7.3 Asymptotic Normalization Coefficient in Mirror Nuclei
331
two-neutron halos. On the other hand, 10 He and 26,28 O are weakly unbound. Pairing
correlations in this case are not sufficiently strong to bind these nuclei mitigating
only their resonance character.
It is clear that halo nuclei have to be described within a quantum framework comprising both nucleon–nucleon correlations and continuum coupling. The Gamow
shell model is thus the tool of choice, as continuum coupling is present at the onebody level through the use of the Berggren basis, and nucleon–nucleon correlations
are exactly taken into account by configuration mixing. Moreover, as the wave
functions are explicitly calculated, one can directly assess the structure of halo
nuclear states with overlap functions and spectroscopic factors, which, as one will
see, provide with the weights and radial formfactors of two-cluster configurations,
such as 5 He + neutron in 6 He. The exact consideration of the Coulomb interaction
in the Gamow shell model also allows to precisely determine the isospin mixing
features of the nuclear Hamiltonian, prominent to understand physics at particleemission threshold, and thus halo nuclei.
7.3
Asymptotic Normalization Coefficient in Mirror Nuclei
The one-nucleon overlap functions and the resulting spectroscopic factors are
standard tools in the theory of direct reactions [66–71]. The fundamental interest
of asymptotic normalization coefficients is that they are not sensitive to the changes
of short-range physics (see Ref. [72], the discussion in Ref. [73] about observables
and non-observables, and also Ref. [74] for an overview of unitary transformations
and their impact on nuclear structure operators). Indeed, asymptotic normalization
coefficients are invariant under unitary transformations, so that they are observables,
contrary to spectroscopic factors. This arises because a unitary transformation of the
Hamiltonian only affects the interior region of the wave function, so that asymptotic
normalization coefficients remain unchanged.
Experimentally, asymptotic normalization coefficients have been measured for
s-shell and p-shell nuclei (see Ref. [75] and the references therein). Theoretical
calculations of asymptotic normalization coefficients have been performed using
the hyperspherical harmonics approach [76], the Gamow shell model and the shell
model embedded in the continuum [55], the Green’s function Monte Carlo [77–79]
(see also Ref. [80]).
The spectroscopic factors are deduced from experimental cross sections and
measure configuration mixing in the A-body wave functions. However, the determination of spectroscopic factors depends on the models used, which implies
problems in the experimental determination of spectroscopic factors. The common
assumptions in standard shell model, i.e. that nucleons occupy localized states,
and degrees of freedom associated with the continuum are neglected, can been
tested using the Gamow shell model [12, 81]. In the following, we shall discuss
the radiative capture reaction b + c → a + γ , and define the radial overlap function
I
a
bc for a process a → b + c. The radial overlap function I
a
bc can be written in the
331
two-neutron halos. On the other hand, 10 He and 26,28 O are weakly unbound. Pairing
correlations in this case are not sufficiently strong to bind these nuclei mitigating
only their resonance character.
It is clear that halo nuclei have to be described within a quantum framework comprising both nucleon–nucleon correlations and continuum coupling. The Gamow
shell model is thus the tool of choice, as continuum coupling is present at the onebody level through the use of the Berggren basis, and nucleon–nucleon correlations
are exactly taken into account by configuration mixing. Moreover, as the wave
functions are explicitly calculated, one can directly assess the structure of halo
nuclear states with overlap functions and spectroscopic factors, which, as one will
see, provide with the weights and radial formfactors of two-cluster configurations,
such as 5 He + neutron in 6 He. The exact consideration of the Coulomb interaction
in the Gamow shell model also allows to precisely determine the isospin mixing
features of the nuclear Hamiltonian, prominent to understand physics at particleemission threshold, and thus halo nuclei.
7.3
Asymptotic Normalization Coefficient in Mirror Nuclei
The one-nucleon overlap functions and the resulting spectroscopic factors are
standard tools in the theory of direct reactions [66–71]. The fundamental interest
of asymptotic normalization coefficients is that they are not sensitive to the changes
of short-range physics (see Ref. [72], the discussion in Ref. [73] about observables
and non-observables, and also Ref. [74] for an overview of unitary transformations
and their impact on nuclear structure operators). Indeed, asymptotic normalization
coefficients are invariant under unitary transformations, so that they are observables,
contrary to spectroscopic factors. This arises because a unitary transformation of the
Hamiltonian only affects the interior region of the wave function, so that asymptotic
normalization coefficients remain unchanged.
Experimentally, asymptotic normalization coefficients have been measured for
s-shell and p-shell nuclei (see Ref. [75] and the references therein). Theoretical
calculations of asymptotic normalization coefficients have been performed using
the hyperspherical harmonics approach [76], the Gamow shell model and the shell
model embedded in the continuum [55], the Green’s function Monte Carlo [77–79]
(see also Ref. [80]).
The spectroscopic factors are deduced from experimental cross sections and
measure configuration mixing in the A-body wave functions. However, the determination of spectroscopic factors depends on the models used, which implies
problems in the experimental determination of spectroscopic factors. The common
assumptions in standard shell model, i.e. that nucleons occupy localized states,
and degrees of freedom associated with the continuum are neglected, can been
tested using the Gamow shell model [12, 81]. In the following, we shall discuss
the radiative capture reaction b + c → a + γ , and define the radial overlap function
I
a
bc for a process a → b + c. The radial overlap function I
a
bc can be written in the