6.6 Comparison of Gamow Shell Model and Hamiltonian Complex Scaling. . .
299
6.6.3 Numerical Comparison of Eigenenergies of Gamow Shell
Model and Hamiltonian Complex Scaling Method
One will now compare Gamow shell model and Hamiltonian complex scaling
frameworks in a numerical example. The considered model space must be sufficiently small to allow for exact calculations with both approaches, while the
considered system must bear weakly bound or unbound eigenstates. The 6 He and
6 Be isotopes are ideal testing grounds for that matter [152]. Indeed, their 0 + ground
state and 2 + first excited state are either halo or resonance states. Moreover, 6 He
and 6 Be can be modelled as two valence nucleons above an inert 4 He core, so that
model spaces are not excessively large.
The used Hamiltonian consists of a KKNN potential mimicking a 4 He core,
to which a residual inter-neutron interaction is added. The cluster orbital shell
model formalism is utilized (see Sect. 5.2), so that no center-of-mass excitation can
occur. The used nuclear interaction consists of the Minnesota force [23], to which a
phenomenological Gaussian interaction is added to account for missing three-body
[152]. The Coulomb interaction is also included when considering 6 Be. Hence, all
radial form factors of the Hamiltonian are of Coulomb or Gaussian type, so that
matrix elements can be calculated analytically in the basis of Gaussian functions
[152].
Different model spaces have been considered to study 6 He and 6 Be eigenstates
with the Gamow shell model and Hamiltonian complex scaling method (see Ref.
[152] for details). In particular, the maximal orbital momentum of valence partial
waves l max varies from 1 to 5. Results issued from the diagonalization in the Gamow
shell model and Hamiltonian complex scaling frameworks are displayed in the upper
and lower panels of Fig. 6.21 for 6 He and 6 Be, respectively.
As can be seen on Fig. 6.21), the eigenenergies provided by Gamow shell model
and Hamiltonian complex scaling diagonalizations are close to each other. On the
one hand, the halo 0 + ground state energy of 6 He is almost exactly the same in
the Gamow shell model and Hamiltonian complex scaling frameworks. On the
other hand, the resonance energies of the ground state 0 + in 6 Be, and the first
excited states 2 + in 6 He and 6 Be, differ by about 10–20 keV in both approaches.
As a consequence, the calculations depicted in Fig. 6.21 have proved that Gamow
shell model and Hamiltonian complex scaling method are equivalent in practice.
It is, nevertheless, important to understand the relatively larger discrepancy of
resonance energies compared to bound state energies. This discrepancy has two
main origins. The first one comes from the different methods used to calculate the
two-body matrix elements. While a harmonic oscillator basis expansion is used in
the Gamow shell model for that matter (see Sect. 5.6)), two-body matrix elements
are analytical within the Hamiltonian complex scaling framework. Consequently, a
slight dependence on the number of harmonic oscillator basis states N max arises in
Gamow shell model calculations. Another reason is the dependence in practice on
the used rotation angle θ for the diagonalization of complex-scaled Hamiltonians
(see Sect. 6.6.2.2).
299
6.6.3 Numerical Comparison of Eigenenergies of Gamow Shell
Model and Hamiltonian Complex Scaling Method
One will now compare Gamow shell model and Hamiltonian complex scaling
frameworks in a numerical example. The considered model space must be sufficiently small to allow for exact calculations with both approaches, while the
considered system must bear weakly bound or unbound eigenstates. The 6 He and
6 Be isotopes are ideal testing grounds for that matter [152]. Indeed, their 0 + ground
state and 2 + first excited state are either halo or resonance states. Moreover, 6 He
and 6 Be can be modelled as two valence nucleons above an inert 4 He core, so that
model spaces are not excessively large.
The used Hamiltonian consists of a KKNN potential mimicking a 4 He core,
to which a residual inter-neutron interaction is added. The cluster orbital shell
model formalism is utilized (see Sect. 5.2), so that no center-of-mass excitation can
occur. The used nuclear interaction consists of the Minnesota force [23], to which a
phenomenological Gaussian interaction is added to account for missing three-body
[152]. The Coulomb interaction is also included when considering 6 Be. Hence, all
radial form factors of the Hamiltonian are of Coulomb or Gaussian type, so that
matrix elements can be calculated analytically in the basis of Gaussian functions
[152].
Different model spaces have been considered to study 6 He and 6 Be eigenstates
with the Gamow shell model and Hamiltonian complex scaling method (see Ref.
[152] for details). In particular, the maximal orbital momentum of valence partial
waves l max varies from 1 to 5. Results issued from the diagonalization in the Gamow
shell model and Hamiltonian complex scaling frameworks are displayed in the upper
and lower panels of Fig. 6.21 for 6 He and 6 Be, respectively.
As can be seen on Fig. 6.21), the eigenenergies provided by Gamow shell model
and Hamiltonian complex scaling diagonalizations are close to each other. On the
one hand, the halo 0 + ground state energy of 6 He is almost exactly the same in
the Gamow shell model and Hamiltonian complex scaling frameworks. On the
other hand, the resonance energies of the ground state 0 + in 6 Be, and the first
excited states 2 + in 6 He and 6 Be, differ by about 10–20 keV in both approaches.
As a consequence, the calculations depicted in Fig. 6.21 have proved that Gamow
shell model and Hamiltonian complex scaling method are equivalent in practice.
It is, nevertheless, important to understand the relatively larger discrepancy of
resonance energies compared to bound state energies. This discrepancy has two
main origins. The first one comes from the different methods used to calculate the
two-body matrix elements. While a harmonic oscillator basis expansion is used in
the Gamow shell model for that matter (see Sect. 5.6)), two-body matrix elements
are analytical within the Hamiltonian complex scaling framework. Consequently, a
slight dependence on the number of harmonic oscillator basis states N max arises in
Gamow shell model calculations. Another reason is the dependence in practice on
the used rotation angle θ for the diagonalization of complex-scaled Hamiltonians
(see Sect. 6.6.2.2).