Introduction

Since the original 1997 work of Frauendorf and Meng [1], the phenomenon of chiral rotation in atomic nuclei attracts quite a significant attention. The effect is expected to occur in nuclei having stable triaxial deformation, and in which there are a few high-$j$ valence particles and a few high-$j$ valence holes. The former drive the nucleus towards prolate, and the latter towards oblate shapes. The interplay of these opposite tendencies may favor a stable triaxial deformation. For such a shape, the valence particles and holes align their angular momenta along the short and long axes of the density distribution, respectively. However, the nuclear-bulk moment of inertia with respect to the medium axis is the largest, which favors collective rotation about that axis. Thus, the particle, hole, and collective angular momentum vectors are aplanar, and may form either a left-handed or a right-handed set. In this way, the two enantiomeric forms may give rise to pairs of rotational bands, which are called chiral doublets. It is expected that the energy splitting between the partners in such doublets is very small, and in fact the authors of Ref. [2], who have analyzed the experimental results, have used essentially the argument 'by elimination' - the bands were suggested to be chiral partners mainly because their properties could not be explained within other scenarios used by the authors.

The first doublet band, later reinterpreted as chiral [1], was found in 1996 by Petrache et al. in $^{134}$Pr [3]. Now, about 15 candidate chiral doublet bands are known in the $A\approx130$ region, and about 10 in the $A\approx100$ region. Most bands in the $A\approx130$ nuclei are assigned to the simplest chiral configuration, with one proton particle on the $\pi h_{11/2}$ orbital, and one neutron hole on the $\nu h_{11/2}$ orbital. Configurations for $A\approx100$ nuclei usually involve one $\pi g_{9/2}$ proton hole and one $\nu h_{11/2}$ neutron particle orbitals. A few cases with more than one active particle or hole were also found [4,5,6]. So far, experimental information about absolute B(E2) and B(M1) values for transitions within the observed bands is available only for $^{128}$Cs and $^{132}$La, from recent lifetime measurements by Grodner [7,8] and Srebrny [9], and collaborators.

On the theoretical side, chiral rotation has been extensively studied by using various versions of the Particle-Rotor Model (PRM); see, e.g., Refs. [12,10,11]. In PRM, the nucleus is represented by the valence particles and holes coupled via the quadrupole-quadrupole interaction to a rotator, often described within the Davydov-Filippov model [13] with moments of inertia given by the irrotational-flow formula [14]. However, the main concept of rotational chirality summarized above, came from considerations within the Frauendorf's mean-field Tilted-Axis-Cranking (TAC) model [15,16,17], which is used in parallel with the PRM. That model is a straightforward generalization of the standard cranking approach to situations where the axis of rotation does not coincide with any principal axis of the mass distribution. First chiral TAC solution for a fixed triaxial shape was obtained already in 1987 by Frisk and Bengtsson [18], and then in 1997 by Frauendorf and Meng [1,19]. In 2000, Dimitrov, Frauendorf and Dönau [20] obtained first such solution by minimizing the energy over deformation. From the TAC results, Frauendorf [21,22], Dimitrov [23] and collaborators reproduced or predicted the chiral rotation in several nuclei from the 130 and 100 mass regions, and also in $^{79}$Br, $^{162}$Tm, and $^{188}$Ir. Other authors analyzed many experimental data within that approach [2,4,5,6,24,25], and generally a good correlation was found between the existence of chiral TAC solutions and the appearance of candidate chiral bands.

Up to now, all the TAC calculations for chiral rotation were based on simple model single-particle (s.p.) potentials [26] combined either with the Pairing + Quadrupole-Quadrupole model (PQTAC), or the Strutinsky Shell Correction (SCTAC) [16]. A more fundamental description requires self-consistent methods, which would provide a strong test of the stability of the proposed chiral configurations with respect to the core degrees of freedom. Self-consistent methods are also necessary to take into account all kinds of polarization of the core by the valence particles and full minimization of the underlying energies with respect to all deformation degrees of freedom, including deformations of the current and spin distributions. Application of self-consistent methods to the description of chiral rotation is the subject of the present paper.

Our study concerns four $N=75$ isotones, $^{130}$Cs, $^{132}$La, $^{134}$Pr, and $^{136}$Pm, which are the first nuclei in which candidate chiral bands were systematically studied [2]. We used the Hartree-Fock (HF) method with the Skyrme effective interaction. The results were obtained for two Skyrme parameter sets, SLy4 [27] and SKM* [28], and the role of terms in the mean field that are odd under the time reversal was examined. Calculations were carried out by using a new version of the code HFODD (v2.05c) [29,30,31], which was specially constructed by the authors for the purpose of the present study. From among the considered four isotones, self-consistent chiral solutions were obtained in $^{132}$La. A brief report on the results obtained in $^{132}$La was given in Ref. [32].

The paper is organized as follows. In Section 2 we discuss some characteristic aspects of the TAC calculations within the self-consistent framework. Section 3.1 briefly recalls previous studies on chiral rotation in the concerned four $N$=75 isotones. Section 3.2 describes all technical details of the present calculations - in particular the way in which the role of time-odd nucleonic densities was examined. Energy minima obtained for non-rotating states are listed in Section 3.3. In Section 3.4, rotational properties of the valence nucleons and of the core are examined within standard Principal-Axis Cranking (PAC). In Section 3.5 we solve a simple classical model of chiral rotation and show that such a rotation cannot exist below a certain critical angular frequency. The HF solutions for planar and chiral rotation are presented in Sections 3.6 and 3.7, respectively. In Section 3.8 we demonstrate that our results obtained for the three-dimensional rotation can actually be represented as a sum of three independent one-dimensional rotations about the principal axes. The values obtained for the critical frequency and the agreement of our results with experimental level energies are discussed in Section 4. In Appendix A, we study response of the s.p. angular momenta to three-dimensional rotation, and introduce the notions of soft and stiff alignments.