the SIF terms in Eq. (46), whereas $\mathrm{RPA}+\mathrm{SI}$ denotes both SIF and SI contributions combined.

Figure 6

The SIF and SI contributions to the polarization corrections of Eq. (46), calculated in ${}^{100}$Sn for the Skyrme EDF SLy5.

Figure 7

Same as in Fig. 3, but for the contributions to the RPA SIF polarization corrections of $|m_{\lambda }|=j_{\lambda }$ orbitals, determined for the Skyrme EDF SLy5.

Figure 8

Same as in Fig. 7, but for the $|m_{\lambda }|=\frac{1}{2}$ orbitals.

Figure 9

Comparison of the QRPA $\mathrm{SIF}+\mathrm{SI}$ (symbols) and HFB (lines) polarization corrections to quasiparticle energies in tin isotopes.

Figure 10

Same as in Fig. 6, but for ${}^{110}$Sn.

Figure 11

Same as in Fig. 6, but for ${}^{120}$Sn.

Figure 12

Same as in Fig. 7, but for ${}^{110}$Sn.

Figure 13

Same as in Fig. 8, but for ${}^{110}$Sn.

Figure 14

Average QRPA ($\mathrm{SIF}+\mathrm{SI}$) polarization corrections $\delta E_{\text{SIF}}+E_{\text{SI}}$, Eqs. (pp1), in tin isotopes.

Figure 15

Same as in Fig. 14, but for the QRPA SIF polarization corrections only.

Figure 16

Average SI corrections $E_{\text{SI}}$ (A18a) plotted for quasiparticles closest to the Fermi level in open-shell tin isotopes.