430
9 The Unification of Structure and Reaction Frameworks
where N CMmin is chosen so as to include the cluster eigenstates of ˆ
H P sizably
occupying the core. ˆ
H CM is then redefined similarly to H in the one-nucleon
projectile case (see Sect. 9.1.1.1):
ˆ
H CM → ˆ
P CM ˆ
H CM ˆ
P CM = ˆ
H CM − ˆ
Q CM ˆ
H CM − ˆ
H CM ˆ
Q CM + ˆ
Q CM ˆ
H CM ˆ
Q CM ,
(9.93)
which thereby generates an additional short-range interaction to be added to ˆ
H P in
Eq. (9.15).
As the ˆ
Q CM and ˆ
P CM projectors (see Eqs. (9.91) and (9.92)) are written in centerof-mass and relative coordinates, they cannot be exactly equal to ˆ
Q and ˆ
P written
in orbital shell model coordinates (see Eqs. (9.16) and (9.17)). However, as it has
been checked in practical applications, ˆ
Q ˆ
Q CM and ˆ
P CM ˆ
P , so that the use of
ˆ
Q CM and ˆ
P CM allows the orthogonalization condition model to be verified. Hence,
in practical calculations, similarly to the one-nucleon case, composite states can
be considered to be orthogonal to the core occupied states, even though a strict
orthogonality is no longer exactly fulfilled with many-body projectiles.
9.2.6 Asymptotic Cancellation of Correlations at High Projectile
Energy
Contrary to the one-nucleon projectile case, the nucleon–nucleon interaction
between target and projectile does not become negligible when the energy of
the projectile nucleons becomes very large (see Sect. 9.1.1). Indeed, even though
the nucleons inside a rapidly moving cluster do not interact with the nucleons of
the target, they still strongly interact with each other. Thus, the description of a
many-nucleon projectile of high velocity will demand to include two-body matrix
elements involving one-body states of high energy. Nevertheless, as the projectile
almost does not interact with the target, the Hamiltonian describing target–projectile
channels of high energy will greatly simplify. In fact, it will decouple similarly to
the one-nucleon case (see Sect. 9.1.1), with a low-energy part, containing target–
projectile correlations, and a high energy part, where target and projectile are
decoupled.
Let us define the projectile state as a product of low-energy pairs of particles,
which embodies its compact cluster structure:
|Ψ
J P
P =
⎛
⎜
⎝A
k rel ≤k
(max)
rel
|k rel
⎞
⎟
⎠ |K CM ,
(9.94)
where |k rel stands for a particle pair, k
(max)
rel
is a maximal linear momentum allowed
for pairs, related to the average relative velocity of nucleons inside the cluster, and
9 The Unification of Structure and Reaction Frameworks
where N CMmin is chosen so as to include the cluster eigenstates of ˆ
H P sizably
occupying the core. ˆ
H CM is then redefined similarly to H in the one-nucleon
projectile case (see Sect. 9.1.1.1):
ˆ
H CM → ˆ
P CM ˆ
H CM ˆ
P CM = ˆ
H CM − ˆ
Q CM ˆ
H CM − ˆ
H CM ˆ
Q CM + ˆ
Q CM ˆ
H CM ˆ
Q CM ,
(9.93)
which thereby generates an additional short-range interaction to be added to ˆ
H P in
Eq. (9.15).
As the ˆ
Q CM and ˆ
P CM projectors (see Eqs. (9.91) and (9.92)) are written in centerof-mass and relative coordinates, they cannot be exactly equal to ˆ
Q and ˆ
P written
in orbital shell model coordinates (see Eqs. (9.16) and (9.17)). However, as it has
been checked in practical applications, ˆ
Q ˆ
Q CM and ˆ
P CM ˆ
P , so that the use of
ˆ
Q CM and ˆ
P CM allows the orthogonalization condition model to be verified. Hence,
in practical calculations, similarly to the one-nucleon case, composite states can
be considered to be orthogonal to the core occupied states, even though a strict
orthogonality is no longer exactly fulfilled with many-body projectiles.
9.2.6 Asymptotic Cancellation of Correlations at High Projectile
Energy
Contrary to the one-nucleon projectile case, the nucleon–nucleon interaction
between target and projectile does not become negligible when the energy of
the projectile nucleons becomes very large (see Sect. 9.1.1). Indeed, even though
the nucleons inside a rapidly moving cluster do not interact with the nucleons of
the target, they still strongly interact with each other. Thus, the description of a
many-nucleon projectile of high velocity will demand to include two-body matrix
elements involving one-body states of high energy. Nevertheless, as the projectile
almost does not interact with the target, the Hamiltonian describing target–projectile
channels of high energy will greatly simplify. In fact, it will decouple similarly to
the one-nucleon case (see Sect. 9.1.1), with a low-energy part, containing target–
projectile correlations, and a high energy part, where target and projectile are
decoupled.
Let us define the projectile state as a product of low-energy pairs of particles,
which embodies its compact cluster structure:
|Ψ
J P
P =
⎛
⎜
⎝A
k rel ≤k
(max)
rel
|k rel
⎞
⎟
⎠ |K CM ,
(9.94)
where |k rel stands for a particle pair, k
(max)
rel
is a maximal linear momentum allowed
for pairs, related to the average relative velocity of nucleons inside the cluster, and