Pairing Nambu-Goldstone Modes within Nuclear Density Functional Theory

We show that the Nambu-Goldstone formalism of the broken gauge symmetry in the presence of the $T=1$ pairing condensate offers a quantitative description of the binding-energy differences of open-shell superfluid nuclei. We conclude that the pairing-rotational moments of inertia are excellent pairing indicators, which are free from ambiguities attributed to odd-mass systems. We offer a new, unified interpretation of the binding-energy differences traditionally viewed in the shell model picture as signatures of the valence nucleon properties. We present the first systematic analysis of the off-diagonal pairing-rotational moments of inertia and demonstrate the mixing of the neutron and proton pairing-rotational modes in the ground states of even-even nuclei. Finally, we discuss the importance of mass measurements of neutron-rich nuclei for constraining the pairing energy density functional.

Figure 1

Neutron pairing-rotational energy measured from a reference state in ${}^{116}\mathrm{Sn}$. The parabolic expression Eq. (1) with ${\lambda }_n$ and ${\mathcal{J}}_{nn}$ evaluated for the reference nucleus is shown as a solid line. The HFB (squares) and experimental (circles) values have been extracted from binding energies according to $E_{\text{rot}}^{\text{pair}}=E(N)-E(66)-{\lambda }_n(66)(N-66)$.

Figure 2

Chemical potential (top panels) and pairing-rotational moment of inertia (bottom panels) for (a) Sn, (b) Pb, and (c) Ca isotopes, and (d) $N=50$ isotones. Spherical UNEDF1-HFB solutions with mixed (squares), volume (circles), and surface (triangles) pairing are compared to experimental data from Ref. [44]. For $N=50$ isotones, Belyaev moments of inertia are also shown by lines without symbols.

Figure 3

Pairing-rotational moments of inertia ${\mathcal{J}}_{nn}$ (a,d), ${\mathcal{J}}_{np}$ (b,e), and ${\mathcal{J}}_{pp}$ (c,f) for open-shell Er isotopes (left) and $N=100$ isotones (right) exhibiting static neutron and proton pairing.