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-
valence particles and a few high-
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 Pr
[3]. Now, about 15 candidate chiral doublet bands are known in
the
region, and about 10 in the
region.
Most bands in the
nuclei are assigned to the simplest
chiral configuration, with one proton particle on the
orbital, and one neutron hole on the
orbital.
Configurations for
nuclei usually involve one
proton
hole and one
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
Cs and
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
Br,
Tm, and
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 isotones,
Cs,
La,
Pr, and
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
La. A brief report on the results obtained in
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 =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.