7.4 Near-Threshold Behavior of Wave Functions
343
where the decay constant is given by
κ =
2m ˜
S 1n / ¯
h .
(7.50)
One should add in passing that if the unbound states are involved in the extraction
of the effective matrix elements of nucleon–nucleon interaction from experimental
binding energies of neighboring nuclei, one should use complex and not real binding
energies because in this case the two-body matrix elements calculated in Gamow
shell model become complex as well.
For bound states, the real overlap function can be modelled by a Woods–Saxon
single-particle state of energy −S 1n [27]. However, a real Woods–Saxon potential
cannot model complex one neutron separation energy ˜
S 1n . Hence, the radial overlap
functions of Fig. 7.10 have been compared to one-body states generated by a real
Woods–Saxon potential whose depth fixes the real-energy −S 1n , or by a Woods–
Saxon potential whose depth is complex in order to reproduce the complex-energy
− ˜
S 1n . One can see that only a complex potential can reproduce almost exactly the
radial overlap function if S 1n is complex, as a real potential cannot provide, for
example, a state of zero or negative energy and nonzero width, as is the case in
Figs. 7.10.
A numerical example showing that cusps occur in the context of the spectroscopic factors is done in Exercise III for the pair of nuclei 9 He/ 10 He.
Exercise III
In this numerical example, one will consider the spectroscopic factors of the pair
of nuclei 9 He/ 10 He and one will show that a cusp occurs at particle-emission
threshold.
A. We will use a core of 8 He. The Hamiltonian is otherwise the same as discussed
in this section.
Explain why the use of a core of 8 He to describe the unbound eigenstates
of 9 He and 10 He is sound from both physical and numerical perspective.
Considering the small widths of the 1/2 − state of 9 He and 0 + state of
10 He, explain why the study of the p 1/2 spectroscopic factors of 9 He/ 10 He
is meaningful.
B. Run the two-body Gamow shell model code using cluster orbital shell model
coordinates to calculate p 1/2 spectroscopic factors of 9 He/ 10 He.
Plot the spectroscopic factors as a function of the energy, similarly to Fig. 7.8.
Notice the Wigner cusp at S 1n =0 MeV occurring using full Berggren
basis and compare results with the 5 He/ 6 He case.
343
where the decay constant is given by
κ =
2m ˜
S 1n / ¯
h .
(7.50)
One should add in passing that if the unbound states are involved in the extraction
of the effective matrix elements of nucleon–nucleon interaction from experimental
binding energies of neighboring nuclei, one should use complex and not real binding
energies because in this case the two-body matrix elements calculated in Gamow
shell model become complex as well.
For bound states, the real overlap function can be modelled by a Woods–Saxon
single-particle state of energy −S 1n [27]. However, a real Woods–Saxon potential
cannot model complex one neutron separation energy ˜
S 1n . Hence, the radial overlap
functions of Fig. 7.10 have been compared to one-body states generated by a real
Woods–Saxon potential whose depth fixes the real-energy −S 1n , or by a Woods–
Saxon potential whose depth is complex in order to reproduce the complex-energy
− ˜
S 1n . One can see that only a complex potential can reproduce almost exactly the
radial overlap function if S 1n is complex, as a real potential cannot provide, for
example, a state of zero or negative energy and nonzero width, as is the case in
Figs. 7.10.
A numerical example showing that cusps occur in the context of the spectroscopic factors is done in Exercise III for the pair of nuclei 9 He/ 10 He.
Exercise III
In this numerical example, one will consider the spectroscopic factors of the pair
of nuclei 9 He/ 10 He and one will show that a cusp occurs at particle-emission
threshold.
A. We will use a core of 8 He. The Hamiltonian is otherwise the same as discussed
in this section.
Explain why the use of a core of 8 He to describe the unbound eigenstates
of 9 He and 10 He is sound from both physical and numerical perspective.
Considering the small widths of the 1/2 − state of 9 He and 0 + state of
10 He, explain why the study of the p 1/2 spectroscopic factors of 9 He/ 10 He
is meaningful.
B. Run the two-body Gamow shell model code using cluster orbital shell model
coordinates to calculate p 1/2 spectroscopic factors of 9 He/ 10 He.
Plot the spectroscopic factors as a function of the energy, similarly to Fig. 7.8.
Notice the Wigner cusp at S 1n =0 MeV occurring using full Berggren
basis and compare results with the 5 He/ 6 He case.