7.4 Near-Threshold Behavior of Wave Functions
339
where σ is the cross section, k ∼ 0 is the transferred momentum, and is the
transferred orbital angular momentum. Equation (7.45) is for a capture process and
Eq. (7.46) for an emission process. The generalization of the Wigner threshold law
to three particles have been studied in Refs. [102–104]
Wigner cusps are particularly well visible for neutron = 0, 1 waves, while
their manifestation is subtle in neutron waves with ≥ 2, as we will see in the next
section. One can notice in Eqs. (7.45), (7.46) that the cross sections at the particleemission threshold are mainly different in partial waves of low angular momentum.
No Wigner cusp appears for charged particles, as in this case Eqs. (7.45), (7.46)
become
σ ∼ exp (−2π η) k
−2
(7.47)
σ ∼ exp (−2π η) .
(7.48)
Here, Eq. (7.47) describes the capture process and Eq. (7.48) the emission process.
One can notice that the energy dependence in Eq. (7.46) is identical to that of
Eq. (2.199). This feature is not accidental and arises from the peripheral character
of cross sections when k ∼ 0. Indeed, one-body bound or resonance states are
extended in space close to the particle-emission threshold, so that cross sections
are proportional to the modulus square of their asymptotic wave function, equal
to |C + H
+
,η (kR)| 2 . Nuclear structure is entirely contained in C + , which is the
asymptotic normalization coefficient of Eq. (7.32). Consequently, Eqs. (7.45), (7.46)
are obtained similarly to Eq. (2.199), by separating k-dependence for k ∼ 0 from
factors depending on Hamiltonian parameters and kinetic operator only.
Charged particle cross sections are proportional to the Gamow factor (see
Eq. (2.31)), which strongly suppresses the cross section at low energies, hence
preventing from a visible Wigner cusp to develop at k = 0. One can see as well the
analogy between Eqs. (7.48) and (2.201), arising for the same reason as for neutral
particles.
7.4.2 Spectroscopic Factors and Radial Overlap Integrals
One neutron spectroscopic factors in the ground state of 6 He, 7 He, and excited
state of 18 O ∗ are shown in Figs. 7.8 and 7.9. In the oxygen case, it is the excited
state of dominant configuration 17 O ⊗ (0d 3/2 ) 2 coupled to J = 0 which will be
studied. The involved partial waves in these spectroscopic factors are ((, j ) = p 3/2
for helium isotopes and ((, j ) = d 3/2 for oxygen isotopes. The Hamiltonian consists
of the Woods–Saxon potential mimicking the 4 He or 16 O core (see Eq. (3.83))
and the finite-range modified surface Gaussian interaction (see Eq. (9.174)). The
Hamiltonian parameters are varied in such a way that the ground states of the 6 He
nucleus (upper and lower panels of Fig. 7.8), and 5 He nucleus (middle panel of
Fig. 7.8), as well as the 3/2 + excited state of 17 O, denoted as 17 O ∗ , i.e. the d 3/2
Woods–Saxon states, (see Fig. 7.9) vary from bound to unbound, continuously. This
339
where σ is the cross section, k ∼ 0 is the transferred momentum, and is the
transferred orbital angular momentum. Equation (7.45) is for a capture process and
Eq. (7.46) for an emission process. The generalization of the Wigner threshold law
to three particles have been studied in Refs. [102–104]
Wigner cusps are particularly well visible for neutron = 0, 1 waves, while
their manifestation is subtle in neutron waves with ≥ 2, as we will see in the next
section. One can notice in Eqs. (7.45), (7.46) that the cross sections at the particleemission threshold are mainly different in partial waves of low angular momentum.
No Wigner cusp appears for charged particles, as in this case Eqs. (7.45), (7.46)
become
σ ∼ exp (−2π η) k
−2
(7.47)
σ ∼ exp (−2π η) .
(7.48)
Here, Eq. (7.47) describes the capture process and Eq. (7.48) the emission process.
One can notice that the energy dependence in Eq. (7.46) is identical to that of
Eq. (2.199). This feature is not accidental and arises from the peripheral character
of cross sections when k ∼ 0. Indeed, one-body bound or resonance states are
extended in space close to the particle-emission threshold, so that cross sections
are proportional to the modulus square of their asymptotic wave function, equal
to |C + H
+
,η (kR)| 2 . Nuclear structure is entirely contained in C + , which is the
asymptotic normalization coefficient of Eq. (7.32). Consequently, Eqs. (7.45), (7.46)
are obtained similarly to Eq. (2.199), by separating k-dependence for k ∼ 0 from
factors depending on Hamiltonian parameters and kinetic operator only.
Charged particle cross sections are proportional to the Gamow factor (see
Eq. (2.31)), which strongly suppresses the cross section at low energies, hence
preventing from a visible Wigner cusp to develop at k = 0. One can see as well the
analogy between Eqs. (7.48) and (2.201), arising for the same reason as for neutral
particles.
7.4.2 Spectroscopic Factors and Radial Overlap Integrals
One neutron spectroscopic factors in the ground state of 6 He, 7 He, and excited
state of 18 O ∗ are shown in Figs. 7.8 and 7.9. In the oxygen case, it is the excited
state of dominant configuration 17 O ⊗ (0d 3/2 ) 2 coupled to J = 0 which will be
studied. The involved partial waves in these spectroscopic factors are ((, j ) = p 3/2
for helium isotopes and ((, j ) = d 3/2 for oxygen isotopes. The Hamiltonian consists
of the Woods–Saxon potential mimicking the 4 He or 16 O core (see Eq. (3.83))
and the finite-range modified surface Gaussian interaction (see Eq. (9.174)). The
Hamiltonian parameters are varied in such a way that the ground states of the 6 He
nucleus (upper and lower panels of Fig. 7.8), and 5 He nucleus (middle panel of
Fig. 7.8), as well as the 3/2 + excited state of 17 O, denoted as 17 O ∗ , i.e. the d 3/2
Woods–Saxon states, (see Fig. 7.9) vary from bound to unbound, continuously. This