7.4 Near-Threshold Behavior of Wave Functions
337
7.4
Near-Threshold Behavior of Wave Functions
Due to the unitarity of the scattering matrix and the resulting flux conservation, an
opening reaction channel can generate the so-called Wigner cusps in other open
channels with lower thresholds. Wigner cusps are the non-analyticities appearing
in cross sections at energies when a new particle-emission channel opens. They
translate into very sharp changes of cross sections, discontinuities of the derivative
of cross sections or of the cross sections themselves, depending on the angular
momentum content of the wave functions involved in cross sections.
In order to study Wigner cusps, one will evaluate spectroscopic factors of weakly
bound and resonance systems, whose energy will vary smoothly by changing the
parameters of the Hamiltonian. Contrary to the evaluation of energies, a precise
calculation of spectroscopic factors with the Gamow shell model demands a very
large number of discretized scattering basis states. In the following analysis, one
will concentrate on systems, such as 5–7 He and 17–18 O isotopes, as they one can
be described with only a few valence nucleons in the presence of a core of 4 He
or 16 O core, respectively. In these systems, the Gamow shell model calculations
can be realized without any model space truncation. It will be shown that the
spectroscopic factors exhibit qualitatively different features according to the bound
or unbound nature of involved nuclear states. Moreover, due to the different partial
waves involved in the studied isotopes, p shells for 5–7 He isotopes and sd shells
for 17–18 O isotopes, spectroscopic factors will exhibit Wigner cusps of different
nature. The radial overlap functions associated with the calculated spectroscopic
factors will also be evaluated. In particular, it will be shown that the calculated radial
overlap functions resemble one-body states generated by Woods–Saxon potential,
whose depth parameter is real or complex.
Another aspect of near-threshold properties of many-body wave functions is the
isospin mixing which has been estimated to be very small, of the order of a few
percent at most in well-bound light nuclei [98,99]. This arises because the Coulomb
interaction has an infinite range, so that its main effect is to change the energy of
valence shells by a nearly constant shift. Consequently, well-bound nuclear states
possess an almost exact isospin, the Coulomb Hamiltonian translating only into a
charge dependence of binding energies of nuclear states.
Conversely, in the vicinity of particle-emission threshold, the asymptotic behavior of the wave functions of mirror nuclei is very different. Neutron wave functions
are only subject to the centrifugal barrier at large distance, while proton wave
functions also depend on the Coulomb barrier. While this effect is negligible in wellbound states, it is prominent in near-threshold wave functions, whose asymptote
extends far away from the nuclear zone. In particular, the isobaric analog states
of mirror nuclei near particle-emission threshold bear different excitation energies
and widths, whereas they should be identical in the presence of exact isospin
symmetry. In order to exhibit isospin mixing close to particle-emission threshold,
one will consider the isobaric analogue states of 5–6 He, 5–6 Li, and 6 Be. Indeed,
these states lie close to the particle-emission threshold. Isospin symmetry breaking
337
7.4
Near-Threshold Behavior of Wave Functions
Due to the unitarity of the scattering matrix and the resulting flux conservation, an
opening reaction channel can generate the so-called Wigner cusps in other open
channels with lower thresholds. Wigner cusps are the non-analyticities appearing
in cross sections at energies when a new particle-emission channel opens. They
translate into very sharp changes of cross sections, discontinuities of the derivative
of cross sections or of the cross sections themselves, depending on the angular
momentum content of the wave functions involved in cross sections.
In order to study Wigner cusps, one will evaluate spectroscopic factors of weakly
bound and resonance systems, whose energy will vary smoothly by changing the
parameters of the Hamiltonian. Contrary to the evaluation of energies, a precise
calculation of spectroscopic factors with the Gamow shell model demands a very
large number of discretized scattering basis states. In the following analysis, one
will concentrate on systems, such as 5–7 He and 17–18 O isotopes, as they one can
be described with only a few valence nucleons in the presence of a core of 4 He
or 16 O core, respectively. In these systems, the Gamow shell model calculations
can be realized without any model space truncation. It will be shown that the
spectroscopic factors exhibit qualitatively different features according to the bound
or unbound nature of involved nuclear states. Moreover, due to the different partial
waves involved in the studied isotopes, p shells for 5–7 He isotopes and sd shells
for 17–18 O isotopes, spectroscopic factors will exhibit Wigner cusps of different
nature. The radial overlap functions associated with the calculated spectroscopic
factors will also be evaluated. In particular, it will be shown that the calculated radial
overlap functions resemble one-body states generated by Woods–Saxon potential,
whose depth parameter is real or complex.
Another aspect of near-threshold properties of many-body wave functions is the
isospin mixing which has been estimated to be very small, of the order of a few
percent at most in well-bound light nuclei [98,99]. This arises because the Coulomb
interaction has an infinite range, so that its main effect is to change the energy of
valence shells by a nearly constant shift. Consequently, well-bound nuclear states
possess an almost exact isospin, the Coulomb Hamiltonian translating only into a
charge dependence of binding energies of nuclear states.
Conversely, in the vicinity of particle-emission threshold, the asymptotic behavior of the wave functions of mirror nuclei is very different. Neutron wave functions
are only subject to the centrifugal barrier at large distance, while proton wave
functions also depend on the Coulomb barrier. While this effect is negligible in wellbound states, it is prominent in near-threshold wave functions, whose asymptote
extends far away from the nuclear zone. In particular, the isobaric analog states
of mirror nuclei near particle-emission threshold bear different excitation energies
and widths, whereas they should be identical in the presence of exact isospin
symmetry. In order to exhibit isospin mixing close to particle-emission threshold,
one will consider the isobaric analogue states of 5–6 He, 5–6 Li, and 6 Be. Indeed,
these states lie close to the particle-emission threshold. Isospin symmetry breaking