Fitting procedure used to constrain the coupling constants
and
was described in detail in Ref. [22]
and we only recall it here very briefly.
The idea is to reproduce experimental
SO splittings in
three key nuclei:
Ca,
Ca, and
Ni.
Since
Ca is, as discussed above, a SS system, the
conventional SO term of Eq. (5) is the only source of the SO
splitting. Hence, this nucleus is used to set the isoscalar strength of
the SO term
.
Having set
, we next constrain the
strength by using
the
SO splitting in the isoscalar (
)
SUS nucleus
Ni. Finally, we move to
Ca, where protons and neutrons constitute a SS and SUS system, respectively.
This nucleus is used to fit the isovector coupling constants or, more precisely,
to fit
, because the ratio
is
kept equal to the value characteristic for
the given Skyrme parametrization. There is
one piece of data on the
SO splittings,
preferably in
Ni or
Ni,
which is badly needed to fit the tensor and SO terms uniquely.
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The procedure outlined above is illustrated in Fig. 2
for the case of the SLy4 functional [28] but it is
qualitatively independent of the initial parameterization. As shown in the
figure, a good agreement with empirical data requires, for this
low-effective-mass force, circa 35% reduction of
as
compared to the original value, a large attractive isoscalar tensor coupling
constant
of about
MeVfm
, and
of about
MeVfm
.
It appears that the resulting tensor coupling
constants
(as well as the SO strengths
) are, to
large extent, independent on the initial parameterization.
This is illustrated in Fig. 1 where different functionals modified
according to our prescription, see
Refs. [22,23,29,24], are collected. They are
labeled by a subscript
following the force acronym and marked by black
triangles. Note, that the SkO
and SkO
parameterizations
have slightly larger (smaller) values of the
(
) coupling constants as compared to the other parameterizations,
respectively. These values are needed in order to compensate for non-standard,
very strong isovector spin-orbit strength characterizing the SkO based
functionals. The SkO
functional has an interesting property.
It appears to reproduce reasonably well both the SO splittings and masses
of spherical nuclei [29,24]. This
result seems to be at variance not only with
our SLy4
functional fitted in Ref. [22],
which reproduces masses of spherical nuclei much worse than the original
SLy4 functional, but also with the results of systematic study in
Ref. [11].