6.4 Three-Body Model in Berggren Basis
271
that the diverging diagonal Coulomb matrix element becomes a large but converging
off-diagonal matrix element.
6.4.2 Convergence Properties of Eigenvalues Calculated in Jacobi
and Cluster Orbital Shell Model Coordinates
Let us now compare Gamow shell model results obtained using Jacobi coordinates and cluster orbital shell model coordinates. In cluster orbital shell model
coordinates, one defines a truncated model space according to a given maximum
orbital angular momentum max of particle orbital angular momenta 1 , 2 . In Jacobi
coordinates, the model space is truncated according to the maximum value of ( x ,
y ) (see Fig. 6.11).
In the cluster orbital shell model framework, one uses the Berggren basis for
the s, p, d orbits and harmonic oscillator basis states for partial waves bearing
≥ 3. Indeed, high orbital angular momentum components play a small role in
wave function asymptotes so that they can be restricted to a few harmonic oscillator
shells. The complex contours of the Berggren basis start from k = 0 fm −1 and
end in k max = 3 fm −1 . In the framework using Jacobi coordinates, all calculations
have been done taking into account the maximum hyperspherical quantum number
K max = 20. Similarly to cluster orbital shell model, one uses the Berggren basis
for the K ≤ 6 channels, where K is the hyperspherical quantum number of
the considered partial wave, while harmonic oscillator basis states are used when
K ≥ 7. Note that the energy range covered by the three-body model using Jacobi
coordinates is about twice as large as that of cluster orbital shell model. In Jacobi
coordinates, one has k 2
ρ = k 2
x + k 2
y , where k x , k y , and k ρ are the conjugate linear
momenta of the hyperspherical coordinates x, y, and ρ =
x 2 + y 2 , respectively
(see Fig. 6.11). Let k represents the linear momentum of a single nucleon in cluster
orbital shell model. From Eq. (6.27), the energy of two-body cluster is ¯
h 2 k 2
ρ /2m,
while that of one nucleon is ¯
h
2 k 2 /2m from Eq. (3.66). Consequently, at large kinetic
energy, k 2
ρ 2k 2 , as the binding energy of the cluster can be hereby neglected.
Maximal linear momenta thus have to be chosen differently for k ρ and k if one
wants model spaces to be as close as possible in both models.
As an example, let us consider 6 He, 6 Li, and 6 Be. Their structure is relatively
simple because they can be treated as an α-particle core and two valence nucleons.
For the calculation both in Jacobi coordinates and in cluster orbital shell model
coordinates, details of the contours in the complex plane and their discretization
are given in Ref. [95]. The original Minnesota interaction [111] is used to mimic
the nuclear interaction between valence nucleons. Interaction between core and
valence nucleons describes the Woods–Saxon potential whose parameters are fitted
to the resonances of the core+n system. The Coulomb potential is added for charged
particles. Parameters of the Woods–Saxon potential for A = 6 nuclei are detailed in
Ref. [95]).
271
that the diverging diagonal Coulomb matrix element becomes a large but converging
off-diagonal matrix element.
6.4.2 Convergence Properties of Eigenvalues Calculated in Jacobi
and Cluster Orbital Shell Model Coordinates
Let us now compare Gamow shell model results obtained using Jacobi coordinates and cluster orbital shell model coordinates. In cluster orbital shell model
coordinates, one defines a truncated model space according to a given maximum
orbital angular momentum max of particle orbital angular momenta 1 , 2 . In Jacobi
coordinates, the model space is truncated according to the maximum value of ( x ,
y ) (see Fig. 6.11).
In the cluster orbital shell model framework, one uses the Berggren basis for
the s, p, d orbits and harmonic oscillator basis states for partial waves bearing
≥ 3. Indeed, high orbital angular momentum components play a small role in
wave function asymptotes so that they can be restricted to a few harmonic oscillator
shells. The complex contours of the Berggren basis start from k = 0 fm −1 and
end in k max = 3 fm −1 . In the framework using Jacobi coordinates, all calculations
have been done taking into account the maximum hyperspherical quantum number
K max = 20. Similarly to cluster orbital shell model, one uses the Berggren basis
for the K ≤ 6 channels, where K is the hyperspherical quantum number of
the considered partial wave, while harmonic oscillator basis states are used when
K ≥ 7. Note that the energy range covered by the three-body model using Jacobi
coordinates is about twice as large as that of cluster orbital shell model. In Jacobi
coordinates, one has k 2
ρ = k 2
x + k 2
y , where k x , k y , and k ρ are the conjugate linear
momenta of the hyperspherical coordinates x, y, and ρ =
x 2 + y 2 , respectively
(see Fig. 6.11). Let k represents the linear momentum of a single nucleon in cluster
orbital shell model. From Eq. (6.27), the energy of two-body cluster is ¯
h 2 k 2
ρ /2m,
while that of one nucleon is ¯
h
2 k 2 /2m from Eq. (3.66). Consequently, at large kinetic
energy, k 2
ρ 2k 2 , as the binding energy of the cluster can be hereby neglected.
Maximal linear momenta thus have to be chosen differently for k ρ and k if one
wants model spaces to be as close as possible in both models.
As an example, let us consider 6 He, 6 Li, and 6 Be. Their structure is relatively
simple because they can be treated as an α-particle core and two valence nucleons.
For the calculation both in Jacobi coordinates and in cluster orbital shell model
coordinates, details of the contours in the complex plane and their discretization
are given in Ref. [95]. The original Minnesota interaction [111] is used to mimic
the nuclear interaction between valence nucleons. Interaction between core and
valence nucleons describes the Woods–Saxon potential whose parameters are fitted
to the resonances of the core+n system. The Coulomb potential is added for charged
particles. Parameters of the Woods–Saxon potential for A = 6 nuclei are detailed in
Ref. [95]).