7.2 Halo Nuclei
329
distribution is concentrated inside the nucleus and negligible in the classically
forbidden region, represent states of closed quantum systems.
While pairing interaction provides with only a small additional binding in wellbound nuclei, it is responsible for binding of weakly bound nuclear states close to
the particle-emission threshold. Indeed, in a sole presence of the mean-field, nuclei
close to the particle-emission threshold would bear nucleons in unbound singleparticle orbits. The residual interaction among these nucleons yields an additional
binding so that the nucleus becomes weakly bound and nucleons in originally
unbound orbits may form halo in the classically forbidden region.
The simplest systems in which pairing correlations are determinant for halo
formation are the three-cluster systems. The asymptotic behavior of halo states
can be analyzed therein under specific conditions [58, 59]. Rigorous results can
be obtained if one lifts antisymmetrization in subsystems. The diverging character
of neutron density, occurring when = 0, 1 (see Sect. 7.2.1), is typically absent
from the three-body systems. On the other hand, if one subsystem is bound and
has an energy almost equal to the total energy of a three-body system, the bound
subsystem behaves as one particle in the presence of the field generated by the other
two subsystems [59]. No halo can occur if two subsystems of the three-body system
are bound as then the root-mean-square radius of the ground state is always finite at
zero binding energy.
When considering two nucleons interacting with a core, the exponential decay
of the one- and two-body densities of valence neutrons is mainly a function of
the energy of the two correlated valence nucleons, while there is basically no
dependence on single-particle energies of the combined system. This model is,
however, oversimplified as the pairing interaction is not treated self-consistently.
The antisymmetrization in the total system can be restored at the price of adding
complexity in the initially concise cluster description [60]. The main result obtained
in Ref. [60] using fully antisymmetrized wave functions is that the exponential
decay constant of the total density is equal to κ =
√
2mE 3 / ¯
h, where E 3 is the
binding energy of the three-cluster system relatively to the three-body break-up
threshold. The asymptotic behavior of the three-particle density is of two-body
character if a two-body subsystem is bound therein,
The previous discussion applies to nuclei having two- or three-cluster structure.
Therefore, it would be desirable to devise a model which can be applied in various
mass regions in order to numerically assess if a ground state wave function can
give rise to a halo. In such an approach, it is necessary to incorporate both
pairing correlations and the proper asymptotic behavior of single-particle wave
functions. Moreover, the resulting model should be applicable in the whole nuclear
chart. Therefore, one has to recur to mean-field methods. A convenient tool for
that matter is the Hartree–Fock-Bogolyubov approach in coordinate space [61].
For even particle number N, where N is the number of protons or neutrons, the
asymptote of neutron density in the Hartree–Fock-Bogolyubov approach is provided
by the total binding energy relative to the two-body break-up threshold E 2 . As a
consequence, the aforementioned decay constant becomes κ =
√
2mE 2 / ¯
h, where
329
distribution is concentrated inside the nucleus and negligible in the classically
forbidden region, represent states of closed quantum systems.
While pairing interaction provides with only a small additional binding in wellbound nuclei, it is responsible for binding of weakly bound nuclear states close to
the particle-emission threshold. Indeed, in a sole presence of the mean-field, nuclei
close to the particle-emission threshold would bear nucleons in unbound singleparticle orbits. The residual interaction among these nucleons yields an additional
binding so that the nucleus becomes weakly bound and nucleons in originally
unbound orbits may form halo in the classically forbidden region.
The simplest systems in which pairing correlations are determinant for halo
formation are the three-cluster systems. The asymptotic behavior of halo states
can be analyzed therein under specific conditions [58, 59]. Rigorous results can
be obtained if one lifts antisymmetrization in subsystems. The diverging character
of neutron density, occurring when = 0, 1 (see Sect. 7.2.1), is typically absent
from the three-body systems. On the other hand, if one subsystem is bound and
has an energy almost equal to the total energy of a three-body system, the bound
subsystem behaves as one particle in the presence of the field generated by the other
two subsystems [59]. No halo can occur if two subsystems of the three-body system
are bound as then the root-mean-square radius of the ground state is always finite at
zero binding energy.
When considering two nucleons interacting with a core, the exponential decay
of the one- and two-body densities of valence neutrons is mainly a function of
the energy of the two correlated valence nucleons, while there is basically no
dependence on single-particle energies of the combined system. This model is,
however, oversimplified as the pairing interaction is not treated self-consistently.
The antisymmetrization in the total system can be restored at the price of adding
complexity in the initially concise cluster description [60]. The main result obtained
in Ref. [60] using fully antisymmetrized wave functions is that the exponential
decay constant of the total density is equal to κ =
√
2mE 3 / ¯
h, where E 3 is the
binding energy of the three-cluster system relatively to the three-body break-up
threshold. The asymptotic behavior of the three-particle density is of two-body
character if a two-body subsystem is bound therein,
The previous discussion applies to nuclei having two- or three-cluster structure.
Therefore, it would be desirable to devise a model which can be applied in various
mass regions in order to numerically assess if a ground state wave function can
give rise to a halo. In such an approach, it is necessary to incorporate both
pairing correlations and the proper asymptotic behavior of single-particle wave
functions. Moreover, the resulting model should be applicable in the whole nuclear
chart. Therefore, one has to recur to mean-field methods. A convenient tool for
that matter is the Hartree–Fock-Bogolyubov approach in coordinate space [61].
For even particle number N, where N is the number of protons or neutrons, the
asymptote of neutron density in the Hartree–Fock-Bogolyubov approach is provided
by the total binding energy relative to the two-body break-up threshold E 2 . As a
consequence, the aforementioned decay constant becomes κ =
√
2mE 2 / ¯
h, where