Summary

To conclude, the first Skyrme-Hartree-Fock (Skyrme-HF) calculations with the Tilted-Axis Cranking were performed in $^{130}$Cs, $^{132}$La, $^{134}$Pr, and $^{136}$Pm in the search for self-consistent solutions corresponding to nuclear chiral rotation. Only the configuration $\pi{}h_{11/2}^1~\nu{}h_{11/2}^{-1}$, earlier assigned to the observed candidate chiral bands in those isotones, was considered. Two Skyrme parametrizations, SLy4 and SkM*, were used. Terms depending on time-odd nucleonic densities were either kept or excluded from the Skyrme energy functional.

From the Principal-Axis-Cranking analysis it was concluded that the system in question can be modeled by two gyroscopes, representing the valence particle and hole, with spins stiffly aligned with the short and long axes of a triaxial rigid rotor, which stands for the core. Such a model was analyzed in the classical framework. This led to an important conclusion that chiral rotation can exist only above a critical angular frequency, given by a simple expression.

The HF solutions representing planar rotation were found in all the considered $N=75$ isotones, and chiral solutions were obtained in $^{132}$La. These solutions provide the first proof based on fully self-consistent methods that nuclear rotation can attain a chiral character. In all cases, the self-consistent solutions agree surprisingly well with the results of the classical model, which means that the model faithfully represents salient features of the examined phenomenon.

It was found that the time-odd densities in the energy functional have no qualitative influence on the results, and change mainly the moments of inertia. The HF values of the critical frequency are rather high as compared to the spin range in which the candidate chiral bands were observed in $^{132}$La. The HF energies agree satisfactorily with experiment only in the low-spin parts of the bands, where the rotation is supposed to be planar. In the chiral regime, the mean field is unable to reproduce the data precisely, and the agreement is only qualitative. It seems, also, that the experimentally observed bands actually represent a transition from planar to chiral rotation.

The criteria used so far when attributing the chirality-partnership interpretation to the experimentally found rotational bands that are based on the 'small energy splitting' argument are clearly unsatisfactory on a long run. Numerous superdeformed band studies (followed by the normal-deformation studies) have shown that different intrinsic configurations and thus strictly speaking different-shape nuclei may manifest nearly identical rotational bands. It is therefore necessary, to provide the experimental evidence going beyond just the energy measurements, first of all the accompanying electromagnetic-transition information. This could help excluding the mistake of interpreting e.g. the shape coexistence phenomena in terms of chirality - without providing extra sufficient conditions. In this paper we were not able to propose any clear-cut necessary-and-sufficient condition criteria to attribute the chirality label to the experimental bands either. However, we do believe that the concept of the critical frequency discussed in detail in this paper provides a useful tool in interpreting the experimental results. The very fact that in some nuclei the self-consistent HF calculations do provide chiral solutions is highly non-trivial and very encouraging message in this field of research.

We would like to thank H. Flocard, J. Bartel, J. Styczen, and W. Satua for valuable discussions. This work was supported in part by the Polish Committee for Scientific Research (KBN) under Contract No. 1 P03B 059 27, by the Foundation for Polish Science (FNP), and by the French-Polish integrated actions program POLONIUM.