9.4 Calculation of Spectra and Reaction Cross Sections in the Coupled-. . .
465
The mirror radiative capture reaction 7 Li(n, γ ) at high energies constrains
those of the 7 Be(p, γ ) at solar energies, which are not accessible experimentally
[81,82]. The 7 Li(n, γ ) radiative capture reaction is not found in usual thermonuclear
cycles [78, 83] but is, however, a part of the inhomogeneous models of Big Bang
nucleosynthesis [84]. Added to that, it is expected that 6,7 Li isotopes, similarly
to hydrogen and helium, have been created almost entirely during the Big Bang
nucleosynthesis [85]. Consequently, the determination of the universal abundance
of 6,7 Li isotopes might allow to better understand the early universe.
This problem is challenging theoretically as at present the calculated abundances
of 6,7 Li do not reproduce experimental data [85]. The experimentally inferred
primordial abundance of 6 Li from the baryon-to-photon ratio issued from the
Wilkinson Microwave Anisotropy Probe [86] is about 1000 times larger than that
provided by theoretical estimates [85]. That same value for 7 Li is about 3 times
smaller than the theoretical predictions [85]. Consequently, a detailed coupledchannel Gamow shell model studies of the reactions involving 6,7 Li isotopes might
help understanding this state of affairs.
9.4.2.1 7 Be(p, γ ) and 7 Li(n, γ ) Mirror Radiative Capture Reactions
The model space in 7 Be and 8 B is limited by the core of 4 He. The core is
described by a Woods–Saxon potential with -dependent ( ≤ 2) depth for protons
and neutrons, supplemented by the Coulomb potential of radius r c = 2.8 fm
[64]. The two-body interaction between valence nucleons is described using the
Furutani–Horiuchi–Tamagaki effective interaction (Eq. (6.18)). Parameters of the
Hamiltonian have been fitted to reproduce binding energies of low-lying states in
7 Be and 8 B [64].
The valence nucleons can occupy the 0p 3/2 and 0p 1/2 discrete single-particle
states and several nonresonant single-particle continuum states on discretized
contours: L +
s 1/2
, L +
p 1/2
, L +
p 3/2
, L
+
d 3/2
, and L
+
d 5/2
. Each contour consists of three
segments and is discretized with 30 points [64].
The Gamow shell model basis is truncated so as to reduce the size of the
Gamow shell model Hamiltonian matrix. For this, the occupation of p 3/2 and p 1/2
scattering states in basis Slater determinants is limited to two particles, while the
occupation of s 1/2 , d 5/2 , and d 3/2 scattering states is limited to one particle only.
The latter truncation is justified by the fact that the p 3/2 and p 1/2 partial waves
play a dominant role in the Gamow shell model target states, s 1/2 , d 5/2 , and d 3/2
states occurring only in the partial wave decomposition of the proton or neutron
projectile. In this approximation, the ground state of 7 Be in Gamow shell model
calculations is bound with respect to 4 He by 9.378 MeV, close to the experimental
value E exp = 9.304 MeV. Reaction channels in the coupled-channel Gamow shell
model calculations are obtained by the coupling of the ground state 3/2
− and the
first excited state 1/2
− of 7 Be with the proton partial waves: s 1/2 , p 1/2 , p 3/2 , d 3/2 ,
and d 5/2 .
465
The mirror radiative capture reaction 7 Li(n, γ ) at high energies constrains
those of the 7 Be(p, γ ) at solar energies, which are not accessible experimentally
[81,82]. The 7 Li(n, γ ) radiative capture reaction is not found in usual thermonuclear
cycles [78, 83] but is, however, a part of the inhomogeneous models of Big Bang
nucleosynthesis [84]. Added to that, it is expected that 6,7 Li isotopes, similarly
to hydrogen and helium, have been created almost entirely during the Big Bang
nucleosynthesis [85]. Consequently, the determination of the universal abundance
of 6,7 Li isotopes might allow to better understand the early universe.
This problem is challenging theoretically as at present the calculated abundances
of 6,7 Li do not reproduce experimental data [85]. The experimentally inferred
primordial abundance of 6 Li from the baryon-to-photon ratio issued from the
Wilkinson Microwave Anisotropy Probe [86] is about 1000 times larger than that
provided by theoretical estimates [85]. That same value for 7 Li is about 3 times
smaller than the theoretical predictions [85]. Consequently, a detailed coupledchannel Gamow shell model studies of the reactions involving 6,7 Li isotopes might
help understanding this state of affairs.
9.4.2.1 7 Be(p, γ ) and 7 Li(n, γ ) Mirror Radiative Capture Reactions
The model space in 7 Be and 8 B is limited by the core of 4 He. The core is
described by a Woods–Saxon potential with -dependent ( ≤ 2) depth for protons
and neutrons, supplemented by the Coulomb potential of radius r c = 2.8 fm
[64]. The two-body interaction between valence nucleons is described using the
Furutani–Horiuchi–Tamagaki effective interaction (Eq. (6.18)). Parameters of the
Hamiltonian have been fitted to reproduce binding energies of low-lying states in
7 Be and 8 B [64].
The valence nucleons can occupy the 0p 3/2 and 0p 1/2 discrete single-particle
states and several nonresonant single-particle continuum states on discretized
contours: L +
s 1/2
, L +
p 1/2
, L +
p 3/2
, L
+
d 3/2
, and L
+
d 5/2
. Each contour consists of three
segments and is discretized with 30 points [64].
The Gamow shell model basis is truncated so as to reduce the size of the
Gamow shell model Hamiltonian matrix. For this, the occupation of p 3/2 and p 1/2
scattering states in basis Slater determinants is limited to two particles, while the
occupation of s 1/2 , d 5/2 , and d 3/2 scattering states is limited to one particle only.
The latter truncation is justified by the fact that the p 3/2 and p 1/2 partial waves
play a dominant role in the Gamow shell model target states, s 1/2 , d 5/2 , and d 3/2
states occurring only in the partial wave decomposition of the proton or neutron
projectile. In this approximation, the ground state of 7 Be in Gamow shell model
calculations is bound with respect to 4 He by 9.378 MeV, close to the experimental
value E exp = 9.304 MeV. Reaction channels in the coupled-channel Gamow shell
model calculations are obtained by the coupling of the ground state 3/2
− and the
first excited state 1/2
− of 7 Be with the proton partial waves: s 1/2 , p 1/2 , p 3/2 , d 3/2 ,
and d 5/2 .