368
8 No-Core Gamow Shell Model
Posterior probability distributions also test the dependence on prior distributions.
Error propagation on predictions of effective field theory shows the dependence on
the uncertainties of the experimental data used to fit the low-energy constants.
In the domain of the light nuclei, with a number of nucleons typically smaller
than about 15, the most fundamental approaches are that of the no-core shell
model [27, 28] and of the Green-Function Monte Carlo approach [6]. They are
based on the quasi-exact resolution of the many-body Schrödinger equation. The
Green-Function Monte Carlo approach starts from an approximate many-body wave
function of Jastrow type provided by the variational principle, which is then evolved
in configuration space via stochastic paths generated by a Monte Carlo process
[6]. The no-core shell model makes use of a large basis of Slater determinants,
generated by harmonic oscillator states, with which completeness is achieved [28].
Both approaches are rather successful in describing binding energies and spectra of
light nuclei [29, 30].
The no-core shell model can also calculate particle-emission widths and reaction
cross sections by including continuum degrees of freedom with the resonating
group method approach [31–40]. This approach is very close to that of the shell
model embedded in the continuum, developed about 20 years ago in the context of
nucleon and cluster direct reactions [41–44]. The resulting no-core shell model with
continuum was applied to calculate the S-matrix of reactions populating 11 Be [45],
(d, t) fusion reactions [46], the calculation of 6 He and 9 He nuclear states [34, 47],
elastic scattering of protons on 10 C [48] and the 7 Li(d,p) 8 Li transfer reaction [36].
While very powerful, the no-core shell model and the Green-Function Monte
Carlo approach are, however, hindered by computational limitations when the
number of nucleons increases, upon which their numerical cost scales exponentially
[6, 28]. The often necessary use of three-body forces in the no-core shell model
worsens the situation, as the latter greatly augments the number of nonzero matrix
elements of the Hamiltonian, so that nuclei with more than 10 nucleons can hardly
be considered therein. Moreover, the Green-Function Monte Carlo approach can
only provide with bound states, so that resonance states cannot be calculated in
principle, except through extrapolation methods [49]. Hence, different methods have
to be used to consider other nuclei of the nuclear chart.
For this, the coupled-cluster and Gorkov–Green-function frameworks have been
developed, where the many-body wave function is no longer calculated, but is
replaced by an exponential ansatz of a simple many-body wave function in coupledcluster approach [50–54], and by many-body particle and pairing densities in
Gorkov–Green-function framework [55–57], which iteratively converge to their
exact value starting from a Hartree–Fock Bogoliubov many-body state via the use of
Gorkov propagators. Consequently, medium and heavy nuclei can be treated, with
the examples of Sn and Ni isotopes in the coupled-cluster framework [53,58] and of
the Ca, Ti, and Ni isotopes in the Gorkov–Green-function framework [56, 57]. The
neutron dripline can also be reached by the coupled-cluster method and unbound
states can be calculated therein using the Berggren basis to include continuum
coupling [51].
8 No-Core Gamow Shell Model
Posterior probability distributions also test the dependence on prior distributions.
Error propagation on predictions of effective field theory shows the dependence on
the uncertainties of the experimental data used to fit the low-energy constants.
In the domain of the light nuclei, with a number of nucleons typically smaller
than about 15, the most fundamental approaches are that of the no-core shell
model [27, 28] and of the Green-Function Monte Carlo approach [6]. They are
based on the quasi-exact resolution of the many-body Schrödinger equation. The
Green-Function Monte Carlo approach starts from an approximate many-body wave
function of Jastrow type provided by the variational principle, which is then evolved
in configuration space via stochastic paths generated by a Monte Carlo process
[6]. The no-core shell model makes use of a large basis of Slater determinants,
generated by harmonic oscillator states, with which completeness is achieved [28].
Both approaches are rather successful in describing binding energies and spectra of
light nuclei [29, 30].
The no-core shell model can also calculate particle-emission widths and reaction
cross sections by including continuum degrees of freedom with the resonating
group method approach [31–40]. This approach is very close to that of the shell
model embedded in the continuum, developed about 20 years ago in the context of
nucleon and cluster direct reactions [41–44]. The resulting no-core shell model with
continuum was applied to calculate the S-matrix of reactions populating 11 Be [45],
(d, t) fusion reactions [46], the calculation of 6 He and 9 He nuclear states [34, 47],
elastic scattering of protons on 10 C [48] and the 7 Li(d,p) 8 Li transfer reaction [36].
While very powerful, the no-core shell model and the Green-Function Monte
Carlo approach are, however, hindered by computational limitations when the
number of nucleons increases, upon which their numerical cost scales exponentially
[6, 28]. The often necessary use of three-body forces in the no-core shell model
worsens the situation, as the latter greatly augments the number of nonzero matrix
elements of the Hamiltonian, so that nuclei with more than 10 nucleons can hardly
be considered therein. Moreover, the Green-Function Monte Carlo approach can
only provide with bound states, so that resonance states cannot be calculated in
principle, except through extrapolation methods [49]. Hence, different methods have
to be used to consider other nuclei of the nuclear chart.
For this, the coupled-cluster and Gorkov–Green-function frameworks have been
developed, where the many-body wave function is no longer calculated, but is
replaced by an exponential ansatz of a simple many-body wave function in coupledcluster approach [50–54], and by many-body particle and pairing densities in
Gorkov–Green-function framework [55–57], which iteratively converge to their
exact value starting from a Hartree–Fock Bogoliubov many-body state via the use of
Gorkov propagators. Consequently, medium and heavy nuclei can be treated, with
the examples of Sn and Ni isotopes in the coupled-cluster framework [53,58] and of
the Ca, Ti, and Ni isotopes in the Gorkov–Green-function framework [56, 57]. The
neutron dripline can also be reached by the coupled-cluster method and unbound
states can be calculated therein using the Berggren basis to include continuum
coupling [51].