Discussion

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, $\mathcal{J}_s$, $\mathcal{J}_m$, $\mathcal{J}_l$, $s_s$, and $s_l$, extracted from the HF PAC results in Section 3.4, of the classical critical frequency, $\omega _{\text{crit}}^{\text{clas}}$, calculated from Eq. (22), and of the critical frequency, $\omega _{\text{crit}}^{\text{HF}}$, obtained from the HF TAC calculations as defined in Section 3.7. In addition, the Table gives the corresponding values of critical spins, $I_{\text{crit}}^{\text{clas}}$ and $I_{\text{crit}}^{\text{HF}}$, respectively.

First of all, it can be seen from the examples of $^{130}$Cs and $^{132}$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 $G$ increases all the moments of inertia, but particularly $\mathcal{J}_m$, with respect to the case $N$. This causes a decrease in $\omega _{\text{crit}}^{\text{clas}}$, but the corresponding $I_{\text{crit}}^{\text{clas}}$ does not change much, because $\mathcal{J}_m$ is larger. Switching on the fields $T$ results in values of $\mathcal{J}_m$ between those obtained for the cases $N$ and $G$. The critical frequency always becomes higher than for the $G$ fields, and the resulting critical spin is always the highest among all the examined sets of time-odd fields. These variations in $I_{\text{crit}}^{\text{clas}}$ are of the order of a few spin units.

In $^{132}$La, where the HF chiral solutions were found in Section 3.7, values of the HF critical frequency and spin, $\omega _{\text{crit}}^{\text{HF}}$ and $I_{\text{crit}}^{\text{HF}}$, are slightly higher than the classical estimates, $\omega _{\text{crit}}^{\text{clas}}$ and $I_{\text{crit}}^{\text{clas}}$. 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 $I_m/\omega _m$, representing the moment of inertia with respect to the medium axis, $\mathcal{J}_m$, 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 $\omega _{\text{crit}}^{\text{HF}}$ and $I_{\text{crit}}^{\text{HF}}$ vary similarly to those of $\omega _{\text{crit}}^{\text{clas}}$ and $I_{\text{crit}}^{\text{clas}}$.

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 $\omega_{\rm crit}$, which corresponds to non-uniform classical rotations [23], while above $\omega_{\rm crit}$, 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 $^{132}$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, $I_{\text{crit}}^{\text{HF}}=15.9-18.5\hbar$, are rather high as compared to the spin range, in which the bands B1 and B3 are observed. Yet, the classical estimate, $I_{\text{crit}}^{\text{clas}}=9.2\hbar$, 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 $I_{\text{crit}}$ 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 $^{130}$Cs, similar conclusions can be drawn for that isotone on the basis of the classical estimates of the critical spin. In case of $^{134}$Pr and $^{136}$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 $^{132}$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.