Shape coexistence and \alpha-decay chains of {}^{{}^{\text{293}}}Lv

Abstract

Two recently observed ²⁹³Lv (Z=116) α-decay chains [Eur. Phys. J. A 48, 62 (2012)] are investigated in the framework of covariant density functional theory with PC-PK1, where the pairing correlations are treated by the Bardeen–Cooper–Schrieffer method with a density-independent zero-range force. From the calculated potential energy curves, it is found that two minima always occur, with one having an almost spherical shape and the other exhibiting a large deformed prolate shape. Originating from the ground state and the shape-isomeric state of ²⁹³Lv, the two observed α-decay chains are constructed and the calculated Q_{\alpha } values are found to be in good agreement with the data.

Graphical abstract

Keywords

shape coexistence / α decay / ²⁹³Lv / covariant density functional theory

Due to quantum shell effects, the existence of an island of stability in the region of the superheavy nuclei (SHN) at around Z=114 and N=184 was first predicted in the 1960’s [1–8]. Since then, investigation of the properties of SHN has been a topic of considerable interest and also a significant challenge in nuclear physics. In experimental studies, which have benefited from the improvement of beam intensities and detection efficiencies, superheavy elements with Z⩽118 have been synthesized via cold and/or hot fusion reactions [9–13]. Theoretically, a number of investigations (for reviews, see, e.g., Ref. [14]) have been devoted to the SHN structure, which were primarily based on either traditional macroscopic-microscopic or microscopic self-consistent approaches. In particular, self-consistent mean-field models based on nuclear energy density functionals [15] have been extensively applied to investigate the various properties of SHN, such as their ground-state properties [16–18], shell structures [19–22], α-decay energies and half-lives [23, 24], fission barriers [25–29], and shape transitions [30]. As regards nuclear reaction, the nucleosynthesis of SHN has been extensively investigated using the dinuclear system model [31–37], the fluctuation-dissipation model [38], the fusion-by-diffusion model [39–42], the quantum molecular dynamics model [43, 44], the two-step model [45], and other models [46, 47].

Focusing on Lv isotopes, ^290-293Lv were first discovered at the Joint Institute for Nuclear Research Flerov Laboratory of Nuclear Reactions (JINR-FLNR) in Dubna (Russia) [48, 49]. For ²⁹³Lv in particular, the decay properties were measured through the fusion-evaporation reaction, {}_{}{}^{248}\mathrm{Cm}{{(}^{48}\mathrm{Ca},\mathrm{xn})}^{296-x}\mathrm{Lv} [49]. A number of theoretical studies concentrating on the α-decay of ²⁹³Lv have since been performed based on various models, including the relativistic [50, 51] and nonrelativistic [50] mean-field models, the generalized liquid drop model [52], the Wigner–Kirkwood method [53], the two-center shell model [54], the cluster model [55–57], and the unified fission model [58]. Recently, similar to the previous JINR-FLNR experiment [49], the irradiation of ²⁴⁸Cm targets with ⁴⁸Ca ions has been conducted [59] using the separator for heavy ion reaction products (SHIP) velocity filter at GSI in Darmstadt (Germany). In this experiment, the previously measured α-decay data were independently confirmed. Moreover, a new α-decay chain was observed, which was tentatively assigned to isomeric states. In fact, it has been shown that the existence and prevalence of long-lived isomers at low excitation energies in SHN enhances nuclear stability [60]. Moreover, in Ref. [61], it was pointed out that α-decay may not proceed through ground-state-to-ground-state chains, but rather through excited states. Therefore, it would be very interesting to investigate whether or not the newly observed α-decay chain in ²⁹³Lv originates from its isomeric state.

The covariant density functional theory (CDFT) has attracted widespread attention because of its considerable success in describing both the ground-state and excited-state properties of nuclei throughout the nuclide chart [62–66]. Recently, a new point-coupling density functional, PC-PK1, was proposed by fitting the masses, charge radii, and empirical pairing gaps of 60 selected spherical nuclei [67]. The powerful predictive ability of PC-PK1 has been illustrated through its descriptions of magnetic rotation [68], antimagnetic rotation [69, 70], low-lying exited states [65, 71, 72], and nuclear masses [73–75].

In this paper, two ²⁹³Lv α-decay chains are investigated in the framework of CDFT with PC-PK1. Here, axial symmetry is assumed in the calculations, since the triaxiality and the reflection asymmetry are negligible in this region [76]. The pairing correlations are treated by the Bardeen–Cooper–Schrieffer (BCS) method with a density-independent zero-range force, where smooth cutoff is introduced to simulate the finite-range effects [77, 78]. Note that, since the nuclei under consideration in this study have odd neutron numbers, the unpaired neutron in each nucleus will block its occupied level in the BCS calculations, i.e., it will be prevented from participating in the nucleon pair scattering process due to the Pauli principle [79]. After blocking the possible single-particle levels (SPLs) near the Fermi surface, a series of nuclear states corresponding to different blocking levels will be obtained. The lowest-energy state is the ground state. Thus, in each iteration step of the CDFT calculation, one should ensure that the unpaired particle sits in the blocked SPL. A microscopic center-of-mass correction [80, 81] is adopted and the rotational correction is not considered. The Dirac equation for nucleons is solved in a harmonic oscillator basis using 20 major shells [82].

The potential energy curves (PECs) for the nuclei in the ²⁹³Lv α-decay chains obtained by the adiabatic CDFT [83] are shown in Fig. 1. It can be seen that, for ²⁹³Lv, two energy minima with total energies E=-2082.45 MeV and -2081.06 MeV undergoing prolate (quadrupole deformation, \beta =0.55) and oblate (\beta =-0.09) deformations are obtained, which correspond to the ground state and the shape-isomeric state, respectively. Note that the isomeric state in ²⁹³Lv is rather soft with respect to the β degree of freedom, although it has an energy that is approximately 1.39 MeV higher than its ground state. By following the ²⁹³Lv α-decay chains, one can always find two energy minima at \beta \sim 0.10 and \beta \sim 0.55 for ²⁸⁹Fl, ²⁸⁵Cn, and ²⁸¹Ds. Further, in contrast to ²⁹³Lv, their ground states have \beta \sim 0.10. With decreasing mass number A, the energy differences between the isomeric and ground states become increasingly large and, in the case of ²⁷⁷Hs, the isomeric state at \beta \sim 0.55 finally disappears. Therefore, assuming similar deformations in the parent and daughter nuclei, one can clearly build two α-decay chains with similar deformations from the ground state (Chain-1) and shape-isomeric state (Chain-2) of ²⁹³Lv. This assignment is consistent with experimental data [59], which shows that the two observed α-decay chains are terminated after three and four α decays through spontaneous fission. It should be noted that Chain-2 was tentatively assigned to high-spin isomers in Ref. [59], following analysis of the low-lying excited one-quasiparticle states using the Skyrme–Hartree–Fock–Bogoliubov method [84].

To understand the minima in the PECs in a microscopic manner, taking ²⁹³Lv as an example, the proton and neutron SPLs near the Fermi surfaces as functions of β are shown in Fig. 2(a) and Fig. 2(b), respectively. For protons, it can be seen that two significant gaps occur near the Fermi surface at \beta \sim -0.10 and \beta \sim 0.55. For neutrons, a similar conclusion can also be drawn. Apparently, the two energy minima in the PEC of ²⁹³Lv, as shown in Fig. 1, are connected to the two energy gaps observed in the SPLs.

Theoretically, by blocking each possible SPL near the Fermi surfaces, a series of E and \beta values corresponding to different blocking levels are obtained, which are tabulated in Table 1 and Table 2 for ²⁹³Lv, ²⁸⁹Fl, ²⁸⁵Cn, ²⁸¹Ds, and ²⁷⁷Hs at \beta \sim 0.5 for Chain-1 and |\beta |\sim 0.1 for Chain-2, respectively. The blocked levels are labeled by the projection of the angular momentum on the symmetry axis \Omega and parity π. To give a more intuitive picture, the E calculated by blocking different {\Omega }^{\pi } levels for Chain-1 and Chain-2 are shown in Fig. 3 and Fig. 4, respectively, where the positive (negative) parity levels are represented by short solid (dashed) lines.

To facilitate a comparison with the experimental data for Chain-1 (Chain-2), eleven (ten) α-decay chains are theoretically built as follows:

(i) Assuming both the parent and daughter nuclei have the same {\Omega }^{\pi }, for Chain-1 (Chain-2) eight (seven) α-decay chains are constructed, as shown in the rows in Table 1 (Table 2);

(ii) Assuming the parent and daughter nuclei are in the lowest-energy states regardless of their {\Omega }^{\pi }, one α-decay chain can be constructed, which is represented by the black solid arrows in Fig. 3 (Fig. 4) for Chain-1 (Chain-2);

(iii) Assuming the parent and daughter nuclei are in the lowest-energy states with the same parity, two additional α-decay chains can be constructed, which are represented by the red dotted and green dashed arrows in Fig. 3 (Fig. 4) for Chain-1 (Chain-2).

For the first case, one can calculate the root mean square deviation \mathrm{\Delta }{Q}_{\alpha }, as shown in the last column of Table 1 (Table 2) for Chain-1 (Chain-2), from

\begin{array}{}\text{(1)}& \mathrm{\Delta }{Q}_{\alpha }=\sqrt{{\displaystyle \sum _{i=1}^{N-1}}{\left(Q_{{\alpha }_i}^{\mathrm{Cal}.}-Q_{{\alpha }_i}^{\mathrm{Exp}.}\right)}^2/(N-1)},\end{array}

where N is the number of nuclei involved in a given α-decay chain. It can be seen that the α-decay chain with 5/2^{-} can best reproduce the data for Chain-1, while 5/2^{+} yields the optimum agreement with the Chain-2 data.

For the latter two cases, where the emitted α particle has a nonzero angular momentum due to the different {\Omega }^{\pi } values of the parent and daughter nuclei (meaning that this α-decay is quenched by the centrifugal barrier), the calculated Q_{\alpha } values are compared with the data in Figs. 5(a) and (b) for Chain-1 and Chain-2, respectively. One can see that it was possible for the experimental data for both α-decay chains to be well reproduced within 1 MeV. However, it is difficult to determine whether or not the parity is changed in the present mean-field framework.

An accurate description of the Q_{\alpha } value is related to the description of the total energies for the parent and daughter nuclei. Previous systematic investigation has revealed that, although the accuracy for nuclear mass is approximately 2.3 MeV, the accuracy for one-neutron separation energy is approximately 0.6 MeV [85]. For the density functional PC-PK1, the accuracy for nuclear mass is improved to 1.4 MeV [66]. The accuracy for one-neutron separation energy and the Q_{\alpha } value should, therefore, be improved accordingly.

In summary, in this paper, ²⁹³Lv α-decay chains are investigated in the framework of covariant density functional theory with PC-PK1, where the pairing correlations are treated by the Bardeen-Cooper-Schrieffer method with a density-independent zero-range force. From the obtained potential energy curves, it is found that the two α-decay chains observed in ²⁹³Lv may originate from the ground state (Chain-1) and the shape-isomeric state (Chain-2) of ²⁹³Lv. By assuming that both the parent and daughter nuclei have the same {\Omega }^{\pi }, the experimental Q_{\alpha } values for Chain-1 and Chain-2 can be reproduced quite well by the two α-decay chains built on 5/2^{-} and 5/2^{+}, respectively. Further, by assuming that both the parent and daughter nuclei are in the lowest-energy states, the experimental data can be reproduced with the same accuracy regardless of whether or not the parity is changed. Finally, it may be possible to distinguish between these α-decay chains by further comparing the resultant α-decay half-lives, which depend not only on the Q_{\alpha } value, but also on the angular momentum of the emitted α particle.

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