344
7 Wave Functions in the Vicinity of Particle-Emission Threshold
7.4.2.1 Spectroscopic Factors and Pairing Correlations in 11 Li
Pairing correlations are an essential ingredient of the two-neutron halos. In the last
section, one considered the case of the ground state of 6 He, which was bound by
about 1 MeV, while the two subsystems in its partition into 5 He and dineutron are
unbound by about 900 keV and 50 keV, respectively. By considering 6 He composite
system close to neutron emission threshold, one could point out to the importance of
continuum via the study of spectroscopic factors of the form [ 5 He(g.s.) ⊗ νp 3/2 ] 0 + .
Indeed, large Wigner cusps appear in spectroscopic factors, indicating an important
role of coupling to the nonresonant continuum occurring in the structure of 6 He
composite at low energy. In this study, one could only obtain information about
correlations in neutron = 1 partial wave. It would be interesting to determine also
the effects of neutron = 0 partial wave at neutron emission threshold. Indeed, as
clearly shown by the Wigner power laws of Eqs. (7.45), (7.46), both = 0 (s wave)
and = 1 (p wave) partial waves are likely to create cusps in transfer cross sections,
as they induce an inverse-square-root- or square-root-dependence of transfer cross
sections, respectively, as a function of linear momentum k for k ∼ 0.
A good candidate for the study of near-threshold properties involving neutron
= 0 partial wave is 11 Li. The ground state of 11 Li is a two-neutron halo state of
two-neutron separation energy close to 350 keV [28]. Hence, it can be modelled
by a 9 Li core surrounded by two valence neutrons. 11 Li can be considered as
a Borromean system because all subsystems, namely dineutron in the partition
[2n⊕ 9 Li], or 10 Li in the partition [n⊕ 10 Li] are unbound.
The wave function of the ground state of 10 Li in the three-body model is
an antibound = 0 neutron state 1s 1/2 coupled to the 9 Li core. Consequently,
the ground state of 11 Li possesses large components of neutron-s basis states.
Theoretical calculations typically predict a many-body wave function of 11 Li made
of about 50% neutron-s states, 50% neutron-p states, possibly along with a minor
part built from neutron-d states [105–108], in accordance with experimental studies
[109]. The wave function components involving = 1 neutron basis states can be
explained from the vicinity of a one-body 0p 1/2 neutron state lying close to neutron
emission threshold. Consequently, besides the 3/2
−
1 ground state of 11 Li, one can
also expect a low-lying 3/2
+
1 excited state of 11 Li, as suggested in Ref. [110]. One
can reasonably assume that the 3/2
+
1 state of 11 Li would then mainly consist of two
neutrons occupying 1s 1/2 and 0p 1/2 one-body states and coupled to 0 − .
One will represent the 9 Li core by a Woods–Saxon potential, whose 1s 1/2 and
0p 1/2 one-body states have an energy of −0.1 MeV and 0.236 MeV, respectively.
The energy of the 0p 1/2 one-body state is fitted to that of the 1
+
1 excited state of 10 Li.
This state can be approximated by the 9 Li core coupled to a neutron in a 0p 1/2 shell.
One can also note that the width of the calculated 0p 1/2 one-body state is 122 keV,
very close to the experimental value of 100 keV. Consequently, the 1
+
1 excited state
of 10 Li is well described in a single-particle model, as it mainly consists of a neutron
occupying a 0p 1/2 resonance state, generated by a well-bound 9 Li core. The ground
state of 10 Li is unbound, and is very close to a 9 Li core coupled to an antibound 1s 1/2
neutron state, of energy around −25 keV. It would then seem logical to consider an
7 Wave Functions in the Vicinity of Particle-Emission Threshold
7.4.2.1 Spectroscopic Factors and Pairing Correlations in 11 Li
Pairing correlations are an essential ingredient of the two-neutron halos. In the last
section, one considered the case of the ground state of 6 He, which was bound by
about 1 MeV, while the two subsystems in its partition into 5 He and dineutron are
unbound by about 900 keV and 50 keV, respectively. By considering 6 He composite
system close to neutron emission threshold, one could point out to the importance of
continuum via the study of spectroscopic factors of the form [ 5 He(g.s.) ⊗ νp 3/2 ] 0 + .
Indeed, large Wigner cusps appear in spectroscopic factors, indicating an important
role of coupling to the nonresonant continuum occurring in the structure of 6 He
composite at low energy. In this study, one could only obtain information about
correlations in neutron = 1 partial wave. It would be interesting to determine also
the effects of neutron = 0 partial wave at neutron emission threshold. Indeed, as
clearly shown by the Wigner power laws of Eqs. (7.45), (7.46), both = 0 (s wave)
and = 1 (p wave) partial waves are likely to create cusps in transfer cross sections,
as they induce an inverse-square-root- or square-root-dependence of transfer cross
sections, respectively, as a function of linear momentum k for k ∼ 0.
A good candidate for the study of near-threshold properties involving neutron
= 0 partial wave is 11 Li. The ground state of 11 Li is a two-neutron halo state of
two-neutron separation energy close to 350 keV [28]. Hence, it can be modelled
by a 9 Li core surrounded by two valence neutrons. 11 Li can be considered as
a Borromean system because all subsystems, namely dineutron in the partition
[2n⊕ 9 Li], or 10 Li in the partition [n⊕ 10 Li] are unbound.
The wave function of the ground state of 10 Li in the three-body model is
an antibound = 0 neutron state 1s 1/2 coupled to the 9 Li core. Consequently,
the ground state of 11 Li possesses large components of neutron-s basis states.
Theoretical calculations typically predict a many-body wave function of 11 Li made
of about 50% neutron-s states, 50% neutron-p states, possibly along with a minor
part built from neutron-d states [105–108], in accordance with experimental studies
[109]. The wave function components involving = 1 neutron basis states can be
explained from the vicinity of a one-body 0p 1/2 neutron state lying close to neutron
emission threshold. Consequently, besides the 3/2
−
1 ground state of 11 Li, one can
also expect a low-lying 3/2
+
1 excited state of 11 Li, as suggested in Ref. [110]. One
can reasonably assume that the 3/2
+
1 state of 11 Li would then mainly consist of two
neutrons occupying 1s 1/2 and 0p 1/2 one-body states and coupled to 0 − .
One will represent the 9 Li core by a Woods–Saxon potential, whose 1s 1/2 and
0p 1/2 one-body states have an energy of −0.1 MeV and 0.236 MeV, respectively.
The energy of the 0p 1/2 one-body state is fitted to that of the 1
+
1 excited state of 10 Li.
This state can be approximated by the 9 Li core coupled to a neutron in a 0p 1/2 shell.
One can also note that the width of the calculated 0p 1/2 one-body state is 122 keV,
very close to the experimental value of 100 keV. Consequently, the 1
+
1 excited state
of 10 Li is well described in a single-particle model, as it mainly consists of a neutron
occupying a 0p 1/2 resonance state, generated by a well-bound 9 Li core. The ground
state of 10 Li is unbound, and is very close to a 9 Li core coupled to an antibound 1s 1/2
neutron state, of energy around −25 keV. It would then seem logical to consider an