176
4 Two-Particle Systems in the Berggren Basis
positive-parity ground-state band of 11 Be can be viewed as a weakly bound/unbound
neutron coupled to a deformed core of 10 Be [117, 118]. The nonadiabatic approach
developed in Refs. [88, 89, 114–116] (see also Sect. 4.3.4.1) in the context of proton
emitters is now applied to the neutron + deformed core approach.
The exact form of the deformed pseudo-potential representing the nucleon-core
interaction is not essential for the purpose of following discussion, as long as
the one-nucleon threshold is correctly reproduced. One approximates the pseudopotential by a deformed Woods-Saxon potential with a spherical spin-orbit term
[115]. The total angular momentum of the system is J = j + j r , where j = + s
is the angular momentum of the valence nucleon and j r is that of the rotor. In
the coupled-channel formalism, eigenstates of the decaying nucleus |Ψ J π are
expanded in the basis of channel wave functions labeled by channel quantum
numbers c = ((jj r ). Each channel state is given by the cluster radial wave function
u c (r)/r representing the relative radial motion of the particle and the core, and the
orbital-spin part |j ((, s)j r ; J M J
The rotational structure of nuclear states can be interpreted in terms of the
intrinsic density of the valence particle in the core reference frame. One assumes that
the core is associated with the rigid rotor axially deformed around the z-axis. When
expressed in the deformed reference frame, the eigenstates |Ψ J π can be expanded
in the basis |Ψ
J π
K , where K is the projection of the total angular momentum J on
the symmetry axis in the core frame. The total density in intrinsic frame is obtained
by summing up of all the K-components of ρ J K (r, θ ) densities [83]. If only one Kcomponent is nonzero, K becomes a good quantum number and the strong coupling
(adiabatic) limit is strictly obeyed [89, 90, 114, 115].
Coupled-channel equations of the particle-rotor model are solved up to a
maximal radius of R max = 30 fm and the rotation radius for the exterior complex
scaling is R rot = 15 fm. Since the studied rotational bands have positive parity, one
takes partial waves with = 0, 2, 4, 6 in the Berggren ensemble, and the maximum
angular momentum of the core that is large enough to guarantee that the number of
included states in the ground state band of the daughter nucleus does not impact
the calculated widths [83, 115]. The neutron 0s 1/2 shell in the core is excluded
from the construction of the coupled-channel basis, in order that the Pauli principle
between core and valence particles is approximately satisfied [120]. For the core
energies E
j π
r
d , one takes known experimental energies of j π
r = 0 + , 2 + , and 4 +
members of the ground-state band of 10 Be [108]. The higher-lying band members
are approximated by means of the rigid rotor expression with the moment of inertia
corresponding to the 4 + level. The particle width of the core states with j r ≥ 4 will
be ignored in the following. Experimentally, the 2
+
1 state in 10 Be is particle-bound,
and the 4
+
1 level has a fairly small width of 121 keV.
Parameters of the deformed pseudo-potential are fitted to the 1/2 + and 5/2 +
members of the yrast band of 11 Be [108]. These states collapse to the same bandhead energy in the adiabatic limit (I → ∞), i.e., they are members of the same
rotational band. The experimental energy of higher-lying band members is uncertain, although candidates for the 3/2
+
1 , 7/2
+
1 , and 9/2
+
1 states have been suggested
4 Two-Particle Systems in the Berggren Basis
positive-parity ground-state band of 11 Be can be viewed as a weakly bound/unbound
neutron coupled to a deformed core of 10 Be [117, 118]. The nonadiabatic approach
developed in Refs. [88, 89, 114–116] (see also Sect. 4.3.4.1) in the context of proton
emitters is now applied to the neutron + deformed core approach.
The exact form of the deformed pseudo-potential representing the nucleon-core
interaction is not essential for the purpose of following discussion, as long as
the one-nucleon threshold is correctly reproduced. One approximates the pseudopotential by a deformed Woods-Saxon potential with a spherical spin-orbit term
[115]. The total angular momentum of the system is J = j + j r , where j = + s
is the angular momentum of the valence nucleon and j r is that of the rotor. In
the coupled-channel formalism, eigenstates of the decaying nucleus |Ψ J π are
expanded in the basis of channel wave functions labeled by channel quantum
numbers c = ((jj r ). Each channel state is given by the cluster radial wave function
u c (r)/r representing the relative radial motion of the particle and the core, and the
orbital-spin part |j ((, s)j r ; J M J
The rotational structure of nuclear states can be interpreted in terms of the
intrinsic density of the valence particle in the core reference frame. One assumes that
the core is associated with the rigid rotor axially deformed around the z-axis. When
expressed in the deformed reference frame, the eigenstates |Ψ J π can be expanded
in the basis |Ψ
J π
K , where K is the projection of the total angular momentum J on
the symmetry axis in the core frame. The total density in intrinsic frame is obtained
by summing up of all the K-components of ρ J K (r, θ ) densities [83]. If only one Kcomponent is nonzero, K becomes a good quantum number and the strong coupling
(adiabatic) limit is strictly obeyed [89, 90, 114, 115].
Coupled-channel equations of the particle-rotor model are solved up to a
maximal radius of R max = 30 fm and the rotation radius for the exterior complex
scaling is R rot = 15 fm. Since the studied rotational bands have positive parity, one
takes partial waves with = 0, 2, 4, 6 in the Berggren ensemble, and the maximum
angular momentum of the core that is large enough to guarantee that the number of
included states in the ground state band of the daughter nucleus does not impact
the calculated widths [83, 115]. The neutron 0s 1/2 shell in the core is excluded
from the construction of the coupled-channel basis, in order that the Pauli principle
between core and valence particles is approximately satisfied [120]. For the core
energies E
j π
r
d , one takes known experimental energies of j π
r = 0 + , 2 + , and 4 +
members of the ground-state band of 10 Be [108]. The higher-lying band members
are approximated by means of the rigid rotor expression with the moment of inertia
corresponding to the 4 + level. The particle width of the core states with j r ≥ 4 will
be ignored in the following. Experimentally, the 2
+
1 state in 10 Be is particle-bound,
and the 4
+
1 level has a fairly small width of 121 keV.
Parameters of the deformed pseudo-potential are fitted to the 1/2 + and 5/2 +
members of the yrast band of 11 Be [108]. These states collapse to the same bandhead energy in the adiabatic limit (I → ∞), i.e., they are members of the same
rotational band. The experimental energy of higher-lying band members is uncertain, although candidates for the 3/2
+
1 , 7/2
+
1 , and 9/2
+
1 states have been suggested