9.4 Calculation of Spectra and Reaction Cross Sections in the Coupled-. . .
475
This feature of the FHT(E) calculation follows directly from the phase shifts
obtained for this interaction (see the dashed line in Fig. 9.7). In particular, the 3 P 0
phase shifts below 8 MeV are too large in absolute value, what implies the too large
values of differential cross section therein. While the 3 S 1 phase shifts are correct
close to 3 MeV, they rapidly become too large in absolute value at higher energies.
Also the 3 D 2 phase shifts increase too quickly around 4 MeV, to remain too high
afterwards due to the too small width of the 2
+
1 state. Both these effects seem to
induce wrong positions of minima and maxima in differential cross sections above
4 MeV.
On the contrary, differential cross sections calculated with the FHT(ANC)
interaction are always close to experimental data. The 3 D 2 phase shifts are also
very close to experimental data, as the width of the 2
+
1 state is larger, of 840 keV
instead of 1.3 MeV. Hence, the noticed discrepancies with experimental data in
elastic deuteron cross sections (see Fig. 9.8) should arise due to the 3 P 0 and 3 D 1
phase shifts, which are further away from the experimental data.
9.4.4 Inclusion of Nonresonant Channels: The Resonant Spectrum
of 42 Sc
In previous applications, reaction channels have been built within a scheme similar
to the pole approximation. Indeed, while the projectile can be of scattering character,
the target was always a bound or resonance many-body state. This approximation
is justified because in practical applications the combined target–projectile systems
have low energy, so that the probability to excite high lying nonresonant manybody target states is small. However, in theory, all target states, whether resonant
or scattering, enter the coupled-channel representation of Eq. (9.19). The absence
of nonresonant target states in reaction channels has already been noticed, as the
many-body eigenstate energies are slightly different in Gamow shell model in
Slater determinant representation and coupled-channel Gamow shell model (see
Sect. 9.4.3 and Exercise X). Thus, in practice, the absence of nonresonant target
states is compensated by the corrective factors, i.e. one multiplies all two-body
matrix elements by a constant so that the calculated eigenvalues are almost the
same in the Gamow shell model and in the coupled-channel Gamow shell model
(see Sect. 9.4.3 and Exercise X). However, this phenomenological procedure does
not provide with information about the physical importance of the reaction channels
built from nonresonant target states. Consequently, it is interesting to calculate the
energy spectrum using a basis of reaction channels either possessing both resonant
and scattering target states or defined only from resonant target states.
For this, one can only consider two valence nucleons, as otherwise calculations
cannot be effected due to the large number of occurring nonresonant target states.
Hence, one will consider 42 Sc for that matter, as it can modelled by one valence
proton and one valence neutron outside of the 40 Ca core. The used Hamiltonian has
the same structure as in Sects. 9.4.1.2 and 9.4.2.1, as it consists of Woods–Saxon
potential mimicking the 40 Ca core, to which a residual nucleon–nucleon interaction
of Furutani–Horiuchi–Tamagaki type is added [74]. All partial waves bearing an
475
This feature of the FHT(E) calculation follows directly from the phase shifts
obtained for this interaction (see the dashed line in Fig. 9.7). In particular, the 3 P 0
phase shifts below 8 MeV are too large in absolute value, what implies the too large
values of differential cross section therein. While the 3 S 1 phase shifts are correct
close to 3 MeV, they rapidly become too large in absolute value at higher energies.
Also the 3 D 2 phase shifts increase too quickly around 4 MeV, to remain too high
afterwards due to the too small width of the 2
+
1 state. Both these effects seem to
induce wrong positions of minima and maxima in differential cross sections above
4 MeV.
On the contrary, differential cross sections calculated with the FHT(ANC)
interaction are always close to experimental data. The 3 D 2 phase shifts are also
very close to experimental data, as the width of the 2
+
1 state is larger, of 840 keV
instead of 1.3 MeV. Hence, the noticed discrepancies with experimental data in
elastic deuteron cross sections (see Fig. 9.8) should arise due to the 3 P 0 and 3 D 1
phase shifts, which are further away from the experimental data.
9.4.4 Inclusion of Nonresonant Channels: The Resonant Spectrum
of 42 Sc
In previous applications, reaction channels have been built within a scheme similar
to the pole approximation. Indeed, while the projectile can be of scattering character,
the target was always a bound or resonance many-body state. This approximation
is justified because in practical applications the combined target–projectile systems
have low energy, so that the probability to excite high lying nonresonant manybody target states is small. However, in theory, all target states, whether resonant
or scattering, enter the coupled-channel representation of Eq. (9.19). The absence
of nonresonant target states in reaction channels has already been noticed, as the
many-body eigenstate energies are slightly different in Gamow shell model in
Slater determinant representation and coupled-channel Gamow shell model (see
Sect. 9.4.3 and Exercise X). Thus, in practice, the absence of nonresonant target
states is compensated by the corrective factors, i.e. one multiplies all two-body
matrix elements by a constant so that the calculated eigenvalues are almost the
same in the Gamow shell model and in the coupled-channel Gamow shell model
(see Sect. 9.4.3 and Exercise X). However, this phenomenological procedure does
not provide with information about the physical importance of the reaction channels
built from nonresonant target states. Consequently, it is interesting to calculate the
energy spectrum using a basis of reaction channels either possessing both resonant
and scattering target states or defined only from resonant target states.
For this, one can only consider two valence nucleons, as otherwise calculations
cannot be effected due to the large number of occurring nonresonant target states.
Hence, one will consider 42 Sc for that matter, as it can modelled by one valence
proton and one valence neutron outside of the 40 Ca core. The used Hamiltonian has
the same structure as in Sects. 9.4.1.2 and 9.4.2.1, as it consists of Woods–Saxon
potential mimicking the 40 Ca core, to which a residual nucleon–nucleon interaction
of Furutani–Horiuchi–Tamagaki type is added [74]. All partial waves bearing an