6.4 Three-Body Model in Berggren Basis
267
6.4
Three-Body Model in Berggren Basis
The cluster orbital shell model formalism has been employed in the Gamow
shell model to remove spurious center-of-mass excitations (see Sect. 5.2). In this
section, we shall compare Gamow shell model calculations realized with cluster
orbital shell model coordinates and Jacobi coordinates in a three-body model.
A three-body Hamiltonian can be conveniently solved with Jacobi coordinates
using hyperspherical harmonics. As the latter framework is translationally invariant,
both cluster orbital shell model coordinates and Jacobi coordinates are equivalent
and hence, the eigenvalues of a Hamiltonian expressed using both approaches
are formally identical. However, due to the different truncation schemes used in
practical applications, the eigenvalues may not be equal in both frameworks.
6.4.1 Berggren Basis Expansion in the Three-Body Model
The three-body model discussed in this section describes nuclei in the approximation of a frozen core and two valence nucleons or clusters of nucleons interacting
via a residual nucleon–nucleon interaction. The Hamiltonian can be written as:
ˆ
H =
3
i=1
ˆ
p
2
i
2m i
+
3
i>j =1
ˆ
V ij (r ij ) − ˆ
T CM ,
(6.20)
where V ij is the interaction between clusters i and j , including central, spin–orbit,
and Coulomb terms, and ˆ
T CM is the kinetic energy of the center-of-mass.
The main drawback of three-body models is the appearance of Pauli-forbidden
states arising from the lack of antisymmetrization between core and valence
particles. In order to eliminate these states, one can implement the projection
technique [99]:
ˆ
H −→ ˆ
H + Λ
c
|ϕ
j c m c
j c m c | ,
(6.21)
where Λ is a constant and |ϕ j c m c is a two-body state of the core with angular
quantum numbers j c m c . At large values of Λ, Pauli-forbidden states appear at high
energies, so that they are effectively suppressed.
When one uses cluster orbital shell model coordinates, the finite mass of the
core induces a recoil term (see Eq. (5.42)) in the Hamiltonian. In order to describe
asymptotics and to eliminate the spurious center-of-mass motion exactly, one
expresses the three-body model in the Jacobi relative coordinates [100, 101]:
x =
√
μ ij (r i − r j ),
y =
√
μ (ij )k
r k −
A i r i + A j r j
A i + A j
,
(6.22)
267
6.4
Three-Body Model in Berggren Basis
The cluster orbital shell model formalism has been employed in the Gamow
shell model to remove spurious center-of-mass excitations (see Sect. 5.2). In this
section, we shall compare Gamow shell model calculations realized with cluster
orbital shell model coordinates and Jacobi coordinates in a three-body model.
A three-body Hamiltonian can be conveniently solved with Jacobi coordinates
using hyperspherical harmonics. As the latter framework is translationally invariant,
both cluster orbital shell model coordinates and Jacobi coordinates are equivalent
and hence, the eigenvalues of a Hamiltonian expressed using both approaches
are formally identical. However, due to the different truncation schemes used in
practical applications, the eigenvalues may not be equal in both frameworks.
6.4.1 Berggren Basis Expansion in the Three-Body Model
The three-body model discussed in this section describes nuclei in the approximation of a frozen core and two valence nucleons or clusters of nucleons interacting
via a residual nucleon–nucleon interaction. The Hamiltonian can be written as:
ˆ
H =
3
i=1
ˆ
p
2
i
2m i
+
3
i>j =1
ˆ
V ij (r ij ) − ˆ
T CM ,
(6.20)
where V ij is the interaction between clusters i and j , including central, spin–orbit,
and Coulomb terms, and ˆ
T CM is the kinetic energy of the center-of-mass.
The main drawback of three-body models is the appearance of Pauli-forbidden
states arising from the lack of antisymmetrization between core and valence
particles. In order to eliminate these states, one can implement the projection
technique [99]:
ˆ
H −→ ˆ
H + Λ
c
|ϕ
j c m c
j c m c | ,
(6.21)
where Λ is a constant and |ϕ j c m c is a two-body state of the core with angular
quantum numbers j c m c . At large values of Λ, Pauli-forbidden states appear at high
energies, so that they are effectively suppressed.
When one uses cluster orbital shell model coordinates, the finite mass of the
core induces a recoil term (see Eq. (5.42)) in the Hamiltonian. In order to describe
asymptotics and to eliminate the spurious center-of-mass motion exactly, one
expresses the three-body model in the Jacobi relative coordinates [100, 101]:
x =
√
μ ij (r i − r j ),
y =
√
μ (ij )k
r k −
A i r i + A j r j
A i + A j
,
(6.22)