7.4 Near-Threshold Behavior of Wave Functions
347
an infinitely repulsive neutron-neutron interaction becomes that of one neutron
occupying the 1s 1/2 single-particle state with an energy of −0.1 MeV, and of a
second neutron, of almost zero energy with vanishing probability in the nuclear zone
(see the above discussion about the replacement of a neutron antibound state 1s 1/2
by a weakly bound 1s 1/2 neutron state). Consequently, energy of the 11 Li ground
state converges to −0.1 MeV and the spectroscopic factor tends to half its maximal
value, i.e. becomes equal to 1.
This unusual situation occurs because there is no centrifugal barrier in a neutrons partial wave, so that a neutron can spread throughout all space. Consequently,
it is impossible for a correlated two-body state to be unbound and have a decay
width generated by neutron s states only. Indeed, the zero neutron separation energy
corresponds to the energy of the weakly bound 1s 1/2 shell for the 11 Li wave
function, so that having an energy larger than this one is impossible even with an
infinitely repulsive interaction.
If a wave function of one neutron is negligible in the nuclear zone, then also
the matrix elements of interaction involving both valence neutrons are negligible.
This occurs if the valence neutrons occupy only the neutron s states. Thus, the
pairing correlation vanishes in a vicinity of particle-emission threshold if the manybody wave function mainly consist of neutron s basis states. Note that due to
the disappearance of pairing correlations close to particle-emission threshold, the
pairing anti-halo effect [63] cannot develop. Therefore, the wave function generates
a halo of neutron s waves at low energy, which fills all space at the limit of zero
neutron separation energy.
The last study concerns the spectroscopic factors where the 11 Li(3/2
+
1 )
excited state is involved. Due to the parity conservation, the two valence
neutrons of 11 Li(3/2
+
1 in a pole approximation are in the [0p 1/2 1s 1/2 ]
configuration and are coupled to 0 − . Consequently, two different pole states
are occupied and therefore two different one neutron spectroscopic factors
are important in this configuration. In the first spectroscopic factor, equal to
11 Li(3/2
+
1 )|[ 10 Li(2
−
1 ) ⊗ p 1/2 ] 3/2 +
2 , the weakly bound 1s 1/2 basis neutron state is
coupled to the 9 Li ground state (see Fig. 7.12). In the second spectroscopic factor,
equal to 11 Li(3/2
+
1 )|[ 10 Li(2
+
1 ) ⊗ s 1/2 ] 3/2 +
2 , the 0p 1/2 neutron state is coupled
to the 9 Li ground state (see Fig. 7.13). Due to the presence of the 0p 1/2 state, the
many-body wave function of 11 Li(3/2
+
1 ) has components depending on = 1
partial waves, so that the current situation is much closer to that of Fig. 7.8.
Indeed, one can clearly see a Wigner cusp in Fig. 7.12 at the energy of −0.1 MeV,
when the state 11 Li(3/2
+
1 ) becomes unbound. The dependencies seen in Wigner
expressions of Eqs. (7.45), (7.46) are also visible. Indeed, denoting as S 1n the
separation energy of one neutron, the spectroscopic factor behaves as S
1/2
1n for
S 1n > 0 and has an infinite derivative for S 1n → 0 + , whereas it becomes regular
for S 1n < 0, in accordance with the S
3/2
1n law. In Fig. 7.12, one can see the
opening of a p 1/2 neutron emission channel, occurring at E =0.236 MeV. The
spectroscopic factor 11 Li(3/2
+
1 )|[ 10 Li(2
−
1 ) ⊗ p 1/2 ] 3/2 +
2 starts to increase quickly
347
an infinitely repulsive neutron-neutron interaction becomes that of one neutron
occupying the 1s 1/2 single-particle state with an energy of −0.1 MeV, and of a
second neutron, of almost zero energy with vanishing probability in the nuclear zone
(see the above discussion about the replacement of a neutron antibound state 1s 1/2
by a weakly bound 1s 1/2 neutron state). Consequently, energy of the 11 Li ground
state converges to −0.1 MeV and the spectroscopic factor tends to half its maximal
value, i.e. becomes equal to 1.
This unusual situation occurs because there is no centrifugal barrier in a neutrons partial wave, so that a neutron can spread throughout all space. Consequently,
it is impossible for a correlated two-body state to be unbound and have a decay
width generated by neutron s states only. Indeed, the zero neutron separation energy
corresponds to the energy of the weakly bound 1s 1/2 shell for the 11 Li wave
function, so that having an energy larger than this one is impossible even with an
infinitely repulsive interaction.
If a wave function of one neutron is negligible in the nuclear zone, then also
the matrix elements of interaction involving both valence neutrons are negligible.
This occurs if the valence neutrons occupy only the neutron s states. Thus, the
pairing correlation vanishes in a vicinity of particle-emission threshold if the manybody wave function mainly consist of neutron s basis states. Note that due to
the disappearance of pairing correlations close to particle-emission threshold, the
pairing anti-halo effect [63] cannot develop. Therefore, the wave function generates
a halo of neutron s waves at low energy, which fills all space at the limit of zero
neutron separation energy.
The last study concerns the spectroscopic factors where the 11 Li(3/2
+
1 )
excited state is involved. Due to the parity conservation, the two valence
neutrons of 11 Li(3/2
+
1 in a pole approximation are in the [0p 1/2 1s 1/2 ]
configuration and are coupled to 0 − . Consequently, two different pole states
are occupied and therefore two different one neutron spectroscopic factors
are important in this configuration. In the first spectroscopic factor, equal to
11 Li(3/2
+
1 )|[ 10 Li(2
−
1 ) ⊗ p 1/2 ] 3/2 +
2 , the weakly bound 1s 1/2 basis neutron state is
coupled to the 9 Li ground state (see Fig. 7.12). In the second spectroscopic factor,
equal to 11 Li(3/2
+
1 )|[ 10 Li(2
+
1 ) ⊗ s 1/2 ] 3/2 +
2 , the 0p 1/2 neutron state is coupled
to the 9 Li ground state (see Fig. 7.13). Due to the presence of the 0p 1/2 state, the
many-body wave function of 11 Li(3/2
+
1 ) has components depending on = 1
partial waves, so that the current situation is much closer to that of Fig. 7.8.
Indeed, one can clearly see a Wigner cusp in Fig. 7.12 at the energy of −0.1 MeV,
when the state 11 Li(3/2
+
1 ) becomes unbound. The dependencies seen in Wigner
expressions of Eqs. (7.45), (7.46) are also visible. Indeed, denoting as S 1n the
separation energy of one neutron, the spectroscopic factor behaves as S
1/2
1n for
S 1n > 0 and has an infinite derivative for S 1n → 0 + , whereas it becomes regular
for S 1n < 0, in accordance with the S
3/2
1n law. In Fig. 7.12, one can see the
opening of a p 1/2 neutron emission channel, occurring at E =0.236 MeV. The
spectroscopic factor 11 Li(3/2
+
1 )|[ 10 Li(2
−
1 ) ⊗ p 1/2 ] 3/2 +
2 starts to increase quickly