404
9 The Unification of Structure and Reaction Frameworks
As the Gamow shell model can be defined accurately in a core + valence
particle approach, it can be considered in wider sections of the nuclear chart.
Moreover, realistic nucleon–nucleon forces issued from many-body perturbation
theory [41, 42] or the in-medium similarity renormalization group method [43–
47] can be included in this framework as well [48]. However, the Gamow shell
model methods presented in previous chapters cannot be used directly to calculate
reaction cross sections. Indeed, the many-body scattering states arising from the
diagonalization of the Gamow shell model Hamiltonian matrix do not have a well
defined asymptotic behavior so the many-body eigenstates are generally a linear
combination of many reaction channels. Hence, the scattering eigenstates of the
Gamow shell model cannot be identified, contrary to resonant states, for which the
overlap method can be utilized (see Sect. 5.4).
Therefore, one has to build a new theoretical framework to be able to use the
Gamow shell model therein and obtain in this way a unified theory of nuclear
structure and reactions, which remains valid in both bound and unbound atomic
nuclei. To achieve this objective, a route similar to that used in the no-core shell
model with the resonating group method [24, 28–31, 33] has been followed. One
constructs target–projectile reaction channels using the resonating group method,
where target and projectile wave functions are issued from the diagonalization of the
Gamow shell model Hamiltonian. In this way, the considered targets and projectiles
can be weakly bound or resonance nuclei, and nevertheless, the reaction channels
are well defined. Many-body scattering states are obtained from the solution of a
Gamow shell model Hamiltonian expressed in the coupled-channel representation.
Reaction observables, such as, for example, the elastic/inelastic scattering and
radiative capture cross sections, can then be calculated.
In this approach, the Gamow shell model unifies structure and reaction degrees
of freedom as both spectra and reaction cross sections can be calculated using the
same Hamiltonian. In particular, the calculation of the coupling potentials entering
coupled-channel equations includes complete information about structure of target
and projectile, and the wave functions of target and projectile are expanded on a
basis of Berggren many-body states. Added to that, continuum degrees of freedom
are taken into account to calculate spectra, as resonant states in the coupled-channel
Gamow shell model are linear combination of reaction channels. As all degrees of
freedom are included in the coupled-channel Gamow shell model, it is similar to nocore shell model with continuum used with the resonating group method [26–29],
or the shell model embedded in the continuum in S-matrix formulation [15, 19–21].
The fact that target nuclei can be loosely bound or even unbound is a very
interesting feature of the coupled-channel Gamow shell model. As targets are
directly expanded with the Berggren basis, it is not necessary to expand their
asymptotes with reaction channels, as is necessary in the no-core shell model
with resonating group method [26–29]. Therefore, a smaller number of reaction
channels are needed to represent the same many-body scattering wave functions.
For example, the description of scattering reaction p + 11 Li in the no-core shell
model with continuum demands three nucleons in the asymptotic region, whereas
it is a two-body reaction using coupled-channel Gamow shell model, as the two-
9 The Unification of Structure and Reaction Frameworks
As the Gamow shell model can be defined accurately in a core + valence
particle approach, it can be considered in wider sections of the nuclear chart.
Moreover, realistic nucleon–nucleon forces issued from many-body perturbation
theory [41, 42] or the in-medium similarity renormalization group method [43–
47] can be included in this framework as well [48]. However, the Gamow shell
model methods presented in previous chapters cannot be used directly to calculate
reaction cross sections. Indeed, the many-body scattering states arising from the
diagonalization of the Gamow shell model Hamiltonian matrix do not have a well
defined asymptotic behavior so the many-body eigenstates are generally a linear
combination of many reaction channels. Hence, the scattering eigenstates of the
Gamow shell model cannot be identified, contrary to resonant states, for which the
overlap method can be utilized (see Sect. 5.4).
Therefore, one has to build a new theoretical framework to be able to use the
Gamow shell model therein and obtain in this way a unified theory of nuclear
structure and reactions, which remains valid in both bound and unbound atomic
nuclei. To achieve this objective, a route similar to that used in the no-core shell
model with the resonating group method [24, 28–31, 33] has been followed. One
constructs target–projectile reaction channels using the resonating group method,
where target and projectile wave functions are issued from the diagonalization of the
Gamow shell model Hamiltonian. In this way, the considered targets and projectiles
can be weakly bound or resonance nuclei, and nevertheless, the reaction channels
are well defined. Many-body scattering states are obtained from the solution of a
Gamow shell model Hamiltonian expressed in the coupled-channel representation.
Reaction observables, such as, for example, the elastic/inelastic scattering and
radiative capture cross sections, can then be calculated.
In this approach, the Gamow shell model unifies structure and reaction degrees
of freedom as both spectra and reaction cross sections can be calculated using the
same Hamiltonian. In particular, the calculation of the coupling potentials entering
coupled-channel equations includes complete information about structure of target
and projectile, and the wave functions of target and projectile are expanded on a
basis of Berggren many-body states. Added to that, continuum degrees of freedom
are taken into account to calculate spectra, as resonant states in the coupled-channel
Gamow shell model are linear combination of reaction channels. As all degrees of
freedom are included in the coupled-channel Gamow shell model, it is similar to nocore shell model with continuum used with the resonating group method [26–29],
or the shell model embedded in the continuum in S-matrix formulation [15, 19–21].
The fact that target nuclei can be loosely bound or even unbound is a very
interesting feature of the coupled-channel Gamow shell model. As targets are
directly expanded with the Berggren basis, it is not necessary to expand their
asymptotes with reaction channels, as is necessary in the no-core shell model
with resonating group method [26–29]. Therefore, a smaller number of reaction
channels are needed to represent the same many-body scattering wave functions.
For example, the description of scattering reaction p + 11 Li in the no-core shell
model with continuum demands three nucleons in the asymptotic region, whereas
it is a two-body reaction using coupled-channel Gamow shell model, as the two-