9.1 Nucleon–Nucleus Reactions
411
reads from the results of Exercise I
V cc (n i
HO , n f
HO ) =
n max
n i n f α i α f
c
(i)
α i
c
(f)
α f
SD α f | ˆ
H |SD α i i
HO
|n i f
HO
|n f
−δ cc’
n≤n max
(E T i + e n ) i
HO
|n f
HO
|n ,
(9.24)
where |n HO
i j τ and |n HO
f j τ are one-body harmonic oscillator states,
|n i j τ f j τ , and |nnj τ are one-body Berggren basis states, E T i is the energy of
|Ψ
J T i
T i
e n is the single-particle energy of |nnj τ , and the expansion coefficients c
(i)
α
and c
(f)
α are those of Eq. (9.7) for the incoming and outgoing channels, respectively.
The expression for matrix element of the N cc is similar to that of V cc and can be
calculated from Eqs. (9.23) and (9.24)
N cc (r, r
) = δ(r − r
) δ cc + ΔN cc (r, r
) ,
(9.25)
where
ΔN cc (n i
HO , n f
HO ) =
n max
n i n f α
c
(i)
α c
(f)
α i
HO
|n i f
HO
|n f − δ cc
n≤n max
i
HO
|n f
HO
|n
(9.26)
A fundamental property of the coupled-channel Gamow shell model Hamiltonian
of Eq. (9.23) is that both resonant and scattering states can be calculated by
solving this equation. Resonant states are indeed eigenstates of Eq. (9.23) by
imposing an outgoing wave condition therein, whereas scattering states are solution
of Eq. (9.23) for all positive energies. Therefore, Eq. (9.23) unifies structure and
reaction formalisms. Hence, it is possible to study the mutual dependence between
structure observables and reaction cross sections with the coupled-channel Gamow
shell model. For example, a sudden increase of reaction cross section at a given
energy can be explained by the presence of a narrow resonance state in the energy
spectrum. Conversely, one can assess the single-particle or collective nature of
weakly bound and resonance states, as they are linear combination of Berggren basis
reaction channels.
9.1.2 Electromagnetic Transitions in a Gamow Coupled-Channel
Formalism
Radiative capture is a very important reaction process in nuclear and astrophysical
applications. Indeed, a very large number of isotopes are generated in stars from
radiative nucleon capture. For example, the reactions entering the CNO cycles
411
reads from the results of Exercise I
V cc (n i
HO , n f
HO ) =
n max
n i n f α i α f
c
(i)
α i
c
(f)
α f
SD α f | ˆ
H |SD α i i
HO
|n i f
HO
|n f
−δ cc’
n≤n max
(E T i + e n ) i
HO
|n f
HO
|n ,
(9.24)
where |n HO
i j τ and |n HO
f j τ are one-body harmonic oscillator states,
|n i j τ f j τ , and |nnj τ are one-body Berggren basis states, E T i is the energy of
|Ψ
J T i
T i
e n is the single-particle energy of |nnj τ , and the expansion coefficients c
(i)
α
and c
(f)
α are those of Eq. (9.7) for the incoming and outgoing channels, respectively.
The expression for matrix element of the N cc is similar to that of V cc and can be
calculated from Eqs. (9.23) and (9.24)
N cc (r, r
) = δ(r − r
) δ cc + ΔN cc (r, r
) ,
(9.25)
where
ΔN cc (n i
HO , n f
HO ) =
n max
n i n f α
c
(i)
α c
(f)
α i
HO
|n i f
HO
|n f − δ cc
n≤n max
i
HO
|n f
HO
|n
(9.26)
A fundamental property of the coupled-channel Gamow shell model Hamiltonian
of Eq. (9.23) is that both resonant and scattering states can be calculated by
solving this equation. Resonant states are indeed eigenstates of Eq. (9.23) by
imposing an outgoing wave condition therein, whereas scattering states are solution
of Eq. (9.23) for all positive energies. Therefore, Eq. (9.23) unifies structure and
reaction formalisms. Hence, it is possible to study the mutual dependence between
structure observables and reaction cross sections with the coupled-channel Gamow
shell model. For example, a sudden increase of reaction cross section at a given
energy can be explained by the presence of a narrow resonance state in the energy
spectrum. Conversely, one can assess the single-particle or collective nature of
weakly bound and resonance states, as they are linear combination of Berggren basis
reaction channels.
9.1.2 Electromagnetic Transitions in a Gamow Coupled-Channel
Formalism
Radiative capture is a very important reaction process in nuclear and astrophysical
applications. Indeed, a very large number of isotopes are generated in stars from
radiative nucleon capture. For example, the reactions entering the CNO cycles