6.4 Three-Body Model in Berggren Basis
273
Nevertheless, obtained results become very close when max reaches a value of 7–8
in both models.
One can see from Fig. 6.12 that all calculations done with Jacobi coordinates
converge slightly faster than those done using cluster orbital shell model coordinates. The discrepancy between results of these two frameworks is largest when the
cluster formed by two valence nucleons is well bound, as for 6 Li. This comes from
the attractive character of the proton–neutron interaction in the T = 0 channel.
Indeed, the energy difference between ground states of 6 Li in both approaches is
∼2 MeV for max = 2, whereas it is less than 200 keV for 6 He and 6 Be ground
states for the same truncation.
The T-type Jacobi coordinates pertain to a cluster, as they consist of a center-ofmass coordinate and of a relative coordinate (see Fig. 6.11). In particular, a localized
basis wave function in these coordinates represents a well-bound cluster close to
the core. Consequently, inter-nucleon correlations are built in already in the basis
wave functions, contrary to cluster orbital shell model, whose basis is made of
Slater determinants, which possess no cluster structure. Therefore, three-body wave
functions in Jacobi coordinates have to converge faster with the number of basis
states than those calculated in cluster orbital shell model coordinates. This is in
agreement with the conclusion of Ref. [112], where calculations using complex
scaling method with cluster orbital shell model coordinates provide a slightly less
bound wave function than that of Ref. [106], which used Jacobi coordinates.
This comparison demonstrates that one obtains very similar results for ground
state energies when continuum effects and sufficiently large model spaces are
considered. The typical energy differences between states in the three-body model
using Jacobi and cluster orbital shell model coordinates are less than 100 keV.
These results support also the argument [113] that cluster orbital shell model
can successfully eliminate center-of-mass energy but at the price of a slower
convergence due to the appearance of a recoil term.
Examples presented in Fig. 6.12 demonstrate that a standard three-body model in
Jacobi coordinates formulated in Berggren formalism is an excellent theoretical tool
to describe both weakly bound and resonance nuclei consisting of a two-body cluster
above a well-bound core. The model is particularly advantageous for a calculation of
emission of correlated two nucleons, e.g. two-proton emission in light and medium
nuclei [74, 114].
6.4.3 Correlation Densities in a Three-Body Gamow Model
Correlations between the two valence neutrons can be conveniently studied using
correlation density (see Eq. (6.19)) and Refs. [93, 94]): The two-nucleon angular
correlation is obtained by integrating the correlation density of Eq. (6.19) over r 1
and r 2 :
ρ(θ) =
+∞
0
8π
2 r
2
1 r
2
2 sin(θ )ρ(r 1 , r 2 )dr 1 dr 2 .
(6.30)
273
Nevertheless, obtained results become very close when max reaches a value of 7–8
in both models.
One can see from Fig. 6.12 that all calculations done with Jacobi coordinates
converge slightly faster than those done using cluster orbital shell model coordinates. The discrepancy between results of these two frameworks is largest when the
cluster formed by two valence nucleons is well bound, as for 6 Li. This comes from
the attractive character of the proton–neutron interaction in the T = 0 channel.
Indeed, the energy difference between ground states of 6 Li in both approaches is
∼2 MeV for max = 2, whereas it is less than 200 keV for 6 He and 6 Be ground
states for the same truncation.
The T-type Jacobi coordinates pertain to a cluster, as they consist of a center-ofmass coordinate and of a relative coordinate (see Fig. 6.11). In particular, a localized
basis wave function in these coordinates represents a well-bound cluster close to
the core. Consequently, inter-nucleon correlations are built in already in the basis
wave functions, contrary to cluster orbital shell model, whose basis is made of
Slater determinants, which possess no cluster structure. Therefore, three-body wave
functions in Jacobi coordinates have to converge faster with the number of basis
states than those calculated in cluster orbital shell model coordinates. This is in
agreement with the conclusion of Ref. [112], where calculations using complex
scaling method with cluster orbital shell model coordinates provide a slightly less
bound wave function than that of Ref. [106], which used Jacobi coordinates.
This comparison demonstrates that one obtains very similar results for ground
state energies when continuum effects and sufficiently large model spaces are
considered. The typical energy differences between states in the three-body model
using Jacobi and cluster orbital shell model coordinates are less than 100 keV.
These results support also the argument [113] that cluster orbital shell model
can successfully eliminate center-of-mass energy but at the price of a slower
convergence due to the appearance of a recoil term.
Examples presented in Fig. 6.12 demonstrate that a standard three-body model in
Jacobi coordinates formulated in Berggren formalism is an excellent theoretical tool
to describe both weakly bound and resonance nuclei consisting of a two-body cluster
above a well-bound core. The model is particularly advantageous for a calculation of
emission of correlated two nucleons, e.g. two-proton emission in light and medium
nuclei [74, 114].
6.4.3 Correlation Densities in a Three-Body Gamow Model
Correlations between the two valence neutrons can be conveniently studied using
correlation density (see Eq. (6.19)) and Refs. [93, 94]): The two-nucleon angular
correlation is obtained by integrating the correlation density of Eq. (6.19) over r 1
and r 2 :
ρ(θ) =
+∞
0
8π
2 r
2
1 r
2
2 sin(θ )ρ(r 1 , r 2 )dr 1 dr 2 .
(6.30)