462
9 The Unification of Structure and Reaction Frameworks
from the experimental value by about 300 keV. All states in 19 Na are narrow
resonances that decay by proton emission. The last column in Table 9.1 contains
excitation energies of 19 Na calculated in the coupled-channel Gamow shell model.
A small difference between results of Gamow shell model and coupled-channel
Gamow shell model excitation energies of 19 Na resonances is due to neglecting
higher-lying discrete and scattering states of 18 Ne, above the first excited state
J π = 2
+
1 , in constructing the reaction channels.
9.4.1.2 14 O(p,p) Elastic Scattering Cross Section and 15 F Spectrum
The 15 F isotope is unbound, and a few resonance states are known in its spectrum.
In particular, it possesses a narrow resonance state above the Coulomb barrier in
the vicinity of the two-proton emission threshold. In fact, the unbound spectrum
of 15 F and the excitation function of the 14 O(p,p) elastic scattering reaction have
been studied experimentally in Ref. [73]. The coupled-channel Gamow shell model
approach has been used therein to analyze the experimental data [73].
In the theoretical analysis, we use the Hamiltonian that consists of the Woods–
Saxon potential with the spin–orbit term that describes the field of the 12 C core
acting on valence nucleons in 13 N, 14 O, and 15 F, the Furutani–Horiuchi–Tamagaki
finite-range two-body interaction [74] between valence nucleons, and the recoil
term. Parameters of the Hamiltonian are adjusted to reproduce the binding energies
of low-lying states, and the one- and two-proton separation energies in 15 F.
Parameters of the Hamiltonian are adjusted to reproduce the binding energies of
low-lying states and the one- and two-proton separation energies in 15 F. GSM-CC
calculations are performed in three resonant shells: 0p 1/2 , 0d 5/2 , and 1s 1/2 , and
several shells in the nonresonant continuum along the discretized contours: L
+
d 5/2
and L +
s 1/2
in the complex momentum k plane.
The eigenstates of the coupled-channel Gamow shell model are expanded in the
basis of channel states that are built by coupling the Gamow shell model wave
functions for ground state 0
+
1 and excited states 1
−
1 , 0
+
2 , 3
−
1 , 2
+
1 , 0
−
1 , 2
+
2 , 2
−
1 of
14 O with the proton wave functions in partial waves: s 1/2 , p 1/2 , p 3/2 , d 3/2 , and
d 5/2 . Calculated binding energies of 1/2
+
1 , 5/2
+
1 , and 1/2
−
1 states in 15 F, relative to
the energy of 12 C, are, respectively, (E, Γ ) GSM-CC = (−8.29 MeV, 0.437 MeV),
(−6.66 MeV, 0.211 MeV), and (−4.48 MeV, 0.031MeV) and well reproduce
experimental energies of these resonances: (E, Γ ) exp = (−8.21 MeV, 0.376 MeV),
(−6.57 MeV, 0.305 MeV), and (−4.48 MeV, 0.036 MeV). The narrow resonance
1/2
−
1 in the vicinity of 13 N + 2p decay channel can decay either by 1p- or 2pemission. Its structure corresponds to an almost pure configuration: [ 13 N(1/2
−
1 )
⊗π(p 1/2 )π(s 1/2 ) 2 ] of one proton in p 1/2 and two protons in s 1/2 shells. This is
the reason why the width of this state is so narrow. The 2p-decay is hindered by
the smallness of the available phase space for this decay. The 1p-decay in the
channel 14 O(0
+
1 ) + p is suppressed due to the nature of the 1/2
−
1 resonance, which is
dominated by the diproton configuration due to the proximity of the 13 N + 2p-decay
channel. This is a splendid example of the continuum-coupling collectivization of a
near-threshold shell model state by which the shell model state aligns with a nearby
9 The Unification of Structure and Reaction Frameworks
from the experimental value by about 300 keV. All states in 19 Na are narrow
resonances that decay by proton emission. The last column in Table 9.1 contains
excitation energies of 19 Na calculated in the coupled-channel Gamow shell model.
A small difference between results of Gamow shell model and coupled-channel
Gamow shell model excitation energies of 19 Na resonances is due to neglecting
higher-lying discrete and scattering states of 18 Ne, above the first excited state
J π = 2
+
1 , in constructing the reaction channels.
9.4.1.2 14 O(p,p) Elastic Scattering Cross Section and 15 F Spectrum
The 15 F isotope is unbound, and a few resonance states are known in its spectrum.
In particular, it possesses a narrow resonance state above the Coulomb barrier in
the vicinity of the two-proton emission threshold. In fact, the unbound spectrum
of 15 F and the excitation function of the 14 O(p,p) elastic scattering reaction have
been studied experimentally in Ref. [73]. The coupled-channel Gamow shell model
approach has been used therein to analyze the experimental data [73].
In the theoretical analysis, we use the Hamiltonian that consists of the Woods–
Saxon potential with the spin–orbit term that describes the field of the 12 C core
acting on valence nucleons in 13 N, 14 O, and 15 F, the Furutani–Horiuchi–Tamagaki
finite-range two-body interaction [74] between valence nucleons, and the recoil
term. Parameters of the Hamiltonian are adjusted to reproduce the binding energies
of low-lying states, and the one- and two-proton separation energies in 15 F.
Parameters of the Hamiltonian are adjusted to reproduce the binding energies of
low-lying states and the one- and two-proton separation energies in 15 F. GSM-CC
calculations are performed in three resonant shells: 0p 1/2 , 0d 5/2 , and 1s 1/2 , and
several shells in the nonresonant continuum along the discretized contours: L
+
d 5/2
and L +
s 1/2
in the complex momentum k plane.
The eigenstates of the coupled-channel Gamow shell model are expanded in the
basis of channel states that are built by coupling the Gamow shell model wave
functions for ground state 0
+
1 and excited states 1
−
1 , 0
+
2 , 3
−
1 , 2
+
1 , 0
−
1 , 2
+
2 , 2
−
1 of
14 O with the proton wave functions in partial waves: s 1/2 , p 1/2 , p 3/2 , d 3/2 , and
d 5/2 . Calculated binding energies of 1/2
+
1 , 5/2
+
1 , and 1/2
−
1 states in 15 F, relative to
the energy of 12 C, are, respectively, (E, Γ ) GSM-CC = (−8.29 MeV, 0.437 MeV),
(−6.66 MeV, 0.211 MeV), and (−4.48 MeV, 0.031MeV) and well reproduce
experimental energies of these resonances: (E, Γ ) exp = (−8.21 MeV, 0.376 MeV),
(−6.57 MeV, 0.305 MeV), and (−4.48 MeV, 0.036 MeV). The narrow resonance
1/2
−
1 in the vicinity of 13 N + 2p decay channel can decay either by 1p- or 2pemission. Its structure corresponds to an almost pure configuration: [ 13 N(1/2
−
1 )
⊗π(p 1/2 )π(s 1/2 ) 2 ] of one proton in p 1/2 and two protons in s 1/2 shells. This is
the reason why the width of this state is so narrow. The 2p-decay is hindered by
the smallness of the available phase space for this decay. The 1p-decay in the
channel 14 O(0
+
1 ) + p is suppressed due to the nature of the 1/2
−
1 resonance, which is
dominated by the diproton configuration due to the proximity of the 13 N + 2p-decay
channel. This is a splendid example of the continuum-coupling collectivization of a
near-threshold shell model state by which the shell model state aligns with a nearby