6.2 T = 0, 1 Nuclear Matrix Elements in the Berggren Basis
245
or from the semi-classical arguments. Indeed, when one considers the one-body
states of high orbital angular momenta, the geometrical features of their spherical
harmonics prevail, whereas the different components of the nuclear interaction, of
central, tensor, spin–orbit type for the most important ones, are integrated out and
become unimportant as compared to the dependence both on the angular momenta
of one-body states and on the total angular momentum of the pair of nucleons.
It is particularly interesting to study the generic features of nuclear matrix
elements involving proton and neutron shells of a fixed angular momentum, and
coupled to the total angular momentum J . Spherical harmonics are indeed simple
when represented in coordinate space and their radial parts have very few nodes, so
that the main features of nuclear matrix elements can be analyzed therein without
problem. To study these matrix elements in well-bound nuclei, one can use a
simple surface delta interaction [22] which captures main features of the realistic
interactions and can be fitted to reproduce empirical nuclear matrix elements.
This interaction is not well suited in weakly bound and resonance nuclei
due to its zero-range, which ignores the asymptotes of nuclear wave functions.
Consequently, a finite-range interaction is better suited, as it depends on radial
wave functions at all radii, and not only at the surface radius. In the following,
we will use the Minnesota interaction [23] as nucleon–nucleon interaction for that
matter. Minnesota interaction is a rather simple effective force, depending on a few
parameters and bearing only central terms, so that the analysis of the its T = 0, 1
matrix elements in Gamow shell model can be made similarly as in the standard
shell model [22].
In the study of generic properties of two-body matrix elements, one will consider
various pairs of valence nucleons, consisting of either two neutrons, or one proton
and one neutron. Ignoring the Coulomb interaction, i.e. assuming exact isospin
symmetry, implies that pairs of protons or pairs of neutrons behave identically.
The considered closed-shell cores, namely 4 He, 16 O, 40 Ca, and 132 Sn, are standard
and are widely used in various spectroscopic studies. Bound and resonance valence
proton and neutron shells consist of 0p 3/2 , 0p 1/2 for 4 He, 0d 5/2 , 1s 1/2 , and 0d 3/2
for 16 O, and 0f 7/2 , 1p 3/2 , 1p 1/2 , 0f 5/2 for 40 Ca. For nuclei above the core of
132 Sn, one will consider only the shells of largest angular momenta, namely 0g 7/2 ,
0h 11/2 for protons and 0h 9/2 , 0i 13/2 , 1g 9/2 for neutrons. The strengths of spin–orbit
and Woods–Saxon potentials describing the used cores are first fitted to reproduce
experimental single-particle energies. Then, in order to emphasize effects of the
continuum coupling, the strengths of the Woods–Saxon core central potentials are
modified keeping the spin–orbit strength unchanged so that the two-body states have
a total binding energy of −10, −1, or +1 MeV with respect to the core. Coupling
to the continuum states, which is rather negligible at −10 MeV, becomes visible
at −1 MeV while the nuclear state is still bound. Finally, the continuum coupling
becomes important for unbound states with energy +1 MeV above the core. These
three examples then allow to assess the importance of continuum coupling on the
considered nuclear matrix elements.
245
or from the semi-classical arguments. Indeed, when one considers the one-body
states of high orbital angular momenta, the geometrical features of their spherical
harmonics prevail, whereas the different components of the nuclear interaction, of
central, tensor, spin–orbit type for the most important ones, are integrated out and
become unimportant as compared to the dependence both on the angular momenta
of one-body states and on the total angular momentum of the pair of nucleons.
It is particularly interesting to study the generic features of nuclear matrix
elements involving proton and neutron shells of a fixed angular momentum, and
coupled to the total angular momentum J . Spherical harmonics are indeed simple
when represented in coordinate space and their radial parts have very few nodes, so
that the main features of nuclear matrix elements can be analyzed therein without
problem. To study these matrix elements in well-bound nuclei, one can use a
simple surface delta interaction [22] which captures main features of the realistic
interactions and can be fitted to reproduce empirical nuclear matrix elements.
This interaction is not well suited in weakly bound and resonance nuclei
due to its zero-range, which ignores the asymptotes of nuclear wave functions.
Consequently, a finite-range interaction is better suited, as it depends on radial
wave functions at all radii, and not only at the surface radius. In the following,
we will use the Minnesota interaction [23] as nucleon–nucleon interaction for that
matter. Minnesota interaction is a rather simple effective force, depending on a few
parameters and bearing only central terms, so that the analysis of the its T = 0, 1
matrix elements in Gamow shell model can be made similarly as in the standard
shell model [22].
In the study of generic properties of two-body matrix elements, one will consider
various pairs of valence nucleons, consisting of either two neutrons, or one proton
and one neutron. Ignoring the Coulomb interaction, i.e. assuming exact isospin
symmetry, implies that pairs of protons or pairs of neutrons behave identically.
The considered closed-shell cores, namely 4 He, 16 O, 40 Ca, and 132 Sn, are standard
and are widely used in various spectroscopic studies. Bound and resonance valence
proton and neutron shells consist of 0p 3/2 , 0p 1/2 for 4 He, 0d 5/2 , 1s 1/2 , and 0d 3/2
for 16 O, and 0f 7/2 , 1p 3/2 , 1p 1/2 , 0f 5/2 for 40 Ca. For nuclei above the core of
132 Sn, one will consider only the shells of largest angular momenta, namely 0g 7/2 ,
0h 11/2 for protons and 0h 9/2 , 0i 13/2 , 1g 9/2 for neutrons. The strengths of spin–orbit
and Woods–Saxon potentials describing the used cores are first fitted to reproduce
experimental single-particle energies. Then, in order to emphasize effects of the
continuum coupling, the strengths of the Woods–Saxon core central potentials are
modified keeping the spin–orbit strength unchanged so that the two-body states have
a total binding energy of −10, −1, or +1 MeV with respect to the core. Coupling
to the continuum states, which is rather negligible at −10 MeV, becomes visible
at −1 MeV while the nuclear state is still bound. Finally, the continuum coupling
becomes important for unbound states with energy +1 MeV above the core. These
three examples then allow to assess the importance of continuum coupling on the
considered nuclear matrix elements.