434
9 The Unification of Structure and Reaction Frameworks
All functions are bounded in Eq. (9.106), due to the normalizations of basis
states and the low-energy character of ˆ
V res − ˆ
U 0 . Therefore, in order to show that
a f b | ˆ
V res − ˆ
U 0 |f c f d 0, it is sufficient to find one negligible factor in the
integrand of Eq. (9.106). One will show in Exercise III that ab| ˆ
V res − ˆ
U 0 |cd 0
when Eq. (9.96) is fulfilled using Eq. (9.106). This exercise is technical, so that
readers can admit its main results if they find the exercise too difficult.
Exercise III
One will demonstrate that the matrix elements ab| ˆ
V res − ˆ
U 0 |cd vanish when
Eq. (9.96) is verified. This will be necessary to derive the coupled-channel
Hamiltonian equations for many-body projectiles.
A. Let us firstly consider that the two conditions k
i k max and k
j k max are
not verified ∀(i,j) ∈ {a, b, c, d}. Express these conditions in terms of k max and
k
(H)
max . Show that the integrand of Eq. (9.106) vanishes in this case.
B. Let us now suppose that ∃(i,j) ∈ {a, b, c, d} so that k
i k max and k
j k max .
Expand the matrix element
a a m a k
b b m b | ˆ
V res − ˆ
U 0 |k
c c m c k
d d m d B
using plane waves. Deduce from the plane wave expansion that Eq. (9.106)
vanishes in this situation.
C. Conclude that Eq. (9.96) is fulfilled by real-energy neutron Berggren basis
states.
9.2.6.3 Cancellation of High Energy Matrix Elements with Real-Energy
Proton States
For the cases involving proton states, one can consider the screening method in
order to avoid divergences due to the infinite range of the Coulomb potential. This
is justified by the fact that reactions occurring far from the target are only sensitive
to the Coulomb interaction. Hence, one can consider that the Coulomb Hamiltonian
vanishes after a radius R. Proton states for which k ∼ 0 can be neglected as well,
as they can only induce Rutherford scattering at large distances. Consequently, as
proton states generated by a screened Coulomb Hamiltonian can be expanded with
planes waves, one obtains (see Sect. 2.6.5)
ab| ˆ
V res − ˆ
U 0 |cd = =a s b s | ˆ
V res − ˆ
U 0 |c s d s + O(R
−1 ) ,
(9.107)
where the index s stands for “screening.” Consequently, as R can be arbitrarily large,
one can consider that ab| ˆ
V res − ˆ
U 0 |cd 0 in Eq. (9.107) with both proton and
neutron states when the conditions in Eq. (9.96) are verified.
9.2.6.4 Suppression of Antisymmetry at High Projectile Energy
The results previously obtained imply that the only significant two-body matrix
elements ab| ˆ
V res − ˆ
U 0 |cd verify k i k max ∀i ∈ {a, b, c, d}, assuming K CM
K CMmax , with K CMmax = 2k max . As the target is always in a low-energy eigenstate,
9 The Unification of Structure and Reaction Frameworks
All functions are bounded in Eq. (9.106), due to the normalizations of basis
states and the low-energy character of ˆ
V res − ˆ
U 0 . Therefore, in order to show that
a f b | ˆ
V res − ˆ
U 0 |f c f d 0, it is sufficient to find one negligible factor in the
integrand of Eq. (9.106). One will show in Exercise III that ab| ˆ
V res − ˆ
U 0 |cd 0
when Eq. (9.96) is fulfilled using Eq. (9.106). This exercise is technical, so that
readers can admit its main results if they find the exercise too difficult.
Exercise III
One will demonstrate that the matrix elements ab| ˆ
V res − ˆ
U 0 |cd vanish when
Eq. (9.96) is verified. This will be necessary to derive the coupled-channel
Hamiltonian equations for many-body projectiles.
A. Let us firstly consider that the two conditions k
i k max and k
j k max are
not verified ∀(i,j) ∈ {a, b, c, d}. Express these conditions in terms of k max and
k
(H)
max . Show that the integrand of Eq. (9.106) vanishes in this case.
B. Let us now suppose that ∃(i,j) ∈ {a, b, c, d} so that k
i k max and k
j k max .
Expand the matrix element
a a m a k
b b m b | ˆ
V res − ˆ
U 0 |k
c c m c k
d d m d B
using plane waves. Deduce from the plane wave expansion that Eq. (9.106)
vanishes in this situation.
C. Conclude that Eq. (9.96) is fulfilled by real-energy neutron Berggren basis
states.
9.2.6.3 Cancellation of High Energy Matrix Elements with Real-Energy
Proton States
For the cases involving proton states, one can consider the screening method in
order to avoid divergences due to the infinite range of the Coulomb potential. This
is justified by the fact that reactions occurring far from the target are only sensitive
to the Coulomb interaction. Hence, one can consider that the Coulomb Hamiltonian
vanishes after a radius R. Proton states for which k ∼ 0 can be neglected as well,
as they can only induce Rutherford scattering at large distances. Consequently, as
proton states generated by a screened Coulomb Hamiltonian can be expanded with
planes waves, one obtains (see Sect. 2.6.5)
ab| ˆ
V res − ˆ
U 0 |cd = =a s b s | ˆ
V res − ˆ
U 0 |c s d s + O(R
−1 ) ,
(9.107)
where the index s stands for “screening.” Consequently, as R can be arbitrarily large,
one can consider that ab| ˆ
V res − ˆ
U 0 |cd 0 in Eq. (9.107) with both proton and
neutron states when the conditions in Eq. (9.96) are verified.
9.2.6.4 Suppression of Antisymmetry at High Projectile Energy
The results previously obtained imply that the only significant two-body matrix
elements ab| ˆ
V res − ˆ
U 0 |cd verify k i k max ∀i ∈ {a, b, c, d}, assuming K CM
K CMmax , with K CMmax = 2k max . As the target is always in a low-energy eigenstate,