470
9 The Unification of Structure and Reaction Frameworks
D. Run the Gamow shell model coupled-channel code to generate the radiative
capture cross sections of 7 Be(p, γ ) and 7 Li(n, γ ).
Used proton and neutron effective charges in the calculation for 7 Be(p, γ ) and
7 Li(n, γ ) are larger than their standard theoretical values by factors 2.5 and 2,
respectively. Explain the physical significance of this modification.
9.4.3 4 H e(d, d) Cross Section and Deuteron Emission Width
The 4 He(d,d) elastic scattering reaction is interesting from many points of views.
Firstly, it is the simplest reaction involving a many-nucleon projectile. Indeed, it
can be modelled by two valence nucleons, one proton and one neutron, above a
well-bound 4 He core, so that dimensions of model space are minimal. Moreover,
experimental projectile energies are smaller than 10 MeV, so that the number of
reaction channels to consider remains tractable.
6 Li isotope also has a very low deuteron emission threshold, and its proton and
neutron emission thresholds are 3–4 MeV higher. Consequently, all T = 0 states of
6 Li below 10 MeV are deuteron emitters so deuteron continuum degrees of freedom
must be included to properly describe the spectrum of 6 Li. The cross sections of the
4 He(d,d) elastic scattering reaction are then dominated by the deuteron channels,
and one can neglect other emission channels. T = 1 excited states are not important
hereby as isospin is almost entirely conserved in the 4 He(d,d) reaction.
In this study using the unified framework of the coupled-channel Gamow shell
model, one will describe with the same Hamiltonian the energy spectrum of 6 Li, the
asymptotic normalization coefficient of the 6 Li ground state, and the 4 He(d,d) elastic
scattering cross sections. To generate intrinsic states of deuteron, the realistic N 3 LO
interaction [92], fitted on phase shifts properties of proton–neutron elastic scattering
reactions [93], is firstly diagonalized in a two-body Berggren basis written in relative
coordinates and generated by a Woods–Saxon potential. Diffuseness, radius, central,
and spin–orbit strengths of the potential are 0.65 fm, 1.5 fm, 40 MeV, and 7.5 MeV,
respectively. The internal structure of deuteron is taken into account using the N 3 LO
interaction [92], fitted on phase shifts properties of proton–neutron elastic scattering
reactions (see Ref.[93] for details about basis space truncations).
The basis of relative channel two-body states is generated by Woods–Saxon
potential, whose radius is 1.5 fm and whose central and spin–orbit potential depths
are equal to 40 MeV and 7.5 MeV, respectively. There is only one bound basis
state, which is the ground state of the 1 S 0 partial wave, of energy –0.457 MeV.
Consequently, the basis of relative channel two-body states can be expected to be
well adapted to expand intrinsic deuteron wave functions because deuteron has only
one bound state, which is dominated by the 1 S 0 channel. Contours along the real
k-axis in Berggren basis are discretized with 10 points. It has been checked in
Ref.[93] that a finer discretization of contours does not change results significantly.
The deuteron ground state energy obtained in this way is −2.061 MeV, close to
9 The Unification of Structure and Reaction Frameworks
D. Run the Gamow shell model coupled-channel code to generate the radiative
capture cross sections of 7 Be(p, γ ) and 7 Li(n, γ ).
Used proton and neutron effective charges in the calculation for 7 Be(p, γ ) and
7 Li(n, γ ) are larger than their standard theoretical values by factors 2.5 and 2,
respectively. Explain the physical significance of this modification.
9.4.3 4 H e(d, d) Cross Section and Deuteron Emission Width
The 4 He(d,d) elastic scattering reaction is interesting from many points of views.
Firstly, it is the simplest reaction involving a many-nucleon projectile. Indeed, it
can be modelled by two valence nucleons, one proton and one neutron, above a
well-bound 4 He core, so that dimensions of model space are minimal. Moreover,
experimental projectile energies are smaller than 10 MeV, so that the number of
reaction channels to consider remains tractable.
6 Li isotope also has a very low deuteron emission threshold, and its proton and
neutron emission thresholds are 3–4 MeV higher. Consequently, all T = 0 states of
6 Li below 10 MeV are deuteron emitters so deuteron continuum degrees of freedom
must be included to properly describe the spectrum of 6 Li. The cross sections of the
4 He(d,d) elastic scattering reaction are then dominated by the deuteron channels,
and one can neglect other emission channels. T = 1 excited states are not important
hereby as isospin is almost entirely conserved in the 4 He(d,d) reaction.
In this study using the unified framework of the coupled-channel Gamow shell
model, one will describe with the same Hamiltonian the energy spectrum of 6 Li, the
asymptotic normalization coefficient of the 6 Li ground state, and the 4 He(d,d) elastic
scattering cross sections. To generate intrinsic states of deuteron, the realistic N 3 LO
interaction [92], fitted on phase shifts properties of proton–neutron elastic scattering
reactions [93], is firstly diagonalized in a two-body Berggren basis written in relative
coordinates and generated by a Woods–Saxon potential. Diffuseness, radius, central,
and spin–orbit strengths of the potential are 0.65 fm, 1.5 fm, 40 MeV, and 7.5 MeV,
respectively. The internal structure of deuteron is taken into account using the N 3 LO
interaction [92], fitted on phase shifts properties of proton–neutron elastic scattering
reactions (see Ref.[93] for details about basis space truncations).
The basis of relative channel two-body states is generated by Woods–Saxon
potential, whose radius is 1.5 fm and whose central and spin–orbit potential depths
are equal to 40 MeV and 7.5 MeV, respectively. There is only one bound basis
state, which is the ground state of the 1 S 0 partial wave, of energy –0.457 MeV.
Consequently, the basis of relative channel two-body states can be expected to be
well adapted to expand intrinsic deuteron wave functions because deuteron has only
one bound state, which is dominated by the 1 S 0 channel. Contours along the real
k-axis in Berggren basis are discretized with 10 points. It has been checked in
Ref.[93] that a finer discretization of contours does not change results significantly.
The deuteron ground state energy obtained in this way is −2.061 MeV, close to