5.2 Translationally Invariant Shell Model Scheme: The Cluster Orbital Shell. . .
195
is complex, E is stationary with respect to small variations of |Ψ , so that it should
not vary significantly in the vicinity of the exact energy of the eigenstate in the
complex plane. As a consequence, the generalized variational principle allows to
prove that the resonance states calculated from the diagonalization of Hamiltonian
matrices in a finite model space are as precise as bound states (see Exercise II for a
numerical illustration of the generalized variational principle).
5.2
Translationally Invariant Shell Model Scheme: The Cluster
Orbital Shell Model
In the standard shell model, it is customary to remove center-of-mass excitations
using the Lawson method [11]. In the Gamow shell model, however, this method
is precluded because Berggren basis states are not eigenstates of the harmonic
oscillator potential. Therefore, in order to eliminate center-of-mass excitations,
while avoiding the numerical difficulties arising from the use of Jacobi coordinates
in describing nuclei with many valence nucleons, one chose to work in the cluster
orbital shell model framework [12]. In cluster orbital shell model, one considers
a core plus valence particles, so that coordinates of valence particles are defined
relatively to the center-of-mass of the core. This allows to work in a translationally
invariant many-body framework.
Let us define the cluster orbital shell model coordinates:
r i = r i,lab − R CM,core if i ∈ val
(5.24)
r i = r i,lab
if i ∈ core
(5.25)
where r i,lab is the coordinate of a nucleon in the laboratory system and
R CM,core =
1
M core
i∈core
m i r i,lab
(5.26)
is the coordinate of the center-of-mass of the core. In this expression, m i is the mass
of the ith particle and M core =
i∈core
m i is the mass of the core. The cluster orbital
shell model momentum reads:
p i = −i ¯
h∇ i ,
(5.27)
where ∇ i is the gradient associated to r i .
It is possible to write the momentum in the laboratory frame p i,lab as a function
of the cluster orbital shell model operator p i of Eq. (5.27). For this, the partial
derivative operator
∂
∂x i,lab
needs to be expressed in cluster orbital shell model
coordinates:
195
is complex, E is stationary with respect to small variations of |Ψ , so that it should
not vary significantly in the vicinity of the exact energy of the eigenstate in the
complex plane. As a consequence, the generalized variational principle allows to
prove that the resonance states calculated from the diagonalization of Hamiltonian
matrices in a finite model space are as precise as bound states (see Exercise II for a
numerical illustration of the generalized variational principle).
5.2
Translationally Invariant Shell Model Scheme: The Cluster
Orbital Shell Model
In the standard shell model, it is customary to remove center-of-mass excitations
using the Lawson method [11]. In the Gamow shell model, however, this method
is precluded because Berggren basis states are not eigenstates of the harmonic
oscillator potential. Therefore, in order to eliminate center-of-mass excitations,
while avoiding the numerical difficulties arising from the use of Jacobi coordinates
in describing nuclei with many valence nucleons, one chose to work in the cluster
orbital shell model framework [12]. In cluster orbital shell model, one considers
a core plus valence particles, so that coordinates of valence particles are defined
relatively to the center-of-mass of the core. This allows to work in a translationally
invariant many-body framework.
Let us define the cluster orbital shell model coordinates:
r i = r i,lab − R CM,core if i ∈ val
(5.24)
r i = r i,lab
if i ∈ core
(5.25)
where r i,lab is the coordinate of a nucleon in the laboratory system and
R CM,core =
1
M core
i∈core
m i r i,lab
(5.26)
is the coordinate of the center-of-mass of the core. In this expression, m i is the mass
of the ith particle and M core =
i∈core
m i is the mass of the core. The cluster orbital
shell model momentum reads:
p i = −i ¯
h∇ i ,
(5.27)
where ∇ i is the gradient associated to r i .
It is possible to write the momentum in the laboratory frame p i,lab as a function
of the cluster orbital shell model operator p i of Eq. (5.27). For this, the partial
derivative operator
∂
∂x i,lab
needs to be expressed in cluster orbital shell model
coordinates: