Superheavy nuclei in relativistic effective Lagrangian model

Isotopic and isotonic chains of superheavy nuclei are analyzed to search for spherical double shell closures beyond Z=82 and N=126 within the new effective field theory model of Furnstahl, Serot, and Tang for the relativistic nuclear many-body problem. We take into account several indicators to identify the occurrence of possible shell closures, such as two-nucleon separation energies, two-nucleon shell gaps, average pairing gaps, and the shell correction energy. The effective Lagrangian model predicts N=172 and Z=120 and N=258 and Z=120 as spherical doubly magic superheavy nuclei, whereas N=184 and Z=114 show some magic character depending on the parameter set. The magicity of a particular neutron (proton) number in the analyzed mass region is found to depend on the number of protons (neutrons) present in the nucleus.


I. INTRODUCTION
In the last thirty years a continuing effort has been devoted to the investigation of superheavy nuclei both in experiments and in theoretical research. A fascinating challenge in the study of these nuclei is the quest for the islands of stability where the next magic numbers beyond N = 126 and Z = 82 may be located. Experiments made at GSI, Dubna and Berkeley have allowed the synthesis and detection of some superheavy nuclei. For instance, light isotopes of the elements Z = 110, 111 and 112 have been obtained at GSI and Dubna [1][2][3]. They have been identified by their characteristic α-decay chains which lead to already known isotopes. These new nuclei are expected to be deformed, consistently with the predicted occurrence of a deformed magic shell closure at Z = 108 and N = 162 (see e.g. Refs. [4][5][6]). First data of some heavier and more neutron-rich isotopes of atomic number Z = 112 (N = 171), Z = 114 (N = 173-175) and Z = 116 (N = 176) produced by means of fusion reactions have also been measured at Dubna [7].
Theoretical predictions made at the end of the sixties pointed towards the existence of an island of long-lived superheavy elements (SHE) centered around N = 184 and Z = 114 [8][9][10][11]. The nuclei around the hypothetical doubly magic element 298 114 were expected to be nearly spherical with longer half-lives. Such superheavy nuclei, having a negligible liquiddrop fission barrier, would be stabilised mostly by quantal shell effects. Many of the more recent theoretical works on superheavy nuclei are based on the nuclear mean field approach and can be classified in two main groups. On the one hand, we have the macroscopicmicroscopic models which include a liquid-drop contribution for the part of the energy which varies smoothly with the number A of nucleons, and a shell correction contribution obtained from a suitable single-particle potential for the fine tuning. On the other hand, there are the self-consistent Hartree-Fock or Hartree calculations based on Skyrme forces or on the relativistic non-linear σ − ω model, respectively.
The nuclei in the range around Z ≈ 110 already detected in experiments bridge the gap between the known actinides and the unknown superheavy elements. With the advent of more experimental data, a commendable endeavor has been undertaken in nuclear structure research [6,[12][13][14][15][16][17][18][19][20][21][22][23][24][25] aimed at verifying the reliability of the present theoretical models in the regime of the heavier actinides and of the discovered superheavy nuclei around Z = 110, which requires deformed calculations. The fact that many of the observed data for SHE are for odd-even decay chains renders the calculations and the comparison with experiment even more complicated, since the deformed level density is high and the observed nuclei may be in isomeric states. Calculations with self-consistent models of some α-decay chains [20], deformation energy curves along the fission path [25], and shell structures [6] find that there is a gradual transition from well-deformed nuclei around the deformed Z = 108 and N = 162 shell closures to spherical shapes approaching larger superheavy nuclei around the putative N = 184 magic neutron gap, in qualitative agreement with the earlier studies in mac-mic and semiclassical models. Still, the Hartree-Fock model mass formula of Ref. [26] predicts large deformations in many of the isotopes of Z = 114 and in almost all of the Z = 120 isotopes. As pointed out in some recent works, the description of deformed SHE may require to consider triaxial deformations and reflection-asymmetric shapes [25,27,28] (Ref. [29] pioneered the relativistic mean field triaxial calculations). It is even possible that there exist isolated islands of stability associated with exotic (semi-bubble, bubble, toroidal, and band-like) topologies in nuclei with very large atomic numbers [20,30,31].
Another longstanding goal of the nuclear structure studies in the field of superheavy nuclei has been to establish the location in N and Z of the next spherical double shell closures for elements heavier than 208 Pb, and of the largest shell effects which are a necessary condition for the stability of SHE against fission. In this context, most of the calculations published in the literature are performed in spherical symmetry. It is well established that the macroscopic-microscopic calculations predict spherical shell closures at Z = 114 and N = 184 [4]. In self-consistent calculations, however, the proton and neutron shell structures strongly affect each other and other N and Z values can appear as candidates for shell closures depending on the model interaction. For example, Hartree-Fock calculations with a variety of Skyrme forces show the most pronounced spherical shell effects at Z = 124, 126 and N = 184 [32][33][34][35][36]. As an exception to this rule, Skyrme parametrizations such as SkI3 and SkI4 which have a modified spin-orbit interaction prefer Z = 120 and Z = 114, respectively, for the proton shell closure [33,34]. Hartree-Fock-Bogoliubov calculations with the finite range Gogny force predict Z = 120, 126 and N = 172, 184 as possible spherical (or nearly spherical) shell closures [30,31]. At variance with Skyrme Hartre-Fock, the relativistic mean field (RMF) theory with the conventional scalar and vector meson field couplings typically prefers Z = 120 and N = 172 as the best candidates for spherical shell closures [32][33][34][35][36]. Of course, the different nature of the spin-orbit interaction in the Skyrme and RMF models is pivotal in deciding the location of the stronger shell effects. Detailed comparisons between Skyrme Hartree-Fock and RMF calculations of SHE can be found in Refs. [33] and [35].
The discrepancies in the predicted spherical shell closures for SHE motivate us to reinvestigate them using the more general RMF model derived from the chiral effective Lagrangian proposed by Furnstahl, Serot and Tang [37][38][39]. In this first attempt to apply the effective field theory (EFT) model to the region of superheavy nuclei we will restrict ourselves to analyze the occurrence of double shell closures and the shell stabilizing effect in spherical symmetry, as done e.g. in Refs. [32][33][34][35][36]. We want to learn whether in this respect the EFT approach shows a different nature compared to the usual RMF theory or not. The possible extension of the calculations to deformed geometries and the subsequent application of the EFT model to the heavier actinides and the lighter transactinides where experimental data have been measured is left for future consideration.
The relativistic model of Refs. [37][38][39] is a new approach to the nuclear many-body problem which combines the modern concepts of effective field theory and density functional theory (DFT) for hadrons. An EFT assumes that there exist natural scales to a given problem and that the only degrees of freedom relevant for its description are those which can unravel the dynamics at the scale concerned. The unresolved dynamics corresponding to heavier degrees of freedom is encoded in the coupling constants of the theory, which are determined by fitting them to known experimental data. The Lagrangian of Furnstahl, Serot and Tang is intended as an EFT of low-energy QCD. As such, its main ingredients are the lowest-lying hadronic degrees of freedom and it has to incorporate all the infinite (in general non-renormalizable) couplings consistent with the underlying symmetries of QCD.
To endow the model with predictive power the Lagrangian is expanded and truncated.
Terms that contribute at the same level are grouped together with the guidance of naive dimensional analysis. Truncation at a certain order of accuracy is consistent only if the coupling constants eventually exhibit naturalness (i.e., if they are of order unity when in appropriate dimensionless form). In the nuclear structure problem the basic expansion parameters are the ratios of the scalar and vector meson fields and of the Fermi momentum to the nucleon mass M, as these ratios are small in normal situations. To solve the equations of motion that stem from the constructed effective Lagrangian one applies the relativistic mean field approximation in which the meson fields are replaced by their classical expectation values.
EFT and DFT are bridged by interpreting the expansion of the effective Lagrangian as equivalent to an expansion of the energy functional of the many-nucleon system in terms of nucleon densities and auxiliary meson fields. The RMF theory is then viewed as a covariant formulation of DFT in the sense of Kohn and Sham [40]. That is, the mean field model approximates the exact, unknown energy functional of the ground-state densities of the nucleonic system, which includes all higher-order correlations, using powers of auxiliary classical meson fields. This merger of EFT and DFT provides an approach to the nuclear problem which retains the simplicity of solving variational Hartree equations with the bonus that further contributions, at the mean field level or beyond, can be incorporated in a systematic and controlled manner.
If the chiral effective Lagrangian is truncated at fourth order, in mean field approach one recovers the same couplings of the usual non-linear σ − ω model plus additional non-linear scalar-vector and vector-vector meson interactions, besides tensor couplings [38,39]. The free parameters of the resulting energy functional have been fitted to ground-state observables of a few doubly-magic nuclei. The fits, parameter sets named G1 and G2 [38], do display naturalness and are not dominated by the last terms retained; an evidence which confirms the usefulness of the EFT concepts and validates the truncation of the effective Lagrangian at the first lower orders. The ideas of EFT have been fruitful [41], moreover, to elucidate the empirical success of previous RMF models, like the original σ − ω model of Walecka [42] and its non-linear extensions with cubic and quartic scalar self-interactions [43]. However, these conventional RMF models truncate the effective Lagrangian at some level without further physical rationale or symmetry arguments. The introduction of new interaction terms in the effective model pursues an improved representation of the relativistic energy functional [38,39].
Previous works have shown that the EFT model is able to describe in a unified manner the properties of nuclear matter, both at normal and at high densities [44,45], as well as the properties of finite nuclei near and far from the valley of β stability [46,47], with similar and even better quality to standard RMF force parameters. With this positive experience at hand, in the present paper we want to investigate the predictability of the new effective Lagrangian approach to the nuclear many-body problem in extrapolations to superheavy nuclei. Concretely, we shall focus on analyzing the model predictions for spherical shell closures. Our calculations will be performed in spherical symmetry. Though deformation is an important degree of freedom for SHE [5,6,14,18,25], we are searching for spherical shell stability around 298 184 114 and 292 172 120 where deformation is expected to be small and where the shell structure has often been analyzed in the spherical approximation [5,[32][33][34][35][36]. For exploration, we also compute hyperheavy nuclei around N ∼ 258 which spherical calculations have found to correspond to a possible region of increased shell stability [36]. Deformation would certainly change the picture in the details and add deformed shell closures, e.g., like those predicted around Z = 108 and N = 162 [4][5][6], but it should not change drastically the predictions for the values of N and Z where the strongest shell effects show up already in the spherical calculation. Of course, for a quantitative discussion, one needs to account for deformation effects which will serve to extend the island of shell stabilized superheavy nuclei and to decide on the specific form of the ground-state shapes of these nuclei.
Our analysis uses the EFT parameter sets G1 and G2 [38]. The results are compared with those obtained with the NL3 parameter set [48], taken as one of the best representatives of the usual RMF model with only scalar self-interactions. The paper is organized as follows. In the second section we briefly summarize the RMF model derived from EFT and our modified BCS approach to pairing. The third section is devoted to the study of several properties of superheavy nuclei such as two-particle separation energies and shell gaps, average pairing gaps, single-particle energy spectra, and shell corrections. The summary and conclusions are laid in the fourth section.

A. The model
The EFT model used here has been developed in Ref. [38]. Further insight into the model and the concepts underlying it can be gained from Refs. [37,39,41,49]. For our purposes, the basic ingredient is the EFT energy density functional for finite nuclei. It reads [38,39] The coupling constants have been written so that in the present form they should be of order unity according to the naturalness assumption. The index α runs over all occupied nucleon states ϕ α (r) of the positive energy spectrum. The meson fields are Φ ≡ g s φ 0 (r), W ≡ g v V 0 (r) and R ≡ g ρ b 0 (r), and the photon field is A ≡ eA 0 (r). Variation of the energy density (1) with respect to ϕ † α and the meson fields gives the Dirac equation fulfilled by the nucleons and the Klein-Gordon equations obeyed by the mesons (see Refs. [38,44] for the detailed expressions). We solve the Dirac equation in coordinate space by transforming it into a Schrödinger-like equation.
In this work we shall employ the EFT parameter sets G1 and G2 of Ref. [38] that were fitted by a least-squares optimization procedure to 29 observables (binding energies, charge form factors and spin-orbit splittings near the Fermi surface) of the nuclei 16 O, 40 Ca, 48 Ca, 88 Sr and 208 Pb. A satisfactory feature of the set G2 is that it presents a positive value of κ 4 , as opposed to G1 and to most of the successful RMF parametrizations such as NL3. We note that the value of the effective mass at saturation M * ∞ /M in the EFT sets (∼ 0.65) is somewhat larger than the usual value in the RMF parameter sets (∼ 0.60), which is due to the presence of the tensor coupling f v of the ω meson to the nucleon [46,50]. Also, the bulk incompressibility of G1 and G2 is K = 215 MeV, while the NL3 set has K = 271 MeV.

B. Pairing
In order to describe open-shell nuclei the pairing correlations have to be explicitly taken into account. The most popular approach for well-bound isotopes has been the BCS method.
However, the BCS approximation breaks down for exotic nuclei near the drip lines because it does not treat the coupling to the continuum properly. This difficulty is disposed of either by the non-relativistic Hartree-Fock-Bogoliubov theory, with Skyrme [51] and Gogny [52] forces, or by the relativistic Hartree-Bogoliubov (RHB) theory [53][54][55][56].
Pairing correlations are another important ingredient in the study of superheavy elements. Furthermore, some of the predicted regions of shell stability in superheavy nuclei lie close to the drip point and a suitable treatment is required. Many calculations of SHE have often used a zero-range two-body pairing force V pair = V 0,p/n δ(r − r ′ ), with adjustable strengths for protons and neutrons (see Refs. [33,35]). A study of SHE using the RHB approach, with the NL-SH parameter set, was carried out in Ref. [57].
To deal with the pairing correlations we use here a simplified prescription which we have previously found to be in acceptable agreement with RHB calculations [46]. The procedure is similar to the one employed for Skyrme forces in Ref. [58]. For each kind of nucleon we assume a constant pairing matrix element G q , which simulates the zero range of the pairing force, and we include quasibound levels in the BCS calculation as done in Ref. [58]. These levels of positive single-particle energy, retained by their centrifugal barrier (neutrons) or by their centrifugal-plus-Coulomb barrier (protons), mock up the influence of the continuum in the pairing calculation. The wave functions of the considered quasibound levels are mainly localized in the classically allowed region and decrease exponentially outside it. As a consequence, the unphysical nucleon gas which surrounds the nucleus if continuum levels are included in the normal BCS approach is eliminated [46]. We restrict the space of states involved in the pairing correlation to one harmonic oscillator shell above and below of the Fermi level, to avoid the unrealistic pairing of highly excited states and to confine the region of influence of the pairing potential to the vicinity of the Fermi level.
As described in Ref. [46], the solution of the pairing equations allows us to find the average pairing gap ∆ q for each kind of nucleon. We write the pairing matrix elements as G q = C q /A. We have fixed the constants C q by looking for the best agreement of our calculation with the known experimental binding energies of Ni and Sn isotopes for neutrons, and of N = 28 and N = 82 isotones for protons [46]. The values obtained from these fits are C n = 21 MeV and C p = 22.5 MeV for the G1 set, C n = 19 MeV and C p = 21 MeV for the G2 set, and C n = 20.5 MeV and C p = 23 MeV for the NL3 interaction.
In Ref. [46] we applied this improved BCS approach with the G1 and G2 parametrizations to study one-and two-neutron (proton) separation energies for several chains of isotopes (isotones) from stability to the drip lines. We found a reasonable agreement with the available experimental data, similar to the one obtained using the NL3 set. The analysis showed that the parameters sets based on EFT are able to describe nuclei far from the β-stability valley when a pairing residual interaction is included.

III. RESULTS AND DISCUSSION
Traditionally a large gap in the single-particle spectrum has been interpreted as an indicator of a shell closure, at least for nuclei of atomic number Z < 100. However, for a large nucleus like a superheavy element, it may not be sufficient to simply draw the single-particle level scheme and to look for the gaps, due to the complicated structure of the spectrum and the presence of levels with a high degree of degeneracy. Moreover, in a self-consistent calculation, a strong coupling between the neutron and proton shell structure takes place. Therefore, when dealing with SHE it is imperative to look for other quantities to reliably identify the shell closures and magic numbers, apart from the analysis of the single-particle level structure.
Here we shall consider the following observables as indicators for shell closures: a) A sudden jump in the two-neutron (two-proton) separation energies of even-even nuclei, defined as where N q is the number of neutrons (protons) in the nucleus for q = n (q = p). A sharp drop in S 2q means that a very small amount of energy is required to remove two more nucleons from the remnant of the parent nucleus. Thus, the parent nucleus is more stable which is a character of magicity. This observable is an efficient tool to quantify the shell effect because of the absence of odd-even effects [33].
b) The size of the gap in the neutron (proton) spectrum is determined by half of the difference in Fermi energy when going from a closed shell nucleus to a nucleus with two additional neutrons (protons). This quantity is very well accounted for by the two-neutron (two-proton) shell gap which is defined as the second difference of the binding energy [32,33]: This quantity measures the size of the step found in the two-nucleon separation energy and, therefore, it is strongly peaked at magic shell closures.
c) The neutron and proton average pairing gaps ∆ q of open-shell nuclei can be related to the odd-even mass difference, from where the empirical law ∆ ∼ 12/ √ A can be derived [59].
However, for closed shell nuclei ∆ q should vanish. Thus, we shall use the vanishing of the average pairing gap obtained from our calculations as another signal for identifying closed shell nuclei.
We next calculate the above observables for the isotopic chain of Z = 120 and for several isotonic chains, assuming spherical symmetry. We employ the parameter sets G1 and G2 due to the EFT formalism and compare the results with those obtained from the standard RMF parametrization NL3, which is well established as a successful interaction for nuclei at and away from the line of β stability.
It is to be mentioned that the previous indicators correspond to energy differences between neighbouring nuclei. However, they do not have a direct connection with the shell corrections which stabilize a given (N, Z) superheavy nucleus against fission [35]. The shell corrections are related to the difference between the nuclear binding energies and the predictions of a liquid-drop model. As a complementary study, after our search for spherical shell closures, we shall analyze the shell corrections for the discussed chains of SHE.
A. Isotopic chain of Z = 120 We first consider the chain of isotopes with atomic number Z = 120, which is found as a magic number in recent relativistic mean field calculations of nuclei in the superheavy mass region [32][33][34][35]. Figure 1  In order to analyze the force dependence of the location of the shell closures for the superheavy nuclei, we calculate the quantities S 2n , δ 2n and ∆ n with the EFT set G1 and with the NL3 parameter set for the same isotopic chain Z = 120 and display the results in Figures 2 and 3 and NL3 sets, indicating that the occupancy of the single-particle levels is diffused across the Fermi level, contrarily to the case of G2.
One expects a relatively large energy gap to appear between the last occupied and the first unoccupied single-particle levels for the neutron numbers corresponding to the shell closures detected above. Let us now look into the neutron single-particle spectra, displayed in Figure  two particular levels is strongly modified along the isotopic chain. Consequently, an analysis of the spectra alone would not suffice and the use of the discussed energy indicators becomes mandatory in order to make predictions for shell closures in superheavy nuclei.

B. Isotonic chains
We now proceed to discuss the isotonic chains of the neutron numbers which we have detected as candidates for spherical shell closures in the preceding study of the isotopic chain of Z = 120. We start with the N = 172 isotonic chain in Figure 5, which displays the two-proton separation energy S 2p , the two-proton shell gap δ 2p , and the average pairing gaps ∆ p and ∆ n in the superheavy region from Z = 100 up to the proton drip line, for the EFT model G2 and for the conventional RMF model NL3. For brevity we do not present the results from the G1 set, since the previous section has shown that the predictions of G2 differ from NL3 more than in the case of G1.
From Figure 5 one realizes that all the indicators signal a very robust shell closure at Z = 120, and a much weaker shell closure at Z = 114. The proton gap δ 2p (∼ 5 MeV) of the nucleus 292 120 is nearly twice as large as the corresponding neutron gap δ 2n (∼ 3 MeV, Figures 1 and 3). For the NL3 set, in addition, a little jump in S 2p and a small peaked structure in δ 2p indicates the possibility of a weak shell closure taking place at Z = 106. It is nevertheless known that the region around Z = 106 is deformed [36] and thus the spherical solution does not correspond to the ground state. Moreover, from the bottom panel of  Looking at the proton spectra for the systems with N = 172, a very large gap can be observed for 120 protons (between the 2f 5/2 and 3p 3/2 levels for G2, and between the 2f 5/2 and 1i 11/2 levels for NL3). Instead, practically no gap exists for 114 protons (between the 2f 7/2 and 2f 5/2 levels), specially for the NL3 set. This is consistent with the very weak signals of magicity of Z = 114 in the case of the N = 172 isotonic chain shown by the S 2p and δ 2p indicators in Figure 5.
With the addition of only 12 neutrons, the proton spectra for the systems with N = 184 exhibit a different pattern than for N = 172 near the Fermi energy (cf. Figure 8). The gaps occurring between the levels corresponding to 114 protons and to 120 protons are now comparable in magnitude. This fact is in agreement with the relatively magic character of the 298 114 and 304 120 nuclei predicted by the indicators plotted in Figure 6. In any case, even for N = 184, the magicity of Z = 114 is always smaller than the one shown by Z = 120, as one can see from the comparison of S 2p and δ 2p in Figures 5 and 6. This discussion shows again the strong dependence of the proton (neutron) shell closures of SHE on the neutron (proton) numbers and thus the importance of using the energy indicators.

C. Shell corrections
The stability of superheavy elements with an atomic number larger than Z ∼ 100 is possible thanks to the shell effects. In the liquid droplet model picture these superheavy nuclei are unstable against spontaneous fission because the large Coulomb repulsion can no longer be compensated by the nuclear surface tension. However, SHE may still exist because the quantal shell corrections generate local minima in the nuclear potential energy surface which provide additional stabilization.
In our context the shell correction energy is also useful as a different test for checking the robustness of the shell closures. For experimentally known shell closures, i.e., up to Z = 82 and N = 126, the shell corrections are strongly peaked around the magic numbers (see, e.g., Ref. [60]), providing enhanced binding for magic nuclei. However, in the superheavy mass region, instead of displaying sharp jumps, the shell corrections depict a landscape of rather broad areas of shell stabilized nuclei [34,36]. Still, in these areas the closed shell nuclei show a larger stabilization (i.e., more negative shell corrections) than their neighbors.
In the present subsection we want to study the shell corrections around our selected nuclei with Z = 114 and Z = 120, and N = 172, 184, and 258.
The calculation of the shell correction energy is based on the Strutinsky energy theorem [61] which states that the total quantal energy can be divided in two parts: The largest pieceẼ is the average part of the energy which depends in a smooth way on the number of nucleons (namely, the part well represented by the liquid droplet model).
The smaller piece, the shell correction E shell , has instead an oscillating behaviour. The oscillations are due to the grouping of levels into shells and display maxima at the shell closures. According to the idea of Strutinsky, the average part of the ground-state energy of a shell model potential can be obtained by replacing the Hartree-Fock occupation numbers n α (1 or 0 for occupied or empty states) with occupation numbersñ α smoothed by an averaging function [59]. The shell correction E shell is computed as the difference of the exact energy to that average part.
The Strutinsky smoothing procedure requires the use of several major shells. This faces the problem of the treatment of the continuum when realistic finite depth potentials are employed [34,36,62,63]. Our strategy here, working in coordinate space, and consistently with our approach to the treatment of pairing, is to perform the Strutinsky smoothing including the quasibound levels which are retained by their centrifugal barrier (centrifugalplus-Coulomb barrier for protons). We have taken 7 major shells above the Fermi energy (i.e., states up to around 50 MeV above the Fermi level) and have considered curvature corrections up to 2M = 10 [59]. We have found that the plateau condition of the averaged energy [59] is fulfilled for a smoothing parameter γ ∼ 1.3 − 1.6 MeV for both protons and neutrons. As we have discussed, the quasibound levels included in our calculation do not depend on the size of the box where the calculation is performed. These levels, usually with high angular momentum, lie close in energy to the RHB canonical levels [46]. Of course, one limitation of our approach is that some resonant levels with low angular momentum can be missed, more easily for neutrons, and then their contribution is shared among the higher angular momentum levels which we include in the calculation.
The total (neutron-plus-proton) shell corrections stemming from our calculations for the isotopic chains with Z = 114 and Z = 120 are displayed in Figure 9. The equivalent graph for the isotonic chains with N = 172, 184, and 258 is presented in Figure 10. In Figure 9 the isotopic chain of Z = 120 shows a large negative shell correction at N = 172, due to the presence of low angular momentun levels near the Fermi energy for both neutrons (4s 1/2 , 3d 3/2 and 3d 5/2 levels) and protons (3p 1/2 and 3p 3/2 levels). These levels imply a comparatively lower level density and thus a more negative shell correction.
The isotopic chain also shows another local minimum around N = 182−184, but in this case the shell correction energy is less negative than for N = 172. The pattern exhibited by the total shell correction for the Z = 120 isotopes looks very similar to that of the neutron shell correction displayed in Figure 5 of Ref. [34] for the NL3 parameter set, which was computed by means of the Green's function procedure. Looking at the curves for the Z = 114 isotopic chain represented in Figure 9 one realizes that the shell corrections are globally weaker than that for Z = 120 chain, which means less stability. They also present minima at N = 172 and at N = 184, although in this case the situation is reversed and the largest corrections correspond to N = 184 instead of N = 172.
In the upper panel of Figure  show similar patterns to the proton shell corrections of NL3 which are depicted in Figure 6 of Ref. [34] for these same isotonic chains. The curves of the shell correction for the N = 258 hyperheavy nuclei (lower panel of Figure 10)

IV. SUMMARY AND CONCLUSIONS
We have investigated the predictions of the G1 and G2 parametrizations of Ref. [38] obtained from the modern effective field theory approach to relativistic nuclear phenomenology for the occurrence of spherical double shell closures and the shell stabilizing effect in superheavy nuclei. Within an isotopic or isotonic chain of SHE the possible shell closures are identified by a simultaneous occurrence at a given Z or N of a large jump in the corresponding two-nucleon separation energy S 2q , a pronounced peak in the two-nucleon shell gap δ 2q , and the vanishing of the average pairing gaps ∆ n and ∆ p . To treat the pairing correlations we have employed an improved BCS model that was used successfully in Ref. [46] in calculations of isotopic and isotonic chains with magic proton or neutron numbers.
First we have studied the isotopic chain of Z = 120, which is found to be a magic number To summarize, in previous works [45,46] we showed that the parameter sets derived from the effective field theory approach to the low-energy nuclear many-body problem [38] work nicely for both β-stable and β-unstable nuclei. This is in addition to their ability to yield a realistic equation of state at densities above saturation which compares very favorably with microscopic Dirac-Brueckner-Hartree-Fock calculations [45].