Covariant density functional theory: Reexamining the structure of superheavy nuclei.
A systematic investigation of even-even superheavy elements in the region of proton numbers and in the region of neutron numbers from the proton-drip line up to neutron number is presented. For this study we use five most up-to-date covariant energy density functionals of different types, with a non-linear meson coupling, with density dependent meson couplings, and with density-dependent zero-range interactions. Pairing correlations are treated within relativistic Hartree-Bogoliubov (RHB) theory based on an effective separable particle-particle interaction of finite range and deformation effects are taken into account. This allows us to assess the spread of theoretical predictions within the present covariant models for the binding energies, deformation parameters, shell structures and -decay half-lives. Contrary to the previous studies in covariant density functional theory, it was found that the impact of spherical shell gap on the structure of superheavy elements is very limited. Similar to non-relativistic functionals some covariant functionals predict the important role played by the spherical gap. For these functionals (NL3*, DD-ME2 and PC-PK1), there is a band of spherical nuclei along and near the and lines. However, for other functionals (DD-PC1 and DD-ME) oblate shapes dominate at and in the vicinity of these lines. Available experimental data are in general described with comparable accuracy and do not allow to discriminate these predictions.
pacs:21.10.Dr, 21.10.Pc, 21.60.Jz, 27.90.+b
Science is driven by the efforts to understand unknowns. In low-energy nuclear physics many of such unknowns are located at the extremes of the nuclear landscape Erler et al. (2012); Afanasjev et al. (2013); Agbemava et al. (2014a). The region of superheavy elements (SHE), characterized by the extreme values of proton number , is one of such extremes. Contrary to other regions of the nuclear chart, the SHEs are stabilized only by quantum shell effects. Because of this attractive feature and the desire to extend the nuclear landscape to higher values, this region is an arena of active experimental and theoretical studies.
Currently available experimental data reach proton number Oganessian et al. (2006, 2012) and dedicated experimental facilities such as the Dubna Superheavy Element Factory will hopefully allow to extend the region of SHEs up to and for a wider range on neutron numbers at lower values. Unfortunately, even this facility will not be able to reach the predicted centers of the island of stability of SHEs at , and as given by microscopic+macroscopic (MM) approaches Nilsson et al. (1968, 1969); Patyk and Sobiczewski (1991); Ćwiok et al. (1996); Möller and Nix (1994) or by covariant Rutz et al. (1997); Bender et al. (1999); Afanasjev et al. (2003); Zhang et al. (2005) and Skyrme Ćwiok et al. (1996); Bender et al. (1999) density functional theories (DFT), respectively.
One has to recognize that the majority of systematic DFT studies of the shell structure of SHEs has been performed in the 90ies of the last century and at the beginning of the last decade. These studies indicate that the physics of SHEs is much richer in the DFT framework than in MM approaches. This is due to self-consistency effects which are absent in the MM approaches. For example, they manifest themselves by a central depression in the density distribution of spherical SHEs Bender et al. (1999); Afanasjev and Frauendorf (2005) which has not been seen in the MM approaches. Moreover, besides the successful covariant energy density functionals (CEDFs) NL1 Reinhard et al. (1986) and NL3 Lalazissis et al. (1997), during the last ten years a new generation of energy density functionals has been developed both in covariant Lalazissis et al. (2005); Nikšić et al. (2008); Lalazissis et al. (2009a); Zhao et al. (2010); Roca-Maza et al. (2011) and in non-relativistic Kortelainen et al. (2014); Goriely et al. (2009) frameworks; they are characterized by an improved global performance Kortelainen et al. (2014); Goriely et al. (2009); Agbemava et al. (2014a). In addition, the experimental data on SHEs became much richer Herzberg and Greenlees (2008); Oganessian and Utyonkov (2015) in these years.
In such a situation it is necessary to reanalyze the structure of superheavy nuclei using both the full set of available experimental data on SHEs and the new generation of energy density functionals. There are several goals of this study. First, we will investigate the accuracy of the description of known SHEs with the new generation of covariant energy density functionals and find whether the analysis of existing experimental data allows to distinguish the predictions of different functionals for nuclei beyond the known region of SHEs. Second, the comparative analysis of the results obtained with several state-of-the-art functionals will allow to estimate the spreads of theoretical predictions when extending the region of SHEs beyond the presently known, to establish their major sources and to define the physical observables and regions of SHEs which are less affected by these spreads.
This is a very ambitious goal and, as a consequence, several restrictions are imposed. This study is performed only in the framework of covariant density functional theory (CDFT) Vretenar et al. (2005). We use the covariant energy density functionals (CEDF) NL3* Lalazissis et al. (2009a), DD-ME2 Lalazissis et al. (2005), DD-ME Roca-Maza et al. (2011), DD-PC1 Nikšić et al. (2008) and PC-PK1 Zhao et al. (2010). They are the state-of-the-art functionals representing the major classes of CDFTs (for more details see the discussion in Sect. II of Ref. Agbemava et al. (2014a))111Note that the functional PC-PK1 has not been used and discussed in Ref. Agbemava et al. (2014a) because global studies with it have been performed by the Peking group in Ref. Zhang et al. (2014). It is a state-of-the-art functional for point coupling models with cubic and quartic interactions of zero range Zhao et al. (2010).. Moreover, their global performance has recently been analyzed in Refs. Agbemava et al. (2014a); Zhang et al. (2014) and they are characterized by an improved description of experimental data as compared with previous generation of CEDFs. Moreover, the study of Refs. Agbemava et al. (2014a); Afanasjev et al. (2015) provides theoretical spreads in the description of known nuclei and their propagation towards the neutron-drip line obtained with four CEDFs (NL3*, DD-ME2, DD-ME and DD-PC1). This is in contrast with earlier studies of SHEs in the CDFT framework based on the functionals whose performance was tested only in very restricted regions of the nuclear chart and for which no analysis of theoretical spreads has been carried out.
Contrary to many earlier CDFT studies of the shell structure in superheavy nuclei restricted to spherical shape, in this investigation the effects of deformation are taken into account. As it will be shown later, neglecting deformation can lead to wrong conclusions since many SHEs are characterized by soft potential energy surfaces with coexisting minima. Considering the numerical complexity of global investigations we restrict our calculations to axial reflection symmetric shapes. Such calculations are realistic for the absolute majority of the ground states. Octupole deformation does not play a role in the ground states of SHEs Prassa et al. (2012); Agbemava et al. (2014b) but it affects the properties of the outer fission barriers Abusara et al. (2012); Prassa et al. (2012). In the current manuscript those are not considered in detail (see the discussion in Sect. IV.1). Although triaxiality may play a role in the ground states of some SHEs Ćwiok et al. (2005), such cases are rather exceptions than the rule Abusara et al. (2012); Ćwiok et al. (2005); Prassa et al. (2013, 2012). Moreover, model predictions for stable triaxial deformation in the ground states vary drastically between the models, even in experimentally known nuclei Möller et al. (2008), and are frequently not supported by the analysis of experimental data Snyder et al. (2013). In addition, triaxial RHB Afanasjev et al. (2000) calculations are at present too time-consuming to be undertaken on a global scale. These arguments justify neglecting of triaxiality in the description of the ground states in this investigation.
In addition we restrict our investigation to even-even nuclei. Unfortunately, no reliable configuration assignments exist for ground states of experimentally known odd-mass SHEs to be confronted with the theory. However, systematic studies of the accuracy of the reproduction of energies of deformed one-quasiparticle states in actinides are available for the CEDF NL3* Afanasjev and Shawaqfeh (2011); Dobaczewski et al. (2015).
The paper is organized as follows. Section II describes the details of the calculations. The single-particle structure and shell gaps at spherical shape together with the spreads in their description are discussed in Sec. III. The impact of deformation on the properties of SHEs is considered in Sect. IV. Section V contains the systematics of calculated charge quadrupole deformations. The validity of the and quantities as the indicators of shell gaps is discussed in Sec. VI. We report on masses and separation energies in Sec. VII. The -decay properties are considered in Sec. VIII. Finally, Sec. IX summarizes the results of our work.
Ii The details of the theoretical calculations
In the present manuscript, the RHB framework is used for systematic studies of all even-even actinides and SHEs from the proton-drip line up to neutron number . The details of this formalism have been discussed in Secs. II-IV of Ref. Agbemava et al. (2014a) and Sec. II of Ref. Afanasjev et al. (2015). Thus, we only provide a general outline of the features specific for the current RHB calculations.
We consider only axial and parity-conserving intrinsic states and solve the RHB-equations in an axially deformed oscillator basis Koepf and Ring (1988); Gambhir et al. (1990); Ring et al. (1997); Nikšić et al. (2014). The truncation of the basis is performed in such a way that all states belonging to the shells up to fermionic shells and bosonic shells are taken into account. As tested in a number of calculations with and for heavy neutron-rich nuclei, this truncation scheme provides sufficient numerical accuracy. For each nucleus the potential energy curve is obtained in a large deformation range from up to in steps of by means of a constraint on the quadrupole moment . Then, the correct ground state configuration and its energy are defined; this procedure is especially important for the cases of shape coexistence.
In order to avoid the uncertainties connected with the definition of the size of the pairing window, we use the separable form of the finite range Gogny pairing interaction introduced by Tian et al Tian et al. (2009). Its matrix elements in -space have the form
with and being the center of mass and relative coordinates. The form factor is of Gaussian shape
The two parameters fm and fm of this interaction are the same for protons and neutrons and have been derived in Ref. Tian et al. (2009) by a mapping of the S pairing gap of infinite nuclear matter to that of the Gogny force D1S Berger et al. (1991).
As follows from the RHB studies with the CEDF NL3* of odd-even mass staggerings, moments of inertia and pairing gaps the Gogny D1S pairing and its separable form (Eq. (II)) work well in actinides (Refs. Afanasjev and Abdurazakov (2013); Agbemava et al. (2014a); Dobaczewski et al. (2015)). The weak dependence of its pairing strength on the CEDF has been seen in the studies of pairing and rotational properties of actinides in Refs. Afanasjev et al. (2003); Afanasjev and Abdurazakov (2013), of pairing gaps in spherical nuclei in Ref. Agbemava et al. (2014a) and of pairing energies in Ref. Afanasjev et al. (2015). Thus, the same pairing (Eq. (II)) is used also in the calculations with DD-PC1, DD-ME2, DD-ME, and PC-PK1. Considering the global character of this study, this is a reasonable choice.
Any extrapolation beyond the known region requires some estimate of theoretical uncertainties. This issue has been discussed in the details in Refs. Reinhard and Nazarewicz (2010); Dobaczewski et al. (2014) and in the context of global studies within CDFT in the introduction of Ref. Agbemava et al. (2014a). In the present manuscript, we concentrate on the uncertainties related to the present choice of energy density functionals which can be relatively easily deduced globally Agbemava et al. (2014a). We therefore define spreads of theoretical predictions for a given physical observable as Agbemava et al. (2014a)
where and are the largest and smallest values of the physical observable obtained with the employed set of CEDFs for the nucleus. Note that these spreads are only a crude approximation to the systematic theoretical errors discussed in Ref. Dobaczewski et al. (2014) since they are due to a very small number of functionals which do not form an independent statistical ensemble. Despite this fact they provide an understanding which observables/aspects of many-body physics can be predicted with a higher level of confidence than others for density functionals of the given type. Moreover, it is expected that they will indicate which aspects of many-body problem have to be addressed with more care during the development of next generation of EDFs.
Iii Single-particle structures at spherical shape
As discussed in the introduction, superheavy nuclei are stabilized by shell effects, i.e. by a large shell gap or at least a considerably reduced density of the single-particle states. Therefore, since a long time an island of stability has been predicted in the CDFT for very heavy nuclei in the region around the proton number Rutz et al. (1997); Bender et al. (1999); Afanasjev et al. (2003); Zhang et al. (2005). Fig. 1 shows neutron and proton single-particle spectra of the nuclei 120 and 120 obtained in spherical relativistic mean field (RMF) calculations. Similar figures have been given for the functionals NL3 and DD-ME2 in Refs. Bender et al. (1999); Li et al. (2014). Note that a detailed comparison of several other covariant and non-relativistic Skyrme functionals is presented in Ref. Bender et al. (1999). In Fig. 1 we show the results for an extended set of 8 CEDFs. The global performance of the CEDFs NL3*, DD-ME2, DD-ME and DD-PC1 has been studied in Ref. Agbemava et al. (2014a). The shell gaps at and at are especially pronounced in the nucleus 120 (left panels of Fig. 1). This is a consequence of the presence of a central depression in the density distribution generated by a predominant occupation of the high- orbitals above the occupied single-particle states in Pb. Because of their large -values these orbitals produce density at the surface of the nucleus. Filling up the low- neutron orbitals above the Fermi surface of the 120 nucleus on going from up to leads to a flatter density distribution in the system Afanasjev and Frauendorf (2005) . As a consequence, the and shell gaps are reduced and gap is increased (right panels of Fig. 1). As one can see in Fig. 1, these features are rather general and do not depend much on the specific density functional.
Of course, as shown in Fig. 1, there are theoretical uncertainties in the description of the single-particle energies and in their relative positions. The precise size of the large shell gaps depend on the functional. The corresponding spreads are summarized in Fig. 2a, which shows the average sizes of these shell gaps and the spreads in their predictions. These gaps in the superheavy region are compared with the calculated gaps in lighter doubly magic nuclei, such as Ni, Sn, Sn and Pb.
Since the nuclear radius , i.e. the average width of the potential, increases with the mass number , the shell gaps decrease with and by this reason we show in Fig. 2b the shell gaps and their spreads scaled with a factor . These scaled shell gaps are considerably more constant with , but there is still a tendency that even the scaled gaps decrease in general with . This is probably related to the spin-orbit coupling, which is proportional to the orbital angular momentum , since it causes an increasing downward shift of the high- intruder levels. The spreads give some information on the theoretical uncertainties of the sizes of the calculated gaps. Definitely, the impact of these spreads on the model predictions depends on the ratio of their size with respect of the size of calculated shell gaps. The presence of theoretical spreads has less severe consequences on the predictions of spherical nuclei around magic gaps at and than on similar predictions for SHEs since the former typically have larger shell gaps for comparable theoretical uncertainties. Of course, this is only true in general. The shell gap in the nucleus 120 forms an exception. It is more or less the same for all the CEDFs under consideration and therefore its uncertainty deduced from these spreads is relatively small.
It is evident that the predictive power for new shell gaps in the superheavy region depends on the quality of the description of the single-particle energies of the various CEDFs. Considering Figs. 1 and 2 one can hope that an improvement in the DFT description of the energies of the single-particle states in known nuclei will also reduce the uncertainties in the prediction of the shell structure of the SHEs. It is well known that in DFT the single-particle energies are auxiliary quantities and there are problems in their precise description within this framework. It is generally assumed that this has two reasons. First the coupling of the single-particle motion to low-lying surface vibrations has to be taken into account, and second there is not enough known about the influence of additional terms in the Lagrangian, such as tensor forces Lalazissis et al. (2009b). Particle-vibrational coupling is particularly large in spherical nuclei. So far, its influence on the accuracy of the description of the single-particle energies and on the sizes of shell gaps has been studied in relativistic particle-vibration Litvinova and Ring (2006); Litvinova and Afanasjev (2011) and quasiparticle-vibration Afanasjev and Litvinova (2015) coupling models with the CEDFs NL3 Lalazissis et al. (1997) and NL3* Lalazissis et al. (2009a) only. The experimentally known gaps of Ni, Sn and Pb are reasonably well described in the relativistic particle-vibration calculations of Ref. Litvinova and Afanasjev (2011). Also, the impact of particle-vibration coupling on spherical shell gaps in SHEs has been investigated in Refs. Litvinova and Afanasjev (2011); Litvinova (2012). Although this effect, in general, decreases the size of shell gaps, the gap still remains reasonably large but there is a competition between the smaller and gaps. The accuracy of the description of the energies of one-quasiparticle deformed states in the rare-earth and actinide region has been statistically evaluated in Ref. Afanasjev and Shawaqfeh (2011) within the framework of relativistic Hartree-Bogoliubov theory. On the one hand, these studies have proven some success of CDFT: the covariant functionals provide a reasonable description of the single-particle properties despite the fact that such observables were not used in their fit. On the other hand, they illustrate the need for a better description of the single-particle energies.
Iv The impact of deformation on the properties of superheavy nuclei.
Although it is commonly accepted that the large spherical shell gaps at and define the center of the island of stability of SHEs for the majority of the covariant functionals Bender et al. (1999); Afanasjev et al. (2003), these conclusions were mostly obtained in investigations restricted to spherical shapes. In addition, some calculations suggest Li et al. (2014); Bender et al. (2001), or do not exclude Afanasjev et al. (2003), the existence of a spherical shell gap at the neutron number . However, as discussed below, the inclusion of deformation can change the situation drastically for some functionals.
To illustrate this fact, the deformation energy curves of the isotopes and the isotones are presented in Figs. 3 and 4. Here we restrict our considerations to five CEDFs, namely, NL3*, DD-ME2, DD-ME, DD-PC1 and PC-PK1, whose global performance is well established Agbemava et al. (2014a); Zhang et al. (2014). In the following discussion we neglect the prolate superdeformed minimum, which is sometimes even lower than the spherical or oblate minimum, because of the reasons discussed in detail in Sec. IV.1. In Figs. 3 and 4 the lowest spherical or oblate minimum is considered as the ground state and indicated by a dashed horizonal line. In Fig. 3 we see that the ground states of the isotopes with are spherical for NL3*, DD-ME2, and PC-PK1. This is a consequence of the presence of the large spherical shell gap (see Fig. 1). For these three functionals, the increase of neutron number leads to softer potential energy curves for values between and . As a result, for an oblate minimum either becomes lowest in energy (for NL3*) or competes in energy with the spherical solution (for DD-ME2 and PC-PK1). This softness of the potential energy curves is even more pronounced for the DD-ME and DD-PC1, for which the oblate solution is lower in energy than the spherical solution in all displayed nuclei apart from (Fig. 3).
Although it is tempting to relate this feature to the fact that the size of the gap is smallest among the employed functionals for DD-ME in the nucleus and for DD-PC1 in the nucleus (see Fig. 1a and b), this explanation is too simplistic. This is because even for the cases when the sizes of the gap are very similar (compare, for example, their sizes for DD-ME2 and DD-ME in the nucleus [Fig. 1b]), the deformations of their minima in the ground state are different. This strongly suggest that the evolution of the single-particle structure with deformation, which leads to negative shell correction energies at oblate shape, is responsible for the observed features. Thus, not only the size of the spherical shell gaps but also the location of the single-particle states below and above these gaps is responsible for the observed features.
This is clearly seen in the Nilsson diagrams presented in Fig. 5 for the nucleus 120. Pronounced deformed shell gaps in the proton subsystem are clearly seen for at , for at , and for at . Although, in detail, the size of these deformed gaps, some of which are comparable in magnitude with the spherical gap, depends on the functional, they are present both for DD-PC1 and for NL3*. The most pronounced deformed neutron shell gap is seen at for ; the size of this gap is comparable with the spherical shell gap at . At similar deformations somewhat smaller deformed shell gaps are seen at .
It is important to recognize that contrary to the spherical states with a degeneracy of , deformed states are only two-fold degenerate. This will also impact the shell correction energy since it depends on the averaged density of the single-particle states in the vicinity of the Fermi surface Strutinsky (1966, 1967, 1968). As a result, close to the above discussed deformed shell gaps the negative shell correction energy can be larger in absolute value than the one at spherical shape even for similar sizes of the respective deformed and spherical shell gaps. In the language of the microscopic+macroscopic approach, this difference can be sufficient to counteract the increase of the energy of the liquid drop with increasing oblate deformation in SHEs Nilsson and Ragnarsson . The consequences of this interplay between shell correction and liquid drop energies and the role played by the low level density of the single-particle states in the vicinity of above discussed deformed shell gaps are clearly visible in the potential energy curves of the 120 nucleus presented in Fig. 3 for DD-PC1 and NL3*. For DD-PC1, the ground state is oblate with deformation . However, two excited minima are also seen at and . Although the ground state of the nucleus 120 is spherical for NL3*, three minima at , , and are seen at excitation energies of around 1 MeV. These local minima are the consequence of the fact that the corresponding minima in the proton and neutron shell correction energies correspond to different deformations.
Similar features are also observed for the isotones in Fig. 4. The nucleus Hs () has a well pronounced spherical minimum for NL3*, DD-ME2, and PC-PK1. For these functionals, the increase of proton number leads to an increase of softness in the potential energy surface for . However, the ground state remains spherical up to . On the other side, the ground state of the nucleus becomes oblate in these three functionals. The situation is completely different for the functionals DD-ME and DD-PC1 for which all nuclei shown in Fig. 4 are characterized by soft potential energy curves in the range and by oblate ground states.
iv.1 Comment on superdeformed minima and outer fission barriers
The axial RHB calculations restricted to reflection symmetric shapes show that there exists a second (superdeformed [SD]) minimum with deformation of or higher for all the nuclei under investigation (Figs. 3 and 4). In the nucleus it is in energy close to the ground state for NL3* and DD-ME2 but lower in energy than the spherical or oblate minimum for DD-ME, DD-PC1 and PC-PK1 (see Fig. 3). With the increase of neutron number the SD minimum becomes the lowest in energy in all nuclei (see also Ref. Ren et al. (2001)). A similar situation is also observed for the isotones. In the and isotopes, the spherical minimum is the lowest in energy for NL3*, DD-ME2 and PC-PK1 (see Fig. 4). With increasing proton number , the superdeformed minimum becomes the lowest in energy. The situation is different for DD-ME and DD-PC1 because (i) in these functionals the superdeformed minimum is the lowest in energy in all nuclei shown in Fig. 4 and (ii) the potential energy curves of the and 116 nuclei are much softer [with relatively small inner and outer fission barriers] than in other functionals. It is necessary to conclude that the relative energies of the spherical/oblate and superdeformed minima strongly depend on the functional.
Whether these superdeformed states are stable, metastable, or unstable should be defined by the height (with respect of the SD minimum) and the width of the outer fission barrier. The results presented in Figs. 3 and 4 show that while in some nuclei this barrier is appreciable, it is extremely small in others. Moreover, it was demonstrated in systematic RMF+BCS calculations with NL3* for the nuclei that the inclusion of triaxial or octupole deformation decreases this barrier substantially by 2 to 4 MeV so it is around or less than 2 MeV in the nuclei studied in Ref. Abusara et al. (2012). Calculations with DD-PC1 and DD-ME2 for six nuclei centered around led to similar results Abusara et al. (2012).
The impact of octupole deformation on the outer fission barriers of SHEs has also been studied in the RMF+BCS calculations with the CEDFs NL3 and NL-Z2 in Ref. Bürvenich et al. (2004). In this work, the SD minima exist in the calculations without octupole deformation (see Fig. 5 in Ref. Bürvenich et al. (2004)). Since the heights of outer fission barriers in the axial reflection symmetric calculations is lower in NL-Z2 than in NL3, the inclusion of octupole deformation completely eliminates the outer fission barriers in NL-Z2 but keeps their heights around 2.5 MeV in the NL3 functional. In addition, it was demonstrated in actinides in the RMF+BCS calculations of Ref. Lu et al. (2014) that non-axial octupole deformation can further reduce the height of outer fission barrier by MeV. Similar effect may be expected also in superheavy nuclei.
Thus, a detailed analysis of the outer fission barriers requires symmetry unrestricted calculations in the RHB framework which are extremely time-consuming. So far no such studies exist. Even the symmetry unrestricted RMF+BCS calculations (which are at least by one order of magnitude less time consuming than the RHB calculations) have been performed only for actinides Lu et al. (2012, 2014); Zhao et al. (2015) in which experimental data on outer fission barriers exist. Global symmetry unrestricted RHB calculations for SHE have to be left for the future. On the other hand, from the discussion above it is clear that outer fission barriers are expected to be around 2 MeV or less. The results of Refs. Bürvenich et al. (2004); Abusara et al. (2012) do not cover SHE beyond and lines. However, Figs. 3 and 4 show a clear trend for the decrease of the height of outer fission barrier beyond these lines; for these nuclei its height is less than 2 MeV in many functionals even in axially symmetric calculations restricted to reflection symmetric shapes. Thus, it is already clear that the low outer fission barriers with barrier heights around 2 MeV or less existing in the majority of the CDFT calculations discussed above would translate into a high penetration probability for spontaneous fission, such that most likely these superdeformed states (even if they exist) are metastable. Moreover, non-relativistic calculations usually do not produce a superdeformed minimum and an outer fission barrier in superheavy nuclei Bürvenich et al. (2004); Kowal et al. (2010); Möller et al. (2009); Staszczak et al. (2013); Warda and Egido (2012).
In addition, existing experimental data on SHE (such as total evaporation-residue cross section or spontaneous fission half-lives) Oganessian and Utyonkov (2015) do not show any abrupt deviation from the expected trends which could be interpreted as a transition to a superdeformed ground state.
These are the reasons why we restrict our consideration to the ground states associated with either normal-deformed prolate, oblate or spherical minima.
V The systematics of the deformations
The calculated charge quadrupole deformations of the ground states for five CEDFs are plotted in Fig. 3. They are shown for the nuclei located between the two-proton drip line (see Table IV in Ref. Agbemava et al. (2014a)) and . The impact of the large spherical shell gaps discussed in Sect. III on the structure of superheavy nuclei can be accessed via the analysis of the width of the band of spherical nuclei shown by gray color in the chart of Fig. 3. The width of this gray region along a specific particle number corresponding to a shell closure indicates the impact of this shell closure on the structure of neighbouring nuclei. For example, for NL3* the width of such a band at is on average three even-even nuclei in the direction for and the width of a corresponding band at is on average four even-even nuclei in the direction for . This result is contrary to existing discussions in covariant density functional theory which emphasize the impact of the shell gap over the gap. Our results clearly show that the effect of the spherical shell gap on the equilibrium deformation is more pronounced as compared with the gap. A similar situation exist in the calculations with PC-PK1 (see Fig. 3e). However, the impact of the and the spherical shell gaps becomes less pronounced for DD-ME2 (see Fig. 3b).
The impact of the spherical shell gap is significantly reduced for DD-ME and DD-PC1; only the nuclei with and are spherical for those two functionals. The impact of the shell gap is also considerably decreased; the ground states of the nuclei are spherical only for in DD-ME and for in DD-PC1 (see Figs. 3c and d). Note also that the band of spherical nuclei around is narrow for DD-PC1. These results are in contradiction to the expectation that the large size of the spherical gap in Fig. 1 forces the isotopes with to be spherical for a large range of neutron numbers. Note that proton and neutron shell gaps act simultaneously in the vicinity of a nucleus with proton and neutron numbers corresponding to those gaps. Thus, the effect of a single gap is more quantifiable away from this nucleus.
It is interesting to compare these results with the ones obtained for nuclei in Fig. 17 of Ref. Agbemava et al. (2014a) with NL3*, DD-ME2, DD-ME and DD-PC1. In these nuclei the neutron , and 184 shell gaps have a more pronounced effect on the nuclear deformations as compared with the proton shell gaps at and . This feature was common to all the CEDFs used in Ref. Agbemava et al. (2014a). However, the width of the band of spherical and near-spherical nuclei along these neutron numbers was broader in NL3* as compared with other functionals under consideration. We see the same feature also in superheavy nuclei along the shell closure (see Fig. 3); note that PC-PK1 was not used in Ref. Agbemava et al. (2014a).
The RHB results for superheavy nuclei show the unusual feature that the ground states of the nuclei outside the band of spherical or near-spherical shapes (shown by gray color in Fig. 3) have oblate shapes for NL3*, DD-ME2 and PC-PK1. This is contrary to the usual situation observed in the lighter nuclei (see, for example, Fig. 17 in Ref. Agbemava et al. (2014a)) where, between shell closures, the nuclei change their shape with increasing particle numbers from spherical to prolate, then to oblate, and finally back to spherical. It is interesting to see that the systematic microscopic+macroscopic (MM) calculations of Ref. Jachimowicz et al. (2011) based on the Woods-Saxon potential also show a similar preponderance of oblate shapes in the ground states of superheavy nuclei. These MM results were also confirmed by the calculations for a few nuclei performed within Skyrme DFT with the functional SLy6 Jachimowicz et al. (2011). The situation is even more drastic for the CEDFs DD-ME (DD-PC1) in which no (or very limited) indications of spherical shapes are seen on passing through the nuclei with or . When comparing our results with other calculations one has to keep in mind that not all published results extend to a sufficiently large deformation for oblate shapes (see, for example, Ref. Warda and Egido (2012)).
Fig. 7 shows the map of calculated charge quadrupole deformations with experimentally known nuclei indicated by open circles. One can see that, apart from the nuclei, the predictions of these two functionals (PC-PK1 and DD-PC1) for the equilibrium deformations of experimentally known even-even nuclei are very similar. For these nuclei, PC-PK1 predicts the gradual transition from prolate to spherical shape on going from to . The same happens also for NL3* and DD-ME2 (Fig. 3 a and b). On the contrary, for DD-PC1 the transition from the prolate to oblate minimum is predicted for experimentally known nuclei on going from to and all experimentally known nuclei are expected to be oblate. The same happens also for DD-ME. However, because of the limited scope of experimental data these differences in the description of experimentally known and nuclei between DD-PC1/DD-ME and PC-PK1/NL3*/DD-ME2 cannot be discriminated.
Spreads in the theoretical predictions of the charge quadrupole deformations are shown in left panel of Fig. 8. They are very small in the region of known nuclei and for . Only very few experimentally known nuclei with and 118 are located in the region where substantial theoretical spreads exist (compare Fig. 8 with Fig. 7). However, as discussed above available experimental data on these nuclei does not allow to discriminate different predictions. Quite large spreads exist in the region near the and lines. This is because spherical ground states are predicted in this region by NL3*, DD-ME2 and PC-PK1, while DD-ME and DD-PC1 favor oblate shapes in these nuclei. Very large spreads exist in the region; this is a region where a transition from prolate to oblate shape is seen in the calculations and it takes place at different positions in the chart for the different functionals (see Fig. 3). The theoretical spreads become small again in the upper right corner of the chart; here they are substantial only in several nuclei (shown by green color) which form a “line” parallel to the two-proton drip line. This “line” is a consequence of the fact that the transition from ground state deformations to takes place for different functionals at different positions in the chart (see Fig. 3).
The right panel of Fig. 8 shows theoretical spreads for the case when the functional DD-ME is excluded from consideration. This functional provides unrealistically low heights of inner fission barriers in SHEs Agbemava et al. (tion) and thus it is very unlikely that this functional is appropriate for the region of SHEs. However, its exclusion from consideration reduces only slightly the theoretical spreads.
It is interesting to compare the results of the present analysis of theoretical spreads in the description of ground state deformations with the global analysis presented in Ref. Agbemava et al. (2014a) for nuclei. It is clear that the region of SHEs in the vicinity of the and lines bears the mark of a transitional region characterized either by soft potential energy surfaces or by shape coexistence. This is the source of large theoretical spreads in the prediction of ground state deformations which exist not only in the region of SHEs but also globally (see Ref. Agbemava et al. (2014a)). Both in SHEs and globally these uncertainties are attributable to the deficiencies of the current generation of functionals with respect to the description of single-particle energies.
Vi The quantities and as indicators of shell gaps
Here is the binding energy and is the two-neutron separation energy. The quantity , being related to the second derivative of the binding energy as a function of the neutron number, is a more sensitive indicator of the local decrease in the single-particle density associated with a shell gap than the two-neutron separation energy . This quantity is frequently called as two-neutron shell gap. In a similar way, for protons, is defined as
However, as discussed in detail in Ref. Afanasjev et al. (2003), many factors beyond the size of the single-particle shell gap contribute to and , as for instance deformation and pairing changes. For example, the global analysis of these quantities in Ref. Afanasjev et al. (2015) shows that for some values becomes negative because of deformation changes. Since by definition the shell gap has to be positive, it is clear that the quantities cannot serve as explicit measures of the size of the shell gaps.
Unfortunately, in the majority of cases the analysis of these quantities in superheavy nuclei (and thus the conclusions about the underlying shell structure and the gaps) is based on the results of spherical calculations (see, for example, Refs. Bender et al. (1999); Zhang et al. (2005); Li et al. (2014)). Thus the possibility of a considerable softness of the potential energy surface leading to a deformed minimum is ignored from the beginning. An alternative way to analyze the shell structure is via the microscopic shell correction energy (see, for example, Ref. Bender et al. (2001)). However, such an analysis is also frequently limited to spherical shapes (Ref. Bender et al. (2001)) and, in addition, the comparison with experiment is less straightforward.
The danger of a misinterpretation of the structure of superheavy nuclei based on the analysis of the quantities and obtained in spherical calculations is illustrated in Fig. 9 on the example of DD-PC1. In spherical calculations, the quantity has pronounced maxima at for and less pronounced maxima at for . Note that in this functional the nucleus is located beyond the two-proton drip line. However, it is clear that the impact of the shell gap does not propagate far away from . The quantity is enhanced in a broad region around and has a maximum for which becomes especially pronounced approaching .
However, in deformed RHB calculations, the quantities and show a picture in many respects quite different from the one obtained in spherical calculations. In addition to the maxima in the quantity at for , which are already seen in spherical calculations, deformed RHB calculations show maxima at (for ) and at (for ). The latter gap appears for a number of covariant functionals instead of the experimentally observed gap (see Refs. Dobaczewski et al. (2015) for details). Note that the maxima in seen at in spherical calculations disappear in deformed RHB calculations. In addition, some isolated peaks in the quantity appear across the nuclear chart of Fig. 9d at specific values of and . In many cases, they originate from rapid deformation changes in going from one nucleus to another.
Even more drastic differences are seen when comparing the quantities obtained in spherical and in deformed RHB calculations. Any indication of the spherical shell gap clearly visible in the spherical case (Fig. 9c), disappear in deformed calculations (Fig. 9d). This is a consequence of the fact that apart of the nuclei with which are spherical in the ground state, all other nuclei in the vicinity of the line are oblate in the ground state (see Fig. 3d). The maxima in the quantity obtained in deformed RHB calculations are located at completely different values as compared with spherical calculations indicating a possible lowering of the single-particle level density at these values. For example, the high values of the quantity seen at around are due to deformed the shell gap which exists for a number of CEDFs Afanasjev et al. (2003); Dobaczewski et al. (2015).
These results clearly illustrate the danger of misinterpretation of the structure of superheavy nuclei when using results of spherical calculations. The presence of large spherical shell gaps will definitely manifest itself in the increase of the relevant or quantities. However, the restriction to spherical shapes does not allow to access the softness of the potential energy surfaces and the presence of large shell gaps at deformation.
Vii Masses and separation energies
In Table 1 we list the rms-deviations between theoretical and experimental binding energies for the nuclei with ; experimental masses from the AME2012 mass evaluation Wang et al. (2012) are used here. The masses given in the AME2012 mass evaluation Wang et al. (2012) can be separated into two groups. One represents nuclei with masses defined only from experimental data, the other contains nuclei with masses depending in addition on either interpolation or short extrapolation procedures. These procedures involve some degree of subjectivity but has proven to provide a quite accurate estimate in absolute majority of the cases as seen from the comparison of these estimates with newly measured masses Audi et al. (2012a). For simplicity, we call the masses of the nuclei in the first and second groups as measured and estimated. Estimated masses frequently involve the ones for unknown nuclei and they are estimated from the trends in mass surfaces Audi et al. (2012a). Note that these mass surfaces also incorporate the information on odd and odd-mass SHEs which are more abundant than their even-even counterparts Oganessian and Utyonkov (2015). Experimental physical observables which depend only on measured masses will be shown later by solid symbols in the figures, while the ones which involve at least one estimated mass by open symbols.
For each employed functional the accuracy of the description of the sets of measured and measured+estimated masses is comparable and does not change substantially when the estimated masses are added to the measured ones (see Table 1). The same is true for the quantities which depend on the mass differences such as the two-neutron (two-proton) separation energies and the values. This fact is important because the measured masses represent only 41% in the set of measured+estimated masses used here. It adds additional support to the estimation procedures used in Refs. Audi et al. (2012a) since global studies of Ref. Agbemava et al. (2014a) indicate that CDFT has a good predictive power in the regions of deformed nuclei with no shape coexistence and the absolute majority of the superheavy nuclei for which measured and estimated masses are provided in Ref. Wang et al. (2012) belong to this type of region.
As compared with the global analysis of Refs. Agbemava et al. (2014a); Zhang et al. (2014), the accuracy of the description of masses is better for DD-PC1 and DD-ME2, comparable for DD-ME and PC-PK1 and worse for NL3*. The best accuracy is achieved for DD-PC1. This is not surprising considering that this functional has been carefully fitted to the binding energies of deformed rare-earth nuclei and actinides in Ref. Nikšić et al. (2008). With respect to masses it outperforms other functionals in these regions (see Figs. 6 and 7 in Ref. Agbemava et al. (2014a)).
Since our investigation is restricted to even-even nuclei, we consider two-neutron and two-proton separation energies. Here stands for the binding energy of a nucleus with protons and neutrons. Apart of the proton subsystem in NL3* and DD-ME2 and the neutron subsystem in NL3*, the two-neutron and the two-proton separation energies are described with a typical accuracy of 0.5 MeV (Table 1). This is better by a factor of two than the global accuracy of around 1 MeV obtained for these functionals in Ref. Agbemava et al. (2014a). The accuracy of the description of separation energies depends on the accuracy of the description of mass differences. As a result, not always the functional which provides the best description of masses gives the best description of two-particle separation energies.
Figs. 12 and 13 present a detailed comparison of calculated and experimental two-neutron and two-proton separation energies. While providing in general comparable descriptions of experimental data, the calculated results differ in details. The experimental data for the Rf, Sg, Hs and Ds isotopes clearly show a sharp decrease of the two-neutron separation energies at which is due to the deformed shell gap at this particle number. This decrease is best described by PC-PK1 (Fig. 12e). DD-ME2 and DD-PC1 overestimate this decrease somewhat (Figs. 12b and d) and NL3* underestimates its size. In contradiction to experiment, DD-ME does not show the presence of a gap at but gives a small deformed shell gap at (Fig. 12c).
It is important to recognize that the conclusions about the deformed shell gap are based on the comparison with two-neutron separation energies extracted from estimated masses. However, this gap is present both in the macroscopic+microscopic calculations of Refs. Patyk and Sobiczewski (1991); Sobiczewski et al. (2001); Sobiczewski and Pomorski (2007) and the DFT calculations based on the Gogny D1S force of Ref. Warda and Egido (2012). In addition, there are indications on the presence of this gap from the analysis of experimental data which suggests that the deformed shell gap is much larger than the gap discussed below Audi et al. (2012a).
For higher values there are indications of the presence of the spherical shell gap. However, there is a substantial difference between the functionals on how far this gap propagates into the region of superheavy nuclei. For example, PC-PK1 and DD-ME2 show the propagation of this gap up to (Figs. 12b and e). On the other hand, this gap is visible only up to the Rf/No region for DD-ME and DD-PC1 (Figs. 12b and e). The results for NL3* are in between of these two extremes (Figs. 12a).
Contrary to the neutron subsystem, experimental two-proton separation energies are smoother as a function of proton number without clear indications of pronounced shell gaps (Fig. 13). One should note that there exist small deformed shell gaps at and in heavy actinides/light superheavy nuclei Afanasjev et al. (2003). They are barely visible in the two-particle separation energies (see Figs. 12 and 13 and Ref. Afanasjev et al. (2003)) and are usually seen in the quantities and (see Sect. VI).
Figs. 10 and 11 show that these deformed gaps at and are not reproduced in the CEDFs under consideration. Indeed, the calculations with NL3*, DD-ME2, DD-ME, and DD-PC1 place them at and . These gaps are clearly seen in the nobelium region in the Nilsson diagrams for NL3* (see Fig. 3 in Ref. Dobaczewski et al. (2015)). On the contrary, a neutron gap is seen at and no proton gap exist in the calculations with DD-ME. These problems exist also in the description of experimental deformed gaps with the older generation of the CEDFs used in Ref. Afanasjev et al. (2003). They place a neutron gap either at (NL3, NLRA1 and NL-Z) or at (NL1 and NL-Z) or do not show a gap at all (NLSH). In the same way, a proton gap is placed at (NL3, NL1, and NL-Z) or does not exist in NLRA1. Only NLSH predicts a proton gap at the right value but it is placed between wrong states Afanasjev et al. (2003). Note that this problem is not specific only for covariant functionals; most of the Skyrme functionals also fail to reproduce these gaps (Ref. Bürvenich et al. (1998); Shi et al. (2014)).
Viii -decay properties
In superheavy nuclei spontaneous fission and emission compete and the shortest half-live determines the dominant decay channel and the total half-live. Only in cases where the spontaneous fission half-live is longer than the half-live of emission superheavy nuclei can be observed in experiment. In addition, only nuclei with half-lives longer than s are observed in experiments.
The decay half-live depends on the values which are calculated according to
with MeV Wang et al. (2012) and and representing the parent nucleus.
The RHB results for the values are compared with experiment in Fig. 14 and the corresponding rms-deviations are listed in Table 1. Based on the results presented in this table, the best agreement is obtained for PC-PK1 closely followed by DD-PC1 and DD-ME, and then by DD-ME2 and NL3*. However, a detailed analysis of these results presented in Fig. 14 clearly indicates that DD-ME completely misses both the position in neutron number and the magnitude of the peak at seen in the experimental data for the Rf, Sg, Hs, and Ds isotope chains. Note, however, that the magnitude of the peak in the experimental data is based on the estimated masses. This peak is a consequence of the deformed shell gap which is not reproduced in this functional (see Sec. VII). The other functionals correctly place this peak at . The best reproduction of the magnitude of this peak is obtained for PC-PK1. The CEDFs DD-PC1 and DD-ME2 (NL3*) somewhat overestimate (underestimate) its magnitude reflecting the accuracy of the reproduction of the size of the shell gap in these CEDFs (see Sec. VII).
The comparison of experimental data with theoretical values obtained with the covariant functionals (Fig. 14 of the present manuscript and Fig. 18 of Ref. Bender et al. (2003)), and with those obtained by non-relativistic models (see, for example, Fig. 18 in Ref. Bender et al. (2003) and Figs. 44 and 45 of Ref. Sobiczewski and Pomorski (2007)) clearly indicates that the available experimental data do not allow to distinguish the predictions of different models with respect to the position of the center of the island of stability.
The -decay half-lives were computed using the phenomenological Viola-Seaborg formula V. E. Viola and Seaborg (1966)
with the parameters , , and of Ref. Dong and Ren (2005).
The comparison of calculated and experimental half-lives for the -decays is presented in Fig. 15. One can see that reasonable agreement is obtained for all functionals especially for the case of PC-PK1. However, the local increase above the general trend of the experimental half-lives near visible in the Cf, Fm and No isotope chains, which is due to deformed shell gap, is not reproduced. Neither of the functionals reproduce the position of this gap (see Sec. VII). For higher neutron numbers all functionals predict an increase of the half-lives as a function of neutron number . This trend is however interrupted in the vicinity of the spherical shell gap with . For some isotope chains a drastic decrease of the half-lives is observed. It is a consequence of the well known fact that for nuclei with two neutrons outside a closed shell -particle emission is easier than for the other nuclei in the same isotopic chain Bao et al. (2014). However, above the trend of increasing half-lives with the increase of neutron number is restored. The impact of the shell gap on the -decay half-lives clearly correlates with the impact of this gap of the deformations of the ground states (Sec. V). In SHEs with high values its impact on the -decay half lives is either substantially decreased or completely vanishes.
In the region under investigation the magnitude of the decay half-lives varies in a very wide range from up to s (or even higher for the Cf, Fm and No nuclei with ). For some SHEs with high- values the calculated half-lives fell below the experimental observation limit of s.
Despite the fact that the existing experimental data on the -decay half-lives is described with comparable accuracy by the different functionals, for unknown regions of nuclear chart there are some cases of substantial difference in their predictions. The most extreme difference is seen in the Cf isotopes, where NL3* and DD-ME2 differ from DD-ME and DD-PC1 by approximately 20 orders of magnitude at neutron number and slightly below it. On the other hand, apart from the region the differences in the predictions of different functionals is smaller for SHEs with where it reaches only few orders of magnitude (Fig. 15). In the region of these nuclei the differences between predictions of different functionals increase by additional few orders of magnitude. However, above these differences decrease with increasing proton number because of the diminishing role of the spherical shell gap. For example, the differences in the predicted -decay half-lives do not exceed two orders of magnitude for the and 130 nuclei.
The performance of covariant energy density functionals in the region of superheavy nuclei has been assessed using the state-of-the-art functionals NL3*, DD-ME2, DD-ME, DD-PC1, and PC-PK1. They represent major classes of covariant functionals with different basic model assumptions and fitting protocols. The available experimental data on ground state properties of even-even nuclei have been confronted with the results of the calculations. For the first time, theoretical spreads in the prediction of physical observables have been investigated in a systematic way in this region of the nuclear chart for covariant density functionals. Special attention has been paid to the propagation of these spreads towards unknown regions of higher values and of more neutron-rich nuclei.
The main results of this work can be summarized as follows:
So far, the absolute majority of investigations of the shell structure of SHEs has been performed in spherical calculations. In the framework of covariant density functional theory, these calculations always indicate a large proton shell gap at , a smaller neutron shell gap at , and, for some functionals, a neutron shell gap at . However, the restriction to spherical shapes does not allow to access the softness of the potential energy surfaces and the presence of competing large shell gaps at deformation. As illustrated in the present manuscript, this restriction has led to a misinterpretation of the shell structure of SHEs. The detailed analysis, with deformation included, shows that the impact of the shell gap is very limited in the space for all functionals under investigation. The impact of the and spherical shell gaps depend drastically on the functional. It is most pronounced for NL3* and PC-PK1 and is (almost) completely absent for DD-PC1 and DD-ME.
Available experimental data (separation energies, -values and -decay half-lives) on SHEs are described with comparable accuracy in covariant (current manuscript) and non-relativistic Bender et al. (2003) DFT calculations. Moreover, these data are not very sensitive to the details of the single-particle structure which define the position of the center of the island of stability. Unfortunately, experimental data on single-particle states in odd-mass SHEs are either not available () or scarce (). In addition, in the latter case, the configuration assignments for many nuclei are not fully reliable Jeppesen et al. (2009). However, the analysis of the available data on deformed single-particle states in the actinides performed with different DFTs in Refs. Afanasjev and Shawaqfeh (2011); Dobaczewski et al. (2015) reveals that the problems in the description of the single-particle structure exist in all models. Considering the existing theoretical spreads, it is clear that the available experimental data on SHEs do not allow to distinguish the predictions of different models with respect of the position of the center of the island of stability.
Comparing different functionals one can see that the results obtained with the covariant density functional DD-ME differ substantially from the results of other functionals. This functional is different from all the other functionals used here, because it has been adjusted in Ref. Roca-Maza et al. (2011) using only four phenomenological parameters in addition to some input from ab-inito calculations Baldo et al. (2004); van Dalen et al. (2007). Above it does not predict spherical SHEs. The heights of the inner fission barriers in SHEs with obtained in this functional are significantly lower than the experimental estimates and the values calculated in all other models. In addition, it does not lead to octupole deformation in actinide nuclei which are known to be octupole deformed Agbemava et al. (2014b). All these facts suggest that either the ab-initio input Baldo et al. (2004); van Dalen et al. (2007) for this functional is not precise enough or the number of only four phenomenological parameters (fitted to masses of spherical nuclei) is too small to provide a proper extrapolation to the region of superheavy elements. Thus, this functional is not recommended for future investigations in this area, in spite of the fact that this functional provides a good description of masses and other ground state observables in the nuclei Agbemava et al. (2014a).
Theoretical uncertainties in the predictions of different observables have been quantified. While the uncertainties in the quadrupole deformation of the ground states of known superheavy nuclei are small, they increase on approaching nuclei with and/or . As a result, even the ground state deformations of these nuclei (whether spherical or oblate) cannot be predicted with certainty. Available experimental data do not allow to discriminate between these predictions.
Acknowledgements.This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Number DE-SC0013037 and by the DFG cluster of excellence “Origin and Structure of the Universe” (www.universe-cluster.de). A.A. and T.N. thank the JSPS invitation fellowship in Japan (S-15029) for financial support during completion of the present work. It is also partially supported by JSPS KAKENHI (Grants No. 24105006 and 25287065)) and by ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).
Appendix A Supplemental information on the ground state properties
In addition to the graphical representation of the results, the numerical results for ground state properties obtained with the DD-PC1 and PC-PK1 CEDFs are provided in two tables of the Supplemental Material with this article as Ref. Sup . The structure of these tables is illustrated in Table 2.
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