Nuclear Pairing Energy vs Mean Field Energy: Do They Talk To Each Other For Searching The Energy Minimum?

We study the evolution of the total binding energy (TBE) and pairing energy of Pb, Hg and Ar isotopes, as a function of the nuclear deformation. As for the nuclear model, we exploit a deformed relativistic Hartree-Bogoliubov theory in the continuum (DRHBc), and a deformed Skyrme Hartree-Fock plus BCS model. It is found that the dependence of pairing energy on the deformation is strongly correlated to that of the mean field energy, which is obtained by subtracting the pairing energy from the TBE; in other words, the energy minimum characterized by a large negative mean field energy has a smaller negative pairing energy or, equivalently, a smaller positive pairing gap, while a stronger pairing energy is found in the region away from the minimum of the total energy. Consequently, the two energies show an anti-symmetric feature in their deformation dependence, although the energy scales are very different. Moreover, since the pairing energy has a negative sign with respect to to the pairing gap, the evolution of mean field energy follows closely that of the pairing gap. This implies that the pairing energy (or pairing gap) and the mean field energy talk to each other and work together along the potential energy curve to determine the energy minimum and/or the local minimum.

💡 Research Summary

The authors investigate how the total binding energy (TBE) and the pairing energy of nuclei evolve with quadrupole deformation (β₂). Using two complementary microscopic frameworks – the deformed relativistic Hartree‑Bogoliubov theory in the continuum (DRHBc) with the PC‑PK1 functional and a density‑dependent zero‑range pairing force, and a deformed Skyrme Hartree‑Fock plus BCS (DSHF+BCS) approach – they calculate Pb, Hg and Ar isotopic chains that include spherical, weakly deformed, and shape‑coexistence candidates.

From the TBE they subtract the pairing contribution to obtain the mean‑field energy (E_MF). The central finding is that E_MF and the pairing energy (E_pair) display an almost perfect anti‑symmetric dependence on β₂. At the deformation that minimizes the total energy (usually β₂≈0 for closed‑shell nuclei), the mean‑field energy is most negative while the pairing energy is close to zero. As the nucleus is driven away from this minimum, the level density around the Fermi surface rises, the neutron and proton pairing gaps (Δ_n, Δ_p) increase, and consequently E_pair becomes increasingly negative (more binding from pairing). At the same time, E_MF becomes less negative, producing a complementary “cross‑talk” between the two components.

The authors illustrate this behavior with detailed plots for ²⁰⁸Pb, ¹⁹⁶,¹⁹⁸Pb, and the shape‑coexistence candidates ¹⁸⁴,¹⁸⁶Pb. In ²⁰⁸Pb the mean‑field energy reaches a deep minimum at β₂=0, while the pairing energy is essentially zero; for β₂≈±0.2 the pairing energy drops to about –5 MeV and the mean‑field energy rises by ~20 MeV. Similar anti‑correlated patterns are observed in the other isotopes, regardless of whether the deformation is prolate or oblate.

A schematic BCS expression, E_pair≈–½ ρ (Δ_n²+Δ_p²), explains the quantitative link: the pairing energy is proportional to the level density ρ at the Fermi surface and to the square of the pairing gaps. Because ρ and the gaps both increase with deformation, E_pair becomes more negative, while the mean‑field contribution, which is essentially the rest of the EDF energy, becomes less attractive.

The comparison between the relativistic DRHBc and the non‑relativistic Skyrme‑BCS calculations shows that the anti‑symmetric relationship is robust against the choice of functional and the inclusion of exchange terms. This indicates that the phenomenon is a generic feature of self‑consistent nuclear energy density functional calculations.

The paper argues that pairing should not be treated merely as a small correction to the mean field. Instead, pairing and the mean field “talk to each other” along the deformation path, jointly shaping the potential energy surface, determining both the global minimum and possible local minima associated with shape coexistence. This insight has practical implications for mass predictions, drip‑line studies, and the modeling of shape transitions, where neglecting the feedback between pairing and mean‑field energies could lead to inaccurate energy landscapes.

In summary, the study provides clear evidence that the deformation dependence of pairing energy is strongly correlated (in an opposite sense) with that of the mean‑field energy, and that this interplay is essential for locating energy minima in atomic nuclei.