Solutions to Exercises
493
B. At the Hartree–Fock level, the 1/2 + and 5/2 + resonance states consist of a fully
occupied 0p 1/2 proton shell, with a proton occupying a 1s 1/2 or 0d 5/2 one-body
resonance state, respectively. As there is a gap of several MeVs between the
0p 1/2 shell and the sd shell, the Hartree–Fock configurations can be expected to
form a good approximation of the exact 1/2 + and 5/2 + resonance states.
The 1/2 − has an excitation energy well above the Coulomb barrier. Consequently, one could expect its width to be of several MeVs if its structure is mainly
that of a single proton above an 14 O core. As the width of the 1/2 − resonance is
much smaller, of a few tens of keVs, its many-body wave function must bear a
sizable configuration mixing in the continuum, so that its width is much reduced
compared to its one-particle estimate.
C. About 85% of the many-body wave functions of the 1/2 + and 5/2 + resonance
states consist of configurations of the form [0p 2
1/2 s 1/2 ] or [0p 2
1/2 d 5/2 ],
respectively. This implies that the 1/2 + and 5/2 + resonance states are of singleparticle character.
On the contrary, the 1/2 − resonance state has a fairly large configuration
mixing. Indeed, it consists mainly of [0p 1/2 s 2
1/2 ] and [0p 1/2 d 2
5/2 ] configurations, whose weights are about 75% and 23%, respectively. Therefore, the 1/2 −
resonance state consists mainly of two correlated protons occupying the s 1/2
and d 5/2 resonant and nonresonant shells. Moreover, the pole configurations
[0p 1/2 1s 2
1/2 ] and [0p 1/2 0d 2
5/2 ] have the largest weights, of about 69% and 22%,
respectively. Consequently, one can assume that the two correlated protons are
very likely to be coupled to J π = 0 + in the 1/2 − resonance wave function.
Thus, the 1/2 − resonance state wave function is principally made of a proton
occupying the 0p 1/2 shell and of two protons occupying the sd shells. The
relative stability of the 1/2 − resonance is then explained by the fact that the
0p 1/2 shell is well bound, on the one hand, and because of the smallness of the
phase space available for diproton decay, on the other hand (see Sect. 9.4.1.2).
D. Even though the many-body spaces spanned by Slater determinants and manybody channels are identical, they differ in practice due to the necessary
truncations applied to both model spaces. Consequently, the energies and widths
obtained with the Gamow shell model code and Gamow shell model coupledchannel code are slightly different.
E. From the cross section plot obtained after running the Gamow shell model
coupled-channel code, the energy zones where the effect of 1/2 + , 5/2 + , and
1/2 − resonances are clearly visible.
For projectile energies around 1.5 MeV, the effect of the broad 1/2 + manybody state is dominant. Indeed, the cross section exhibits a sudden increase
around 1.25 MeV. The cross section is significantly spread around 1.5 MeV due
to the large width of the 1/2 + many-body state, of about 400 keV.
Conversely, the cross section takes the form of a narrow peak around 2.8 MeV.
This is entirely due to the presence of the 5/2 + resonance at that energy. The
narrowness of the cross section therein comes from the small width borne by the
5/2 + resonance, of about 100 keV.
493
B. At the Hartree–Fock level, the 1/2 + and 5/2 + resonance states consist of a fully
occupied 0p 1/2 proton shell, with a proton occupying a 1s 1/2 or 0d 5/2 one-body
resonance state, respectively. As there is a gap of several MeVs between the
0p 1/2 shell and the sd shell, the Hartree–Fock configurations can be expected to
form a good approximation of the exact 1/2 + and 5/2 + resonance states.
The 1/2 − has an excitation energy well above the Coulomb barrier. Consequently, one could expect its width to be of several MeVs if its structure is mainly
that of a single proton above an 14 O core. As the width of the 1/2 − resonance is
much smaller, of a few tens of keVs, its many-body wave function must bear a
sizable configuration mixing in the continuum, so that its width is much reduced
compared to its one-particle estimate.
C. About 85% of the many-body wave functions of the 1/2 + and 5/2 + resonance
states consist of configurations of the form [0p 2
1/2 s 1/2 ] or [0p 2
1/2 d 5/2 ],
respectively. This implies that the 1/2 + and 5/2 + resonance states are of singleparticle character.
On the contrary, the 1/2 − resonance state has a fairly large configuration
mixing. Indeed, it consists mainly of [0p 1/2 s 2
1/2 ] and [0p 1/2 d 2
5/2 ] configurations, whose weights are about 75% and 23%, respectively. Therefore, the 1/2 −
resonance state consists mainly of two correlated protons occupying the s 1/2
and d 5/2 resonant and nonresonant shells. Moreover, the pole configurations
[0p 1/2 1s 2
1/2 ] and [0p 1/2 0d 2
5/2 ] have the largest weights, of about 69% and 22%,
respectively. Consequently, one can assume that the two correlated protons are
very likely to be coupled to J π = 0 + in the 1/2 − resonance wave function.
Thus, the 1/2 − resonance state wave function is principally made of a proton
occupying the 0p 1/2 shell and of two protons occupying the sd shells. The
relative stability of the 1/2 − resonance is then explained by the fact that the
0p 1/2 shell is well bound, on the one hand, and because of the smallness of the
phase space available for diproton decay, on the other hand (see Sect. 9.4.1.2).
D. Even though the many-body spaces spanned by Slater determinants and manybody channels are identical, they differ in practice due to the necessary
truncations applied to both model spaces. Consequently, the energies and widths
obtained with the Gamow shell model code and Gamow shell model coupledchannel code are slightly different.
E. From the cross section plot obtained after running the Gamow shell model
coupled-channel code, the energy zones where the effect of 1/2 + , 5/2 + , and
1/2 − resonances are clearly visible.
For projectile energies around 1.5 MeV, the effect of the broad 1/2 + manybody state is dominant. Indeed, the cross section exhibits a sudden increase
around 1.25 MeV. The cross section is significantly spread around 1.5 MeV due
to the large width of the 1/2 + many-body state, of about 400 keV.
Conversely, the cross section takes the form of a narrow peak around 2.8 MeV.
This is entirely due to the presence of the 5/2 + resonance at that energy. The
narrowness of the cross section therein comes from the small width borne by the
5/2 + resonance, of about 100 keV.