Quasiparticle spectra in $Z=99$ (Es) isotopes and $N=151$ isotones

In the left (right) panels of Figs. 5-8, we show experimental and calculated spectra of the even-$N$ $Z=99$ isotopes (even-$Z$ $N=151$ isotones). The results are presented in the convention discussed in Sec. 3.2, with experimental assignments taken from Refs. [67,68]. We see that the experimentally assigned odd-quasiparticle configurations are theoretically always obtained at relevant low excitation energies. However, similarly as in the case of $^{249}$Bk and $^{251}$Cf presented above, the details of the level spacing and ordering vary from one calculation to another and are not very well reproduced.

Figure 6: Same as in Fig. 5 but for the spectra calculated for the Skyrme EDF SLy4 (upper panels) and UNEDF2 (lower panels). Since for SLy4 only the mean values of the single-particle angular momenta $\langle \hat{j}_\parallel \rangle$ and $\langle \hat{j}^2 \rangle$ (and not full Nilsson labels) were available, in this case the labels were assigned by analogy with the results obtained for the Gogny EDF D1S, which provides very similar spectra.

FigSHE06.eps

Figure 7: Same as in Fig. 5 but for the spectra calculated for the Gogny EDF D1S (upper panels) and D1M (lower panels).

FigSHE07.eps

Figure 8: Same as in Fig. 5 but for the spectra calculated for the covariant EDF NL1 (upper panels) and NL3* (lower panels).

FigSHE08.eps

Although in this region of nuclei, the experimental information is richest in the particular isotopic and isotonic chains studied here, it is still quite scarce, and often experimental assignments of configurations are still only tentative [67]. Nevertheless, we can already see several conspicuous experimental trends.

In protons, we see the 7/2[633] ground states and 7/2[514] excited states at fairly constant excitation energies of about 400keV. For the Skyrme EDF SLy4, this feature is very well reproduced, with a significant drop of this excitation energy predicted at $N=156$. For the Skyrme EDF UNEDF2, the 7/2[514] level is also obtained above the Fermi level, with the excitation energy gradually decreasing already at $N=152$. Note, however, that for the Skyrme EDFs, in $^{249}$Bk the relative positions of these two levels were not very precisely reproduced, so the nice agreement obtained in the $Z=99$ isotopes might be fortuitous. For the Gogny EDFs, this pair of the levels is obtained at roughly correct excitation energies, whereas for the covariant EDFs, these levels are fairly well degenerate.

In neutrons, the 9/2[734] ground states are for all EDFs studied here well reproduced, apart from the Skyrme EDF UNEDF2, which gives the 7/2[613] ground state with the hole-character 9/2[734] orbitals appearing at excitation energy of about 400-500keV. Two particle-character excited quasiparticle states, 7/2[613] and 1/2[620], show excitation energies which increase with mass. They are correctly reproduced for the Skyrme SLy4 and covariant EDF's; however, the increase with mass is too fast. Correct trends of these two levels are also determined for the Gogny EDFs. Two other hole-character experimental quasiparticle states, 7/2[624] and 5/2[622], have excitation energies weakly increasing and decreasing with mass, respectively. For the Skyrme SLy4 and covariant NL1 EDFs, their excitations energies both increase quite rapidly with mass, whereas for the Gogny functionals the mass dependence is rather correct. In all models studied here, the energy splitting of these two levels is too large as compared to data.

From the systematics of the one-quasiparticle levels one can draw a number of interesting conclusions about the significance of the Nilsson diagram of single-particle levels for the calculated and observed spectra of one-quasiparticle levels. Already in schematic models there is no quantitative one-to-one correspondence, as the presence of pairing correlations modifies the spectrum of low-lying states. In nuclear EDF calculations, there are additional self-consistency effects from the separate optimization of each one-quasiparticle state.

As already outlined in Sect. 3.1, the various EDFs do not always agree on the size of the energy gaps between spherical subshells in the Nilsson diagrams for $^{254}$ No plotted in Figs. 1, 2 and 3, sometimes even not on the sequence of the levels. For example, the relativistic NL1 and NL3* functionals predict a large spherical $N=138$ gap of about 3MeV. For the non-relativistic D1S, D1M, and SLy4 EDFs this gap is much smaller and even disappears for UNEDF2. With the non-relativistic functionals one finds spherical gaps at $N=148$ and $N=152$ instead. The appearance and size of the $N=138$ gap is controlled by the position of the spherical $1i_{11/2^+}$ level below and the $2g_{9/2^+}$ level above, except for NL3* for which the $1j_{15/2^-}$ is pulled below the $2g_{9/2^+}$. The neutron 9/2[734], 5/2[622], and 9/2[615] single-particle levels that emerge from the spherical $1j_{15/2-}$, $2g_{9/2^+}$, and $i_{11/2^+}$ subshells, respectively, are found just above and below the deformed $N=150$ and $N=152$ gaps in the Nilsson diagrams for $^{254}$ No. Looking at one-quasiparticle spectra of $^{251}$ Cf, as plotted in Fig. 4, one finds that for SLy4, D1S, and D1M the distance between the hole-character 9/2[734] and particle-character 9/2[615] levels is very satisfactorily described within 200keV, whereas for UNEDF2 it is much too large, and for NL1 and NL3* it is much too small. On the other hand, in the $N=151$ isotones shown in Figs. 5-8, the distance between these two levels is on average too large for SLy4, D1S, and D1M, again much too large for UNEDF2, correct for NL1, and again much too small for NL3*. This indicates that the spherical $N=138$ gap of NL3* is probably too large. Still, it is unlikely that it has to be made as small as the one found for Skyrme and Gogny EDFs.

By contrast, the distance between the 9/2[734] and 5/2[622] one-quasiparticle levels in $^{251}$ Cf and $N=151$ isotones, which are connected to the $1j_{15/2^-}$ and $2g_{9/2^+}$ shells, respectively, is overestimated by almost 1MeV for SLy4, D1S, and D1M, by a few hundred keV for NL1 and UNEDF2, but quite well reproduced by NL3*. This indicates first of all that the pronounced deformed $N=150$ gap visible in the Nilsson diagrams for SLy4, D1S, D1M, and NL1 should be much smaller. In fact, the large deformed gap at $N=150$ that is predicted by a substantial number of nuclear EDFs has already quite often been attributed to be one of the main causes for the disagreement between calculated and observed spectroscopic properties of nuclei in the $A \approx 250$ mass region [1,8,21,,43]. It is remarkable that the UNEDF2 and NL3* functionals that both describe well the distance between the 9/2[734] and 5/2[622] levels give very different shell structure at spherical shape in the Nilsson diagrams of Figs. 1 and 3. For UNEDF2, the spherical $1j_{15/2-}$ shell is about 500keV above the spherical $2g_{9/2^+}$, whereas for NL3* it is the other way round, the latter being a unique feature among the functionals studied here. Note that for NL3* the relative position of these levels is quickly moving with particle number: for the slightly lighter $^{244}$ Cm the $1j_{15/2-}$ is already above $2g_{9/2^+}$ level like for the other functionals studied here, cf. Ref. [20].

Altogether, these findings indicate that on cannot expect to find a unique one-to-one correspondence between the spectra of one-quasiparticle levels of deformed nuclei and spherical single-particle levels in a Nilsson diagram. Different shell structure in the Nilsson diagrams might lead to similar one-quasiparticle spectra and vice versa. Indeed, the rearrangement, polarization, pairing and single-particle-mixing effects when constructing self-consistent one-quasiparticles state make the connection quite complex and apparently also slightly EDF-dependent. In addition, it should be recalled that for nuclei in the $A \approx 250$ region the spherical configuration corresponds to a maximum of the deformation energy landscape and therefore should not be associated with a physical state.

Similar cases can be found for proton levels. All Nilsson diagrams for $^{254}$ No plotted in Figs. 1, 2 and 3 exhibit a substantial spherical $Z=92$ gap that is located between the $2f_{7/2^-}$ and $1h_{9/2^-}$ orbitals in non-relativistic functionals and between the $1i_{13/2}$ and $1h_{9/2}$ orbitals in covariant functionals. It has been pointed out in Refs. [69,70] that there is no indication for such a gap, which is also visible in the single-particle spectra of lighter spherical nuclei [63], in the available spectroscopic data for the spherical $^{216}$ Th and $^{218}$ U nuclei. However, the $7/2[514]$, $3/2[521]$, $1/2[521]$, and $7/2[633]$ one-quasiparticle levels of $^{249}$ Bk and the Es isotopes, which originate from the spherical $1h_{9/2}$, $2f_{7/2}$ and $1i_{13/2}$ subshells surrounding the spherical $Z=92$ gap, are reasonably well described by all EDFs employed here, see Figs. 4-8. As a consequence, a large spherical $Z=92$ gap in the Nilsson diagram of $^{254}$ No is not in apparent conflict with the available data for deformed nuclei in the $A \approx 250$ mass region. In particular the non-relativistic Skyrme and Gogny functionals describe well the relative position of the $3/2[512]$ and $7/2[514]$ levels within a few hundred keV. It is only for the relativistic NL1 and NL3* functionals that the spacing between the $7/2[514]$ and $7/2[633]$ one-quasiparticle levels becomes slightly too small, cf. Figs. 4 and 8, which points to a slight overestimation of the spherical $Z=92$ gap for these functionals. Still, the necessary shift of the spherical shells that one would expect to correct for the disagreement between calculation and data will not even reduce the gap to the size found with non-relativistic EDFs.

It is necessary to recognize that the present investigation represents one of first steps in the direction of understanding of the accuracy of the description of one-quasiparticle states in deformed nuclei, which, however, goes beyond previous attempts by directly comparing different classes of the EDF approaches for the same set of experimental data. Based on the current set of experimental data similar accuracy is achieved for employed EDF's. So far statistical analysis of the accuracy of the description of one-quasiparticle states employing full set of available experimental data on proton and neutron states has been performed only in actinides and only in the framework of CDFT using NL1 and NL3* EDF's in Ref. [13]. Although many of the states are described with acceptable accuracy, for some states the deviation of calculated energy from experimental one exceeds 1 MeV.

Less systematic studies have been performed in the non-relativistic EDF approaches. Global survey of the ground state configurations in odd-mass nuclei employing three Skyrme functionals has been performed in Ref. [71]. In Skyrme EDF, the spectra of few actinides and of odd-proton Ho nuclei have been studied in Refs. [7,37]. The spectra of selected Rb, Y, and Nb nuclei have been studied in axial Gogny EDF in Refs. [72,73].

These investigations reveal the same two sources of uncertainties [13]. Although the same set of single-particle states appear in the vicinity of the Fermi level as in experiment, their relative positions and energies are not always correct. This is first source of uncertainties. The second source of uncertainty is related to stretched energy scale in model calculations as compared with experiment which is due to low effective mass of the nucleon at the Fermi level.

While the solution of the first problem can be attempted in the EDF framework, the current analysis with different classes of EDF approaches suggest that it is not likely to remove all existing problems in the description of the single-particle spectra (see also the discussion in Ref. [13]). More comprehensive solution, which would also address the second source of uncertainty, would require taking into account particle-vibration coupling which will explain existing fragmentation of the single-particle states and possibly compress the calculated one-quasiparticle spectra bringing them closer to experiment. Combined with respective re-parametrization of the functionals it may lead to the functionals with better spectroscopic quality. It is clear that existing functionals are biased towards bulk properties since either no (CDFT) or extremely limited information on single-particle properties (Skyrme and Gogny EDF) is used in their fitting protocols. Even in this situation the calculated spectra are not far away from experiment which can be considered as a success and a good starting point for future development.