9.4 Calculation of Spectra and Reaction Cross Sections in the Coupled-. . .
471
the experimental value −2.224 MeV. Deuteron eigenstates are constructed using
the method described in Sect. 9.2.3. Thus, the Berggren eigenstates obtained after
diagonalization are projected on a basis of harmonic oscillator states, written in
cluster orbital shell model coordinates, with n ≤ 5 and ≤ 4.
The combined system [ 4 He + d] is described as 4 He core with two valence
nucleons, using partial waves up to = 4. The 4 He core is mimicked by a Wood–
Saxon potential, fitted on phases shifts of elastic scattering reactions involving a
neutron on a 4 He target [94]). Interaction among valence nucleons is given by the
Furutani–Horiuchi–Tamagaki effective force (see Sect. 6.3.1 and Refs. [74,90]). The
shell model space is that of all proton and neutron harmonic oscillator states having
≤ 4 and n ≤ 5. The use of Berggren basis at this level is not necessary as the
Slater determinants used therein generate only the finite-range Hamiltonian of the
coupled-channel Gamow shell model. The Berggren basis of center-of-mass cluster
states, used to calculate bound, resonance, and scattering states in 6 Li, is generated
by the proton and neutron Woods–Saxon potentials [93] using Eqs. (9.84), (9.85).
The included partial waves, consisting of 3 S 1 , 3 P 0,1,2 , 3 D 1,2,3 , sF 2,3 , and 3 G 3 , bear
3, 2, 2, 1, and 0 bound pole states, respectively. All resonance states lie below the
Berggren basis contours, so that they do not belong to the considered Berggren
bases. The contours consist of three segments, defined by the origin of the Kcomplex plane, with K the linear momentum of the two-body cluster, and the
K-complex points 0.2 − i0.05 fm −1 , 1.0 − i0.05 fm −1 , and 2 fm −1 . Each segment
is discretized with 15 points.
The statistical properties of the Furutani–Horiuchi–Tamagaki interaction parameters for p-shell nuclei have been analyzed in Sect. 6.3.1. Consequently, one can
modify parameters of this interaction within the bounds of calculated statistical
errors. The T = 1 part of interaction is negligible both in the 4 He(d,d) reaction
and in the T = 0 spectrum of 6 Li. Moreover, the dependence of energies on V 00
C is
very weak. Hence, only V 10
C and V 10
T have been considered when fitting parameters
of the Furutani–Horiuchi–Tamagaki interaction. Other parameters of this interaction
remain the same as given in Ref. [94].
It has been found that the 3 S 1 asymptotic normalization coefficient of the 6 Li
ground state is too small if only the energy spectrum of 6 Li is used to fit the
interaction. The 3 D 1 asymptotic normalization coefficient in this case is very
small, similarly as the value reported in Refs. [95–97]. In order to account for the
value of asymptotic normalization coefficients, two different strategies of fitting
the Furutani–Horiuchi–Tamagaki interaction parameters are employed. The first
strategy corresponds to fitting the T = 0 spectrum of 6 Li, as already stated, to
which the 3 D 1 asymptotic normalization coefficient of the 6 Li ground state could be
added without changing the spectrum and scattering observables significantly. The
second strategy corresponds to fitting the T = 0 excited states of 6 Li, along with the
3 S 1 and 3 D 1 asymptotic normalization coefficients of the 6 Li ground state, leaving
the ground state energy of 6 Li out of the fit (see Table 9.2). The second strategy
yields the ratio of 3 D 1 / 3 S 1 = −0.0265, which agrees well with the experimental
value -0.025(6)(10) [95–97]. The sets of interaction parameters obtained in these
471
the experimental value −2.224 MeV. Deuteron eigenstates are constructed using
the method described in Sect. 9.2.3. Thus, the Berggren eigenstates obtained after
diagonalization are projected on a basis of harmonic oscillator states, written in
cluster orbital shell model coordinates, with n ≤ 5 and ≤ 4.
The combined system [ 4 He + d] is described as 4 He core with two valence
nucleons, using partial waves up to = 4. The 4 He core is mimicked by a Wood–
Saxon potential, fitted on phases shifts of elastic scattering reactions involving a
neutron on a 4 He target [94]). Interaction among valence nucleons is given by the
Furutani–Horiuchi–Tamagaki effective force (see Sect. 6.3.1 and Refs. [74,90]). The
shell model space is that of all proton and neutron harmonic oscillator states having
≤ 4 and n ≤ 5. The use of Berggren basis at this level is not necessary as the
Slater determinants used therein generate only the finite-range Hamiltonian of the
coupled-channel Gamow shell model. The Berggren basis of center-of-mass cluster
states, used to calculate bound, resonance, and scattering states in 6 Li, is generated
by the proton and neutron Woods–Saxon potentials [93] using Eqs. (9.84), (9.85).
The included partial waves, consisting of 3 S 1 , 3 P 0,1,2 , 3 D 1,2,3 , sF 2,3 , and 3 G 3 , bear
3, 2, 2, 1, and 0 bound pole states, respectively. All resonance states lie below the
Berggren basis contours, so that they do not belong to the considered Berggren
bases. The contours consist of three segments, defined by the origin of the Kcomplex plane, with K the linear momentum of the two-body cluster, and the
K-complex points 0.2 − i0.05 fm −1 , 1.0 − i0.05 fm −1 , and 2 fm −1 . Each segment
is discretized with 15 points.
The statistical properties of the Furutani–Horiuchi–Tamagaki interaction parameters for p-shell nuclei have been analyzed in Sect. 6.3.1. Consequently, one can
modify parameters of this interaction within the bounds of calculated statistical
errors. The T = 1 part of interaction is negligible both in the 4 He(d,d) reaction
and in the T = 0 spectrum of 6 Li. Moreover, the dependence of energies on V 00
C is
very weak. Hence, only V 10
C and V 10
T have been considered when fitting parameters
of the Furutani–Horiuchi–Tamagaki interaction. Other parameters of this interaction
remain the same as given in Ref. [94].
It has been found that the 3 S 1 asymptotic normalization coefficient of the 6 Li
ground state is too small if only the energy spectrum of 6 Li is used to fit the
interaction. The 3 D 1 asymptotic normalization coefficient in this case is very
small, similarly as the value reported in Refs. [95–97]. In order to account for the
value of asymptotic normalization coefficients, two different strategies of fitting
the Furutani–Horiuchi–Tamagaki interaction parameters are employed. The first
strategy corresponds to fitting the T = 0 spectrum of 6 Li, as already stated, to
which the 3 D 1 asymptotic normalization coefficient of the 6 Li ground state could be
added without changing the spectrum and scattering observables significantly. The
second strategy corresponds to fitting the T = 0 excited states of 6 Li, along with the
3 S 1 and 3 D 1 asymptotic normalization coefficients of the 6 Li ground state, leaving
the ground state energy of 6 Li out of the fit (see Table 9.2). The second strategy
yields the ratio of 3 D 1 / 3 S 1 = −0.0265, which agrees well with the experimental
value -0.025(6)(10) [95–97]. The sets of interaction parameters obtained in these