Solutions to Exercises
359
= 1. All other cases provide with finite root-mean-square radius limits when
κ → 0, similarly to the ≥ 2 case of spherical potentials. The plots of radii
issued from the calculations involving a spherical or deformed potential then
have the same qualitative properties.
Exercise II
A. The overlap asymptote of Eqs. (7.32) and (7.40) for the bound and unbound
cases, respectively, is quickly reached. The overlap functions become equal to
Eq. (7.32) or Eq. (7.40) around 10–15 fm. It is then straightforward to determine
the asymptotic normalization coefficient by fitting the overlap function both in
the case of bound and unbound states.
B. Eqs. (7.32), (7.40) depend on the κ or k value associated with one neutron
separation energy, so that changing the interaction parameters also changes
κ or k. Consequently, the asymptotic normalization coefficient is modified,
as the structure of involved many-body states changes along with interaction
parameters.
Exercise III
A. 8 He is a bound nucleus, where proton 0s 1/2 and neutron 0s 1/2 and 0p 3/2
shells are fully occupied. Added to that, the neutron 0p 1/2 shell is unbound.
Consequently, the use of 8 He as a core to describe 9 He and 10 He allows to
recapture the main features of the eigenstates of 9 He and 10 He, while having
small Gamow shell model space dimensions.
The 1/2 − state of 9 He and 0 + state of 10 He have a small width, close to
100 keV. Therefore, similarly to the study of the p 3/2 spectroscopic factors of
5 He/ 6 He, one can assume that the ground state of 10 He is bound and have the
energy of the 1/2 − state of 9 He vary so that it goes from bound to unbound. A
composite system of 2n+ 8 He is then simulated for different excitation energies.
B. As one considers neutron states, the non-analytic cusps of Eqs. (7.45), (7.46)
must occur at zero neutron separation energy for = 0, 1 only. Moreover, as
wave functions are built from the neutron p 1/2 partial wave, the overall behavior
of spectroscopic factors is similar to the 5 He/ 6 He case.
Exercise IV
A. 12 C has a deformed ground state, so that excitations from 0p 3/2 shells are sizable. The 12 C ground state wave function thus bears an important configuration
mixing in a no-core shell model picture [117]. However, 12 C is a closed-shell
nucleus at Hartree–Fock level, and there is a sizeable gap between 0p 3/2 and
0p 1/2 , of about 5 MeV. Thus, it is reasonable to assume that effective interactions
can be generated by using a 12 C core. Consequently, one can assume that taking
12 C as a core can provide with an effective interaction able to reproduce the main
features of 13,14 C, 13 N, and 14 O. In fact, the Zuker-Buck-McGrory interaction
359
= 1. All other cases provide with finite root-mean-square radius limits when
κ → 0, similarly to the ≥ 2 case of spherical potentials. The plots of radii
issued from the calculations involving a spherical or deformed potential then
have the same qualitative properties.
Exercise II
A. The overlap asymptote of Eqs. (7.32) and (7.40) for the bound and unbound
cases, respectively, is quickly reached. The overlap functions become equal to
Eq. (7.32) or Eq. (7.40) around 10–15 fm. It is then straightforward to determine
the asymptotic normalization coefficient by fitting the overlap function both in
the case of bound and unbound states.
B. Eqs. (7.32), (7.40) depend on the κ or k value associated with one neutron
separation energy, so that changing the interaction parameters also changes
κ or k. Consequently, the asymptotic normalization coefficient is modified,
as the structure of involved many-body states changes along with interaction
parameters.
Exercise III
A. 8 He is a bound nucleus, where proton 0s 1/2 and neutron 0s 1/2 and 0p 3/2
shells are fully occupied. Added to that, the neutron 0p 1/2 shell is unbound.
Consequently, the use of 8 He as a core to describe 9 He and 10 He allows to
recapture the main features of the eigenstates of 9 He and 10 He, while having
small Gamow shell model space dimensions.
The 1/2 − state of 9 He and 0 + state of 10 He have a small width, close to
100 keV. Therefore, similarly to the study of the p 3/2 spectroscopic factors of
5 He/ 6 He, one can assume that the ground state of 10 He is bound and have the
energy of the 1/2 − state of 9 He vary so that it goes from bound to unbound. A
composite system of 2n+ 8 He is then simulated for different excitation energies.
B. As one considers neutron states, the non-analytic cusps of Eqs. (7.45), (7.46)
must occur at zero neutron separation energy for = 0, 1 only. Moreover, as
wave functions are built from the neutron p 1/2 partial wave, the overall behavior
of spectroscopic factors is similar to the 5 He/ 6 He case.
Exercise IV
A. 12 C has a deformed ground state, so that excitations from 0p 3/2 shells are sizable. The 12 C ground state wave function thus bears an important configuration
mixing in a no-core shell model picture [117]. However, 12 C is a closed-shell
nucleus at Hartree–Fock level, and there is a sizeable gap between 0p 3/2 and
0p 1/2 , of about 5 MeV. Thus, it is reasonable to assume that effective interactions
can be generated by using a 12 C core. Consequently, one can assume that taking
12 C as a core can provide with an effective interaction able to reproduce the main
features of 13,14 C, 13 N, and 14 O. In fact, the Zuker-Buck-McGrory interaction