384
8 No-Core Gamow Shell Model
The imaginary part of k can be neglected if:
Γ
2S 1n
→ 0 .
(8.10)
Hence, the validity of Eq. (8.8) does not necessarily mean that the resonance width
is small, but that the ratio of the resonance width and the separation energy is small.
Using the relation (8.8), one obtains the one neutron emission width Γ ANC =
311 keV and Γ ANC = 570 keV for Λ = 1.9 fm −1 and 2.1 fm −1 , respectively. These
values for the widths are consistent with those obtained in the no-core Gamow shell
model, in spite of the fact that the ground state width of 5 He is broad. The ratio
Γ /(2S 1n ) varies in the range from 10% to 15% in the considered interval of Λ
cutoff parameters. The corresponding error is compatible with the deviation between
the width obtained from the no-core Gamow shell model diagonalization and its
extraction from the asymptotic normalization coefficient formula (8.8).
The norm of the radial overlap integral squared I j (r) 2 (see Eqs. (7.33), (7.44)):
S =
+∞
0
I j (r)
2 dr
(8.11)
defines the spectroscopic factor. It is solely a theoretical construction and should not
be considered as a measurable quantity [109]. The value of spectroscopic factor S
depends on the representation of a Hamiltonian, and hence on the interior of the
wave function. Indeed, one has the freedom to change the off-shell behavior of
the potential in any possible manner, while maintaining the NN phase shifts. In
this way, families of phase shift equivalent potentials are obtained. Each method of
changing the off-shell behavior is then considered as a different representation or
scheme. In spite of this shortcoming, the spectroscopic factors can provide useful
information about shell occupancies and hence the nucleon–nucleon correlations,
i.e. how much the nucleus in a given state deviates from the single-particle picture.
Such spectroscopic factors have been calculated by several methods, either in ab
initio or more phenomenological approaches [105, 110–114].
Using no-core Gamow shell model solutions for 4 He/ 5 He in the truncated
model space with at most four nucleons in the nonresonant continuum, one
finds that the spectroscopic amplitude of 5 He corresponding to the channel
4 He(g.s.) ⊗ p 3/2
3/2 −
is 0.787 (S p 3/2 = 0.62) for Λ = 1.9 fm −1 and 0.812
(S p 3/2 = 0.66) for Λ = 2.1 fm −1 . By decreasing the renormalization scale
parameter Λ, i.e. by “softening” the interaction, the spectroscopic factor deviates
further from unity. This tendency is seen also at Λ = 1.5 fm −1 (see Table 8.2) which
yields the energies −28.670 MeV and −27.285 MeV for 4 He and 5 He, respectively.
8 No-Core Gamow Shell Model
The imaginary part of k can be neglected if:
Γ
2S 1n
→ 0 .
(8.10)
Hence, the validity of Eq. (8.8) does not necessarily mean that the resonance width
is small, but that the ratio of the resonance width and the separation energy is small.
Using the relation (8.8), one obtains the one neutron emission width Γ ANC =
311 keV and Γ ANC = 570 keV for Λ = 1.9 fm −1 and 2.1 fm −1 , respectively. These
values for the widths are consistent with those obtained in the no-core Gamow shell
model, in spite of the fact that the ground state width of 5 He is broad. The ratio
Γ /(2S 1n ) varies in the range from 10% to 15% in the considered interval of Λ
cutoff parameters. The corresponding error is compatible with the deviation between
the width obtained from the no-core Gamow shell model diagonalization and its
extraction from the asymptotic normalization coefficient formula (8.8).
The norm of the radial overlap integral squared I j (r) 2 (see Eqs. (7.33), (7.44)):
S =
+∞
0
I j (r)
2 dr
(8.11)
defines the spectroscopic factor. It is solely a theoretical construction and should not
be considered as a measurable quantity [109]. The value of spectroscopic factor S
depends on the representation of a Hamiltonian, and hence on the interior of the
wave function. Indeed, one has the freedom to change the off-shell behavior of
the potential in any possible manner, while maintaining the NN phase shifts. In
this way, families of phase shift equivalent potentials are obtained. Each method of
changing the off-shell behavior is then considered as a different representation or
scheme. In spite of this shortcoming, the spectroscopic factors can provide useful
information about shell occupancies and hence the nucleon–nucleon correlations,
i.e. how much the nucleus in a given state deviates from the single-particle picture.
Such spectroscopic factors have been calculated by several methods, either in ab
initio or more phenomenological approaches [105, 110–114].
Using no-core Gamow shell model solutions for 4 He/ 5 He in the truncated
model space with at most four nucleons in the nonresonant continuum, one
finds that the spectroscopic amplitude of 5 He corresponding to the channel
4 He(g.s.) ⊗ p 3/2
3/2 −
is 0.787 (S p 3/2 = 0.62) for Λ = 1.9 fm −1 and 0.812
(S p 3/2 = 0.66) for Λ = 2.1 fm −1 . By decreasing the renormalization scale
parameter Λ, i.e. by “softening” the interaction, the spectroscopic factor deviates
further from unity. This tendency is seen also at Λ = 1.5 fm −1 (see Table 8.2) which
yields the energies −28.670 MeV and −27.285 MeV for 4 He and 5 He, respectively.