Signature-separation sensitive to the time-odd channels

A different situation takes place in configurations where both
a neutron *and* a proton occupy unbalanced-signature states.
In particular, four near-yrast 3^{1}3^{1} configurations,
shown in Figs. 6(a) and 7, group into
two nearly degenerate pairs of bands having the same signature.
Indeed, the *r*=+1 bands,
3^{1}_{-}3^{1}_{-} and
3^{1}_{+}3^{1}_{+}, are
very close to one another, with the latter one lying
slightly lower in energy, in accordance with the sign of the
small signature splitting of the high-*K* [202]5/2(*r*=)
orbitals. Note that the sudden deviation from regular
behavior, seen in the latter band at
=1.0-1.4MeV,
is due to a poor HF convergence related to strong mixing of
orbitals, see discussion in Sec. 3.4.

The second pair of degenerate 3^{1}3^{1} bands corresponds to the
signature *r*=-1, and is composed of the
3^{1}_{-}3^{1}_{+} and
3^{1}_{+}3^{1}_{-} configurations, that are the mirror partners of
one another in terms of the isospin. Therefore, irrespectively of
the small signature splitting of the [202]5/2(*r*=)
orbitals,
these bands are almost perfectly degenerate. Again, due to
the interactions between orbitals, at *I*10 and
*I*14 one observes small irregularities reflecting
poor HF convergence.

A remarkable feature obtained in the HF calculations is the
fact that the pair of bands just mentioned, with
*r*=-1, lies about 2MeV above the *r*=+1 pair.
As opposed to the standard signature slitting effect, we my
call these bands the signature-separated bands.
Such a separation could not have been obtained in a
phenomenological cranking model,
i.e., the one using the Woods-Saxon or Nilsson potentials, because
there the single-particle degeneracies immediately imply
degeneracies of bands in many-body systems. Indeed, by
putting one neutron and one proton into weakly split and
non-interacting [202]5/2(*r*=)
pair of orbitals, one should have
obtained all the four 3^{1}3^{1} bands strongly degenerate. Note
that the separation of the *r*=-1 and *r*=+1 pairs of bands
cannot be due to the deformation effect, because
deformations of the four bands are very similar, see
Fig. 8(a).

Strong separation of pairs of signature-degenerate bands
results [40,41] from the self-consistent effects related to the
time-odd components [34] in the HF mean fields.
Odd particles induce the time-odd mean fields in odd and
odd-odd nuclei, and similarly, odd particles
in signature-unbalanced states induce the time-odd mean fields
for certain configurations of even-even nuclei. In
particular, when a neutron is put into the [202]5/2(*r*=+*i*)
orbital, it creates, through the strong neutron-proton
interaction which is inherent to any effective nuclear
force, e.g., to the Skyrme force, a strong attractive
component in the proton mean-field corresponding to the same, i.e.
*r*=+*i* symmetry. Therefore, when the proton occupying the
[202]5/2(*r*=+*i*) orbital is
put into such a mean field, the total energy is
significantly lowered. Of course, exactly the same
mechanism applies for two particles occupying the [202]5/2(*r*=-*i*)
orbitals. The proton mean field,
generated by the [202]5/2(*r*=+*i*) neutron, does not influence the states
of the *r*=-*i* symmetry, and therefore, adding then the [202]5/2(*r*=-*i*)
proton does not influence the total energy. Hence, here the
*r*=-1 bands are not affected by the time-odd interactions,
(i.e., the interactions which give the time-odd mean fields through
the HF averaging),
while the *r*=+1 bands are significantly affected, and
acquire an additional binding.

Obviously, the magnitude of the separation between the *r*=-1
and *r*=+1 bands crucially depends on the interaction
strengths in time-odd channels. Unfortunately, the coupling
constants corresponding to these channels [34] are not restricted
by typical ground-state observables (masses, radii, etc.),
that serve as experimental benchmarks with respect to which
the force parameters are adjusted. Therefore, high spin
effects, like the aforementioned
*r*=-1 *vs.* *r*=+1 separation,
that are manifestly sensitive to these unexplored
channels of the interaction, could provide an
extremely important information pertaining to basic properties of
nuclear effective forces. Note that in ^{32}S, the structure of the
yrast line dramatically depends on the strength of the interaction
in these channels, because the 3^{1}3^{1} bands become yrast
mostly due to the time-odd interaction.

By looking at similar quartets of bands, e.g., those
corresponding to the 3^{3}3^{3} and 3^{1}3^{3} configurations,
Figs. 6(a) and (c), respectively, one sees that the
strength of interactions in the time-odd channels depends
on the structure of the underlying orbitals. The signature splitting
of the [211]1/2(*r*=)
orbitals obscures the picture a little
because it gives the splitting of the
3^{3}_{+}3^{3}_{+} and
3^{3}_{-}3^{3}_{-} configurations, however, the centroid of
these two configurations lies visibly below the perfectly
degenerate pair of mirror partners
3^{3}_{-}3^{3}_{+} and
3^{3}_{+}3^{3}_{-}. Hence, within the [211]1/2(*r*=)
orbitals, the
time-odd interaction is significantly weaker. Finally,
there seems to be no such a non-diagonal interaction between
the [211]1/2(*r*=)
and [202]5/2(*r*=)
orbitals, because the
degeneracy pattern of the 3^{1}3^{3} orbitals is completely
different. Indeed, the standard, weakly split, two
signature pairs appear, the lower one composed of the
3^{1}_{+}3^{3}_{+} and
3^{1}_{-}3^{3}_{+} configurations, and the higher
one composed of the
3^{1}_{+}3^{3}_{-} and
3^{1}_{-}3^{3}_{-}configurations, in accordance with the sign of the signature
splitting of the [211]1/2(*r*=)
orbitals.

Results presented in this section indicate that the properly selected
high-spin structures in a SD nucleus reflect the properties
of the effective interaction in the time-odd channel.
Quantitatively, in the restricted set of orbitals
considered in ^{32}S, the time-odd interaction amounts to
an attractive force which acts between
protons and neutrons occupying *the same* orbitals,
i.e., orbitals having the same quantum numbers. Therefore,
the discussed interaction channel has several features of
the *T*=0 pairing interaction, although obviously the
effects discussed here are not related to any collective
pairing channels, but rather pertain to interactions in the
particle-hole channel.