7.4 Near-Threshold Behavior of Wave Functions
345
antibound 1s 1/2 neutron state in the model used. However, the spectroscopic factor
loses its physical meaning in this situation.
Indeed, as demonstrated in Sect. 3.7.2, the inclusion of an antibound state in the
Berggren basis generates a very large coupling to continuum. In particular, the pole
approximation (see Sect. 5.4) including antibound states is not appropriate when
considering many-body bound states. In this case, the scattering components of
many-body bound states would have to suppress the antibound character of pole
components, and generate the bound structure of the considered many-body states.
As the result, the many-body wave function calculated in full model space would
have a very small overlap with that devised in pole approximation. Therefore,
spectroscopic factors which are a measure of correlations in many-body wave
functions, lose their significance in this situation.
Spectroscopic factors are close to one when a resonant valence nucleon is
decoupled from the rest of the nucleus. On the contrary, if the neutron occupies
an antibound 1s 1/2 state, its associated spectroscopic factor is arbitrary and does
not provide with any useful information. Therefore, in the example discussed
below it has been chosen to include a loosely bound 1s 1/2 neutron state instead
of an antibound state. We will see that even if energy of s 1/2 neutron state is
changed by about 100 keV, this choice generates unique features in the pairing
correlations involving the s 1/2 neutron partial wave. In the following, we will use
the Berggren basis generated by the 9 Li core, as in this case the evaluation of 11 Li
wave functions is numerically precise. For the nucleon–nucleon interaction, one will
use the Minnesota interaction [111] whose parameters are varied to achieve either
weakly bound or unbound wave functions for 11 Li. The Berggren basis consist of
the s 1/2 and p 1/2 neutron scattering states, along with the resonant 1s 1/2 and 0p 1/2
states, to which p 3/2 scattering states are added. The 0s 1/2 and 0p 3/2 single-particle
states are not taken into account as they are occupied in the 9 Li core.
Let us first consider the spectroscopic factor S = = 11 Li(3/2
−
1 )|[ 10 Li(2
−
1 ) ⊗
s 1/2 ] 3/2 + 2 , where 10 Li(2
−
1 ) consists of the weakly bound 1s 1/2 neutron singleparticle state of the Berggren basis coupled to the 9 Li ground state. In order to
study the behavior of the 11 Li wave function at small neutron separation energy,
the parameters of Minnesota interaction are adjusted so that the 11 Li ground state
has a binding energy varying from −1 ˙
MeV to −0.1 MeV. Results are shown in
Fig. 7.11. One can see that the energy dependence of spectroscopic factors consists
of two different regions. Firstly, the spectroscopic factor decreases smoothly from
2 at −0.2 MeV to a value close to 1 at −1 MeV. Indeed, if 11 Li ground state is
bound, it has a sizable p components and its spectroscopic factor is smaller than
the maximal value of S = 2. When the interaction becomes less attractive, the
coupling to p waves decreases and the 11 Li wave function becomes very close to the
single-particle [1s 1/2 ] 2 configuration. The spectroscopic factor in this case decreases
abruptly to 1. Moreover, the energy of the 11 Li ground state remains smaller than
−0.1 MeV, which is the energy of the 1s 1/2 neutron state, when the nucleon–nucleon
interaction becomes infinitely repulsive.
345
antibound 1s 1/2 neutron state in the model used. However, the spectroscopic factor
loses its physical meaning in this situation.
Indeed, as demonstrated in Sect. 3.7.2, the inclusion of an antibound state in the
Berggren basis generates a very large coupling to continuum. In particular, the pole
approximation (see Sect. 5.4) including antibound states is not appropriate when
considering many-body bound states. In this case, the scattering components of
many-body bound states would have to suppress the antibound character of pole
components, and generate the bound structure of the considered many-body states.
As the result, the many-body wave function calculated in full model space would
have a very small overlap with that devised in pole approximation. Therefore,
spectroscopic factors which are a measure of correlations in many-body wave
functions, lose their significance in this situation.
Spectroscopic factors are close to one when a resonant valence nucleon is
decoupled from the rest of the nucleus. On the contrary, if the neutron occupies
an antibound 1s 1/2 state, its associated spectroscopic factor is arbitrary and does
not provide with any useful information. Therefore, in the example discussed
below it has been chosen to include a loosely bound 1s 1/2 neutron state instead
of an antibound state. We will see that even if energy of s 1/2 neutron state is
changed by about 100 keV, this choice generates unique features in the pairing
correlations involving the s 1/2 neutron partial wave. In the following, we will use
the Berggren basis generated by the 9 Li core, as in this case the evaluation of 11 Li
wave functions is numerically precise. For the nucleon–nucleon interaction, one will
use the Minnesota interaction [111] whose parameters are varied to achieve either
weakly bound or unbound wave functions for 11 Li. The Berggren basis consist of
the s 1/2 and p 1/2 neutron scattering states, along with the resonant 1s 1/2 and 0p 1/2
states, to which p 3/2 scattering states are added. The 0s 1/2 and 0p 3/2 single-particle
states are not taken into account as they are occupied in the 9 Li core.
Let us first consider the spectroscopic factor S = = 11 Li(3/2
−
1 )|[ 10 Li(2
−
1 ) ⊗
s 1/2 ] 3/2 + 2 , where 10 Li(2
−
1 ) consists of the weakly bound 1s 1/2 neutron singleparticle state of the Berggren basis coupled to the 9 Li ground state. In order to
study the behavior of the 11 Li wave function at small neutron separation energy,
the parameters of Minnesota interaction are adjusted so that the 11 Li ground state
has a binding energy varying from −1 ˙
MeV to −0.1 MeV. Results are shown in
Fig. 7.11. One can see that the energy dependence of spectroscopic factors consists
of two different regions. Firstly, the spectroscopic factor decreases smoothly from
2 at −0.2 MeV to a value close to 1 at −1 MeV. Indeed, if 11 Li ground state is
bound, it has a sizable p components and its spectroscopic factor is smaller than
the maximal value of S = 2. When the interaction becomes less attractive, the
coupling to p waves decreases and the 11 Li wave function becomes very close to the
single-particle [1s 1/2 ] 2 configuration. The spectroscopic factor in this case decreases
abruptly to 1. Moreover, the energy of the 11 Li ground state remains smaller than
−0.1 MeV, which is the energy of the 1s 1/2 neutron state, when the nucleon–nucleon
interaction becomes infinitely repulsive.