4.3 The Particle-Rotor Model in the Berggren Basis
179
width of those states is predicted to be around 0.4 MeV. For J = 1/2, the width
is even totally dominated by s 1/2 . For the J = 5/2 state of this band, the = 0
channel is blocked again; hence, its width is small.
4.3.4.3 Effects of Deformation and Configuration Mixing on E1
Electromagnetic Transition in 11 Be
One will now consider electromagnetic transitions in the particle-rotor model on the
example of electric dipole transition between loosely bound 1/2 − and 1/2 + states
in 11 Be. In the study of this transition, energies of these states must be fitted to their
experimental values. The 1/2 + and 1/2 − states are bound by about −500 keV and
−180 keV, respectively, so the 1/2 − → 1/2 + E1 transition will be enhanced due to
the important nucleon density in the asymptotic zone for both states.
Nuclear interaction in the N ∼ 8 region of nuclear chart leads to an inversion
of the 0p 1/2 and 1s 1/2 shells in 11 Be. This is directly seen in the 11 Be spectrum,
as the 1/2 + state, which is mainly a 10 Be + 1s 1/2 configuration, is more bound
than the 1/2 − state, formed primarily by a 10 Be + 0p 1/2 configuration. Such
shell inversions can be found in different regions of nuclear chart, e.g., inversion
of 1s 1/2 and 0d 3/2 happens for N ∼ 14, whereas the shells 0f 7/2 and 1p 3/2
are inverted at N ∼ 20. Due to the simple Hamiltonian of the particle-rotor
model, where most of nucleon-nucleon interactions are replaced by a deformed
Woods-Saxon potential, it is necessary to fit the depth of Woods-Saxon potential
independently in the s 1/2 and the p 1/2 neutron channels. In the Berggren ensemble,
1s 1/2 single-particle state is weakly bound (E = −0.00702 MeV) and 0p 1/2 is a
broad single-particle resonance (E = 0.00794 MeV, Γ = 0.725 keV). These two
single-particle states form an essential part of the 1/2 + and 1/2 − many-body states
of 11 Be, as they respectively correspond to the channels [ 10 Be(j r = 0)⊗ν1s 1/2 ] 1/2 +
and [ 10 Be(j r = 0) ⊗ ν0p 1/2 ] 1/2 − , where j r is the angular momentum of the 10 Be
rotor state (see Exercise VI for additional numerical applications). The calculated
energies of the 1/2 + and 1/2 − many-body states with respect to one-neutron
emission threshold are equal to −0.503 MeV and −0.184 MeV, respectively, and
closely reproduce the experimental energies of these states, which are −0.502 MeV
and −0.182 MeV, respectively [108].
Exercise VI
One will numerically calculate the B(E1) reduced transition probability
between the ground state 1/2 + and the first excited state 1/2 − of 11 Be, using
different sets of Hamiltonian parameters.
Using the default parameters given in the input file, run the particle-rotor
code in the case of 11 Be to recalculate the binding energies and B(E1) transition
probability discussed above.
Change slightly parameters of the Hamiltonian, e.g., by 10%, with respect to
their default values. Notice that, while energies depend strongly on Hamiltonian
parameters, B(E1) transition probability almost does not change in this case.
179
width of those states is predicted to be around 0.4 MeV. For J = 1/2, the width
is even totally dominated by s 1/2 . For the J = 5/2 state of this band, the = 0
channel is blocked again; hence, its width is small.
4.3.4.3 Effects of Deformation and Configuration Mixing on E1
Electromagnetic Transition in 11 Be
One will now consider electromagnetic transitions in the particle-rotor model on the
example of electric dipole transition between loosely bound 1/2 − and 1/2 + states
in 11 Be. In the study of this transition, energies of these states must be fitted to their
experimental values. The 1/2 + and 1/2 − states are bound by about −500 keV and
−180 keV, respectively, so the 1/2 − → 1/2 + E1 transition will be enhanced due to
the important nucleon density in the asymptotic zone for both states.
Nuclear interaction in the N ∼ 8 region of nuclear chart leads to an inversion
of the 0p 1/2 and 1s 1/2 shells in 11 Be. This is directly seen in the 11 Be spectrum,
as the 1/2 + state, which is mainly a 10 Be + 1s 1/2 configuration, is more bound
than the 1/2 − state, formed primarily by a 10 Be + 0p 1/2 configuration. Such
shell inversions can be found in different regions of nuclear chart, e.g., inversion
of 1s 1/2 and 0d 3/2 happens for N ∼ 14, whereas the shells 0f 7/2 and 1p 3/2
are inverted at N ∼ 20. Due to the simple Hamiltonian of the particle-rotor
model, where most of nucleon-nucleon interactions are replaced by a deformed
Woods-Saxon potential, it is necessary to fit the depth of Woods-Saxon potential
independently in the s 1/2 and the p 1/2 neutron channels. In the Berggren ensemble,
1s 1/2 single-particle state is weakly bound (E = −0.00702 MeV) and 0p 1/2 is a
broad single-particle resonance (E = 0.00794 MeV, Γ = 0.725 keV). These two
single-particle states form an essential part of the 1/2 + and 1/2 − many-body states
of 11 Be, as they respectively correspond to the channels [ 10 Be(j r = 0)⊗ν1s 1/2 ] 1/2 +
and [ 10 Be(j r = 0) ⊗ ν0p 1/2 ] 1/2 − , where j r is the angular momentum of the 10 Be
rotor state (see Exercise VI for additional numerical applications). The calculated
energies of the 1/2 + and 1/2 − many-body states with respect to one-neutron
emission threshold are equal to −0.503 MeV and −0.184 MeV, respectively, and
closely reproduce the experimental energies of these states, which are −0.502 MeV
and −0.182 MeV, respectively [108].
Exercise VI
One will numerically calculate the B(E1) reduced transition probability
between the ground state 1/2 + and the first excited state 1/2 − of 11 Be, using
different sets of Hamiltonian parameters.
Using the default parameters given in the input file, run the particle-rotor
code in the case of 11 Be to recalculate the binding energies and B(E1) transition
probability discussed above.
Change slightly parameters of the Hamiltonian, e.g., by 10%, with respect to
their default values. Notice that, while energies depend strongly on Hamiltonian
parameters, B(E1) transition probability almost does not change in this case.