A consistency check: Superdeformed rotational bands

When a nucleus rotates rapidly, there appear strong current and spin one-body densities along with the usual particle densities that characterize stationary (time-even) states. The time-odd densities are at the origin of strong time-odd mean fields. There are already many self-consistent studies of high-spin states available; see, e.g., reviews in Refs. [12,65,66,67]. The role and significance of the time-odd mean-field terms, however, has not been carefully studied. Basic features of high-spin states can often be well described by models that use phenomenological mean fields of the Woods-Saxon or Nilsson type, where no time-odd terms are explicitly present in the one-body potential. (The time-odd densities are, however, present there through the time-odd cranking term.) For the Gogny interaction [59], or within the standard RMF models [12], they cannot be independently modified; the Gogny interaction is defined as a two-body force (where the time-odd terms show up as exchange terms), while all time-odd terms appearing in standard RMF models are fixed by Lorentz invariance. Within the Skyrme framework, the time-odd terms in superdeformed rotational states were analyzed in an exploratory way in Refs. [3,20].

Unlike the GT response, rotational bands are influenced by both
isoscalar and isovector time-odd channels of the effective
interaction. In fact, the large effects of time-odd coupling constants
found in
[3] are mainly due to the isoscalar channel; the
isovector channel induces corrections that are smaller, though
non-negligible. The SkO' Skyrme parameterization,
which we use for GT calculations,
is unstable when the original parameters from
Eq. (34) are used in the isoscalar spin channel because
*g*_{0} < -1 (a fact that is related
to the unusually high value of *g*_{1} in Table 1).
This leads to unphysical ferromagnetic solutions where all
spins align when the nucleus is cranked.
Of course, the value of *g*_{0} does not influence the GT calculations
for even-even nuclei presented in our study which focuses on
the isovector time-odd coupling constants. Consequently, in the following
we employ a simple spin energy functional using the
Skyrme force value for *C*_{0}^{T}, setting
and neglecting density dependence. We adopt the value *g*_{0} = 0.4
given in [52] (note that a different definition of the
normalization factor is used there) to fix *C*_{0}^{s}.

We perform the calculations in exactly the same way as
in Ref. [3] by using the code HFODD (v1.75r) described
in Ref.[68]. We examine ^{152}Dy, which is a doubly magic
superdeformed system. Pairing has a minor influence and we neglect
it. We focus on the dynamic moment of inertia :

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