494
9 The Unification of Structure and Reaction Frameworks
The effect of the very narrow 1/2 − resonance is seen at the projectile energy
of 4.8 MeV. A similar drop of the cross section at that energy is also seen
experimentally [73].
Exercise XI.
A. The use of four harmonic oscillator states to represent a partial wave heavily
reduces model space dimension. Indeed, a Berggren basis representation would
demand at least 15 discretized scattering states for a given partial wave.
However, one can no longer have correct asymptotes in the sd partial waves
present in target nuclear wave functions, which can be of importance in the cross
sections of radiative capture, even though they are subdominant compared to p
partial waves.
B. If one truncates model space with 2p-2h truncations in the nonresonant continuum, the Berggren basis is no longer complete at the many-body level.
Consequently, the part of configuration mixing that is necessary to have a
precise many-body wave function in the asymptotic region is missing. As a
consequence, the particle-emission width of the first resonance states of 8 Li and
8 B, which is about 30 keV, is equal to zero when using 2p-2h truncations in the
nonresonant continuum. When one has no model space truncation, the manybody completeness is recovered and the particle-emission widths are correctly
reproduced.
C. When one calculates resonances with the Gamow shell model coupled-channel
code, one solves a system of coupled-channel equations involving one-body
nucleon projectiles. As only one-nucleon emission is present, coupled-channel
wave functions can almost exactly represent many-body resonance wave functions in the asymptotic region. Thus, widths are correctly calculated and are
close to those obtained with non-truncated Gamow shell model.
D. The calculation of E2 transitions shows that they are orders of magnitude
smaller than E1 and M1 transitions. All other electromagnetic transitions are
negligible (see Eq. (9.48)). Therefore, as E1 and M1 transitions are dominant in
7 Be(p, γ ) and 7 Li(n, γ ) reactions, electromagnetic transitions of orbital angular
momentum larger than two can be safely neglected.
Obtained cross sections compare well with experimental data. However, an
important effect comes to the fore when considering E1 radiative capture cross
sections. Even though protons have a unit charge and neutrons have no charge,
the effective charge entering Eq. (9.53) is not equal to one for proton or zero for
neutron. Indeed, the effective charge is modified by target recoil [2], so that one
has in fact e p = 1 − Z/A and e n = −Z/A, where e p and e n are the effective
charges of proton and neutron, respectively, and Z and A are the number of protons
and nucleons of the nuclear composite, respectively [2]. Added to that, the effect
of missing configurations in truncated model spaces, such as that induced by core
polarization, is taken into account by a modification of effective charges [2].
9 The Unification of Structure and Reaction Frameworks
The effect of the very narrow 1/2 − resonance is seen at the projectile energy
of 4.8 MeV. A similar drop of the cross section at that energy is also seen
experimentally [73].
Exercise XI.
A. The use of four harmonic oscillator states to represent a partial wave heavily
reduces model space dimension. Indeed, a Berggren basis representation would
demand at least 15 discretized scattering states for a given partial wave.
However, one can no longer have correct asymptotes in the sd partial waves
present in target nuclear wave functions, which can be of importance in the cross
sections of radiative capture, even though they are subdominant compared to p
partial waves.
B. If one truncates model space with 2p-2h truncations in the nonresonant continuum, the Berggren basis is no longer complete at the many-body level.
Consequently, the part of configuration mixing that is necessary to have a
precise many-body wave function in the asymptotic region is missing. As a
consequence, the particle-emission width of the first resonance states of 8 Li and
8 B, which is about 30 keV, is equal to zero when using 2p-2h truncations in the
nonresonant continuum. When one has no model space truncation, the manybody completeness is recovered and the particle-emission widths are correctly
reproduced.
C. When one calculates resonances with the Gamow shell model coupled-channel
code, one solves a system of coupled-channel equations involving one-body
nucleon projectiles. As only one-nucleon emission is present, coupled-channel
wave functions can almost exactly represent many-body resonance wave functions in the asymptotic region. Thus, widths are correctly calculated and are
close to those obtained with non-truncated Gamow shell model.
D. The calculation of E2 transitions shows that they are orders of magnitude
smaller than E1 and M1 transitions. All other electromagnetic transitions are
negligible (see Eq. (9.48)). Therefore, as E1 and M1 transitions are dominant in
7 Be(p, γ ) and 7 Li(n, γ ) reactions, electromagnetic transitions of orbital angular
momentum larger than two can be safely neglected.
Obtained cross sections compare well with experimental data. However, an
important effect comes to the fore when considering E1 radiative capture cross
sections. Even though protons have a unit charge and neutrons have no charge,
the effective charge entering Eq. (9.53) is not equal to one for proton or zero for
neutron. Indeed, the effective charge is modified by target recoil [2], so that one
has in fact e p = 1 − Z/A and e n = −Z/A, where e p and e n are the effective
charges of proton and neutron, respectively, and Z and A are the number of protons
and nucleons of the nuclear composite, respectively [2]. Added to that, the effect
of missing configurations in truncated model spaces, such as that induced by core
polarization, is taken into account by a modification of effective charges [2].