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Shapecurrent orientation
Figure 2:
(Color online) Schematic illustration of relative orientations of shapes and currents
in the three antialigned states
(),
(), and
() discussed in the text.
The long (), intermediate (), and short
() principal axes of the nuclear mass distribution are indicated by thick arrows. The oddneutron () and oddproton () angular momentum
oriented along the , , or axes is shown by thin arrows. Note that in each case the
total angularmomentum alignment,
, is zero.

At variance with the eveneven parent nuclei, the antialigned configurations in oddodd daughter nuclei
are not uniquely defined.
One of the reasons, which was not fully appreciated in our previous work [16],
is related to the relative orientation of the nuclear shapes and currents associated
with the valence neutronproton pairs.
In all signaturesymmetryrestricted calculations for triaxial nuclei, such as ours, there are three antialigned
Slater determinants with the s.p. angular momenta (alignments)
of the valence protons and neutrons pointing, respectively, along
the , , or axes of the
intrinsic shape defined by means of the long (), intermediate (), and short
() principal axes of the nuclear mass distribution.
These solutions, hereafter referred to as
,
, and
, are
schematically illustrated in Fig. 2. Their properties can be
summarized as follows:
 The three solutions are not linearly independent. Their HartreeFock
(HF) binding energies may typically differ by a few hundred keV. The differences come
almost entirely from the isovector correlations in the timeodd
channel, as shown in the lower panel of Fig. 3 for a
representative example of Cl. Let us stress that these
poorlyknown correlations may significantly impact the ISB
corrections, as shown in the upper panel of Fig. 3.
 The type of the isovector timeodd correlations captured by the HF
solutions depends on the relative orientation of the nucleonic
currents with respect to the nuclear shapes. Solutions oriented
perpendicular to the long axis,
and
, are usually similar to one another
(they yield identical correlations for axial systems) and
differ from
, oriented parallel to the long axis,
which captures more correlations due to
the currentcurrent timeodd interactions.
 The three
states projected from the
,
, and
Slater determinants differ in energy by only a few
tens of keV, see the lower panel of Fig. 3. Hence,
energywise, they represent the same physical solution, differing only slightly due
to the polarization effects originating from different components of the
timeodd isovector fields. However,
since these correlations are completely absent in the eveneven
parent nuclei, they strongly impact the calculated
. The largest differences in
have been
obtained for and systems, see Fig. 3 and
Tables 2 and 3.
 Symmetryunrestricted calculations always converge to the
signaturesymmetryconserving solution
which,
rather surprisingly, appears to be energetically unfavored (except
for F). In spite of our persistent
efforts, no selfconsistent tiltedaxis solutions have been found.
Figure 3:
Upper panel: the ISB corrections
for the
superallowed decays
Ar
Cl (open circles) and
Cl
S (full circles) determined for the
shapecurrent orientations , , and depicted schematically in Fig. 2.
Lower panel: differences between the energies of the
and configurations, and the configuration in Cl. Full
triangles correspond to the total HF
energies and open triangles correspond to
contributions from the timeodd isovector
channel. Full dots show the total energy differences obtained for the
angularmomentum and isospinprojected states.

Next: Nearly degenerate orbitals
Up: ISB corrections to the
Previous: ISB corrections to the
Jacek Dobaczewski
20121019