Quasiparticle spectra in $^{251}$Cf and $^{249}$Bk

Figure 4: Experimental and calculated quasiparticle spectra in $^{249}$Bk and $^{251}$Cf, see text for the convention used here. Experimental data are taken from Ref. [67] We label the state with the full Nilsson label of the dominant component of the wave function only if the squared amplitude of this component exceeds 50%. The exception is the 1/2[7] state which is strongly mixed. However, the cumulative squared amplitude of the components of the wave function with $N = 7$ in the structure of this state exceeds 90%. Thus, we label it only by principal quantum number $N$ and $\Omega$.

FigSHE04a.eps FigSHE04b.eps

In Fig. 4, calculated spectra of low-lying band-heads in $^{249}$Bk and $^{251}$Cf are shown along with the experimental data. In these results, the spectra were obtained by individually blocking relevant quasiparticle orbitals and then plotting differences of total many-body energies of obtained nucleonic configurations with respect of the total energy of the ground state, that is, they are not at all equivalent to quasiparticle energies understood as eigenvalues of the HFB Hamiltonian. The ground states were identified as nucleonic configurations with the lowest total energies.

Nuclear configurations of deformed odd nuclei (one-quasiparticle configurations) were labelled by means of the standard asymptotic quantum numbers $\Omega [Nn_z\Lambda ]$ (Nilsson quantum numbers) that correspond to the dominant component in the wave function of the blocked quasiparticle state.

We used the convention of plotting positive (negative) values for the excitation energies of quasiparticle configurations that correspond to blocked quasiparticle states having norms of the second HFB components smaller (larger) than 1/2 before blocking. In this way, the states that are predominantly of a particle (hole) character appear above (below) zero energy. Moreover, these states are always plotted relatively to the ground-state; thus the ground state is plotted identically at the value of zero energy. This convention facilitates the comparison of fully self-consistent results with the Nilsson diagrams. For experimental quasiparticle configurations, we follow the assignments of particle/hole character as presented in Figs. 34 and 35 of Ref. [68].

In these two odd nuclei, prominent intruder configurations correspond to the proton 7/2[633] and neutron 11/2[725] orbitals. In Ref. [21], these two orbitals were used as benchmark states to adjust strengths of the spin-orbit interactions. We see that without such an adjustment, none of the studied standard EDFs places them at the right position. The ground-state proton 7/2[633] orbital, which experimentally is almost degenerate with the 3/2[521] orbital, for Skyrme and Gogny EDFs appears about 500keV above the ground state and for covariant EDFs about 200keV below the ground state, with the calculated ground states corresponding the 3/2[521] orbital (or 7/2[514] for the covariant EDF NL3*).

The neutron 11/2[725] orbital, for covariant, Gogny, and SLy4 EDFs, appears too high and for UNEDF2 EDF too low above its experimental position with respect to the ground-state 1/2[620] orbital. On the one hand, one can say that on the absolute scale these deficiencies are not large. On the other hand, they may point to slightly incorrect positions of spherical intruder orbitals, from which one would like to infer the shell structure of as yet not-reached superheavy nuclei. This analysis shows that detailed structure of very heavy deformed nuclei may depend on extremely fine details of the present-day theoretical models, which very well may be far beyond any reasonable possibility of adjusting them precisely enough to available experimental data.

Similarly as in the analysis presented in Ref. [21], as an attempt to improve the agreement with the experimental values, we have considered variations of the spin-orbit parameter $W_{LS}$ of the Gogny EDF D1S that could influence relative positions of intruder states. Increasing $W_{LS}$ from its nominal value of 130MeVfm$^{5}$ reduces the excitation energy of the $11/2^{-}$ state while it increases the excitation energy of the $9/2^{-}$ state in $^{251}$Cf. These changes improve the agreement with experimental data for larger values of $W_{LS}$. In the $^{249}$Bk case, the $7/2^{+}$ goes down in excitation energy as $W_{LS}$ increases, while the $5/2^{+}$ and $1/2^{-}$ levels go up. As in the $^{251}$Cf case, the comparison with experiment seems to favor larger values of $W_{LS}$. This, however, has to be contrasted with the analysis of the shell structure of heavy spherical nuclei like $^{208}$Pb, which usually calls for weaker spin-orbit interaction [25,63,65].

However, it is necessary to recognize that the studies restricted to spin-orbit potential may have internal limitations that come from the fact that possible deficiencies in the description of the energies of the single-particle states emerging from the central potential, as for example those inferred in Ref. [40], are ignored. The fact that standard Skyrme functionals provide better description of the single-particle states in the $Z=115$ nuclei than the ones with the strength of spin-orbit interaction adjusted to experimental data in nobelium region [64] may be related to such limitation.

Figure 5: Experimental spectra [67] of the even-$N$ $Z=99$ isotopes (left panel) and even-$Z$ $N=151$ isotones (right panel).

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To explore the sensitivity of the results to the amount of pairing correlations in the system, for the Gogny EDFs we performed calculations where the pairing strengths of protons and neutrons were multiplied by factors $f_{p}$ and $f_{n}$, respectively. The first noticeable fact is that increasing the neutron pairing strength does not influence in a significant way the spectrum of $^{249}$Bk (odd $Z$) as it also happens when increasing the proton pairing strength in $^{251}$Cf. Increasing $f_{p}$ reduces the excitation energy of all levels except for the $5/2^{+}$ state that remains more or less constant. The comparison with experimental spectra seems to favor larger proton pairing correlations. In the $^{251}$Cf case, all the levels except the lowest $3/2^{+}$ decrease their excitation energy with increasing pairing strength. As in the $^{249}$Bk case, increasing the pairing correlations in $^{251}$Cf improves the agreement with experimental spectra. The same behavior was observed during the readjustment of the pairing strength for SLy4. However, on should keep in mind that pairing strengths are predominantly defined by the odd-even mass staggering, see Sect. 3.4 below.