Local p-h and p-p densities are basic elements of the HFB-Skyrme
theory: they determine self-consistent fields; hence
the static properties of
the nucleus such as the binding energy, radius, and shape.
The particle and pairing local HFB+SLy4 neutron
densities
and
calculated for several values of
are displayed in Fig. 2 for 150Sn.
With decreasing ,
the p-p density
develops
a long tail extending
towards large distances. This is a direct consequence of the attractiveness
of DDDI at low densities when
is small.
While in the nuclear interior, the p-h density
depends
extremely weakly on the actual form of the pairing interaction; its
asymptotic values are significantly increased
when
gets small (see inset).
Moreover, one observes a clear development of a halo structure,
i.e., a smooth exponential decrease, that for
=1 starts
at r
6fm, for small
is interrupted at
r
9fm, and replaced by a significantly slower decrease of the
density.
This is a direct consequence of
the self-consistent coupling between p-h and p-p parts of the
HFB Hamiltonian. That is, the increased probability of finding a
correlated pair of neutrons at large distances impacts the
probability of finding a single neutron in the halo region.
It is to be noted that the `halo' effect seen for =1/6
solely results from pairing, and it is not related to reduced binding
energy. In contrast, as it is shown below, increased pairing
correlations lead to greater separation energies and lower chemical
potentials, i.e., to increased particle stability.
As emphasized in Ref. [2],
pairing correlations can be enhanced in weakly bound nuclei due to
increased surface effects and the closeness of the particle continuum.
In turn, pairing can influence quite dramatically the asymptotic
properties of density distributions in drip-line systems.
This is nicely illustrated in Fig. 3 which compares
the HFB+SLy4 densities calculated with =1/2
for 120Sn (well bound),
150Sn (weakly bound), and 170Sn (very weakly bound)
drip-line nuclei. One can see that adding neutrons results in a
simultaneous increase of density both in the nuclear interior and in the
surface region. At very large distances, the asymptotic behavior of
reflects the gradual rise of the Fermi level
with a neutron
number. However, this effect is much stronger for the pairing density
[2]. Indeed, as seen in Fig. 3, as compared to
120Sn, there is a dramatic increase in
in the
outer regions of weakly bound nuclei 150Sn and, in particular,
170Sn. These calculations indicate that for small values of
the box size should be chosen as very large if one aims for a very
accurate description of HFB densities at large distances.
The structure of HFB densities determines
the behavior of the self-consistent p-h and p-p potentials.
Figure 4 shows the behavior of
(6) (local part only - see discussion in Ref. [2])
and
(7) obtained for neutrons in the weakly bound
nucleus 150Sn
in the HFB+SLy4 model for several values of
.
The behavior
of the pairing potential
is consistent with the pattern
shown in Fig. 1.
Indeed, for DDDI, the pairing potential is proportional to the
product of the pairing
density
and the pairing strength factor
(11).
Consequently,
is essentially
peaked around the nuclear surface, and both its minimum and range
shift towards
larger values of r with decreasing
.
For
=1/6,
the pairing potential is still sizeable at large distances reaching 14fm
(i.e., twice the nuclear radius).
The central p-h potential U(r) very weakly depends on the form of the
pairing interaction. It is only for small values of
that,
due to a direct contribution from the pairing density to the p-h potential
[see Eq. (A.5a) in Ref. [19]], a
small barrier develops just beyond the nuclear surface. That is, the central
neutron potential becomes slightly repulsive at r
9fm. However,
this effect is more than compensated by the increased pairing potential
and the total binding energy decreases.
Figure 5 compares the
HFB+SLy4 HFB potentials calculated with =1/2
for
120,150,170Sn. With increasing neutron number, the radius of
the p-h potential increases, and the potential becomes more wide
in the outer region (i.e., it becomes more diffused). The p-p potential
becomes more surface-peaked and its range increases. By analyzing
Figs. 4 and 5, one can conclude that it is in
N-rich weakly bound nuclei that the density dependence of the pairing
interaction is most important.