Distribution of poles as a function of deformation

When increasing the quadruple deformation, states with smallest (largest) angular momentum projections onto the symmetry axis, $\Omega$, become more (less) bound on the prolate side, and the opposite holds for the oblate side. For states located above the shell gap, this means that low-$\Omega$ and high-$\Omega$ orbitals become more occupied with increasing prolate and oblate deformation, respectively. Therefore, at some deformation, these orbitals cross the Fermi energy and the corresponding poles cross the unit circle. An analogous situation may also occur for orbitals located below the shell gap, whereupon high-$\Omega$ and low-$\Omega$ Nilsson orbitals become less occupied with increasing prolate and oblate deformation, respectively, and also may cross the Fermi energy. We wish to emphasize that the problem occurs not at the point where the orbitals from above and below the shell gap cross each other, leading to a configuration change, but at deformation where either of these orbitals crosses the Fermi energy.

Figure 7: (color online) Neutron poles $z_n$ in $^{18}$O (28) as functions of quadrupole deformation $\beta$, calculated within the HFB-SIII method with volume pairing interaction.

Such a case is illustrated in Fig. 7 for the nucleus $^{18}$O. In pure HFB calculations (no LN correlations included), this nucleus has neutron pairing only. At the spherical shape, the 1d$_{5/2}$ shell is located above the $N$=8 shell gap, i.e., it has particle character ($\vert z\vert>1$). The three-fold degeneracy of this shell ($q$=3) makes the contribution from this pole to the projected energy vanish. At nonzero deformations, however, the degeneracy is lifted and three individual poles ($q$=1) appear in the complex plane. Moreover, near $\beta=0.12$ and $\beta=-0.12$, poles corresponding to the $\Omega$=1/2 and $\Omega$=5/2 Nilsson levels cross the unit circle $\vert z\vert=1$.

Figure 8: (color online) Neutron (left) and proton (right) poles $z_n$ (28) as functions of quadrupole deformation $\beta$ calculated for $^{32}$Mg with the SIII functional and volume pairing interaction.

Figure 9: (color online) Similar to Fig. 8 except for canonical energies $e_k$.

The situation is much worse for nuclei having more single-particle states with poles close to the unit circle. The neutron-rich nucleus $^{32}$Mg is such a complicated case illustrated in Fig. 8. This example is calculated in the HFB+LN approach, in which both neutron and proton pairing is nonzero. For completeness, canonical single-particle energies $e_n$ associated with the poles $z_n$ are plotted in Fig. 9

As can be seen in Fig. 8, there appear numerous crossings of poles with the unit circle as a function of deformation. On the prolate side, neutron poles 1f $_{7/2,\Omega=1/2}$ (Nilsson level [330]1/2) and 1d$_{3/2,3/2}$ ([202]3/2) cross the unit circle at the same deformation where they cross one another. At larger deformation, the same situation occurs for the 1f$_{7/2,3/2}$ ([321]3/2) and 1d$_{3/2,1/2}$ ([200]1/2) orbitals. For protons, a single 1d$_{5/2,5/2}$ ([202]5/2) orbital crosses the unit circle at small deformations. On the oblate side, neutron orbitals 1f$_{7/2,5/2}$ ([312]5/2) and 1d$_{3/2,1/2}$ cross the unit circle at different deformations, near the point where they cross one another, while the proton 1d$_{3/2,1/2}$ orbital stays near the unit circle for a wide range of deformations.

As discussed in Secs. 3.3 and 4.2, results of the PNP, at least for the density-independent terms, must only depend on the residues of poles that are inside the integration radius $r_0$. However, whenever a given pole crosses the integration contour, the projected energy must undergo a sudden jump as a function of deformation. This jump is, of course, equal to the residue at this pole. The fact that a given pole crosses the integration contour could be without consequence, provided the contour is shifted back to always stay between the same poles. This is always possible, as long as the poles do not cross around the contour. It is obvious that whenever they do, the projected energy may have a sudden jump that cannot be avoided by a contour shift. On the other hand, when two poles cross precisely at the integration contour, the corresponding degeneracy factor $q$ increases by a unity, and the poles may simply disappear (at least for the terms that show polynomial density dependence), in which case the projected energy may stay smooth. Such cases are studied in the next section.