FigSHE04a.eps FigSHE04b.eps |
In Fig. 4, calculated spectra of low-lying
band-heads in Bk and
Cf are shown along with the
experimental data. In these results, the spectra were obtained
by individually blocking relevant quasiparticle orbitals
and then plotting differences of total many-body energies
of obtained nucleonic configurations with respect of the
total energy of the ground state, that is,
they are not at all equivalent to quasiparticle energies understood as
eigenvalues of the HFB Hamiltonian. The ground states were identified as
nucleonic configurations with the lowest total energies.
Nuclear configurations of deformed odd nuclei (one-quasiparticle
configurations) were labelled by means of the standard asymptotic quantum
numbers
(Nilsson quantum numbers) that correspond to the dominant
component in the wave function of the blocked quasiparticle state.
We used the convention of plotting positive (negative) values for the excitation energies of quasiparticle configurations that correspond to blocked quasiparticle states having norms of the second HFB components smaller (larger) than 1/2 before blocking. In this way, the states that are predominantly of a particle (hole) character appear above (below) zero energy. Moreover, these states are always plotted relatively to the ground-state; thus the ground state is plotted identically at the value of zero energy. This convention facilitates the comparison of fully self-consistent results with the Nilsson diagrams. For experimental quasiparticle configurations, we follow the assignments of particle/hole character as presented in Figs. 34 and 35 of Ref. [68].
In these two odd nuclei, prominent intruder configurations correspond to the proton 7/2[633] and neutron 11/2[725] orbitals. In Ref. [21], these two orbitals were used as benchmark states to adjust strengths of the spin-orbit interactions. We see that without such an adjustment, none of the studied standard EDFs places them at the right position. The ground-state proton 7/2[633] orbital, which experimentally is almost degenerate with the 3/2[521] orbital, for Skyrme and Gogny EDFs appears about 500keV above the ground state and for covariant EDFs about 200keV below the ground state, with the calculated ground states corresponding the 3/2[521] orbital (or 7/2[514] for the covariant EDF NL3*).
The neutron 11/2[725] orbital, for covariant, Gogny, and SLy4 EDFs, appears too high and for UNEDF2 EDF too low above its experimental position with respect to the ground-state 1/2[620] orbital. On the one hand, one can say that on the absolute scale these deficiencies are not large. On the other hand, they may point to slightly incorrect positions of spherical intruder orbitals, from which one would like to infer the shell structure of as yet not-reached superheavy nuclei. This analysis shows that detailed structure of very heavy deformed nuclei may depend on extremely fine details of the present-day theoretical models, which very well may be far beyond any reasonable possibility of adjusting them precisely enough to available experimental data.
Similarly as in the analysis presented in Ref. [21], as an
attempt to improve the agreement with the experimental values, we
have considered variations of the spin-orbit parameter of
the Gogny EDF D1S that could influence relative positions of
intruder states. Increasing
from its nominal value of
130MeVfm
reduces the excitation energy of the
state
while it increases the excitation energy of the
state in
Cf. These changes improve the agreement with experimental
data for larger values of
. In the
Bk case, the
goes down in excitation energy as
increases, while
the
and
levels go up. As in the
Cf case,
the comparison with experiment seems to favor larger values of
. This, however, has to be contrasted with the analysis
of the shell structure of heavy spherical nuclei like
Pb, which
usually calls for weaker spin-orbit interaction [25,63,65].
However, it is necessary to recognize that the studies
restricted to spin-orbit potential may have internal limitations that
come from the fact that possible deficiencies in the description of
the energies of the single-particle states emerging from the central
potential, as for example those inferred in Ref. [40], are ignored. The fact that standard Skyrme functionals
provide better description of the single-particle states in the
nuclei than the ones with the strength of spin-orbit
interaction adjusted to experimental data in nobelium region
[64] may be related to such limitation.
FigSHE05a.eps FigSHE05b.eps |
To explore the sensitivity of the results to the amount of pairing
correlations in the system, for the Gogny EDFs we performed calculations where the
pairing strengths of protons and neutrons were multiplied by factors
and
, respectively. The first noticeable fact is
that increasing the neutron pairing strength does not influence in
a significant way the spectrum of
Bk (odd
) as it also happens
when increasing the proton pairing strength in
Cf. Increasing
reduces the excitation energy of all levels except for the
state that remains more or less constant. The comparison
with experimental spectra seems to favor larger proton pairing
correlations. In the
Cf case, all the levels except the
lowest
decrease their excitation energy with increasing
pairing strength. As in the
Bk case, increasing the pairing
correlations in
Cf improves the agreement with experimental
spectra. The same behavior was observed during the readjustment of the pairing strength for SLy4. However, on should keep in mind that pairing strengths are
predominantly defined by the odd-even mass staggering, see
Sect. 3.4 below.
Jacek Dobaczewski 2015-08-21