440
9 The Unification of Structure and Reaction Frameworks
9.2.8 Treatment of Faddeev Equations with the Berggren Basis
As one could see in Sect. 6.4, the three-body approach to nuclear systems is both
conceptually simple and convenient in practical applications. While the approach
detailed in Sect. 6.4 is devoted to nuclear structure, the three-body models are
also widely used for nuclear reaction applications. For example, the three-body
Schrödinger equation can be numerically solved with Lagrange mesh and continuum discretized coupled-channel techniques [58–62]. However, the latter methods
are limited to real-energy states. Consequently, one cannot directly calculate
resonances with these techniques.
It would be convenient to solve the three-body Schrödinger equation for both real
and complex energies, so that the structure and reaction aspects of coupled-channel
Hamiltonians can be unified therein. Therefore, one will present how to solve the
three-body Schrödinger equation with the Berggren basis.
Let us write the Faddeev equations describing an interacting three-body system,
formed by a target, seen as a one-body object, and a projectile formed by two
subsystems. Hamiltonian and Schrödinger equations read
ˆ
H |Ψ = E |Ψ
(9.122)
ˆ
H = ˆ
H P +
P 2
2M CM
+ ˆ
V T (r, R) ,
(9.123)
where P 2 /(2M CM ) is the kinetic energy operator of the center-of-mass of the
projectile, P is the linear momentum of the center-of-mass of the projectile with
respect to that of the target, and M CM is the mass of the projectile. ˆ
H P is the intrinsic
Hamiltonian of the projectile, taking into account nucleon–nucleon correlations in
the projectile, ˆ
V T (r, R) is the potential depending on the intrinsic spatial coordinate
of the projectile r and the coordinate R between the center-of-mass of the target
and that of the projectile, generated by target, which describes the scattering of the
projectile.
In Eqs. (9.122) and (9.123), the target is structureless as its effect on the projectile
is modelled by the sole action of the ˆ
V T (r, R) potential. The potential ˆ
V T (r, R)
induces mixing between the channels formed by target and projectile. Consequently,
Eqs. (9.122) and (9.123) will be expressed as coupled-channel equations, which one
will be solved using the Berggren basis.
The projectile structure is taken into account by the intrinsic Hamiltonian H P ,
which will be detailed in the following. ˆ
H P reads (see Sect. 4.1))
ˆ
H P =
p 2
2μ P
+ ˆ
V P (r) ,
(9.124)
where r is the relative radius of the projectile, p is the intrinsic linear momentum
of the projectile, μ P is the reduced mass of the projectile, and ˆ
V P (r) represents the
interaction between the two subsystems of the projectile.
9 The Unification of Structure and Reaction Frameworks
9.2.8 Treatment of Faddeev Equations with the Berggren Basis
As one could see in Sect. 6.4, the three-body approach to nuclear systems is both
conceptually simple and convenient in practical applications. While the approach
detailed in Sect. 6.4 is devoted to nuclear structure, the three-body models are
also widely used for nuclear reaction applications. For example, the three-body
Schrödinger equation can be numerically solved with Lagrange mesh and continuum discretized coupled-channel techniques [58–62]. However, the latter methods
are limited to real-energy states. Consequently, one cannot directly calculate
resonances with these techniques.
It would be convenient to solve the three-body Schrödinger equation for both real
and complex energies, so that the structure and reaction aspects of coupled-channel
Hamiltonians can be unified therein. Therefore, one will present how to solve the
three-body Schrödinger equation with the Berggren basis.
Let us write the Faddeev equations describing an interacting three-body system,
formed by a target, seen as a one-body object, and a projectile formed by two
subsystems. Hamiltonian and Schrödinger equations read
ˆ
H |Ψ = E |Ψ
(9.122)
ˆ
H = ˆ
H P +
P 2
2M CM
+ ˆ
V T (r, R) ,
(9.123)
where P 2 /(2M CM ) is the kinetic energy operator of the center-of-mass of the
projectile, P is the linear momentum of the center-of-mass of the projectile with
respect to that of the target, and M CM is the mass of the projectile. ˆ
H P is the intrinsic
Hamiltonian of the projectile, taking into account nucleon–nucleon correlations in
the projectile, ˆ
V T (r, R) is the potential depending on the intrinsic spatial coordinate
of the projectile r and the coordinate R between the center-of-mass of the target
and that of the projectile, generated by target, which describes the scattering of the
projectile.
In Eqs. (9.122) and (9.123), the target is structureless as its effect on the projectile
is modelled by the sole action of the ˆ
V T (r, R) potential. The potential ˆ
V T (r, R)
induces mixing between the channels formed by target and projectile. Consequently,
Eqs. (9.122) and (9.123) will be expressed as coupled-channel equations, which one
will be solved using the Berggren basis.
The projectile structure is taken into account by the intrinsic Hamiltonian H P ,
which will be detailed in the following. ˆ
H P reads (see Sect. 4.1))
ˆ
H P =
p 2
2μ P
+ ˆ
V P (r) ,
(9.124)
where r is the relative radius of the projectile, p is the intrinsic linear momentum
of the projectile, μ P is the reduced mass of the projectile, and ˆ
V P (r) represents the
interaction between the two subsystems of the projectile.