The s_1/2 continuum

Figures 5 and 6 show the s_1/2 localizations and phase shifts, respectively, of the upper HFB quasiparticle wave functions calculated numerically (dots), compared with the analytically calculated localizations and phase shifts of the PTG states (lines). We are interested in the low-energy s_1/2 continuum, and therefore the Figures show results well below the broad (39-18i)MeV resonance discussed in Sec. 4.3. The PTG results (no pairing) are shown as functions of the single-particle energy $\epsilon$, while the HFB results are plotted as functions of the quasiparticle energy E shifted by the Fermi energy $\lambda$, i.e., E+$\lambda$. In this way, at large energies the paired and unpaired energy scales coincide.

Three panels presented in Figs. 5 and 6 correspond to the PTG 3s_1/2 states being resonant (a), virtual (b), and weakly bound (c), cf. Table 2. In order to realize these three different physical situations, the bottom of the PTG potential has to be shifted by about 2.75MeV. This illustrates the ``width" of the zone where the 3s_1/2 states are virtual. Apparent shift in the single-particle energies is much smaller; the 3s_1/2state moves then from the -74keV bound-state energy to the 74keV resonance energy (Table 2). Even if the shift in the single-particle states is so small, the corresponding canonical states move by about 1.75MeV (Table 4). These latter states appear to be entirely unaffected by the dramatically different character of the single-particle 3s_1/2 states (Fig. 3). Their wave functions, shown in Fig. 7, are almost identical in the three cases (a)-(c).

Since the canonical states govern the pairing properties of the system, see discussion in Ref. [32], their positions explain the changes in the overall pairing intensity $\langle\Delta_N\rangle$, and in the 3s_1/2 occupation $v^2_{\mbox{\scriptsize {can}}}$, shown in Table 4. It is worth noting that shifting the canonical 3s_1/2level from its $\sim$0.8MeV distance to the Fermi level to a distance of $\sim$2.5MeV decreases the average pairing gap by as much as about 300keV.

Apart from the decrease of the pairing intensity described above, there is no other qualitative change in pairing properties when the 3s_1/2 state becomes unbound. The role of the weakly bound state is simply taken over by the low-energy continuum. The presence and position of the low-energy s_1/2 resonance is not essential for the pairing properties. In particular, it would have been entirely inappropriate to use this resonance, and its energy of 74keV, in any approximate scheme in which the full continuum was replaced by the resonances only.

In all the three cases (a)-(c), in the region of energies between 10 and 20MeV (Fig. 5) one can see the ``background" localization of the order of 0.2. These energies are far away from resonances, and therefore illustrate localizations of a ``typical" non-resonant continuum. Near the resonances (see Figs. 4a and 5a), localizations are larger, up to 0.3, but do not at all dominate over the background value. Therefore, based on estimate (44) one may expect that the pairing coupling of the resonant and non-resonant s_1/2 continuum is fairly similar.

The magnitude of this coupling seems to depend primarily on the s_1/2 single-particle localization in the 2-3MeV zone above the Fermi surface. (Note that cases (a)-(c) differ only by positions of the s_1/2 states; the spectra in other partial waves are kept unchanged.) The weakly bound 3s_1/2 PTG state generates large continuum localization right above the threshold, and the opposite is true for the low-lying 3s_1/2 PTG resonance. Therefore, the average pairing gap $\langle\Delta_N\rangle$ is substantially larger in case (c) than in case (a), and as a consequence, the HFB localizations in case (c) differ strongly from the PTG localizations, while in case (a) they are almost identical. The same pattern clearly also appears for the HFB phase shifts, as compared to the PTG phase shifts, Fig. 6.

Figure 8 shows norms N_n of the lower HFB components, Eq. (45). Since the lower HFB components are not mutually orthogonal, N_n cannot be associated with the occupation probabilities. On the other hand, the canonical occupation factors $v^2_{\mbox{\scriptsize {can}}}$ do play such a role, because the canonical states form an orthogonal basis. Comparing values of N_n (Fig. 8) and $v^2_{\mbox{\scriptsize {can}}}$ (Table 4), one sees that the canonical 3s_1/2 state collects all the occupation strength of the quasiparticle states in the low-energy continuum. Of course, numbers N_n scale with the overall pairing strength, i.e., they are smaller in case (c) than in case (a), but in every case all quasiparticle states below about 5MeV significantly contribute to $v^2_{\mbox{\scriptsize {can}}}$. Moreover, choosing only one quasiparticle state in this zone (call it resonance or not), would have provided only for about one third of the occupation factor. This illustrates again that the s_1/2 continuum has to be taken as a real continuum (discretized, if needed), and not through any single representative.