Before proceeding to readjustments of coupling constants so as to improve the agreement of the SO splittings with data, we analyze the influence of time-even (mass and shape) and time-odd (spin) polarization effects on the neutron SO splittings. Based on results presented in the preceding Section, we calculate the SO splittings as
Figure 2 shows the SO splittings calculated using SLy4 -- the
functional based on the original SLy4 [19] functional
with spin fields readjusted to reproduce empirical Landau parameters
according to the prescription given in
Refs. [21,22]. Plotted values correspond to
results presented in Tables 1 and
2. The results are labeled according to the
following convention: open symbols mark results computed directly
from the s.p. spectra in doubly-magic nuclei (bare s.p. energies).
These bare values contain no polarization effect. Gray symbols label
the SO splittings involving polarization due to the time-even mass-
and shape-driving effects, i.e., those obtained with all time-odd
components in the functional set equal to zero. Black symbols
illustrate fully self-consistent results obtained for the complete
SLy4
functional. Gray and black symbols are shifted slightly to
the left-hand (right-hand) side with respect to the doubly-magic
core in order to indicate the hole (particle) character of the SO
partners. Mixed cases involving the particle-hole SO
partners are also shifted to the right.
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The impact of polarization effects on the SO splittings is indeed
very small, particularly for the cases where both SO partners are of
particle or hole type. Indeed, for these cases, the effect only
exceptionally exceeds 200keV, reflecting a cancellation of
polarization effects exerted on the
partners. The
smallness of polarization effects hardly allows for any systematic
trends to be pinned down. Nevertheless, the self-consistent results
show a weak but relatively clear tendency to slightly enlarge
or diminish the splitting for hole or particle states, respectively.
The situation is clearer when the SO partners are of mixed particle
(
) and hole (
) character. In these cases,
the shape polarization tends to diminish the splitting quite
systematically by about 400-500keV. This behavior follows from
naive deformed Nilsson model picture where the highest-
members of
the
(
) multiplet slopes down (up) as a
function of the oblate (prolate) deformation parameter. As discussed
in the previous Section, the time-odd fields act in the opposite way,
tending to slightly enlarge the gap. The net polarization effect does
not seem to exceed about 300keV. In these cases, however, we deal
with large
orbitals having also quite large SO splittings of
the order of
8MeV. Hence, the relative corrections due to
polarization effects do not exceed about 4%, i.e., they are
relatively small - much smaller than the effects of tensor terms
discussed below and the discrepancy with data, which in Fig. 2
is indicated for the neutron 1f SO splitting in
Ca. These
results legitimate the direct use of bare s.p. spectra in magic
cores for further studies of the SO splittings, which considerably
facilitates the calculations.
As already mentioned, empirical s.p. energies are essentially
deduced from differences between binding energies of doubly-magic core
and their odd- neighbors. Different authors, however, use also one-particle
transfer data, apply phenomenological particle-vibration corrections
and/or treat slightly differently fragmented levels. Hence, published
compilations of the s.p. energies, and in turn the SO splittings, differ slightly
from one another depending on the assumed strategy. The typical
uncertainties in the empirical SO splittings can be inferred from
Table 3, which summarizes the available data on the SO splittings
based on three recent s.p. level compilations published in
Refs. [7,8,9].
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Instead of large-scale fit to the data (see, e.g.,
Ref. [16]), we propose a simple three-step method
to adjust three coupling constants
,
, and
. The entire idea of this procedure is based on the
observation that the empirical
SO splittings in
Ca,
Ni, and
Ca form very
distinct pattern, which cannot be reproduced by using solely the
conventional SO interaction.
The readjustment is done in the following way. First, experimental
data in the spin-saturated (SS) nucleus Ca are used in order
to fit the isoscalar SO coupling constant
. One
should note that in this nucleus, the SO splitting depends only
on
, and not on
(because of the spin
saturation), nor on
(because of the isospin invariance at
), nor on
(because of both reasons above). Therefore,
here one experimental number determines one particular coupling
constants.
Second, once
is fixed, the spin-unsaturated (SUS)
nucleus
Ni is used to establish the isoscalar tensor
coupling constant
. Again here, because of the isospin
invariance, the SO splitting is independent of either of the two
isovector coupling constants,
or
. Finally,
in the third step,
Ca is used to adjust the isovector tensor
coupling constant
. Such a procedure exemplifies the focus of
fit on the s.p. properties, as discussed in the Introduction.
It turns out that current experimental data, and in particular lack
of information in Ni, do not allow for adjusting the fourth
coupling constant,
. For this reason, in the present
study we fix it by keeping the ratio of
equal to that of the given standard Skyrme force. In the
process of fitting, all the remaining time-even coupling constants
are kept unchanged. Variants of the standard functionals
obtained in this way are below denoted by SkP
, SLy4
,
and SkO
. When the time-odd channels, modified so as to reproduce
the Landau parameters, are active, we also use notation SkP
,
SLy4
, and SkO
.
For the SkP functional, the procedure is illustrated in Fig. 3.
We start with the isoscalar nucleus
Ca. The evolution of
the SO splittings in function of
, which is the factor
scaling the original SkP coupling constant
, is shown
in the upper panel of Fig. 3. As it is clearly seen from the
Figure, fair agreement with data requires about 20% reduction in
the conventional SO interaction strength
, cf. results of the recent study in Ref. [38]. It should
also be noted that the reduction in the SO interaction
considerably improves the
and
splittings but slightly spoils the
SO splitting.
Qualitatively, similar results were obtained for the SLy4 and SkO
interactions. Reasonable agreement to the data requires
20%
reduction of the original
in case of the SkO
interaction and quite drastic
35% reduction
of the original
in case of the SLy4 force.
Having fixed
in
Ca we move to the isoscalar
nucleus
Ni.
This nucleus is spin-unsaturated and therefore is very sensitive
to the isoscalar
tensor coupling constant.
The evolution of theoretical s.p. levels versus
is illustrated
in the middle panel of Fig. 3.
As shown in the Figure, reasonable agreement between the empirical and
theoretical
SO splitting is achieved
for
MeVfm
, which by a factor of about five exceeds
the original SkP value for this coupling constant. It is striking
that a similar value of
is obtained
in the analogical analysis performed for the SLy4 interaction.
Finally, the isovector tensor coupling constant
is
established in
nucleus
Ca.
The evolution of theoretical neutron s.p. levels versus
is illustrated in the lowest panel of Fig. 3.
As shown in the Figure, the value of
MeVfm
is needed to reach reasonable agreement for the
SO
splitting in this case. For this value of the
strength one
obtains also good agreement for the proton
SO
splitting (see Figs. 4 and 5 below), without any further
readjustment of the
strength.
Again, very similar value for the
strength is deduced
for the SLy4 force. Note also the improvement in the
splitting caused by the isovector
tensor interaction. Dotted lines show results obtained from the
mass differences, i.e., with all the polarization effects included.
During the fitting
procedure all the remaining functional
coupling constants were kept fixed at their Skyrme values. The ratio
of the isoscalar to the isovector coupling constant in the SO
interaction channel was locked to its standard Skyrme value of
. Since no clear indication for relaxing this
condition is seen (see also Figs. 4 and 5 below),
we have decided to investigate the isovector
degree of freedom in the SO interaction (see [17])
by performing our three-step
fitting process also for the generalized Skyrme interaction
SkO [20], for which
.
All the adopted functional coupling constants resulting from our
calculations are collected in Table 4.
Note, that the procedure leads to essentially identical
SO interaction strengths
for all three forces irrespective of their intrinsic differences,
for example in effective masses. The tensor coupling constants in both
the SkP and the SLy4 functionals are also very similar. In the SkO case, one
observes rather clear enhancement in the isovector tensor coupling constant
which becomes more negative to, most likely, counterbalance the non-standard
positive strength in the isovector SO channel.
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The functionals were modified using only three specific pieces of data
on the neutron
SO splittings. In order to verify the
reliability of the modifications,
we have performed systematic calculations of the
experimentally accessible SO splittings.
The results are depicted in Figs. 4,
5, and 6
for the SkP, SLy4, and SkO functionals,
respectively. Additionally, Fig. 7 shows neutron and proton magic gaps (24)
calculated using the SkP functional.
In all these Figures, estimates taken from Ref. [8] are used as reference
empirical data.
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This global set of the results can be summarized as follows:
Without any doubt the SO splittings are better described by
the modified functionals. It should be stressed that the
improvements were reached using only three additional data points without
any further optimization. The tensor coupling constants
deduced in this work and collected in
table 4 should be therefore considered as reference values.
Indeed, direct calculations show that variations in within
10%
affect the calculated SO splittings only very weakly. The price paid for the
improvements concerns mostly the binding energies, which for the
nuclei
Ni,
Sn, and
Pb become worse as compared
to the original values. This issue is addressed in the next section.