7.4 Near-Threshold Behavior of Wave Functions
341
–0.8
–0.4
0.4
17 O* 18 O*
0
–0.01
–0.06
–0.03
–0.1
0
0.1
0
0.1
GSM-p
GSM
HO-SM
0.94
0.96
0.98
1.00
0
0.8
e d 3/2
(MeV)
(a)
(b)
d Re(S)/de
Im(S)
Spectroscopic factor Re[S]
Fig. 7.9 Real part of the spectroscopic factor S as a function of the energy e d 3/2
of the one-body state 0d 3/2 , equal in the model used to −S 1n [ 17 O ∗ ]. One considers:
18 O(0
+
3 )|[ 17 O(3/2
+
1 ) ⊗ d 3/2 ] 0 +
2 . The first derivative of Re(S) in the vicinity of particle-emission
threshold e d 3/2 = 0 is depicted in the insert (a), while insert (b) depicts the imaginary part of S
(from Ref. [27]).
The Wigner cusp is clearly noticeable in Figs. 7.8 and 7.9. One can see that
the spectroscopic factor S exhibits a cusp at k = 0 which is similar to that of
the transfer cross section of Eqs. (7.45), (7.46). The Wigner cusp can be seen to
originate uniquely from coupling to the nonresonant continuum, as it disappears in
standard shell model calculations utilizing a basis of harmonic oscillator states or
in the pole approximation of Gamow shell model (see Sect. 5.4), where only bound
and resonance states 0d 5/2 , 1s 1/2 , and 0d 3/2 are present in the valence space (see
Fig. 7.9).
Coupling to the nonresonant scattering continuum is essential to describe correctly the spectroscopic factors close to particle-emission threshold. This coupling
is also crucial to preserve unitarity at particle-emission threshold in the Gamow shell
model. Therefore, the salient features of cross sections seen in Eqs. (7.45–7.48 ) are
the direct consequences of respecting unitarity in near-threshold wave functions.
The radial overlap functions connected to the spectroscopic factors of the
6 He/ 5 He pair are studied by fixing the real one neutron separation energy S 1n =
He ] − −[E6 He ] and complex one neutron separation energy ˜
S 1n = E5 He − E6 He
for bound and resonance ground states of 5 He and 6 He, respectively [81] (see
Fig. 7.10). The complex one neutron separation energy:
˜
S 1n ≡ E(N − 1) − E(N) = S 1n −
i
2
[Γ (N − 1) − Γ (N)]
(7.49)
341
–0.8
–0.4
0.4
17 O* 18 O*
0
–0.01
–0.06
–0.03
–0.1
0
0.1
0
0.1
GSM-p
GSM
HO-SM
0.94
0.96
0.98
1.00
0
0.8
e d 3/2
(MeV)
(a)
(b)
d Re(S)/de
Im(S)
Spectroscopic factor Re[S]
Fig. 7.9 Real part of the spectroscopic factor S as a function of the energy e d 3/2
of the one-body state 0d 3/2 , equal in the model used to −S 1n [ 17 O ∗ ]. One considers:
18 O(0
+
3 )|[ 17 O(3/2
+
1 ) ⊗ d 3/2 ] 0 +
2 . The first derivative of Re(S) in the vicinity of particle-emission
threshold e d 3/2 = 0 is depicted in the insert (a), while insert (b) depicts the imaginary part of S
(from Ref. [27]).
The Wigner cusp is clearly noticeable in Figs. 7.8 and 7.9. One can see that
the spectroscopic factor S exhibits a cusp at k = 0 which is similar to that of
the transfer cross section of Eqs. (7.45), (7.46). The Wigner cusp can be seen to
originate uniquely from coupling to the nonresonant continuum, as it disappears in
standard shell model calculations utilizing a basis of harmonic oscillator states or
in the pole approximation of Gamow shell model (see Sect. 5.4), where only bound
and resonance states 0d 5/2 , 1s 1/2 , and 0d 3/2 are present in the valence space (see
Fig. 7.9).
Coupling to the nonresonant scattering continuum is essential to describe correctly the spectroscopic factors close to particle-emission threshold. This coupling
is also crucial to preserve unitarity at particle-emission threshold in the Gamow shell
model. Therefore, the salient features of cross sections seen in Eqs. (7.45–7.48 ) are
the direct consequences of respecting unitarity in near-threshold wave functions.
The radial overlap functions connected to the spectroscopic factors of the
6 He/ 5 He pair are studied by fixing the real one neutron separation energy S 1n =
He ] − −[E6 He ] and complex one neutron separation energy ˜
S 1n = E5 He − E6 He
for bound and resonance ground states of 5 He and 6 He, respectively [81] (see
Fig. 7.10). The complex one neutron separation energy:
˜
S 1n ≡ E(N − 1) − E(N) = S 1n −
i
2
[Γ (N − 1) − Γ (N)]
(7.49)