174
4 Two-Particle Systems in the Berggren Basis
whether such a suppression of collective rotation could also be expected in nuclear
halo systems.
4.3.4.1 Deformed Proton Emitters: The Example of 141 Ho
The particle-rotor model has many advantages for the study of deformed proton emitters, developed in the nonadiabatic approach and applied to the axially
deformed case for that matter [88, 89, 114–116]. Firstly, medium-heavy and heavy
nuclei, especially when they are deformed, cannot be studied with standard shell
model due the very large dimensions of the matrix involved in their description. In
fact, mean-field based methods are currently the only practical frameworks which
can be applied in these regions of the nuclear chart. Moreover, as proton emitters
typically consist of an unbound proton weakly coupled to a well bound core, the
particle-rotor model recaptures the essential physical features of proton emitters
and can be a good theoretical tool for their description. Indeed, deformation degrees
of freedom in this model are exactly taken into account, and pairing correlations act
only in well-bound core, so that they can be taken into account in the renormalized
effective average potential acting on the emitted proton. Added to that, as said
in Sect. 2.6.8, the proton emission widths of proton emitters are very small, of
the order of 10 −22 MeV, with associated lifetimes being of a few milliseconds.
Consequently, the lifetimes of proton emitters are evaluated using Eq. (2.196), where
the u(k, r) state must be replaced by the amplitude of the considered channel. For
this, the asymptotes of the wave functions of proton emitters must be very precisely
evaluated, which is possible in the one-body picture provided by the particle-rotor
model.
The Berggren basis has been applied to the description of the deformed proton
emitter 141 Ho (see Ref. [116] for details). For this, a Hamiltonian generated from
a deformed Woods-Saxon potential, to which a Coulomb potential is added, was
diagonalized in the Berggren basis. Proton emission widths were calculated either
from the imaginary parts of eigenenergies, when width is larger than 1 keV typically,
while the approximate expression of Eq. (2.196), generalized to the coupled-channel
case, was used when widths were much smaller than 1 keV. Results are presented in
Table 4.2.
When proton-emitting states have very narrow widths, coupled-channel calculations effected in coordinate space, diagonalization in harmonic oscillator basis,
and Berggren basis diagonalization provide with similar results, except for a
slightly smaller width for the 7/2 − ground state obtained in the Berggren basis
diagonalization. Indeed, proton one-body states have a quasi-bound structure and
emission widths are calculated from the continuity equation (see Eq. (2.196). The
narrow character of proton one-body states implies that their wave functions vary
smoothly on the real r-axis, so that all used methods are numerically stable. As
could be expected, harmonic oscillator diagonalization can no longer provide with
converged results once the width reaches 10–20 keV because wave functions start to
significantly oscillate in the asymptotic region and, therefore, the wave function
asymptotes cannot be calculated precisely. Consequently, Eq. (2.196) cannot be
used to calculate the proton emission widths. Conversely, the energies and widths
4 Two-Particle Systems in the Berggren Basis
whether such a suppression of collective rotation could also be expected in nuclear
halo systems.
4.3.4.1 Deformed Proton Emitters: The Example of 141 Ho
The particle-rotor model has many advantages for the study of deformed proton emitters, developed in the nonadiabatic approach and applied to the axially
deformed case for that matter [88, 89, 114–116]. Firstly, medium-heavy and heavy
nuclei, especially when they are deformed, cannot be studied with standard shell
model due the very large dimensions of the matrix involved in their description. In
fact, mean-field based methods are currently the only practical frameworks which
can be applied in these regions of the nuclear chart. Moreover, as proton emitters
typically consist of an unbound proton weakly coupled to a well bound core, the
particle-rotor model recaptures the essential physical features of proton emitters
and can be a good theoretical tool for their description. Indeed, deformation degrees
of freedom in this model are exactly taken into account, and pairing correlations act
only in well-bound core, so that they can be taken into account in the renormalized
effective average potential acting on the emitted proton. Added to that, as said
in Sect. 2.6.8, the proton emission widths of proton emitters are very small, of
the order of 10 −22 MeV, with associated lifetimes being of a few milliseconds.
Consequently, the lifetimes of proton emitters are evaluated using Eq. (2.196), where
the u(k, r) state must be replaced by the amplitude of the considered channel. For
this, the asymptotes of the wave functions of proton emitters must be very precisely
evaluated, which is possible in the one-body picture provided by the particle-rotor
model.
The Berggren basis has been applied to the description of the deformed proton
emitter 141 Ho (see Ref. [116] for details). For this, a Hamiltonian generated from
a deformed Woods-Saxon potential, to which a Coulomb potential is added, was
diagonalized in the Berggren basis. Proton emission widths were calculated either
from the imaginary parts of eigenenergies, when width is larger than 1 keV typically,
while the approximate expression of Eq. (2.196), generalized to the coupled-channel
case, was used when widths were much smaller than 1 keV. Results are presented in
Table 4.2.
When proton-emitting states have very narrow widths, coupled-channel calculations effected in coordinate space, diagonalization in harmonic oscillator basis,
and Berggren basis diagonalization provide with similar results, except for a
slightly smaller width for the 7/2 − ground state obtained in the Berggren basis
diagonalization. Indeed, proton one-body states have a quasi-bound structure and
emission widths are calculated from the continuity equation (see Eq. (2.196). The
narrow character of proton one-body states implies that their wave functions vary
smoothly on the real r-axis, so that all used methods are numerically stable. As
could be expected, harmonic oscillator diagonalization can no longer provide with
converged results once the width reaches 10–20 keV because wave functions start to
significantly oscillate in the asymptotic region and, therefore, the wave function
asymptotes cannot be calculated precisely. Consequently, Eq. (2.196) cannot be
used to calculate the proton emission widths. Conversely, the energies and widths