422
9 The Unification of Structure and Reaction Frameworks
In practice, one often shows the astrophysical factor, which removes the exponential
dependence of the cross section at low energies due to the Coulomb potential barrier
S (E CM ) = σ (E CM ) E CM e
2πη CM ,
(9.62)
where η CM is the Sommerfeld parameter (see Sect. 2.1).
9.2
Generalization to Many-Nucleon Projectiles
In this section, one will discuss the coupled-channel Gamow shell model formulation for the case of a complex projectile composed of two or more nucleons. In
principle, all nucleons of the projectile can be scattered in the reaction, so that one
would need to impose three-, four-body, etc., outgoing wave function conditions
in the asymptotic zone. However, reactions involving many-body asymptotes are
very seldom. Indeed, in most cases, the projectile is either the same in incoming
and outgoing channels, as in elastic scattering reactions, or gains/loses one nucleon
through pick-up/stripping reactions, as in (d, p) or (p, d) reactions. Therefore, all
projectiles behave as point particles in the asymptotic region, so that reactions
reduce to a two-body problem therein. This reduction of the number of degrees
of freedom in the asymptotic region, called cluster approximation, considerably
simplifies the theoretical formulation of the reaction framework. The internal
structure of the projectile is nevertheless considered in its entirety in the nuclear
zone, where all the nucleons of the target–projectile composite interact.
In order to treat projectiles in the asymptotic region, it is necessary to separate
their center-of-mass and intrinsic degrees of freedom. Indeed, the trajectory of the
projectile is a function of its center-of-mass coordinates, while its intrinsic part
intervenes in reaction channels and Hamiltonian matrix elements. In order to do
this separation rigorously, one must either use a basis of harmonic oscillator states,
due to the exact factorization of many-body wave functions in center-of-mass and
intrinsic parts therein, or Jacobi coordinates, which are intrinsic by construction.
Both methods have their advantages and disadvantages. On the one hand, one can
conveniently use the basis of harmonic oscillator states in a shell model framework.
Due to the localized character of the basis of harmonic oscillator states, however,
one can only expand well-bound clusters using this method. Hence, it is most useful
for projectiles such as 3 H, 3 He, and especially 4 He particles, which are well bound,
by about 6 MeV for the former and 20 MeV for the latter. On the other hand, it is
not appropriate for deuteron projectiles, which is prone to breakup due to its small
binding energy, of about 2 MeV. Deuteron can be directly treated in an intrinsic
framework, as it has only two particles. Therefore, it will be handled numerically
by diagonalization of a realistic Hamiltonian with the Berggren basis using relative
coordinates. Hence, break-up effects can be included as one can form scattering
deuteron projectile states near the target. Due to their complementary natures, both
aforementioned methods to generate projectile wave functions will be presented.
9 The Unification of Structure and Reaction Frameworks
In practice, one often shows the astrophysical factor, which removes the exponential
dependence of the cross section at low energies due to the Coulomb potential barrier
S (E CM ) = σ (E CM ) E CM e
2πη CM ,
(9.62)
where η CM is the Sommerfeld parameter (see Sect. 2.1).
9.2
Generalization to Many-Nucleon Projectiles
In this section, one will discuss the coupled-channel Gamow shell model formulation for the case of a complex projectile composed of two or more nucleons. In
principle, all nucleons of the projectile can be scattered in the reaction, so that one
would need to impose three-, four-body, etc., outgoing wave function conditions
in the asymptotic zone. However, reactions involving many-body asymptotes are
very seldom. Indeed, in most cases, the projectile is either the same in incoming
and outgoing channels, as in elastic scattering reactions, or gains/loses one nucleon
through pick-up/stripping reactions, as in (d, p) or (p, d) reactions. Therefore, all
projectiles behave as point particles in the asymptotic region, so that reactions
reduce to a two-body problem therein. This reduction of the number of degrees
of freedom in the asymptotic region, called cluster approximation, considerably
simplifies the theoretical formulation of the reaction framework. The internal
structure of the projectile is nevertheless considered in its entirety in the nuclear
zone, where all the nucleons of the target–projectile composite interact.
In order to treat projectiles in the asymptotic region, it is necessary to separate
their center-of-mass and intrinsic degrees of freedom. Indeed, the trajectory of the
projectile is a function of its center-of-mass coordinates, while its intrinsic part
intervenes in reaction channels and Hamiltonian matrix elements. In order to do
this separation rigorously, one must either use a basis of harmonic oscillator states,
due to the exact factorization of many-body wave functions in center-of-mass and
intrinsic parts therein, or Jacobi coordinates, which are intrinsic by construction.
Both methods have their advantages and disadvantages. On the one hand, one can
conveniently use the basis of harmonic oscillator states in a shell model framework.
Due to the localized character of the basis of harmonic oscillator states, however,
one can only expand well-bound clusters using this method. Hence, it is most useful
for projectiles such as 3 H, 3 He, and especially 4 He particles, which are well bound,
by about 6 MeV for the former and 20 MeV for the latter. On the other hand, it is
not appropriate for deuteron projectiles, which is prone to breakup due to its small
binding energy, of about 2 MeV. Deuteron can be directly treated in an intrinsic
framework, as it has only two particles. Therefore, it will be handled numerically
by diagonalization of a realistic Hamiltonian with the Berggren basis using relative
coordinates. Hence, break-up effects can be included as one can form scattering
deuteron projectile states near the target. Due to their complementary natures, both
aforementioned methods to generate projectile wave functions will be presented.