The present study is limited to the simplest chiral s.p. configuration,
, which has been
assigned in the literature to the candidate chiral bands observed in
the
isotones. In order to check whether the existence of the HF
chiral solutions is not a particular feature of one Skyrme parameter
set, all calculations were repeated with two parametrizations, SLy4
[27] and SKM* [28]. The parity was kept as a
conserved symmetry, and the pairing correlations were not included.
In forming the chiral geometry, orientations of several angular-momentum vectors play a crucial role. Their behavior depends, among others, on the nucleonic densities which are odd under time reversal, like the current and spin densities. Therefore, taking those densities and corresponding terms in the energy density functional into account seems important for the microscopic description of the chiral rotation. Self-consistent methods are best suited for such a task, and investigating the role of the time-odd densities was one of our priorities in the present study.
The Skyrme energy density functional depends on time-even and time-odd nucleonic
densities with coupling constants ,
,
,
,
(10 time-even terms) and with coupling constants
,
,
,
,
(10 time-odd
terms) [42]. The index
denotes the isoscalar and isovector
parts. In standard parametrizations, which are used in the present work,
and
additionally depend on the isoscalar particle density. If
the assumption of the local gauge invariance is made [42], there are
several following relations between the time-even and time-odd coupling
constants,
In the present calculations, the coupling constants of the time-even
terms for
were always set to zero, like in the original fits of the
forces SLy4 and SkM*. In order to conform to the local gauge invariance, whose
consequence is Eq. (4), the coupling constants
of the
time-odd terms were set to zero, too. Apart from
, all other time-even
coupling constants were taken as they come from the parameters of the Skyrme
force.
To examine the role of the time-odd densities, we have performed three
variants of calculations, throughout the text denoted as ,
, and
,
and defined in the following way:
The calculations were carried out by using the code HFODD (v2.05c) [29,30,31]. The code expands the s.p. wave-functions onto the HO basis, and uses the iterative method to solve the HF equations. Twelve entire spherical HO shells were included in the basis. It has been verified that increasing this number up to 16, changes the quantities important for the present considerations (deformation, moments of inertia, alignments etc.) by less than 1%.