9 The Unification of Structure and Reaction Frameworks
403
a coupled-channel representation with a basis of target–projectile reaction channels
within the resonating group method.
These theoretical frameworks are exact if all target and projectile many-body
states, as well as all emission channels, are included. However, due to the current
numerical limitations, it is impossible to attain full convergence. Reasonable
convergence can be obtained in lightest many-body systems, but the no-core shell
model, with or without the inclusion of continuum degrees of freedom, to which
the resonating group method is added, cannot treat nuclei in the sd shell [34].
Moreover, the inclusion of many-nucleon projectiles is very difficult therein, due
to the complex character of target–projectile channels. This is due to the necessary
use of Jacobi coordinates to properly describe target–projectile composites in the
asymptotic region.
A similar problem is encountered in standard reaction frameworks, where it is
customary to treat reactions in the frame of three-body or four-body problems,
i.e. with three or four subsystems in the asymptotic region. Indeed, while solving
the five-body and six-body problems is still possible [35, 36], they are at the limit
of current computing capabilities. Hence, the no-core shell model used along the
resonating group method, with or without the inclusion of continuum degrees of
freedom, is very limited and cannot be systematically used to consider reactions
involving nuclei beyond s- and p-shell nuclei.
Another possibility is to use the coupled-cluster method to calculate optical
potentials and cross sections [37–39]. In Ref. [37], composite states, formed by
target state and a one-proton projectile state, are expanded with a hybrid basis,
formed by the Berggren basis for the most important partial waves occurring in
the reaction process, and by a basis of harmonic oscillator states for the other partial
waves. This allows to precisely calculate proton overlap functions and asymptotic
normalization coefficients (see Sect. 7.3), from which phase shifts and hence
reaction cross sections can be calculated [37]. This framework was successfully
applied to the 40 Ca(p,p) elastic scattering reaction at low energies [37]. Another
formalism has been followed in Refs. [38, 39]. In this approach, targets are still
expanded in a hybrid basis formed by harmonic oscillator and Berggren basis states
in order to account for continuum coupling, whereas optical potentials are calculated
from the Green’s function representation of the A-nucleon system [38, 39]. These
optical potentials then provide with scattering states, from which phase shifts are
extracted and reaction cross sections implemented. First applications consisted of
the examples of neutron scattering on 16 O and 40,48 Ca [38, 39]. While the devised
numerical method is stable, the obtained optical potential lacks the absorption, as
its imaginary part is very small. The absence of imaginary potential was attributed
to missing higher order correlation effects in the coupled-cluster formalism using
the singles-and-doubles approximation [38, 39]. Optical potentials describing onenucleon scattering on oxygen and calcium isotopes have recently been calculated
in the context of the Gorkov–Green-function formalism [40]. However, as one has
already seen in Chap. 8, the coupled cluster or the Gorkov–Green-function methods
are also restricted to a small number of targets, near closed-shell nuclei typically.
403
a coupled-channel representation with a basis of target–projectile reaction channels
within the resonating group method.
These theoretical frameworks are exact if all target and projectile many-body
states, as well as all emission channels, are included. However, due to the current
numerical limitations, it is impossible to attain full convergence. Reasonable
convergence can be obtained in lightest many-body systems, but the no-core shell
model, with or without the inclusion of continuum degrees of freedom, to which
the resonating group method is added, cannot treat nuclei in the sd shell [34].
Moreover, the inclusion of many-nucleon projectiles is very difficult therein, due
to the complex character of target–projectile channels. This is due to the necessary
use of Jacobi coordinates to properly describe target–projectile composites in the
asymptotic region.
A similar problem is encountered in standard reaction frameworks, where it is
customary to treat reactions in the frame of three-body or four-body problems,
i.e. with three or four subsystems in the asymptotic region. Indeed, while solving
the five-body and six-body problems is still possible [35, 36], they are at the limit
of current computing capabilities. Hence, the no-core shell model used along the
resonating group method, with or without the inclusion of continuum degrees of
freedom, is very limited and cannot be systematically used to consider reactions
involving nuclei beyond s- and p-shell nuclei.
Another possibility is to use the coupled-cluster method to calculate optical
potentials and cross sections [37–39]. In Ref. [37], composite states, formed by
target state and a one-proton projectile state, are expanded with a hybrid basis,
formed by the Berggren basis for the most important partial waves occurring in
the reaction process, and by a basis of harmonic oscillator states for the other partial
waves. This allows to precisely calculate proton overlap functions and asymptotic
normalization coefficients (see Sect. 7.3), from which phase shifts and hence
reaction cross sections can be calculated [37]. This framework was successfully
applied to the 40 Ca(p,p) elastic scattering reaction at low energies [37]. Another
formalism has been followed in Refs. [38, 39]. In this approach, targets are still
expanded in a hybrid basis formed by harmonic oscillator and Berggren basis states
in order to account for continuum coupling, whereas optical potentials are calculated
from the Green’s function representation of the A-nucleon system [38, 39]. These
optical potentials then provide with scattering states, from which phase shifts are
extracted and reaction cross sections implemented. First applications consisted of
the examples of neutron scattering on 16 O and 40,48 Ca [38, 39]. While the devised
numerical method is stable, the obtained optical potential lacks the absorption, as
its imaginary part is very small. The absence of imaginary potential was attributed
to missing higher order correlation effects in the coupled-cluster formalism using
the singles-and-doubles approximation [38, 39]. Optical potentials describing onenucleon scattering on oxygen and calcium isotopes have recently been calculated
in the context of the Gorkov–Green-function formalism [40]. However, as one has
already seen in Chap. 8, the coupled cluster or the Gorkov–Green-function methods
are also restricted to a small number of targets, near closed-shell nuclei typically.