The main point left for the discussion concerns the values of the critical frequency: where do they come from, whether they depend on the Skyrme force used, how they are altered by the different time-odd terms of the functional, how they would be influenced by the inclusion of pairing, and what their relation to the experimentally observed bands is. In this Section we give some remarks on these and related topics.
Table 1 summarizes the values of the classical-model
parameters,
,
,
,
, and
, extracted
from the HF PAC results in Section 3.4, of the classical
critical frequency,
, calculated from
Eq. (22), and of the critical frequency,
, obtained from the HF TAC calculations as defined in
Section 3.7. In addition, the Table gives the
corresponding values of critical spins,
and
, respectively.
First of all, it can be seen from the examples of Cs and
La that the SLy4 and SkM* forces give quite similar values
for all the concerned quantities. The differences are not larger than
variations within one force due to taking into account different
time-odd fields. For all the four isotones in question, switching on
the fields
increases all the moments of inertia, but particularly
, with respect to the case
. This causes a decrease in
, but the corresponding
does
not change much, because
is larger. Switching on the fields
results in values of
between those obtained for the cases
and
. The critical frequency always becomes higher than for
the
fields, and the resulting critical spin is always the highest
among all the examined sets of time-odd fields. These variations in
are of the order of a few spin units.
In La, where the HF chiral solutions were found in
Section 3.7, values of the HF critical frequency and
spin,
and
, are slightly higher
than the classical estimates,
and
. This can be understood on the basis of results of
the perturbative search for the HF chiral solutions, performed in
Section 3.7. As illustrated in Fig. 9,
the perturbative ratio
, representing the moment of
inertia with respect to the medium axis,
, slightly falls with
rotational frequency, which, according to Eq. (22),
causes the rise of the critical frequency. When the time-odd fields
are included, values of
and
vary
similarly to those of
and
.
The HF method used in the present study does not take into account the pair correlations. In order to include pairing in self-consistent calculations one would have to apply the Hartree-Fock-Bogolyubov (HFB) method in the TAC regime with two quasiparticle states blocked. Present codes do not allow for such a study, and a systematic investigation of pairing has to be left for future analyses. Convergence in presence of blocked states may be more difficult to obtain. Pairing may soften the quasi-particle alignments and render the solution more prone to deformation changes. It remains to be seen how this would influence the pure single-particle picture that we have examined so far. Supposing that the HFB TAC results can be inferred from the HFB PAC calculations through a determination of the parameters of the classical model, one could gain some information on pairing effects by analyzing the HFB PAC moments of inertia and alignments.
The critical frequency represents the transition point between planar
and chiral rotation. The notion of critical frequency was first
introduced in Ref. [32], where the expression
(22) for its value was derived from the simple
classical model presented in Section 3.5. However, the
occurrence of a transition from planar to chiral regime in the
structure of chiral bands was clear already from earlier
investigations. Both within the PRM and TAC, this effect was
obtained numerically but left without comment. This transition is abrupt in
the semi-classical cranking model, but may be rather smooth in real
nuclei. This is because the angular momentum vector oscillates about
the planar equilibrium below
, which corresponds
to non-uniform classical rotations [23], while above
, it can still tunnel between the left and right
chiral minima, which represents chiral vibrations [2].
Because of these reasons, mean-field methods can provide quantitative description of the bands in question only in the low-spin, planar regime, where there is only one minimum. In the chiral region, the mean-field approach does not take into account the interaction between the left and right minima, which are exactly degenerate in energy, and the experimentally observed energy splitting between the chiral partners cannot be calculated. It is argued in the literature that the mean-field chiral solution can be viewed as a kind of average of the two partner bands, and thus mean trends can be compared. One can also speculate about the value of the critical frequency or spin. Description of the transition region is an interesting topic for study invoking techniques beyond the mean field, like the Generating-Coordinate Method.
Figure 1 gives a comparison of experimental and
calculated energies in La. Full symbols denote the
experimental yrast (circles), B1 (squares), and B3 (diamonds) bands,
discussed in Section 3.2. Open circles and black
crosses represent the HF TAC planar and chiral solutions,
respectively. Their classical counterparts are marked with dashed
and solid lines. The HF results for the critical spin,
, are rather high as compared to the
spin range, in which the bands B1 and B3 are observed. Yet, the
classical estimate,
, evaluated for the
Total-Routhian-Surface (TRS)
PAC results is already below that range [32]. This means that the
inclusion of pairing in the calculations may be important for correct
interpretation of the data. However, from the closeness of
experimental spins to the possible values of
one can
suppose that, whichever of the bands B1 and B3 could eventually be interpreted as
the chiral partner of the yrast band, the concerned spin region may
actually represent the transition between planar and chiral rotation.
Although no HF chiral solutions were found in
Cs, similar
conclusions can be drawn for that isotone on the basis of the
classical estimates of the critical spin. In case of
Pr and
Pm it is not clear whether the oblate or triaxial
solutions should be taken for comparison with the bands observed in
those nuclei.
At low spins in La, where the supposed chiral partners have
not been observed, the yrast band is well reproduced by the HF planar
solutions, particularly with the time-odd fields included. This is
consistent with the supposed planar character of rotation at low
spins. Roughly at the spin where the chiral partners commence to be
visible, the yrast band significantly changes its behavior, which can
be attributed to entering into the chiral regime. In
this spin region the HF results agree semi-quantitatively with
experimental energies.