GT strength distributions from existing Skyrme interactions

Residual interactions are often summarized by
the Landau parameters that appear in Eq. (13).
The parameters can be derived as the corresponding coupling constants
when Eq. (14) is evaluated for infinite spin-saturated
symmetric nuclear matter (see Appendix 10).
In the literature, the infinite nuclear matter (INM)
properties of the Skyrme interactions are
usually calculated from Eqs. (34). For the generalized
energy functional (12) discussed here, the
time-even INM properties such as the saturation density, energy per
particle, effective mass, incompressibility, symmetry coefficient,
and the time-even Landau parameters *f*_{i}, *f*_{i}'
are unchanged, but properties of polarized INM and expressions for the
time-odd Landau parameters *g*_{i} and *g*_{i}' are different.
We derive them in Appendix 10. Here we are most
concerned with the Landau parameters in the spin and spin-isospin
channels,

Table 1 also gives values for the Landau parameters calculated for the Gogny forces D1 [59] and D1s [22] from the expressions provided in Appendix 11. In the spirit of the Gogny force as a two-body potential, one has no freedom to choose the time-odd terms independently from the time-even ones. (Note that the Gogny force, however, employs the same local-density approximation for the density-dependence as the Skyrme energy functional that contributes to the Landau parameters.) The higher-order Landau parameters are uniquely fixed by the finite-range part of the Gogny force.

Figures 1 and 2 show the
summed GT strength *B*(*GT*) in ^{208}Pb and ^{124}Sn, calculated with
all the selected Skyrme forces. The ground-state energies are calculated
as described
in Ref. [11], and all strengths are divided by 1.26^{2},
following common practice, to account for GT quenching. Although the GT
resonance in ^{208}Pb comes out at about the right energy for SGII,
SLy4, SkO, and SkO', it is too low for SkP and SLy5. These latter two
interactions also leave too much GT strength at small excitation
energies. It is tempting to interpret these findings in terms of the
Landau parameters for these interactions. Schematic models suggest
[1] that an increase of *g*_{0}' results in an increased resonance
energy and more GT strength in the resonance. The nucleus ^{208}Pb
indeed behaves in this way, as can be seen
in Fig. 1. The forces SkP and SLy5, with small
values of *g*_{0}', yield more low-lying strength and a lower resonance
energy than the remaining forces which correspond to
.

In ^{124}Sn, however, this simple picture does not hold, as
Fig. 2 shows. The resonance energies are
similar (and close to the experimental value) for SkP, SLy5, SkO, and
SkO' forces with very different values of *g*_{0}', while SGII and SLy4
push the resonance energy too high. Only the amount of the low-lying
strength seems to scale with *g*_{0}'. It is interesting, though, that
the related forces SLy4 and SLy5 (which predict very similar
single-particle spectra, but have quite different GT residual
interactions) agree with the schematic model in that SLy4, with larger
*g*_{0}', puts the GT resonance at a higher excitation energy.

It is clear that the scaling predicted by the schematic model is too
simple, and Fig. 3 demonstrates this clearly. There we
show the calculated strengths
in the GT resonances
relative to the sum-rule value
,
and the
calculated GT resonance energies
relative to the
experimental values .
[For ^{90}Zr, ^{112}Sn,
^{124}Sn, and ^{208}Pb we used
MeV,
8.9MeV, 13.7MeV, and 15.5MeV, respectively [46]. Note
that the calculated resonance energy depends on a prescription (see
[11]) not strictly dictated by the QRPA.] The scatter
near
,
in both the resonance energy and in the
amount of low-lying strength, shows that other combinations of parameters
in the residual interaction besides *g*_{0}' affect the GT distribution.
This is not entirely surprising given the complexity of finite nuclei
and of the interaction (14). In
Sect. 4.3 we quantify these other important
combinations and discuss their effects.

But another factor, this one determined by the time-even part of the
Skyrme functional, affects the GT distribution: the underlying
single-particle spectrum. Since GT transitions are especially sensitive
to proton spin-orbit splittings, small changes in the time-even part of
the force can, in principle, move the GT resonance considerably.
Sensitivity to the spin-orbit splitting is particularly obvious in
^{90}Zr, where detailed information has been obtained from a recent
experiment by Wakasa *et al.* [60]. Unlike in ^{124}Sn and
^{208}Pb, which respond to a GT excitation in a collective way, the
^{90}Zr GT spectrum is dominated by two single-particle transitions,
from the neutron 1*g*_{9/2} state to the proton 1*g*_{9/2} and
1*g*_{7/2} states. The difference between the locations of the two
peaks in the GT spectrum is the sum of the proton 1*g* spin-orbit
splitting and a contribution from the residual interaction (which can be
expected to increase the difference). As Fig. 4
shows, all interactions, whatever their value for *g*_{0}', overestimate
this difference; the resonance energy is always too large, even when the
residual interaction is switched off completely.

Most Skyrme interactions give spin-orbit splittings in heavy nuclei that are too large [61]. We can therefore expect errors in their predicted GT strength distributions [45,47]. Figure 5 shows errors in the predicted spin-orbit energies for the same forces as in Fig. 3. Interactions such as SkI3, SkI4, or SLy4 that overestimate the proton spin-orbit splittings give the largest resonance energies (and tend to overestimate them). The best interaction, in view of the combined information from Figs. 3 and 5, appears to be SkO'. Therefore, below, we use its time-even energy functional for further exploration of the time-odd terms.

We have included some new forces in Fig. 5; in a recent
paper [62], Sagawa *et al.* attempt to improve the spin-orbit
interaction for the standard Skyrme forces SIII, SkM^{*}, and SGII, aiming
at better GT-response predictions. They generalize the spin-orbit
interaction through the condition
and include the
term with a coupling given by
Eq. (34). Although the modified forces SkM^{*}-u, and SGII-u
give slightly better descriptions of GT resonances than the original
interactions, they generate unacceptable errors in total binding
energies and do not substantially improve the overall description of
single-particle spectra in ^{208}Pb.

A few remarks are in order before proceeding: (i) The spin-orbit
splittings shown in Fig. 5 are calculated from intrinsic
single-particle energies. Since experimental data are obtained from
binding-energy differences between even-even and adjacent odd-mass
nuclei, core polarization induced by the unpaired nucleon, which depends
partly on time-odd channels of the interaction [7,8],
alters single-particle energies. The effect is largest in small nuclei
(of the order of 20 % in ^{16}O), decreasing rapidly with mass
number [8]. (ii) GT distributions are also affected by the
particle-particle channel of the effective interaction, but mainly at
low energies. The GT resonance is not materially altered [11],
so we can safely neglect the particle-particle interaction here.