The d_3/2 continuum

Since the low-energy s_1/2 resonances are always very broad, the distinction between the resonant and non-resonant continuum is not very clear in this channel. On the other hand, pairing coupling of very narrow high-j resonances is not very different from that of any other bound states, and hence they are not very interesting to look at. Therefore, in addition to investigating the s_1/2resonances, we also study here an intermediate case of low-energy 2d_3/2 resonances, which can be narrow or broad depending on their positions inside the centrifugal barrier. For that we use the PTG' potential which contains the physical centrifugal barrier, Eq. (9), and chose several different values of the depth parameter $\nu_{d{3\over2}}$.

First, we study two cases listed in Table 2, chosen in such a way that the resonances are located at two different positions deep inside the centrifugal barrier. (For the chosen depths, in the PTG potential the 2d_3/2 states are resonant (d) or virtual (e); their analytical energies are given in Table 2.) Although for the physical barrier the analytical results are not available, one can estimate the resonance energies to be about $\epsilon_{\mbox{\scriptsize {res}}}$$\simeq$(1.5-0.4i)MeV (d) and $\epsilon_{\mbox{\scriptsize {res}}}$$\simeq$(0.9-0.2i)MeV (c), respectively. In the two cases, the resonances are located at about 3.5MeV and 4.1MeV below the top of the barrier which is about 5MeV high. One can see that the inclusion of the physical barrier moves the PTG resonance (Table 2) up in energy and decreases its width by a factor of eight, while the PTG virtual state is transformed into a true, rather narrow resonance.

Figure 9 shows the HFB and PTG' localizations of the d_3/2 continuum states. Of course, the low-energy PTG' resonances create narrow peaks of localizations at the resonant energies, and having resonance widths. Insets in Fig. 9 show that the pairing correlations shift the PTG' localizations to slightly higher energies, but otherwise the HFB and PTG' localizations are very similar. Beyond the narrow resonances, the HFB and PTG' localizations reach the non-resonant background values of about 0.3-0.4, which are almost unaffected by the next, very broad d_3/2 resonance at about 40MeV.

Apart from the change in overall pairing intensity, see Table 4, norms of the lower HFB components N_n shown in Fig. 10 closely follow the shape of localizations. In order to further visualize this property, in Figs. 11 and 12 we show results corresponding to the 2d_3/2resonances when they are moved up to the top of the centrifugal barrier. Six panels presented in the Figures have been obtained by using the depth parameters $\nu_{d{3\over2}}$=4.9(0.1)4.4, which corresponds to shifting the bottom of the d_3/2 central potential well from -48.4 to -39.8MeV. As a result, the 2d_3/2 resonances move from about 0.9MeV to 6MeV (at the same time the barrier also slightly increases, from 5 to 6MeV). Hence, the top panels in Figs. 11 and 12 correspond to a broad resonance located right at the top of the centrifugal barrier.

Localizations of the PTG' d_3/2 continuum states (Fig. 11) closely follow the pattern of broadening and rising resonances. Apart from the lowest two panels, corresponding to low-energy resonances, the HFB localizations (dots) do not differ from the PTG' results (lines). This is because the larger the distance of the resonance from the Fermi surface ($\lambda_N$$\simeq$-0.4keV in all cases), the weaker are, of course, modifications induced by pairing correlations. Moreover, the rising resonance leaves behind (at low energies) very low values of localizations, which additionally contributes to a decreasing of the overall pairing intensities.

Interestingly, norms of the lower HFB components N_n, presented in Fig. 12, closely follow the pattern of localizations. It means that values of localizations of the PTG' single-particle states indeed decide about the pairing coupling of the HFB quasiparticle continuum states, cf. Eq. (44). However, again none of the quasiparticle states (close to resonant energies or not) can be used as a single representative of the continuum phase space. Even for the resonances located very deeply inside the barrier (the lowest panels in Fig. 12), norms N_n for quasiparticle states at resonance energies do not exhaust the occupation numbers $v_{\mbox{\scriptsize {can}}}$.

In Figs. 11 and 12, arrows indicate values of the canonical 2d_3/2 energies $\epsilon_{\mbox{\scriptsize {can}}}$. With resonances moving up, these energies increase too, and appear always slightly above the resonant energy. These canonical states represent correctly the whole region of the quasiparticle continuum within, and above the centrifugal barrier. The 1d_3/2 canonical states are deeply bound ($\sim$-25MeV), and the 3d_3/2 canonical states are very high ($\sim$+25MeV), so indeed those corresponding to 2d_3/2 (as shown in the Figures) are well separated representatives of a wide region of the phase space. Therefore, any approximate scheme, aiming at a correct description of pairing correlations near continuum, should vie to find these states at right places. Unfortunately, the HF+BCS method which uses resonant continuum states only, does not seem to have a chance to attain such a goal. However, similarity of localizations and norms N_n found in the present study may give hope for approximate solutions of HFB equations. Of course, when the spherical symmetry is conserved, direct solution of the HFB problem is easy enough that no approximated methods are necessary. However, this is not the case for the deformed systems, where moreover, single-particle resonances split and become much more difficult to treat.