254
6 Physical Applications of the Gamow Shell Model
extrapolations of the used models. The comparison of the quality of different models
becomes also problematic (see Refs. [29–31]). In fact, the systematic calculation of
the uncertainties of theoretical predictions has only started recently [32–39].
When considering light nuclei, it is convenient to separate the model space
in a core and valence part, on the one hand, and, on the other hand, derive a
Hamiltonian built from a one-body potential mimicking the effect of the core
and an effective two-body interaction comprising central, spin–orbit, tensor, and
Coulomb terms. This objective will be handled within two different model spaces.
One will firstly study He, Li, and Be isotopes with A ≤ 9 in a large (psdf)model space, where both resonant and scattering basis states are included. The
aim is to describe structure of nuclei in the A < ∼ 12 region (see Ref. [40] for the
application of the Gamow shell model), with the inclusion of a statistical analysis
of the model results. For this, singular value decomposition is used to estimate
the indeterminacy of parameters. By using the covariance matrix obtained within
the linear regression approach, the statistical uncertainties on the parameters and
observables can be evaluated. One can then apply the Hamiltonian of Gamow shell
model to predict two-nucleon correlation densities and excitation spectra with the
quantified uncertainties. Secondly, one will study the proton-rich and neutron-rich
nuclei with A ≈ 10 nucleons.
To optimize calculations, the single-particle basis is adapted for each nucleus
(see Sect. 5.5). However, it was noticed that the use of the multi-Slater determinant
Hartree–Fock potential in light p-shell nuclei could become numerically imprecise
in the presence of broad 0p 3/2 and 0p 1/2 resonance states. This arises because
of the nonlocal character of the multi-Slater determinant Hartree–Fock potential.
Indeed, resonance states generated by nonlocal potentials are typically less precise
numerically than resonance states generated by local potentials. Consequently,
it was preferred to replace the multi-Slater determinant coupled Hartree–Fock
potential by a Woods–Saxon potential whose eigenstates have the same complex
energy as obtained when using the multi-Slater determinant Hartree–Fock potential.
As the Coulomb potential is directly included in the single-particle basis, one-body
basis states possess the correct asymptotic behavior at infinity (see Sect. 3.7.1).
6.3.1 Study of the Helium, Lithium, and Beryllium Isotope Chains
In the studies of He, Li, and Be isotopes, one is using a core of 4 He as its large
binding energy implies that core excitations can be safely neglected. The Gamow
shell model interaction has two components: the one-body potential of the core
ˆ
U core and the two-body interaction V between the valence nucleons. The core
potential acting on valence particles is modelled by a Woods–Saxon potential with
a Coulomb part, whose charge radius is taken from the experimental value of 4 He:
R ch = 1.681 fm [41]. Different parameters are taken for proton and neutron Woods–
Saxon potentials.
A general form of the two-body effective nuclear potential was derived in Refs.
[42, 43]. In particular, a tensor part was added to central and two-body spin–
6 Physical Applications of the Gamow Shell Model
extrapolations of the used models. The comparison of the quality of different models
becomes also problematic (see Refs. [29–31]). In fact, the systematic calculation of
the uncertainties of theoretical predictions has only started recently [32–39].
When considering light nuclei, it is convenient to separate the model space
in a core and valence part, on the one hand, and, on the other hand, derive a
Hamiltonian built from a one-body potential mimicking the effect of the core
and an effective two-body interaction comprising central, spin–orbit, tensor, and
Coulomb terms. This objective will be handled within two different model spaces.
One will firstly study He, Li, and Be isotopes with A ≤ 9 in a large (psdf)model space, where both resonant and scattering basis states are included. The
aim is to describe structure of nuclei in the A < ∼ 12 region (see Ref. [40] for the
application of the Gamow shell model), with the inclusion of a statistical analysis
of the model results. For this, singular value decomposition is used to estimate
the indeterminacy of parameters. By using the covariance matrix obtained within
the linear regression approach, the statistical uncertainties on the parameters and
observables can be evaluated. One can then apply the Hamiltonian of Gamow shell
model to predict two-nucleon correlation densities and excitation spectra with the
quantified uncertainties. Secondly, one will study the proton-rich and neutron-rich
nuclei with A ≈ 10 nucleons.
To optimize calculations, the single-particle basis is adapted for each nucleus
(see Sect. 5.5). However, it was noticed that the use of the multi-Slater determinant
Hartree–Fock potential in light p-shell nuclei could become numerically imprecise
in the presence of broad 0p 3/2 and 0p 1/2 resonance states. This arises because
of the nonlocal character of the multi-Slater determinant Hartree–Fock potential.
Indeed, resonance states generated by nonlocal potentials are typically less precise
numerically than resonance states generated by local potentials. Consequently,
it was preferred to replace the multi-Slater determinant coupled Hartree–Fock
potential by a Woods–Saxon potential whose eigenstates have the same complex
energy as obtained when using the multi-Slater determinant Hartree–Fock potential.
As the Coulomb potential is directly included in the single-particle basis, one-body
basis states possess the correct asymptotic behavior at infinity (see Sect. 3.7.1).
6.3.1 Study of the Helium, Lithium, and Beryllium Isotope Chains
In the studies of He, Li, and Be isotopes, one is using a core of 4 He as its large
binding energy implies that core excitations can be safely neglected. The Gamow
shell model interaction has two components: the one-body potential of the core
ˆ
U core and the two-body interaction V between the valence nucleons. The core
potential acting on valence particles is modelled by a Woods–Saxon potential with
a Coulomb part, whose charge radius is taken from the experimental value of 4 He:
R ch = 1.681 fm [41]. Different parameters are taken for proton and neutron Woods–
Saxon potentials.
A general form of the two-body effective nuclear potential was derived in Refs.
[42, 43]. In particular, a tensor part was added to central and two-body spin–