bare | T-even | T-even | T-even | T-odd | total | exp. | theory | |||||||||||
pol. | & T-odd | pol. | pol. | [8] | ![]() |
|||||||||||||
(a) | (b) | (b)![]() |
(c) | (c)![]() |
(c)![]() |
(d) | (c)![]() |
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![]() |
![]() |
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0.96 | ![]() |
![]() |
0.28 | ![]() |
1.55 | |||||||||
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0.99 | ![]() |
![]() |
0.68 | ![]() |
1.80 | ||||||||||
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0.92 | ![]() |
0.40 | 1.32 | ![]() |
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0.99 | ![]() |
0.49 | 1.48 | ![]() |
1.05 | ||||||||||
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0.39 | 1.19 | 0.80 | 1.37 | 0.18 | 0.98 | 1.50 | ![]() |
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![]() |
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0.46 | ![]() |
![]() |
0.14 | ![]() |
0.52 | |||||||||
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0.32 | ![]() |
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0.68 | ||||||||||
![]() |
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0.47 | ![]() |
![]() |
0.33 | ![]() |
0.66 | ||||||||||
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0.36 | ![]() |
0.22 | 0.58 | ![]() |
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![]() |
0.39 | ![]() |
0.18 | 0.57 | ![]() |
2.09 | ||||||||||
![]() |
![]() |
![]() |
0.40 | ![]() |
0.11 | 0.51 | ![]() |
2.21 | ||||||||||
![]() |
![]() |
![]() |
0.28 | ![]() |
0.11 | 0.39 | ![]() |
2.39 | ||||||||||
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0.58 | ![]() |
![]() |
0.39 | ![]() |
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0.34 | ![]() |
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0.58 | ![]() |
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0.53 | ![]() |
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0.56 | ![]() |
![]() |
0.45 | ![]() |
0.34 | ||||||||||
![]() |
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0.29 | ![]() |
0.13 | 0.42 | ![]() |
0.13 | ||||||||||
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![]() |
0.33 | ![]() |
0.10 | 0.43 | ![]() |
0.47 | ||||||||||
![]() |
![]() |
![]() |
0.08 | ![]() |
0.04 | 0.12 | ![]() |
0.46 |
bare | T-even | T-even | T-even | T-odd | total | exp. | theory | |||||||||||
pol. | & T-odd | pol. | pol. | [8] | ![]() |
|||||||||||||
(a) | (b) | (b)![]() |
(c) | (c)![]() |
(c)![]() |
(d) | (c)![]() |
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0.33 | ![]() |
![]() |
0.22 | ![]() |
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0.35 | ![]() |
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0.33 | ![]() |
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0.22 | ![]() |
![]() |
0.08 | ![]() |
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0.20 | ![]() |
![]() |
0.11 | ![]() |
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0.33 | ![]() |
![]() |
0.27 | ![]() |
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![]() |
0.14 | ![]() |
0.07 | 0.21 | ![]() |
0.68 | ||||||||||
![]() |
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![]() |
0.23 | ![]() |
0.26 | 0.49 | ![]() |
1.34 | ||||||||||
![]() |
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![]() |
0.37 | ![]() |
0.03 | 0.40 | ![]() |
1.18 | ||||||||||
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0.00 | ![]() |
0.01 | 0.01 | ![]() |
0.88 | ||||||||||
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0.24 | ![]() |
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0.16 | ![]() |
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0.18 | ![]() |
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0.36 | ![]() |
![]() |
0.33 | ![]() |
0.09 | ||||||||||
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![]() |
0.20 | ![]() |
![]() |
0.16 | ![]() |
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![]() |
0.01 | ![]() |
0.05 | 0.06 | ![]() |
0.34 | ||||||||||
![]() |
0.17 | 0.26 | 0.09 | 0.31 | 0.05 | 0.14 | ![]() |
1.62 | ||||||||||
![]() |
0.95 | 0.79 | ![]() |
0.77 | ![]() |
![]() |
![]() |
1.68 | ||||||||||
![]() |
0.97 | 1.08 | 0.11 | 1.12 | 0.04 | 0.15 | ![]() |
1.84 | ||||||||||
![]() |
0.82 | 0.84 | 0.02 | 0.88 | 0.04 | 0.06 | ![]() |
1.23 | ||||||||||
![]() |
![]() |
![]() |
![]() |
0.17 | ![]() |
![]() |
0.12 | ![]() |
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0.23 | ![]() |
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0.22 | ![]() |
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![]() |
0.14 | ![]() |
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0.06 | ![]() |
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![]() |
0.14 | ![]() |
![]() |
0.12 | ![]() |
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![]() |
0.10 | ![]() |
![]() |
0.05 | ![]() |
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0.00 | ![]() |
0.03 | 0.03 | ![]() |
0.57 | ||||||||||
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![]() |
1.44 | ||||||||||
![]() |
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![]() |
0.04 | ![]() |
0.03 | 0.07 | ![]() |
1.79 | ||||||||||
![]() |
0.56 | 0.65 | 0.09 | 0.80 | 0.15 | 0.24 | ![]() |
2.61 | ||||||||||
![]() |
0.08 | 0.10 | 0.02 | 0.11 | 0.01 | 0.03 | ![]() |
1.46 | ||||||||||
![]() |
0.69 | 0.76 | 0.07 | 0.78 | 0.02 | 0.09 | ![]() |
2.11 |
In Tables 1 and 2 we list the
neutron s.p. energies calculated in six doubly-magic nuclei for the
SLy4 interaction. The bare s.p. energies (a) are compared to
those calculated from total energies, Eqs. (21) and
(22), with the mass and shape (b) or mass, shape, and spin
(c) polarizations included.
In order to remove ambiguities associated with occupancy of the valence
particle (hole), binding energies of odd-
nuclei were calculated by
blocking the lowest (highest)
orbitals at oblate (prolate) shape
for particle
(hole) orbitals. The blocked orbitals were selected by performing cranking
calculation with angular-frequency vector parallel to the symmetry axis.
Such a cranking does not affect total energy or wave function, but
splits spherical multiplets into orbitals having good projections of
the angular momentum on the symmetry axis.
Calculations were performed by using the
code HFODD (v2.30a) [49,50,51,52] for the
spherical basis of
harmonic-oscillator shells.
As seen by comparing columns (b) and (a) of Tables 1 and
2, the energy shifts caused by the time-even polarization
effects with respect to bare s.p. spectra are almost always positive, both
for particle and hole states. A few exceptions occur only for
large- unfavored (
) SO partners in heavy nuclei.
These shifts clearly decrease in magnitude
with increasing mass, from about 1MeV in
O to below
0.25MeV in
Pb. As discussed in Sec. 3.1, they are
mainly caused by the mass-polarization effects related to the center-of-mass
correction. Indeed, shifts of s.p. energies (25), calculated
for the six doubly-magic nuclei of Tables 1 and
2, are 0.87, 0.40, 0.36, 0.20, 0.14, and 0.09MeV,
respectively.
It is also clearly visible that shifts of particle states are systematically smaller than those of hole states, i.e., the time-even polarizations tend to slightly decrease shell gaps.
The time-odd polarization effects systematically shift the hole
states down and particle states up in energy, i.e., they result in an
increase of shell gaps, cf. also Fig. 1.
This result is at variance with that
obtained within the RMF approach [4], where the time-odd
fields corresponded to magnetic properties driven by the Lorentz
invariance, while here they are determined by experimental values of
the Landau parameters [21,22]. We note here that
in recent derivations of the time-odd coupling constants within
the relativistic point-coupling model [53], one obtains
values of the Landau parameters compatible with experimental values.
Shifts of s.p. energies due to the time-odd polarization effects
also decrease with mass, from about 0.7(+0.5)MeV in
O to
below
0.1(0.15)MeV in
Pb for hole (particle) states.
The total effect of combined time-even and time-odd polarizations
results in adding up the shifts for particle states and subtracting
those for hole states. In this way, the total shifts of particle and
hole states become mostly positive and (apart from light nuclei)
comparable in magnitude, with quite small net effects on shell gaps.
They also decrease with increasing mass, from up to 1.5MeV in
O to below 0.25MeV in
Pb. Altogether, polarization
effects turn out to be significantly smaller than those obtained in
previous estimates. Although in quantitative analysis they cannot at
all be neglected, discrepancies with experimental data (last columns
in Tables 1 and 2) are still
markedly larger in magnitude. Therefore, bare s.p. energies can be
safely used, at least in all studies that do not achieve any better
overall agreement with data.