350
7 Wave Functions in the Vicinity of Particle-Emission Threshold
because relative weight of the p 1/2 scattering components in the wave function
of 11 Li(3/2
+
1 ) increases. The short-ranged nuclear interaction is then less and less
effective, which leads to a decrease of the spectroscopic factor. However, that effect
is not as pronounced as in Fig. 7.11 due to the presence of a centrifugal barrier
for = 1 basis states. As in Fig. 7.11, the p 1/2 components always augment
when the binding energy of 11 Li(3/2
+
1 ) becomes smaller, so that the corresponding
spectroscopic factor also decreases (see Fig. 7.13).
A Wigner cusp for = 1 neutron states also develops at S 1n = 0, as
imposed by the Wigner laws (7.45), (7.46). However, the spectroscopic factor
11 Li(3/2
+
1 )|[ 10 Li(2
+
1 ) ⊗ s 1/2 ] 3/2 +
2 remains almost constant for S 1n < 0 MeV,
even at the threshold of the 0p 1/2 emission channel (E = 0.236 MeV), where a
cusp might a priori be expected to occur. This arises due to a destructive interference
which occurs between the [0p 1/2 1s 1/2 ] and [0p 1/2 s
(scat)
1/2 ] basis states components.
Indeed, the contribution of the [0p 1/2 s
(scat)
1/2 ] basis states is negative, while that of
[0p 1/2 1s 1/2 ] is larger than 1 by about the same amount. Hence, the spectroscopic
factor does not necessarily change significantly when a new decay channel opens,
as pairing correlations therein can have an enhancing or mitigating effect at particleemission threshold.
The presence of neutron-s waves in many-body wave functions is especially
important when other partial waves play a negligible role. In the example depicted
in Fig. 7.11, one has seen that it is impossible to obtain unbound resonance states,
even though the nucleon–nucleon interaction becomes infinitely repulsive. In this
situation, bound states become more extended in space due to the absence of a
centrifugal barrier, similarly to the one-body case (see Sect. 3.7.2). If one has both s
and p waves in the many-body wave function, Wigner cusps do not radically change
compared to the case where only p waves play a significant role (see Figs. 7.12
and 7.13). Indeed, the passage from bound to resonance many-body state is only
due to the p-wave components, so that Wigner cusps obey the power laws of
Eqs. (7.45), (7.46) for = 1.
The situation changes if the many-body state of A − 1 nucleons is a resonance
(see Fig. 7.12). In this case, the spectroscopic factor does not follow the powerlaw expressions of Eqs. (7.45), (7.46), because the 11 Li(3/2
+
1 ) nuclear many-body
state was already unbound for smaller energies. However, that emission channel
can still generate important change of the neutron spectroscopic factor. This change
is nevertheless dependent on the pairing correlations, which can provide with
constructive or destructive interferences when a channel associated with a manybody resonance state of A − 1 nucleons opens.
7.5
Mirror Nuclei and Isospin Symmetry Breaking
The very different asymptotic behavior of proton and neutron wave functions at
the particle-emission threshold induces breaking of isospin symmetry. Moreover,
in loosely bound and resonance many-body states, a strong coupling to the
7 Wave Functions in the Vicinity of Particle-Emission Threshold
because relative weight of the p 1/2 scattering components in the wave function
of 11 Li(3/2
+
1 ) increases. The short-ranged nuclear interaction is then less and less
effective, which leads to a decrease of the spectroscopic factor. However, that effect
is not as pronounced as in Fig. 7.11 due to the presence of a centrifugal barrier
for = 1 basis states. As in Fig. 7.11, the p 1/2 components always augment
when the binding energy of 11 Li(3/2
+
1 ) becomes smaller, so that the corresponding
spectroscopic factor also decreases (see Fig. 7.13).
A Wigner cusp for = 1 neutron states also develops at S 1n = 0, as
imposed by the Wigner laws (7.45), (7.46). However, the spectroscopic factor
11 Li(3/2
+
1 )|[ 10 Li(2
+
1 ) ⊗ s 1/2 ] 3/2 +
2 remains almost constant for S 1n < 0 MeV,
even at the threshold of the 0p 1/2 emission channel (E = 0.236 MeV), where a
cusp might a priori be expected to occur. This arises due to a destructive interference
which occurs between the [0p 1/2 1s 1/2 ] and [0p 1/2 s
(scat)
1/2 ] basis states components.
Indeed, the contribution of the [0p 1/2 s
(scat)
1/2 ] basis states is negative, while that of
[0p 1/2 1s 1/2 ] is larger than 1 by about the same amount. Hence, the spectroscopic
factor does not necessarily change significantly when a new decay channel opens,
as pairing correlations therein can have an enhancing or mitigating effect at particleemission threshold.
The presence of neutron-s waves in many-body wave functions is especially
important when other partial waves play a negligible role. In the example depicted
in Fig. 7.11, one has seen that it is impossible to obtain unbound resonance states,
even though the nucleon–nucleon interaction becomes infinitely repulsive. In this
situation, bound states become more extended in space due to the absence of a
centrifugal barrier, similarly to the one-body case (see Sect. 3.7.2). If one has both s
and p waves in the many-body wave function, Wigner cusps do not radically change
compared to the case where only p waves play a significant role (see Figs. 7.12
and 7.13). Indeed, the passage from bound to resonance many-body state is only
due to the p-wave components, so that Wigner cusps obey the power laws of
Eqs. (7.45), (7.46) for = 1.
The situation changes if the many-body state of A − 1 nucleons is a resonance
(see Fig. 7.12). In this case, the spectroscopic factor does not follow the powerlaw expressions of Eqs. (7.45), (7.46), because the 11 Li(3/2
+
1 ) nuclear many-body
state was already unbound for smaller energies. However, that emission channel
can still generate important change of the neutron spectroscopic factor. This change
is nevertheless dependent on the pairing correlations, which can provide with
constructive or destructive interferences when a channel associated with a manybody resonance state of A − 1 nucleons opens.
7.5
Mirror Nuclei and Isospin Symmetry Breaking
The very different asymptotic behavior of proton and neutron wave functions at
the particle-emission threshold induces breaking of isospin symmetry. Moreover,
in loosely bound and resonance many-body states, a strong coupling to the