An essential element in understanding the physics of weakly bound fermion systems, is the residual coupling between bound states and continuum. With decreasing particle separation energy, the zone of correlated states around the Fermi surface must incorporate the phase space of particle continuum. One of the most interesting discoveries in loosely bound nuclei is the neutron halo [1,2]. In this low density, high-isospin state of matter, the two-particle correlations, like the pairing modified by the effects of virtual scattering to the continuum, are expected to play a crucial role.
Nuclear pairing correlations are important throughout the periodic table, providing an extra binding for ordinary even-even and odd nuclei, but they become essential for loosely bound nuclei, because without them the systems in question cease to be bound. For their description, two main elements have to be combined. Firstly, the residual interaction which is expected to be different in the nuclear exterior and in the nuclear interior, should include the density dependence in the self-consistent framework. Secondly, the residual coupling between bound and continuum states must be treated consistently. Both elements are present in the Hartree-Fock-Bogolyubov (HFB) coordinate-space, mean-field approach using the Skyrme energy density functional [3].
The asymptotic behavior of halo nuclei, was discussed both for 2-body
[4,5] and 3-body systems
[6,7]. For 2-body systems, the root-mean-square
radius (rms) of a
low-angular-momentum (=0,1) neutron orbital in an average
potential diverges when its energy approaches the continuum.
Rigorous results for 3-body systems have been obtained
neglecting the antisymmetrization. The strong divergencies seen in 2-body
systems are now seldom.
When all 2-body subsystems of the 3-body system
are unbound, the rms radius diverges at most logarithmically with vanishing
binding energy
[6]. When one subsystem has a bound state with energy
close to the total energy of the 3-body system,
the system
behaves as one particle in the common field of the 2-body state
[7]. Finally, when at least two subsystems
have bound states, the ground state (g.s.) rms radius always converges
[7].
The pairing near the neutron drip line was studied first in a schematic 3-body model of two interacting nucleons together with a structureless core [8]. The exponential decay of both the single-particle (s.p.) density and the 2-particle density for valence neutrons was found to be governed rather by the 2-particle energy of the correlated pair than by the s.p. binding energy. This analysis is insufficient because just near the drip lines the particle-hole (p-h) and particle-particle (p-p) channels of the effective interaction are strongly coupled, and the non-self-consistent treatment of the pairing interaction based on computing pairing correlations after determining the s.p. basis is unjustified.
In light nuclei, many-body correlations may essentially modify the mean-field
picture
[9,10]. For that, different
cluster models have been applied [11,12],
and the 3-body dynamics has been investigated
[11,13,14,15]. Much
progress has been made in the treatment of the Pauli exclusion principle
[16], which is needed to remove spurious components
of the halo wave function even at small densities.
The whole problem is subtle because the proper antisymmetrization
of the wave function easily spoils many advantages of the cluster description.
It was found that the 3-body asymptotics of the cluster wave function is
given by the exponential fall-off with a decay constant
=
,
where E(3) is the binding energy of the
system relative to the 3-body break-up threshold.
If a bound 2-body subsystem is present, the asymptotics is given by a binary
channel asymptotics
[17].
In heavier nuclei, the concept of self-consistent mean-field is better justified. Moreover, the one-body densities are always governed by the so-called natural orbitals (the eigenstates of the exact one-body density matrix), and the mean-field approximation can be understood as a method to analyze properties of the natural orbitals. Strong pairing correlations do not allow for the decoupling of weakly bound neutrons from the rest of the system [18,19]. It is therefore a crucial question whether the neutron halo phenomenon can happen close to the neutron drip-line in heavy nuclei.
In the case of no pairing, irrespective of whether the number of neutrons
is even or odd,
the g.s. neutron mean-field asymptotic density equals
for
=
(
is the s.p. energy of the least bound
=0 neutron),
and has a
long tail with the rms radius
for
0.
In the presence of pairing, we have to separately consider the even-N and odd-Nisotopes. For N even,
the asymptotics of the HFB neutron density is
given by the binding energy relative to the 2-body break-up threshold
E(2). Therefore, the decay constant becomes [20,21]:
=
for E(2)=
,
where
is the lowest discrete
(
<
)
quasi-particle (q.p.) energy, and
is the chemical potential.
The HFB q.p. energies
are fairly close to the canonical
q.p. energies[21]:
In odd-N nuclei, the neutron densities are given by one-q.p.
states[22], which gives the asymptotic decay corresponding
to
.
This threshold energy goes to zero
when the minus Fermi energy approaches the pairing gap of a state
at the Fermi surface. Therefore, the pairing correlations do not prevent
the neutron radii of odd-N nuclei from diverging in the limit of
.
Below we give a numerical example illustrating the mechanism
of the pairing anti-halo effect in even-N nuclei,
which takes the most dramatic
form for
0,
because then the no-pairing tail of the neutron density gives
a divergent rms radius, while that for the paired density stays finite.
Before that we must, however, consider that in weakly bound nuclei
the situation is more complicated, because for
0 there can be no discrete HFB quasiparticle
state at all, and we have to take into account the HFB continuum explicitly.
To this end we express the asymptotic density, coming from the
q.p. continuum, in the form of the integral:
These amplitudes are determined by the pairing interaction and cannot
be inferred from any simple geometric considerations. We can
expect that A(E)
0 for
0,
because in this limit, the lower HFB q.p. wave functions
are completely delocalized [18], and
thus become decoupled from the pairing field inside the nucleus.
It is instructive to realize that a
power-law dependence of A2(E)
(E-
gives
for
=0 the density vanishing according to a power law too:
.
At any small non-zero value of
,
the latter power law holds up to
r
,
but then
turns over to the standard exponential dependence of
.
The two-neutron drip line (0) in heavier nuclei
will not probably be reached before
soon, however, the experimental data for neutron halos exist
in light nuclei [23].
Fig. 1(a) compares the calculated
one-neutron separation energies, obtained using the
spherical HFB code of Ref. [20], with the experimental and systematics
values in heavy carbon isotopes[24]. The
Skyrme SLy4 interaction [25] in the p-h channel
together with the zero-range, density-dependent (surface) pairing p-p
interaction, as described in Refs.[21,26],
has been used in these
calculations. The overall agreement is encouraging, if one considers that the
spherical symmetry is assumed and, moreover,
correlations beyond the mean-field are very important
in these light systems.
The HFB code has been improved by employing the analytical wave functions in
the asymptotic region, which allowed for solving the HFB equations in
a very large box, and thus obtaining an arbitrarily dense
discretized-continuum spectrum. The box of
=125fm has been
used in the present study, which gives the lowest
discretized-continuum HF state at
0.013MeV.
Therefore, we may safely treat the mean-field
continuum phenomena for the Fermi energies
at least several
times larger than this limiting value of 0.013MeV. We have
also performed test calculations in the box of
=400fm, for which this limiting value is an order of
magnitude smaller, and we found no modifications of conclusions
reached in the present study.
In Fig. 1(b) the spherical self-consistent Hartree-Fock
(HF) neutron s.p. energies are shown for Z=6 as functions of the
neutron number N. The carbon isotopes are special in that the least
bound spherical neutron orbital at the neutron drip line has =0, i.e.,
infinite rms radius should appear in the no-pairing scenario,
had this orbital approached the zero binding energy. Similarly,
an infinite rms radius would be given by an unpaired weakly bound deformed
orbital, had this orbital contained even a small 2s1/2 component.
In self-consistent calculations within a given effective interaction,
it is extremely difficult to change the relative positions of the
s.p. orbitals without spoiling other global characteristics of the
weakly bound nuclei. Therefore, in order to study dependence of the
results on the 2s1/2 binding energy, we have performed the HFB
calculations for fixed (i.e., non-self-consistent) s.p. wave
functions of the Pöschl-Teller-Ginocchio (PTG)
potential (see the description and references given in
Ref.[18]). Parameters of the PTG potential have been fixed
at =5, s=0.085fm-1. The depth parameter of the PTG potential
for the s1/2, p1/2, p3/2,
d3/2, and d5/2 channels is
=3.251, 2.814,
3.126, 2.747, and 3.176, respectively. At N=16, these values
reproduce the HFB+SLy4 neutron spectrum of Fig. 1(b)
exactly. Using the PTG potential we may now move the 2s1/2 orbital up or
down, by changing the value of the depth parameter
.
Figure 2(a) shows the s.p. energy of the 2s1/2orbital as function of the potential depth
.
With decreasing
,
the 2s1/2 orbital
approaches the zero binding energy at
=3 (full circles),
then turns into a virtual
state (dashed line), and finally reappears at
=2.857
as a very broad resonance
(open circles). The rms radius of this orbital
becomes infinite at
=0, and hence the neutron PTG rms radius
shown in Fig. 2(b) for N=16 (no pairing) becomes infinite too.
On the other hand, the HFB rms radius which is also shown in
Fig. 2(b), is completely unaffected by the transition
of the 2s1/2 level through the point of no s.p. binding.
This constitutes a vivid illustration of the pairing anti-halo effect.
Detailed behavior of the rms radius can be traced back
to the behavior of the 2s1/2 q.p. energies (1) induced
by the canonical s.p. energies
and pairing gaps
shown in
Figs. 2(a) and (c), respectively.
The HFB calculations have to be performed not at a fixed
particle number of N=16, but at a fixed Fermi energy of
=-0.056MeV. This value of
corresponds to
N=16 at
=3.251. At smaller values of
,
the
2s1/2 orbital depopulates, and the particle number
cannot be kept constant for any negative value of
.
Values of the average neutron number
at
=-0.056MeV (see Fig. 2(c)), change from 16
to about 14.7 in the studied interval of
.
Nevertheless, the canonical-basis occupation
factor
[21] of the 2s1/2 canonical state
(open squares in Fig. 2(c)), is far from zero
and decreases only to about 0.16 at
the smallest value of
=2.778.
We have also performed calculations for fixed N=14
(this
value
can be kept constant with the varying position of the
2s1/2 orbital), and we obtained exactly the same qualitative
results as for fixed
and a variable neutron number.
The
occupation of the 2s1/2 canonical state for
0
is entirely due to virtual pair excitations.
In order to visualize the origins of the pairing anti-halo effect, in
Fig. 3 are shown the contributions, N(E)and r2(E), of the =0 continuum HFB q.p. states to the particle
number and radius squared, respectively. They are defined as the
corresponding space integrals of the lower components [21] of
the q.p. wave functions squared,
.
When the 2s1/2 orbital is well bound (see panels (a) and
(d)), the largest contributions to the particle number
=
,
and to the radius
squared
=
,
come
from narrow peaks located near the corresponding s.p. energies.
However, when the 2s1/2 orbital is pushed across
=0, and
up into the continuum, the peaks in N(E) and r2(E) do not at all
approach E=0, but stay at a fairly large values of
E
2MeV. In this sense, the
large pairing gaps prevent the largest contributions to the rms radius from
entering the region of divergencies at E
0, where the lower
components of the HFB q.p. wave functions may develop long tails.
The sequence of curves presented in Figs. 3(d)-(f) is particularly
instructive, because it shows that the 2s1/2 orbital gives either
large or small contribution to the rms radius,
depending on whether it is bound or
resonant, but this contribution is always located around E2MeV
and simply disappears with 2s1/2 being pushed up. On the other hand, the
low-energy continuum contributions, described by the asymptotic density matrices
given in Eq. (2), are almost independent of the 2s1/2 position,
and give well-pronounced, low-energy contributions to the rms radii. However, at any small but non-zero value of
,
the curve r2(E) bends over at small E, and goes to zero, thus
giving a finite, non-diverging values of the HFB rms radii.
In conclusion, we have shown that pairing correlations significantly
modify the asymptotic properties of one-body neutron densities
in weakly bound neutron rich nuclei.
The neutron rms radii in
even-N nuclei are predominantly determined by two types of quasiparticles:
(i) those near the threshold of E=
,
and (ii) those
corresponding to the
=0 s.p. orbitals. We
have checked by numerical calculations that: (i) the density of
the former quasiparticle states vanishes at E=
,
which is due to a
delocalization of the quasiparticle wave functions, and (ii)
the latter quasiparticles
have energies Elarger than the corresponding pairing gaps, and thus cannot give
diverging rms radii. As a result, and contrary to the no-pairing
approximation, the rms neutron radii in even-N nuclei never diverge,
even in the limit of
=0.
In this work, we have analyzed the HFB asymptotics corresponding to the
one-neutron densities. The 3-body asymptotics of two-neutron densities cannot be
consistently described in the HFB. One should notice, however, that the break-up
threshold for two-neutron emission is expected to be
higher than the two-neutron separation energy
S2N
-2,Z)-
by the amount of the 2-particle energy of the correlated neutron pair. Hence, one
expects that the pairing correlations will also act against the two-neutron halo
effect in heavy even-N systems, by preventing the neutron radii from rapid
growth in the limit of S2N
0.
We would like to thank J.-F. Berger for useful discussions and W. Nazarewicz for a critical reading of the manuscript and many comments. This research was completed at the ECT* in Trento, Italy, and was supported in part by the Polish Committee for Scientific Research (KBN) under Contract No. 2 P03B 040 14 and by the French-Polish integrated actions program POLONIUM.