Structure of Superheavy Nuclei Along Element 115 Decay Chains

A recent high-resolution $\alpha$, $X$-ray, and $\gamma$-ray coincidence-spectroscopy experiment offered first glimpse of excitation schemes of isotopes along $\alpha$-decay chains of $Z=115$. To understand these observations and to make predictions about shell structure of superheavy nuclei below $^{288}115$, we employ two complementary mean-field models: self-consistent Skyrme Energy Density Functional approach and the macroscopic-microscopic Nilsson model. We discuss the spectroscopic information carried by the new data. In particular, candidates for the experimentally observed $E1$ transitions in $^{276}$Mt are proposed. We find that the presence and nature of low-energy $E1$ transitions in well-deformed nuclei around $Z=110, N=168$ strongly depends on the strength of the spin-orbit coupling; hence, it provides an excellent constraint on theoretical models of superheavy nuclei. To clarify competing theoretical scenarios, an experimental search for $E1$ transitions in odd-$A$ systems $^{275,277}$Mt, $^{275}$Hs, and $^{277}$Ds is strongly recommended.

Introduction.-Superheavynuclei at the limit of nuclear mass and atomic number pose a formidable challenge to both experiment and theory.The low cross sections for production of these nuclei, in the picobarn range or less, offer limited structural information.Moreover, the α-decay chains of nuclei synthesized in experiments using the 48 Ca beam with actinide targets [1][2][3][4][5][6][7][8][9] terminate by spontaneous fission before reaching the known region of the nuclear chart.This poses a problem with the unambiguous identification of the new isotopes, and more direct techniques to determine Z and A must be employed [9].Theoretical predictions of the shell structure of superheavy nuclei are also difficult, as the interplay between the electrostatic repulsion and nuclear attraction, combined with a very high density of single-particle (s.p.) states, make the results of calculations extremely sensitive to model details [10][11][12][13][14][15].
In a recent experimental study [9,16], unique structural information on low-lying states in superheavy nuclei below 288 115 has been obtained.Of particular interest is the finding that some of the measured transitions in the nucleus assigned to be 276 Mt have E1 character, thus suggesting opposite parities of the connected states.The new data offer an exciting opportunity to constraint theoretical models in this region for the first time.Indeed, previous self-consistent studies [17,18] have shown that the number of opposite-parity s.p. orbitals around the Fermi level is fairly limited, and this is consistent with the Nilsson model analysis of Ref. [9].
Because of the above-mentioned sensitivity to model details, robust predictions in this region are difficult to make as one is dealing with large extrapolations.To this end, when aiming at the spectroscopic quality of predictions, it is advisable to use a model that performs well in the neighboring region where experimental information is more abundant.Furthermore, since the quadrupole deformations of α-decay daughters of 288 115 are expected to increase gradually with decreasing Z and A along the α-decay chain [13,14,17,18], shape polarization is going to play a role when determining the energies of low-lying states.
In this work, we study the low-lying states in the superheavy nuclei below 288 115, using the locallyoptimized self-consistent Skyrme Energy Density Functional (SEDF) and Nilsson-Strutinsky (NS) frameworks.To assess systematic errors, we also carry out calculations using a globally-optimized SEDF model.
Models.-The SEDF approach is a variant of nuclear density functional theory, which offers a global, selfconsistent description of nuclear properties across the nuclear landscape [19,20].The recent self-consistent study of Ref. [15] offers a locally optimized SEDF parameterization unedf1 SO that meets our local-extrapolability requirements: it reproduces one-quasiparticle (1-q.p.) states in 251 Cf and 249 Bk (the two heaviest systems where 1-q.p. energies are experimentally well known), predicts crucial deformed shell gaps at N = 152 and Z = 100, and describes rotational bands in Fm, No, and Rf isotopes.The parameter set unedf1 SO has been obtained by adjusting the spin-orbit coupling constants of a global SEDF parametrization unedf1 [21] that performs well for heavy nuclei and large deformations.We shall also use unedf1 in this study.The calculations follow closely Ref. [15].The Skyrme Hartree-Fock-Bogolyubov (SHFB) equations were solved using the symmetry-unrestricted solver hfodd (v2.52j) [22] by expanding 1-q.p. wave functions in 680 deformed harmonic-oscillator (HO) basis states.To compute 1-q.p. excitations in odd-A nuclei, we blocked relevant orbits around the Fermi level as described in Ref. [23].The strengths of the pairing force for neutrons and protons were adjusted to the odd-even mass staggering in 251 Cf and 249 Bk and the kinematic moment of inertia of 252 No.The SEDF results are compared with those of the Nilsson-Strutinsky (NS) approach of Ref. [24] with the modified harmonic oscillator (MO) potential and pairing as in Ref. [25].The shell-independent MO parameters (κ p = 0.058, µ p = 0.63, κ n = 0.0526, and µ n = 0.457) have been locally optimized to the actinide nuclei [26] and applied to, e.g., 228,230 Pa [27] and 242 Am [28].
Results.-We first discuss properties of the even-even nuclei belonging to the α-decay chain of 296 120.Their ground states form q.p. vacua for neighboring odd-A and odd-odd systems.The calculated quadrupole moments are shown in Fig. 1.Both SEDF models predict a similar smooth increase of quadrupole deformation along the α-chain.In the NS calculations, 296 120 is nearly spherical, 292 118 and 288 Lv are very weakly deformed, 284 Fl and 280 Cn are spherical, and the shapes of the lightest daughters have deformations close to those predicted by SEDF.These results suggest that a direct comparison between SEDF and NS models is most meaningful for Z < 112.It is instructive to begin the discussion from the Nilsson s.p. diagram of the MO potential shown in Fig. 2. The main features of this diagram, such as the appearance of spherical shell gaps at Z = 114 and N = 184, have remained unchanged since the late 1960s [29,30].The deformed shell structure of nuclei at the end of the and protons (bottom) for nuclei along the α-decay chain of 296 120 using the MO potential of Ref. [26].The orbits are labelled by the standard asymptotic Nilsson numbers.The positive/negative parity levels are marked by solid/dashed lines.The Fermi levels of nuclei in Fig. 1 are indicated by dots.The quadrupole moment was determined from shape deformations 2 and 4: Q20 = 0.8AR The spherical shell structure in superheavy nuclei strongly depends on the spin-orbit splitting, which governs the size of the Z = 114 gap (cf.Table 4 of Ref. [10] and discussion therein).Also, the coupling between Coulomb interaction and nuclear interaction is expected to impact the predictions.To consider both effects, we studied s.p. canonical states obtained with unedf1 SO and unedf1 SEDF models, which differ in the spin-orbit sector and treat the electrostatic energy self-consistently.
The s.p. energies of unedf1 SO along the α-decay chain of 296 120 are depicted in Fig. 3 The general pattern of s.p. states predicted by un-edf1 SO is not that far from that in Fig. 2 of the MO potential.However, there are differences in the spherical shell structure, which will impact detailed predictions for deformed superheavy nuclei belonging to Z = 115 αdecay chains.In particular, MO predicts larger spherical shell gaps at Z = 114, N = 148, and N = 178.In unedf1 SO , the splitting between the 1j 15/2 and 1i 11/2 spherical neutron shells is very small.This results in an upward shift of the ([606]11/2, [604]9/2) doublet.
In the case of unedf1 the unique-parity ν1j 15/2 and π1i 13/2 shells are shifted up by a few hundred keV, which results in a significant reduction of spherical N = 164 and Z = 114 shell closures [15].The change in the spin-orbit potential also impacts positions of deformed levels (see   Although s.p. energies are not experimental observables, those around the Fermi level carry information about the low-lying q.p. configurations in neighboring odd-A and odd-odd nuclei.To get more insights, we computed the energies of 1-q.p. excitations for odd-Z, even-N superheavy nuclei that form the α-decay chains of 287  116 Lv 171 and 293 117 176 .These results are shown in Figs. 4 and 5. See Supplemental Material [31] for tabulated results; the theoretical error on 1-q.p. excitations due to the adopted size of the HO basis is less than 60 keV when going from 680 stretched HO states to 969 states.By combining those 1-q.p. excitations, one can deduce possible 2-q.p. states in the odd-odd nuclei that form αdecay chains of 288 115. Let us look into the structure of 276 Mt in some detail.The structural information relevant to this nucleus, predicted by SEDF, is contained in the 1-q.p. spectra of its odd-A neighbors shown in Table I: 275,277 Mt, 275 Hs, and 277 Ds.As seen in tables in Supplemental Material [31], all low-lying 1-q.p. states in these nuclei correspond to very similar quadrupole mass deformation of   [32] both scenarios are equally likely.To analyse the case of 272 Bh, we have calculated the 1-q.p. spectra of 271,273 Bh, 273 Hs, and 271 Sg (see Supplemental Material [31] for the corresponding tables).In this case, there are quite a few candidates for low-energy E2 and M 1 transitions that were seen experimentally.
The calculated Q α values depend, of course, on the structure of parent and daughter states [17] (see Figs. 4  and 5).The agreement with the measured values for the heaviest elements is reasonable, usually better than 1 MeV.This is consistent with other calculations [14,[33][34][35].
Conclusions.-Insummary, we studied shell structure of superheavy nuclei within the self-consistent SHFB approach and macroscopic-microscopic NS model.Detailed predictions have been made for the quasi-proton and quasi-neutron structures of nuclei belonging to the αdecay chains of 287 115, 287 Lv, 289 Lv, and 293 117.The un-edf1 and unedf1 SO SEDF models differ in the strength of the spin-orbit term, and this impacts detailed predictions for the deformed nuclei around Z = 110 and N = 168.The recent observation of low-energy E1 transitions in 276 Mt [9] provides a stringent constraint on theoretical models.Indeed, the recently proposed unedf1 SO parametrization that performs well in the transfermium region does not offer a simple explanation of the E1 data, whereas the global unedf1 parametrization explains the data in terms of the proton π[505]9/2 → π[615]11/2 transition.The MO models suggests two competing scenarios: a proton transition similar to that of unedf1, and an alternative neutron ν[716]13/2 → ν[606]11/2 E1 transition.To confirm or disprove these scenarios, theory strongly recommends a search for E1 transitions in neighboring odd-A systems 275,277 Mt, 275 Hs, and 277 Ds.Experimentally, this calls for high-resolution αphoton coincidence spectroscopy of decay chains starting from 293 117, 287,289 115, or 285,287 Fl, respectively.However, these systems are either hampered by relatively low production cross-sections or large spontenous fission branches on the way to the nuclei of structural interest [1][2][3][4][5][6][7][8][9].A solution to this spectroscopic puzzle will have far-reaching consequences for our understanding of shell structure in superheavy nuclei, and the strength of the spin-orbit splitting in particular.

FIG. 3 .
FIG. 3. (Color online) Single-neutron (top) and single-proton (bottom) canonical energies of unedf1 SO for nuclei along the α-decay chain of 296 120 as in Fig. 1.The orbits are labelled by the standard asymptotic Nilsson numbers corresponding to the dominant components of the SHFB canonical wave functions.The positive/negative parity levels are marked by solid/dashed lines.The Fermi levels are indicated by thick dotted lines.

Fig. SM 1
in Supplemental Material [31]).In particular, the deformed neutron gap at N = 152 is reduced, and that at N = 162 opens up.In the proton sector, the deformed Nilsson state [615]11/2 appears just below the significantly increased Z = 116 gap, close to the [505]9/2 and [510]1/2 levels.The second proton intruder state [624]9/2 shows up just below the deformed proton gap at Z = 108.

FIG. 4 .
FIG. 4. (Color online) 1-q.p. spectra for nuclei forming the α-decay chain 287 Lv → • • • → 275 Ds predicted with unedf1 SO (upper sequence) and unedf1 (lower sequence).Qα values for g.s.→g.s.transitions are marked.The binding energy differences between different nuclei are shifted arbitrarily, whereas the excitation energies within a given nucleus are shown to scale.