9.1 Nucleon–Nucleus Reactions
405
neutron halo in 11 Li is already included in the many-body wave function arising
from the Gamow shell model.
9.1
Nucleon–Nucleus Reactions
One will firstly consider elastic/inelastic scattering and radiative capture cross
sections involving one-nucleon projectiles. They are of important experimental
interest, and many theoretical aspects of the presented formalism are common to
the reactions making use of one-nucleon and many-nucleon projectiles.
The A-body system representing a composite system, i.e. the many-nucleon system formed by the projectile and the target, is described by the Hamiltonian (5.42)
given in the cluster orbital shell model coordinates. To determine reaction cross
sections, one has to calculate the scattering state |Ψ
J A
M A
of the A-body system, which
verifies the Schrödinger equation:
ˆ
H |Ψ
J A
M A
= E |Ψ
J A
M A
.
(9.1)
Here, due to the energy conservation, one has: E = E T + E P , where E P and E T are,
respectively, the total energy of the projectile and the total energy of the target. To
calculate the cross section with a given entrance channel, one has to determine the
A-body scattering state |Ψ
J A
M A
ˆ
A {|Ψ
J T
T ⊗ |Ψ
J P
P
J A
M A
.
(9.2)
Here |Ψ
J T
T and |Ψ
J P
P are, respectively, the target and projectile states with their
momentum J T and J P so that the total angular momentum is: J A = J T + J P . All
A-body wave functions are fully antisymmetrized, as emphasized by the ˆ
A symbol.
The first step is to build the channel basis, and the second step is to determine
the expansion of the A-body scattering state |Ψ
J A
M A
in this channel basis. In order to
consider reactions involving one-proton or neutron scattering processes, one defines
the projectile state as
|Ψ
j
P = |rrj τ ,
(9.3)
where the angular momentum notation J P introduced in Eq. (9.2) has been replaced
by j for convenience. Here, r is the radial coordinate, , j are the angular quantum
numbers, and τ is the isospin quantum number.
To express antisymmetry in a convenient way, the |rrj τ channel is firstly
expanded with the single-particle basis of Gamow shell model wave functions u n (r)
|rrj τ =
n
u n (r)
r
|u n j τ ,
(9.4)
405
neutron halo in 11 Li is already included in the many-body wave function arising
from the Gamow shell model.
9.1
Nucleon–Nucleus Reactions
One will firstly consider elastic/inelastic scattering and radiative capture cross
sections involving one-nucleon projectiles. They are of important experimental
interest, and many theoretical aspects of the presented formalism are common to
the reactions making use of one-nucleon and many-nucleon projectiles.
The A-body system representing a composite system, i.e. the many-nucleon system formed by the projectile and the target, is described by the Hamiltonian (5.42)
given in the cluster orbital shell model coordinates. To determine reaction cross
sections, one has to calculate the scattering state |Ψ
J A
M A
of the A-body system, which
verifies the Schrödinger equation:
ˆ
H |Ψ
J A
M A
= E |Ψ
J A
M A
.
(9.1)
Here, due to the energy conservation, one has: E = E T + E P , where E P and E T are,
respectively, the total energy of the projectile and the total energy of the target. To
calculate the cross section with a given entrance channel, one has to determine the
A-body scattering state |Ψ
J A
M A
ˆ
A {|Ψ
J T
T ⊗ |Ψ
J P
P
J A
M A
.
(9.2)
Here |Ψ
J T
T and |Ψ
J P
P are, respectively, the target and projectile states with their
momentum J T and J P so that the total angular momentum is: J A = J T + J P . All
A-body wave functions are fully antisymmetrized, as emphasized by the ˆ
A symbol.
The first step is to build the channel basis, and the second step is to determine
the expansion of the A-body scattering state |Ψ
J A
M A
in this channel basis. In order to
consider reactions involving one-proton or neutron scattering processes, one defines
the projectile state as
|Ψ
j
P = |rrj τ ,
(9.3)
where the angular momentum notation J P introduced in Eq. (9.2) has been replaced
by j for convenience. Here, r is the radial coordinate, , j are the angular quantum
numbers, and τ is the isospin quantum number.
To express antisymmetry in a convenient way, the |rrj τ channel is firstly
expanded with the single-particle basis of Gamow shell model wave functions u n (r)
|rrj τ =
n
u n (r)
r
|u n j τ ,
(9.4)