318
7 Wave Functions in the Vicinity of Particle-Emission Threshold
nucleon interaction in the continuum. Consequently, halo states in Borromean nuclei
consist of a two-neutron halo surrounding a well-bound A − 2 core. Even more
complicate halo configurations are possible, as can be seen in 6 He and 8 He. These
two nuclei are Borromean, because they are bound, on the one hand, while the 7 He
and 5 He nuclei and dineutron are neutron unbound, on the other hand. Moreover, the
two-neutron separation energy is larger in 8 He than in 6 He, which is an abnormal
situation. Indeed, neutron-rich isotopes are typically less and less bound when
neutrons are added. In fact, the unusually large binding energy of 8 He is due only
to correlations. This arises because 8 He consists mainly of a 4 He +0p 4
3/2 neutron
configuration, so that the neutrons in the 0p 3/2 shell strongly interact due to the
closed-shell character of the p 3/2 shell at Hartree Fock level in the 8 He ground
state.
However, it is not sufficient to have either loosely bound nucleons occupying
low- partial waves or the neutron emission thresholds set as in Borromean nuclei
to form a halo state. In fact, it is also necessary for external nucleons to sufficiently
decouple from the well-bound core, so that pairing correlations between nucleons
inside and outside the core are significantly reduced. Otherwise, as one will see later
in more details, the pairing anti-halo effect arise, which prevents formation of the
giant halo in most states lying close to the particle-emission threshold.
Typical hint of the presence of halos for energies near particle-emission threshold
are important correlations among nucleons in the asymptotic zone, as is the case for
11 Li [48]. Large electromagnetic transitions are also typical features of halo nuclei,
as they are enhanced by the unusually large density of nucleons in the asymptotic
region [46]. A striking example is the strong dipole transition occurring between
the first excited state 1/2 − and the ground state 1/2 + of 11 Be [28]. A simple
but accurate model to describe these states is that of a neutron interacting with a
potential mimicking a 10 Be core. The 1/2
+
1 and 1/2
−
1 states are essentially one
neutron 1s 1/2 and 0p 1/2 states coupled to a 10 Be core (see Sect. 4.3.4.3). Hence, the
E1 electromagnetic transition is mainly proportional to the square of the integral
of r times the radial wave functions of the 1s 1/2 and 0p 1/2 states. Unusually
large observed value of this transition implies that neutron one-body states extend
significantly in space, i.e. 11 Be is a halo nucleus.
Halo states have been discovered in the ground states of light nuclei, such as
6,8 He, 11 Be, and 11 Li. Also 14 Be, 14 B, 15 C, and 19 C have been confirmed to be halo
nuclei, whereas 15 B, 17 B, 19 B, 22 C, and 23 O exhibit only certain features of halo
configuration [46, 49]. Even though most halo nuclei are neutron halo states, proton
halos also exist. One can cite examples of the one-proton halos in the ground states
of 8 B and 13 N and of the first excited state of 17 F, as well as the Borromean twoproton configuration in the ground state of 17 Ne. However, due the presence of the
Coulomb barrier, proton halo states are less probable to occur and bear a reduced
spatial extension [46].
Halo states also exist in other quantum systems, outside the domain of nuclear
physics. In atomic clusters, halo configurations have been observed in 4 He 2 dimer
[50, 51] and 4 He 2 − 3 He trimer [50, 52]. The 4 He 2 dimer has a very large root-mean
7 Wave Functions in the Vicinity of Particle-Emission Threshold
nucleon interaction in the continuum. Consequently, halo states in Borromean nuclei
consist of a two-neutron halo surrounding a well-bound A − 2 core. Even more
complicate halo configurations are possible, as can be seen in 6 He and 8 He. These
two nuclei are Borromean, because they are bound, on the one hand, while the 7 He
and 5 He nuclei and dineutron are neutron unbound, on the other hand. Moreover, the
two-neutron separation energy is larger in 8 He than in 6 He, which is an abnormal
situation. Indeed, neutron-rich isotopes are typically less and less bound when
neutrons are added. In fact, the unusually large binding energy of 8 He is due only
to correlations. This arises because 8 He consists mainly of a 4 He +0p 4
3/2 neutron
configuration, so that the neutrons in the 0p 3/2 shell strongly interact due to the
closed-shell character of the p 3/2 shell at Hartree Fock level in the 8 He ground
state.
However, it is not sufficient to have either loosely bound nucleons occupying
low- partial waves or the neutron emission thresholds set as in Borromean nuclei
to form a halo state. In fact, it is also necessary for external nucleons to sufficiently
decouple from the well-bound core, so that pairing correlations between nucleons
inside and outside the core are significantly reduced. Otherwise, as one will see later
in more details, the pairing anti-halo effect arise, which prevents formation of the
giant halo in most states lying close to the particle-emission threshold.
Typical hint of the presence of halos for energies near particle-emission threshold
are important correlations among nucleons in the asymptotic zone, as is the case for
11 Li [48]. Large electromagnetic transitions are also typical features of halo nuclei,
as they are enhanced by the unusually large density of nucleons in the asymptotic
region [46]. A striking example is the strong dipole transition occurring between
the first excited state 1/2 − and the ground state 1/2 + of 11 Be [28]. A simple
but accurate model to describe these states is that of a neutron interacting with a
potential mimicking a 10 Be core. The 1/2
+
1 and 1/2
−
1 states are essentially one
neutron 1s 1/2 and 0p 1/2 states coupled to a 10 Be core (see Sect. 4.3.4.3). Hence, the
E1 electromagnetic transition is mainly proportional to the square of the integral
of r times the radial wave functions of the 1s 1/2 and 0p 1/2 states. Unusually
large observed value of this transition implies that neutron one-body states extend
significantly in space, i.e. 11 Be is a halo nucleus.
Halo states have been discovered in the ground states of light nuclei, such as
6,8 He, 11 Be, and 11 Li. Also 14 Be, 14 B, 15 C, and 19 C have been confirmed to be halo
nuclei, whereas 15 B, 17 B, 19 B, 22 C, and 23 O exhibit only certain features of halo
configuration [46, 49]. Even though most halo nuclei are neutron halo states, proton
halos also exist. One can cite examples of the one-proton halos in the ground states
of 8 B and 13 N and of the first excited state of 17 F, as well as the Borromean twoproton configuration in the ground state of 17 Ne. However, due the presence of the
Coulomb barrier, proton halo states are less probable to occur and bear a reduced
spatial extension [46].
Halo states also exist in other quantum systems, outside the domain of nuclear
physics. In atomic clusters, halo configurations have been observed in 4 He 2 dimer
[50, 51] and 4 He 2 − 3 He trimer [50, 52]. The 4 He 2 dimer has a very large root-mean