1 Introduction

According to nuclear theory, the limits of the existence of heavy nuclei, as well as their decay properties, are completely determined by nuclear shell effects. The α decay Q values and half-lives of the new nuclides with Z=104-118 and N=162-177 that have been synthesized in the complete-fusion reactions of ²³⁸U, ^{242, 244}Pu, ²⁴³Am, and ^{245, 248}Cm targets with ⁴⁸Ca beams indicate a considerable increase in the stability of super-heavy nuclei (SHN) with an increase in the number of neutrons [1-9]. The observed radioactive properties of these newly synthesized nuclei and the products of its α decay data, such as the trend in the α-decay energies of the already synthesized Z=114 isotopes in Ref. [10, 11], can be considered as experimental proof of the “island of stability” approach of super-heavy elements around Z=114.

Theoretically, nuclear mass calculation with increasing precision has been pursued over the past decades [12-15], and consequently, great improvements have been made. The extrapolations based on the nuclear shell model performed in the 1960s led to the prediction of a magic stability island surrounding the double-magic nuclear number Z=114 and N=184 [16-18]. In purely microscopic relativistic mean-field (RMF) calculations with Z=120, N=172, 184, 228 [19-21] and the Hartree-Fock-Bogoliubov (HFB) model, with Z=124, 126, N=162 et al. [22, 23]. In practice, many model calculations predict the existence of a closed spherical shell at N=184. Nevertheless, there is no consensus among theorists regarding which nuclide should be the next double-magic nucleus after ${}_{82}^{208}$Pb. At present, one of the most reliable methods is to further improve the theoretical calculation accuracy by using constantly enriched and updated experimental data to provide more reliable predictions for the properties of SHN.

In recent years, an updated macroscopic-microscopic Weizsäcker-Skyrme mass formula [24] referred to as WS* was developed, in which the constraint between mirror nuclei was considered. The root mean square (rms) deviation with respect to the 2149 masses [25] is reduced to 441 keV, which is one of the best results to date. Another important advantage is the ability to provide reasonable mass extrapolations for exotic and heavy nuclei. Hence, it is necessary to use the calculations from the WS* to further explore [26] the features of SHN around the predicted “island of stability”, starting with the properties of the isotopic chain at Z=114.

The remainder of this paper is organized as follows. In Sec. 2, the theoretical framework of the mass model and relatively detailed formulas of microscopic potential are presented. The detailed calculations and discussion are presented in Sec. 3. Finally, Sec. 4 summarizes the findings of the study. All the macro- and micro-physical quantities provide a vital signal that ${}_{114}^{298}$Fl as a spherical double-magic nucleus is the center of the SHN.

2 Theoretical descriptions

In Ref. [24], Wang et al. proposed a semiempirical nuclear mass formula based on the macroscopic-microscopic method [27, 28], i.e., WS*. The total energy of a nucleus can be written as the sum of the liquid-drop energy and microscopic energy and the Strutinsky shell correction:

Z and A are the nuclear charge number and mass number, respectively. The liquid-drop energy of a spherical nucleus E_LD is described by a modified Bethe-Weizsäcker mass formula, and more details can be found in Ref [24]. The term ${\displaystyle {\prod }_{k\ge 2}}(1+b_k{\beta }_k^2)$ is the result of the Skyrme energy density functional together with the Thomas-Fermi approximation [15], where βk denotes the deformation parameters.

To obtain the microscopic energy E_mic or the nuclear single-particle energy levels, we take the standard form of the potential with the central Woods-Saxon term, the spin-orbit term, and (in the case of protons) with the electrostatic Coulomb term. The Schrödinger equation was solved using an approximate method [29, 30]. The single-particle Hamiltonian can be written as:

The spin-orbit potential takes the form of

where λ denotes the strength of the spin-orbit potential and considers the isospin dependence in the form $\lambda ={\lambda }_0(1+\frac{N_i}{A})$ with Ni=Z for protons and Ni=N for neutrons. The central Woods-Saxon potential takes the form:

where the depth of the central potential (q=p for protons and q=n for neutrons) can be written as

with the plus sign for neutrons and the minus sign for protons. a is the surface diffuseness parameter of the single-particle potential. 𝒭(θ) is equal to the distance from the origin of the coordinate system to a point on the nuclear surface.

with the scale factor c₀, which represents the incompressibility effect of nuclear matter in the nucleus and is determined by the so-called constant volume condition [29]. R=r₀A^1/3 is the radius of the spherical nucleus of an identical volume (with the same value for protons and neutrons). Ylm(θ,ϕ) are spherical harmonics. For protons, the Coulomb potential was also involved (see Ref. [29, 30] for details). It should be noted that to increase the credibility of extrapolations in the macroscopic-microscopic approach, the symmetry potential V_s in the WS* is the same as the macroscopic symmetry energy coefficient a_sym, which bridges the relationship between the macroscopic and microscopic parts.

The macroscopic term is only able to reproduce the smooth trend of binding energy, but not local fluctuations. Hence, in addition to the smooth trend, the microscopic energy is considered for the local fluctuations [31]. The most important of these fluctuations stems from shell effects. To extract the shell correction, a practical Strutinsky’s prescription [32, 33] is applied in the calculation of E_min, which takes the form of

where E_shell is equal to the sum of the shell corrections for neutrons and protons. ${E^{\prime }}_{\text{shell}}$ denotes the shell correction energy of the mirror nucleus. The term $\left|I\right|{E^{\prime }}_{\text{shell}}$ takes into account the mirror nuclei constraint and isospin-symmetry-breaking effect [24]. c₁ denotes the scale factor.

3 Results and discussion

The one- and two-nucleon separation energies indicate how difficult it is to peel off one and two nucleons from the parent nucleus. Hence, one and two neutron (proton) separation energies can be extracted from the nuclear binding energies (B) by using the definition S_{n; 2n}(N, Z)=B(N, Z)-B(N-n; 2n,Z), S_{p; 2p}(N,Z)=B(N,Z)-B(N,Z-p; 2p). The most striking application is to obtain information about the shell structure. In particular, the separation energies of two nucleons are useful for finding new magic numbers in super-heavy and exotic nuclei [34, 35]. Owing to the powerful extrapolation ability of the macroscopic-microscopic nuclear mass formula, one can extrapolate nuclear masses for unknown nuclei based on theoretical mass models. Here, we calculated the binding energy of the nuclei around Z=114 and N=184 from the WS* to investigate the physical properties of the Z=114 isotopic chain and N=184 isotonic chain.

The one- and two-nucleon separation energies of the Z=114 isotopic and N=184 isotonic chains of the WS* [24] are shown in Fig. 1, labeled as (a) and (b), respectively, and used to find the magic numbers. The experimental data [36] and theoretical calculations using the finite-range droplet model (FRDM2012) [37] are also included for comparison, where red, black, and purple symbols denote the results of the WS*, FRDM2012, and experimental data, respectively. As shown in Fig. 1(a), the agreement between the WS* and experimental data is good. There are only small differences between these two curves, but the trends are consistent. The values of S_n and S_2n show a decreasing trend with an increase in the neutron number. In addition, S_n shows an obvious even-odd stagger effect. Furthermore, the calculations of these two theories are consistent with each other. With careful observation, it can be found that both have a larger downward slope between N=184 and N=185, implying that for the Z=114 isotopic chain, the nucleus is more stable at a neutron number of 184. In addition, the results of the FRDM2012 in Fig. 1(a) have an unusual tendency to jitter violently at neutron numbers 197 and 199 because the deformation value β₂ changes violently from less than 0.1 at 196 to more than 0.54 at neutron numbers 197 and 198 and finally returns to normal after 199. A similar trend can be seen between Z=114 and Z=115 in Fig. 1(b), and the trend of proton separation energy curve of the WS* is also very consistent with the FRDM2012, which are nearly in superposition with each other. For N=184 isotonic chain, Z=114 is the magic proton number and for Z=114 isotopic chain N=184 is the magic neutron number.

Fig. 1.Full-screen View

(Color online) (a) The one- and two-neutron separation energies of Z=114 isotopic chain obtained by the WS* [24], FRDM2012 [37], and experimental [36], labeled with solid red, open black and solid purple, respectively. The lower right corner label indicates whether it counts one- or two-nucleon separations and is accompanied by a different shape. (b) One- and two-proton separation energies of the N=184 isotonic chain obtained by WS* [24] and FRDM2012 [37]. The marking rules are the same as those in (a). (The unit of the ordinate is MeV).

It is also interesting to explore the α-decay energies (Qα=B(α)+B(Z-2,A-4)-B(Z,A)) to determine the magic numbers. Fig. 2 contains the α-decay energies Qα that are calculated with experimental [36], the WS* [24], and FRDM2012 [37] for Z=114 isotopic and N=184 isotonic chains, marked (a) and (b), respectively. It can be seen that the downward trend of WS* in Fig. 2(a) is essentially the same as the experimental trend, but the slopes between them are slightly different. Meanwhile, we find a dramatic turning point of the α-decay energy curve at N=184 for both theories, where the predicted neutron shell closure appears as a local minima in the curve. In addition, these results are very similar to other theoretical calculations, for example, in the second graph of Ref. [10], the trend of the theoretically predicted Q values for α decays between 171<N<175 on the Z=114 isotopic chain remains consistent with the experimental data, and this theoretically predicted Q value also maintains a continuing decreasing trend as the neutron number continues to increase toward N=184. Such a significant decreasing trend is due to the effect of the closed neutron shell, despite the fact that the obtained heavy isotopes of Fl are still far from N=184 by eight neutrons. In Fig. 2(b), there is a tendency for the Qα calculated by the two theories to increase as the number of proton increases, where the number of proton Z continues to increase rapidly from 114 to 116. As a result, the conclusion that ${}_{114}^{298}$Fl is a double-magic nucleus is verified again.

Fig. 2.Full-screen View

(Color online) (a) The α-decay energies Qα of Z=114 isotopic chain obtained by the WS* [24], FRDM2012 [37], and experimental [36], labeled with solid red square, open black circle and solid purple circle, respectively. (b) The same as (a), but for the N=184 isotonic chain. (The unit of the ordinate is MeV).

Based on the conclusions given above, it is necessary to calculate the rms deviations of the binding energies, one- and two-neutron (proton) separation energies, and α-decay energies of heavy and super-heavy nuclei corresponding to WS*. The experimental data needed for calculation here are those derived from the experimental binding energies [36] of Z≥82 and N≥126 (the standard deviation uncertainty of the mass is less than or equal to 150 keV). The corresponding results are listed in Table 1. We find that all rms deviations are very small, which further confirms the conclusion that Z=114 is the magic proton number of the N=184 isotonic chain and N=184 is the magic neutron number of the Z=114 isotopic chain.

From the macro-physical properties above, it can be concluded that ${}_{114}^{298}$Fl is a double-magic nucleus. Moreover, the properties of ${}_{114}^{298}$Fl from the microscopic energy-level structure are worth investigating. The spherical single-particle energy levels of protons and neutrons of ${}_{114}^{298}$Fl corresponding to the WS* parameters are calculated in a Woods-Saxon potential well with the spin-orbit term. The results are shown in Figs. 3, in which the left half belongs to protons and the right half belongs to neutrons. The number in the orange ellipse on both sides represents the number of nucleons that fill the energy level below it, which is the closest to it and has given the energy difference of the corresponding energy levels with blue numbers. The energy level signs with a color corresponding to the energy levels with the same color to facilitate the distinction. From this figure, we can see that the gap of Z=114 with a difference of 1.829 MeV is the largest for the protons of ${}_{114}^{298}$Fl, and no significant gap between the levels at Z=126 appears. Then, turning to the neutron shell structure, the gap N=184 is one of the biggest gaps, corresponding to a difference of 1.712 MeV. The relation between the number of neutrons N or protons Z and the mass number A of a beta-stable nucleus is obtained according to the formula N-Z=0.4A²/(A+200) [38]. The nucleus ${}_{114}^{298}$Fl is also very close to the extrapolated beta-stability line. In addition, the equilibrium deformation β₂, β₄, and β₆ are obtained by searching for the minimum binding energy using the optimized DSC method. Theoretically, the ability to describe SHN properties is enhanced by using a three-dimensional deformation space. A contour map of the microscopic energy Emic of the nucleus ${}_{114}^{298}$Fl calculated from WS* as a function of deformation β₂ and β₄ is shown in Fig. 4. At each point (β₂; β₄), the energy is minimized in β₆ degrees of freedom. As shown in the figure, for nucleus ${}_{114}^{298}$Fl, there is a local spherical minimum, with β₂ and β₄ being almost zero. The calculated β₂, β₄, and β₆ for the nucleus ${}_{114}^{298}$Fl in WS* are equal to -0.0002, 0.0001, and -0.0001, respectively. All these deformation values indicate that the ground state is spherical. From these calculations, it can be seen that ${}_{114}^{298}$Fl is a spherical double-magic nucleus.

Fig. 3.Full-screen View

(Color online) The single-particle levels of the nuclide ²⁹⁸Fl. The left half shows the energy levels of the proton and the right for neutrons (in MeV).

Fig. 4.Full-screen View

(Color online) Contour map of the microscopic energy Emic calculated from WS* for nucleus ²⁹⁸Fl as a function of deformation β₂ and β₄. At each point (β₂; β₄), the energy is minimized in β₆ degrees of freedom (in MeV).

To further demonstrate the accuracy of the theory, a graph of the sum of the shell corrections of protons and neutrons calculated using WS* to the nuclear macroscopic potential energies is shown in Fig. 5, where the deformation values of each nucleus are chosen for its corresponding lowest equilibrium deformation. The black circles in the figure represent the super-heavy nuclei with Z>104 that have been synthesized experimentally to date. Three local minima can be observed in the figure, which is in the vicinity of the double-magic nuclei ²⁰⁸Pb, ²⁷⁰Hs, and ²⁹⁸Fl, for the region above N>176, the minima fall on the nucleus ${}_{114}^{298}$Fl; furthermore, it has been experimentally confirmed by decay energies that the nucleus ²⁷⁰Hs is a double-magic deformed nucleus [39-41]. For the double-magic nucleus ²⁷⁰Hs, as before, the single-particle energy levels obtained using WS* and a contour map of the microscopic energy similar to Fig. 4 are given in Fig. 6 and Fig. 7. First, Fig. 6 shows that the ground state of nucleus ²⁷⁰Hs is deformed and is not a sphere, and WS* calculations show that its lowest equilibrium deformation values are β₂=0.2154, β₄=-0.0553, β₆=-0009. Later, in Fig. 7, the corresponding single-particle energy levels of the nucleus ²⁷⁰Hs are shown at these deformation values, and it can be seen that the energy gaps are relatively significant at proton and neutron numbers filled to 108 and 162, respectively. The theory also confirms that ²⁷⁰Hs is a double-magic deformed nucleus. It is worth noting that when deformations are not considered, that is, the shape of the nucleus if all are treated as spherical, the strong shell correction that the nucleus ²⁷⁰Hs would have had at this point disappears, as can be seen in Fig. 6. By using a multidimensional deformation space, the magic deformed nucleus ²⁷⁰Hs is reproduced, and the region surrounding it connects the double-magic nucleus ²⁰⁸Pb with the predicted SHN around ${}_{114}^{298}$Fl. As shown in Fig. 5, the new image within the region of super-heavy nuclei can be likened to two connected peaks on which these now-synthesized super-heavy nuclei are spread exactly.

Fig. 5.Full-screen View

(Color online) The graph of the sum of the shell corrections Eshell of protons and neutrons to the nuclear macroscopic potential energy calculated using WS* (in MeV). The black circles in the figure represent the super-heavy nuclei with Z>104 that have been synthesized experimentally thus far.

Fig. 6.Full-screen View

(Color online) Contour map of the microscopic energy Emic calculated using WS* for nucleus ²⁷⁰Hs as a function of deformation β₂ and β₄. At each point (β₂; β₄), the energy is minimized in β₆ degrees of freedom (in MeV).

Fig. 7.Full-screen View

(Color online) The single-particle levels of the nuclide ²⁷⁰Hs. The left half shows the energy levels of the proton and the right half shows the energy levels for neutrons (in MeV).

In the absence of super-heavy nuclear data at the time, the possible magic numbers were found to be Z=114 and N=184 in Ref. [16]. This seems to be the first time that the Woods-Saxon potential has been used to describe the structure of a nucleus. Inspired by this article, we calculated the shell structure in a manner similar to Ref. [16] with the WS* parameters used for comparison. The calculation was performed for the mass numbers A=209, 275, 299, and 355 with proton numbers Z=83, 107, 115, and 127, respectively. The results are shown in Fig. 8, where the left half shows energy levels for protons and the right for neutrons. As shown in the left half of the figure, the energy gap at Z=82 becomes narrower as the number of masses increases, but the gap is still relatively large over a large range. Starting from mass number 300, there is no significant gap between the levels at Z=126, indicating that in the super-heavy nuclear region, the well-known magic number 126 is no longer effective for protons. Instead, a larger shell gap appears in the parameter set at Z=114, which can be considered as a magic number. For the neutrons shown in the right half of the figure, the traditional N=126 and N=184 magic numbers still exist throughout the heavy and super-heavy nuclei region. In addition, for neutrons, it is found that the shell gap of N=228 increases as the mass number increases, which can also be considered as a magic number. These findings are in good agreement with those reported in Ref. [16], where they both give consistent magic numbers. In other words, this study can be regarded as the verification and expansion of Ref. [16]. In particular, we reproduce all the results of WS* to ensure our correctness, such as the single-particle levels.

Fig. 8.Full-screen View

(Color online) Energies levels of the proton (left) and neutron (right) versus mass number A. These calculations were performed for the mass numbers A=209, 275,299, and 355 with proton numbers Z=83, 107,115, and 127, respectively.

For heavy and super-heavy nuclei, α-decay is the main decay mode [42]. Moreover, it is also a probe for experimentally exploring the nature of SHN. Accordingly, we calculate the half-lives of the α-decay for the Z=114 isotopic chain and the N=184 isotonic chain. To compute the α-decay half-lives, the latest improved formula proposed by Deng et al. [43-45] was used, which is expressed as:

where the first three terms are the same as those in the original Royer formulas. The fourth term represents the contribution of the centrifugal potential. l is the angular momentum removed by the α particle. The fifth term represents the blocking effect of unpaired nucleons. More details can be found in Ref. [45]. Moreover, a recent study calculated the uncertainties in the empirical formulae for the α decay half-lives of heavy and super-heavy nuclei[46]. The values of spin and parity of nuclei are obtained from calculations of odd-nucleon spin and parity at the nuclear ground state by P. Möller [47]. The results are shown in Fig. 9, where the red, black, purple symbols denote the calculations using the WS* [24], FRDM2012 [37], and experimental data [36], respectively. In addition, the blue symbols in Fig.9(a) denote the calculated half-lives with experimental Qα to test the reliability of Eq. (8). It can be seen that the calculated α-decay half-lives from experimental Qα coincide almost perfectly with the experimental half-lives, implying that, as long as we have the right Qα, the presently used method can give precise results for α-decay half-lives. It can be observed from Fig. 9(a) that the predicted half-lives curves of α-decay obtained using the two theories basically have the same trend for Z=114 isotopic chain, but the results of FRDM2012 are generally higher for N=182. The WS* results were generally in good agreement with the experimental data. In particular, when the neutron number N crosses N=184 for WS*, the predicted α decay half-lives decrease sharply, and at N=186, α decay half-lives are reduced by more than four orders of magnitude. This indicates that strong shell effects were reflected. Its half-life does not decrease much at N=185 compared with N=184 because of the blocking effect of unpaired neutrons and the fact that the angular momentum carried by the α particle during decay is as high as 5, which extends its half-life to some extent. The α-decay half-life of ${}_{114}^{298}$Fl is approximately 42.5 days with Qα of WS* and approximately 6 h with Qα of the FRDM2012. Fig. 9(b) plots the logarithms of calculated half-lives of the two theories for N=184 isotonic chain. From this figure, we can see that when the proton number Z>114 for WS*, the predicted α decay half-lives drop dramatically by eight orders of magnitude at Z=116. As for FRDM2012, its theoretical predictions are very similar to the results given by WS* but have an overall trend opposite to that of Fig. 2(b).

Fig. 9.Full-screen View

(Color online) (a) The predicted logarithmic α-decay half-lives of Z=114 isotopic chain using Eq. (8) with Qα obtained by the WS* [24], FRDM2012 [37], and experimental [36], labeled with solid red squares, open black circles, and solid blue triangles, respectively. (b) The same as (a), but for the N=184 isotonic chain. (The unit of the ordinate is second) The solid purple circle in figure (a) denotes the experimental α-decay half-lives.

A systematic study of shell energy gaps calculated using a variety of theoretical models was recently launched in Ref. [48], where WS* similarly gives results for Z=114 and N=184 as the two obvious magic numbers. Moreover, experimental results show the existence of nuclear shell effects near Z=114, as given in the fourth figure of Ref. [41], where the experimentally measured trend of the total evaporation-residue cross section of the hot fusion becomes locally maximum near Z=114 supports this point. Similarly, the fourth figure in Ref. [49] shows that the maximum cross-section measured for the experimentally synthesized lighter isotopes of Fl decreases sharply with smaller neutron numbers. The fifth figure in Ref. [49] shows the experimental and theoretical spontaneous fission half-lives of even-even nuclei in the super-heavy region, and it can be seen that both shells N=152 and 162 have a strong influence on the spontaneous fission of the isotopes of No and Rf-Hs, whereas a similar stabilizing influence of the N=184 shell also seems apparent for Cn and Fl. Recent experiments have also shown that the element Fl is a volatile metal[50]. In addition, because the combination of stable nuclei does not provide such neutron-rich nuclei for the synthesis of super-heavy nuclei with N>184, it is hoped that in the future it will be possible to make use of secondary beams and, if possible, to synthesize ${}_{114}^{298}$Fl by obtaining compound nuclei with neutron numbers above 300 [51]. Some theoretical studies on super-heavy nuclei Z = 110 125 have also been reported [52-61].

4 Summary

In summary, the existence of an “island of stability” among the super-heavy elements is a fundamental prediction of modern nuclear theory, and this stability island has not yet been localized experimentally. Inspired by Ref. [16], we have approached the study of stability island in super-heavy elements starting from the properties of the Z=114 isotopic chain. The experimental binding energies, one- and two-neutron (proton) separation energies, and α-decay energies were compared with the calculations obtained using the WS* for heavy and super-heavy nuclei, and the agreement was excellent. The features of one- and two-nucleon separation energies and α-decay energies of the WS* and FRDM2012 both indicate that Z=114 is the magic proton number of the N=184 isotonic chain and N=184 is the magic neutron number of the Z=114 isotopic chain. Concurrently, the spherical single-particle levels of nucleus ${}_{114}^{298}$Fl were given using a Woods-Saxon potential well with a spin-orbit term [29, 30] and combined with the equilibrium deformation values (β₂, β₄, β₆) of nucleus ${}_{114}^{298}$Fl calculated by the WS* are essentially equal to zero, it can be seen that ${}_{114}^{298}$Fl is a spherical double-magic nucleus. Then, we calculated the shell structure in a manner similar to Ref. [16] with the WS* parameters, where the results are in good agreement with the results of Ref. [16]. In the final part, the latest formula is applied to predict the α-decay half-lives of the Z=114 isotopic and N=184 isotonic chains. In general, these microscopic and macroscopic physical properties obtained by exploring the nature of isotopic chains provide a positive signal that ${}_{114}^{298}$Fl is a spherical double-magic nucleus and also the center of the SHN.