276
6 Physical Applications of the Gamow Shell Model
in 6 He than in 6 Be. This is due to the repulsive Coulomb interaction between valence
protons in 6 Be.
As for 6 Li, the valence proton–neutron pair is very strongly correlated because
the T = 0 interaction is much stronger than the T = 1 interaction. This is related
to the structure of the ground state of 6 Li, which mainly consists of a deuteron
cluster orbiting around a 4 He core. Due to the strong binding character of the T = 0
interaction, the cluster structure of the valence nucleons in 6 Li is more prevalent than
in 6 He and 6 Be (see Fig. 6.14). While the cluster shape component is about twice as
large as the cigar shape component in 6 He and 6 Be, the cluster shape component of
6 Li is three times larger than its cigar shape component. Consequently, the valence
nucleon pair is more clusterized in 6 Li than in 6 He and 6 Be, which is entirely due to
the different isospin of their ground state wave functions.
For all nuclei, the two-nucleon angular correlations obtained with the Gamow
shell model in cluster orbital shell model coordinates and with the three-body model
in Jacobi coordinates are close one to another. All these different features seen in
Figs. 6.13 and 6.14 demonstrate that the nucleon–nucleon angular correlations contain valuable information about the interaction of valence nucleons and their spatial
configuration. Indeed, the cluster and cigar shapes directly appear from Fig. 6.14,
as well as the averaged spin and isospin content of valence nucleons. Nevertheless,
due to the integration over the radial r coordinate, the asymptotic decrease of twonucleon densities cannot be seen on two-nucleon angular correlations. To state the
radial dependence of nucleon pairs in a nuclear wave function, it is necessary to
consider the full correlation density, as done in Fig. 6.10.
6.5
Inclusion of Continuum Couplings in the Pairing Model
The pairing part of the nucleon–nucleon interaction is responsible for correlations
and fluctuations related to the superfluid character of the nuclear medium, in both
finite nuclei and neutron stars [116]. Therefore, Hamiltonians consisting of a onebody part and of a two-body part built from a pairing interaction are widely studied.
Moreover, exact solutions of the pairing Hamiltonian for a constant pairing strength
and a discrete set of single-particle levels are known since the seminal work of
Richardson [117, 118].
It is possible to derive three classes of solvable models for fermions and bosons
involving pairing Hamiltonians [119], if one combines the exact solution of the
Richardson pairing model to that proposed by Gaudin for quantum spin systems
[119]. The Richardson pairing model, i.e. that of a constant g pairing Hamiltonian,
is a particular case of the rational class of integrable models. Added to that, arbitrary
combinations of the integrals of motion within each of the classes can generate
more general exactly solvable pairing models. In particular, the pairing Hamiltonian
encompassing the main features of heavy nuclei can be derived from the hyperbolic
family of Gaudin models [120]. One can also note that the rational Gaudin model
has been generalized to larger Lie algebras, with, for example, the SO(5) for
6 Physical Applications of the Gamow Shell Model
in 6 He than in 6 Be. This is due to the repulsive Coulomb interaction between valence
protons in 6 Be.
As for 6 Li, the valence proton–neutron pair is very strongly correlated because
the T = 0 interaction is much stronger than the T = 1 interaction. This is related
to the structure of the ground state of 6 Li, which mainly consists of a deuteron
cluster orbiting around a 4 He core. Due to the strong binding character of the T = 0
interaction, the cluster structure of the valence nucleons in 6 Li is more prevalent than
in 6 He and 6 Be (see Fig. 6.14). While the cluster shape component is about twice as
large as the cigar shape component in 6 He and 6 Be, the cluster shape component of
6 Li is three times larger than its cigar shape component. Consequently, the valence
nucleon pair is more clusterized in 6 Li than in 6 He and 6 Be, which is entirely due to
the different isospin of their ground state wave functions.
For all nuclei, the two-nucleon angular correlations obtained with the Gamow
shell model in cluster orbital shell model coordinates and with the three-body model
in Jacobi coordinates are close one to another. All these different features seen in
Figs. 6.13 and 6.14 demonstrate that the nucleon–nucleon angular correlations contain valuable information about the interaction of valence nucleons and their spatial
configuration. Indeed, the cluster and cigar shapes directly appear from Fig. 6.14,
as well as the averaged spin and isospin content of valence nucleons. Nevertheless,
due to the integration over the radial r coordinate, the asymptotic decrease of twonucleon densities cannot be seen on two-nucleon angular correlations. To state the
radial dependence of nucleon pairs in a nuclear wave function, it is necessary to
consider the full correlation density, as done in Fig. 6.10.
6.5
Inclusion of Continuum Couplings in the Pairing Model
The pairing part of the nucleon–nucleon interaction is responsible for correlations
and fluctuations related to the superfluid character of the nuclear medium, in both
finite nuclei and neutron stars [116]. Therefore, Hamiltonians consisting of a onebody part and of a two-body part built from a pairing interaction are widely studied.
Moreover, exact solutions of the pairing Hamiltonian for a constant pairing strength
and a discrete set of single-particle levels are known since the seminal work of
Richardson [117, 118].
It is possible to derive three classes of solvable models for fermions and bosons
involving pairing Hamiltonians [119], if one combines the exact solution of the
Richardson pairing model to that proposed by Gaudin for quantum spin systems
[119]. The Richardson pairing model, i.e. that of a constant g pairing Hamiltonian,
is a particular case of the rational class of integrable models. Added to that, arbitrary
combinations of the integrals of motion within each of the classes can generate
more general exactly solvable pairing models. In particular, the pairing Hamiltonian
encompassing the main features of heavy nuclei can be derived from the hyperbolic
family of Gaudin models [120]. One can also note that the rational Gaudin model
has been generalized to larger Lie algebras, with, for example, the SO(5) for