Fig. 3 shows the contribution to the total nuclear binding
energy due to the tensor term, calculated by
using the spherical Hartree-Fock-Bogoliubov (HFB)
code HFBRAD [30] with the SLy4 functional.
Contributions due to the
isovector and isoscalar parts are depicted separately in the upper and middle
panels, respectively. The total contribution,
, is shown in the lowest
panel of the figure.
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From these results one can see that the isovector component is rather weak.
Hence, the topology of the total contribution to the energy is mostly determined
by the isoscalar term that shows a strong shell dependence. Following the
argumentation presented in Sec. 2, the strongest
tensor effects are expected to appear for =14, 32,
56, and 90. They correspond to nucleons filling the
,
,
and
shells, respectively, which creates a maximum
SUS filling. Since the level (shell) ordering, at least in well bound
near-spherical nuclei, is rather well established experimentally and
since it serves as a natural constrain for effective forces,
the maximum SUS configurations are, to a large extent, parameterization
independent. Hence, the
numbers at which they appear are
robust and can be therefore regarded as
tensorial magic numbers.
However, as it is seen from Fig. 3, the
does not follow the expected pattern exactly.
This is due to (i) pairing-induced configuration mixing and
(ii) changes in the s.p. ordering of levels caused by
the combination of strong attractive tensor fields and strongly reduced
SO field. Two such situations are visible in Fig. 3. For
the tensor contribution is, as expected, largest for
Ge.
For
, however, the minimum on the
plot is shifted toward the
Ni isotopes,
which suggests a change in the order of the
and
proton
sub-shells with increasing neutron excess. Of course, the sub-shell filling
pattern in these nuclei is, to a large extent, determined by pairing.
Nevertheless, the HF calculations confirm that for the
chain of
isotopes inversion of the
and
orbitals indeed
takes place around
as predicted from Fig. 3. The figure
also indicates that on the proton side,
rather than
is the
tensorial magic number. Again, this suggests that the
proton
sub-shell is filled before
. Consequently, the tensorial magic
numbers may slightly differ for neutrons (
= 14, 32, and 56) and for
protons (
= 14, 28, and 50). This effect, however, may strongly depend
upon a rather delicate balance between the tensor and SO strengths and needs
to be studied in detail.
The second major source of configuration mixing in atomic nuclei - nuclear deformation - is beyond the scope of this work. Our recent calculations [24] show, however, that although it further smears out the tensorial effects major topological features of Fig. 3 including the tensorial magic structure are, to a large extent, retained.