Microscopic study of higher-order deformation effects on the ground states of superheavy nuclei around 270Hs

  • We investigate the effects of higher-order deformations βλ (λ=4,6,8, and 10) on the ground state properties of superheavy nuclei (SHN) near the doubly magic deformed nucleus 270Hs using the multidimensionally-constrained relativistic mean-field (MDC-RMF) model with five effective interactions: PC-PK1, PK1, NL3*, DD-ME2, and PKDD. The doubly magic properties of 270Hs include large energy gaps at N=162 and Z=108 in the single-particle spectra. By investigating the binding energies and single-particle levels of 270Hs in the multidimensional deformation space, we find that, among these higher-order deformations, the deformation β6 has the greatest impact on the binding energy and influences the shell gaps considerably. Similar conclusions hold for other SHN near 270Hs. Our calculations demonstrate that the deformation β6 must be considered when studying SHN using MDC-RMF.
  • One of the challenges in modern nuclear physics is exploring the mass and charge limits of atomic nuclei [1-8]. The prediction of the existence of an “island of stability” of superheavy nuclei (SHN) was made in the 1960s [9-14]. Currently, only the elements with Z118 have been synthesized [15-17]. Various predictions of the center of the “island of stability” have been made [9-14, 18-20], but the position of this island is not well established. In contrast to the “island of stability,” the existence of a “shallow” of SHN has been well established theoretically and experimentally. This “shallow” connects the continent of stable nuclei to the “island of stability.” The center of this shallow is predicted to be around Z=108 and N=162 and consists of deformed SHN [21-24]. 270108Hs162 is a doubly magic deformed nucleus [25, 26] and offers a prototype for exploring the structure of SHN.

    There are currently two kinds of theoretical approaches for studying the structures and properties of SHN, the macroscopic-microscopic method (MMM) and the microscopic method. Generally, the surface of a nucleus is parameterized as [27]

    R(θ,φ)=R0[1+β00+λ=1λμ=λβλμYλμ(θ,φ)],

    (1)

    where βλμ is the deformation parameter and R0 is the radius of a sphere with the same volume. However, there is an interesting consideration: how large should the dimension of the deformation space be when studying deformed SHN? In 1991, Patyk and Sobiczewski investigated the ground state properties of the heaviest even-even nuclei with proton numbers Z=90–114 and neutron numbers N=136–168 using the MMM and found that the β6 degree of freedom is important for binding energies and the formation of deformed shells [23, 28]. Additionally, β6 has a considerable influence on the moments of inertia [29, 30] and high-K isomers [31, 32]. The microscopic description of the structure of SHN can be achieved using density functional theories, with few studies investigating the influence of β6 on the binding energy and shell structure of SHN to date.

    Covariant density functional theory (CDFT) is one of the most successful self-consistent approaches and has been used to describe ground and excited states of nuclei throughout the nuclear chart [33-41]. To investigate the ground state properties, potential energy surfaces (PESs), and fission barriers of heavy nuclei and SHN, multidimensionally-constrained (MDC) CDFTs have been developed [41-44]. MDC-CDFTs have been applied to investigate hypernuclei [45-48], the fission barriers and the PESs of actinide nuclei [42, 43, 49], the ground state properties and PESs of 270Hs [50], the nonaxial octupole Y32 correlations in N=150 isotones [51] and Zr isotopes [52], octupole correlations in MχD of 78Br [53] and Ba isotopes [54], etc. In MDC-CDFTs, reflection and axial symmetry are both broken, and the shape degrees of freedom βλμ, where µ is an even number, are self-consistently included, such as β20,β22,β30,β32,β40,β42, and β44. Either the Bardeen-Cooper-Schrieffer (BCS) approach or the Bogoliubov transformation have been implemented to consider the pairing effects. With two different approaches to treat pairing correlations, there are two types of MDC-CDFTs: when using the BCS approach, the MDC relativistic mean-field (RMF) model is used, and for the Bogoliubov transformation, the MDC relativistic Hartree-Bogoliubov (RHB) theory is employed.

    In this study, we use the MDC-RMF model to investigate the ground state properties of SHN around the doubly magic deformed nucleus 270Hs and focus on the influence of the higher-order deformations. This paper is organized as follows. The MDC-CDFTs is introduced in Sec. II. In Sec. III, the results and discussions are presented. Finally, we summarize the study in Sec. IV.

    In the CDFT, nucleons interact with each other through the exchange of mesons and photons or point-coupling interactions. To obtain accurate saturation properties of nuclear matter, the non-linear coupling terms or the density dependence of the coupling constants are introduced. Subsequently, there are four kinds of covariant density functionals: either meson exchange (ME) or point-coupling (PC) combined with the non-linear (NL) or density dependent (DD) couplings. In this study, both the ME and PC density functionals are used. The main formulae of the MDC-CDFTs can be found in Refs. [41, 43, 50, 52]. For convenience, we only introduce the MDC-RMF with the NL-PC effective interactions briefly.

    The NL-PC Lagrangian is

    L=ˉψ(iγμμM)ψLlinLnlLderLCou,

    (2)

    where the linear, nonlinear, derivative couplings, and the Coulomb terms respectively are

    Llin=12αSρ2S+12αVρ2V+12αTSρ2TS+12αTVρTV2,

    (3)

    Lnl=13βSρ3S+14γSρ4S+14γV[ρ2V]2,

    (4)

    Lder=12δS[νρS]2+12δV[νρV]2+12δTS[νρTS]2+12δTV[νρTV]2,

    (5)

    LCou=14FμνFμν+e1τ32A0ρV,

    (6)

    where M represents the nucleon mass, e is the unit charge, and αS, αV, αTS, αTV, βS, γS, γV, δS, δV, δTS, and δTV are coupling constants. The isoscalar density ρS, isovector density ρTS, the time-like components of isoscalar current ρV, and the time-like components of isovector currents ρTV are defined as

    ρS=ˉψψ,ρTS=ˉψτψ,ρV=ˉψγ0ψ,ρTV=ˉψτγ0ψ.

    (7)

    The single particle wave function ψk(r) with energy ϵk of a nucleon is obtained by solving the Dirac equation

    ˆhψk(r)=ϵkψk(r),

    (8)

    with the Dirac Hamiltonian

    ˆh=αp+β[M+S(r)]+V(r),

    (9)

    where the scalar potential S(r) and vector potential V(r) are

    S=αSρS+αTSρTSτ+βSρ2S+γSρ3S+δSΔρS+δTSΔρTSτ,V=αVρV+αTVρTVτ+γVρ2VρV+δVΔρV+δTVΔρTVτ+e1τ32A0.

    (10)

    In the MDC-CDFTs, the wave functions are expanded using the axially deformed harmonic oscillator (ADHO) basis [55, 56], which is obtained by solving the Schrödinger equation

    [22M2+VB(z,ρ)]Φα(rσ)=EαΦα(rσ),

    (11)

    where r=(z,ρ), where ρ=x2+y2, and

    VB(z,ρ)=12M(ω2ρρ2+ω2zz2),

    (12)

    is the ADHO potential with the oscillator frequency, represented by ωρ (ωz), perpendicular to (along) the z axis. More detailed formulae on the applications of ADHO in MDC-RMF can be found in Refs. [41, 43, 50, 52].

    After obtaining the ADHO basis, the single-particle wave functions can be expanded using the basis

    ψi(rσ)=(αfαiΦα(rσ)αgαiΦα(rσ)),

    (13)

    where α denotes a set of quantum numbers of the ADHO basis function, α{nz,nρ,ml,ms}, and fαi and gαi are the expansion coefficients. For the truncation of the ADHO basis, we follow Refs. [55, 57]. Finally, the wave functions are obtained by self-consistent iterations.

    270Hs is an axially deformed nucleus in the ground state [50, 58, 59]. Reflection-asymmetric deformations normally occur only for ultra-neutron-rich nuclei with N182 in the SHN region [59]. Thus, we only consider the axially symmetric deformations βλ, where λ is an even number, in this study. To investigate the influence of each shape degree of freedom on the bulk properties of SHN, constraint calculations on mass multipole moments are performed [27]. In MDC-CDFTs, a modified linear-constraint method is implemented [42, 43] and the Routhian reads

    E=ERMF+λ12CλQλ.

    (14)

    After the nth iteration, the variable C(n+1)λ is determined by

    C(n+1)λ=C(n)λ+kλ(β(n)λβλ),

    (15)

    where C(n)λ is the value of the nth iteration, kλ is a constant, and βλ is the desired value of the deformation parameter.

    The intrinsic multipole moments are calculated as

    Qλ,τ=d3rρτ(r)rλYλ0(Ω),

    (16)

    where τ represents the nucleon (the neutron or the proton), and ρτ is the corresponding vector density. The deformation parameter βλ,τ is given by

    βλ,τ=4π3NτRλQλ,τ,

    (17)

    where R=r0A1/3, with r0=1.2 fm, and Nτ represents the corresponding particle's number, A, N, or Z.

    To investigate the influence of higher-order deformations on the ground state properties of SHN, the doubly magic deformed nucleus 270Hs, even-even Hs isotopes from 264Hs to 276Hs, and N=162 isotones from 266Rf to 272Ds are analyzed. When investigating the ground state properties and the influence of higher-order deformations, the ADHO basis with Nf=20 shells is adopted, leading to an accuracy of 0.1 MeV in total energy of 270Hs [50] in the MDC-RMF caclulations. In the particle-particle channel, a separable pairing force is adopted. Here, the strength and effective range of this force are taken to be the same as those in Ref. [50]: G=1.1G0, where G0=728 MeV fm3, and a=0.644 fm. The effective interactions PC-PK1 [60], PK1 [61], PKDD [61], DD-ME2 [62], and NL3* [63] are employed in the particle-hole channel.

    The ground state properties, including deformation parameters βλ (λ=2,4,6,8, and 10), radii, and binding energies of even-even Hs isotopes with the above-mentioned five effective interactions are given in Table 1, and even-even isotones with N=162 are listed in Table 2. The binding energies of one nucleus with five effective interactions differ from each other, e.g., the largest binding energy of 270Hs is 1973.77 MeV with PK1 and the smallest is 1967.41 MeV with PC-PK1. Such results are relatively close to the empirical values in AME2020, EB=1969.65 MeV [64-66], and are also comparable to the predictions of other models, such as MMM, with EB=1969.20 MeV [23], the Skyrme Hartree-Fock Bogoliubov mass model (HFB-24), with EB=1968.45 MeV [67], the Weizsäcker-Skyrme (WS) mass formula WS4, with EB=1970.27 MeV [68], the finite range droplet model (FRDM(2012)), with EB=1971.48 MeV [58], and several RMF calculations [69-73]. For other nuclei, similar conclusions can also be drawn. From these two tables, it is clear that all considered nuclei are deformed in the MDC-RMF calculations with five effective interactions. This is consistent with the results obtained from MMM calculations [23, 58, 59] and other global studies [74, 75]. Additionally, it has been shown that the inclusion of the rotational energy correction (REC) can improve the description of binding energies with PC-PK1 [60]. In this study, after considering RECs in PC-PK1 calculations, the binding energy of 270Hs changes from 1967.45 to 1969.76 MeV, which is closer to the value provided in AME2020.

    Table 1

    Table 1.  Ground state properties, including quadrupole deformation parameters, of neutrons and protons (β2,n and β2,p), deformation parameters β2, β4, β6, β8, and β10, mass radius Rt, radii of protons and neutrons (Rp and Rn), charge radius Rc, and binding energy EB of Hs isotopes using MDC-RMF with five effective interactions: PC-PK1, PK1, PKDD, DD-ME2, and NL3*.
    β2,nβ2,pβ2β4β6β8β10Rn/fmRp/fmRt/fmRc/fmEB/MeV
    PC-PK1
    264Hs0.2700.2800.274−0.002−0.060−0.0130.0116.2456.0906.1826.1381924.415
    266Hs0.2660.2760.271−0.021−0.063−0.0040.0146.2676.1016.2006.1481939.205
    268Hs0.2620.2730.266−0.040−0.0630.0040.0156.2886.1116.2176.1581953.554
    270Hs0.2570.2690.261−0.057−0.0610.0120.0156.3066.1206.2326.1671967.408
    272Hs0.2450.2580.250−0.060−0.0490.0130.0106.3306.1316.2526.1781979.303
    274Hs0.2160.2280.221−0.053−0.0380.0090.0066.3446.1356.2636.1821990.951
    276Hs0.1880.1980.192−0.049−0.0270.0070.0036.3576.1396.2736.1852002.778
    PK1
    264Hs0.2530.2580.2550.006−0.058−0.0160.0116.2286.0586.1596.1051934.074
    266Hs0.2530.2580.255−0.014−0.065−0.0060.0166.2536.0706.1796.1181947.952
    268Hs0.2560.2610.258−0.034−0.0700.0050.0196.2786.0846.2016.1311961.285
    270Hs02450.2510.248−0.044−0.0620.0100.0166.2976.0916.2166.1381973.766
    272Hs0.2110.2160.213−0.029−0.0530.0050.0106.3056.0906.2216.1371985.924
    274Hs0.1940.1980.195−0.038−0.0400.0060.0106.3226.0976.2346.1441997.412
    276Hs0.1780.1820.180−0.047−0.0280.0070.0076.3426.1056.2506.1512008.356
    PKDD
    264Hs0.2500.2550.2520.001−0.060−0.0150.0116.2076.0536.1456.1011932.544
    266Hs0.2530.2580.255−0.020−0.066−0.0040.0166.2336.0676.1666.1151946.294
    268Hs0.2580.2640.260−0.041−0.0720.0090.0216.2596.0826.1886.1291959.686
    270Hs0.2520.2610.256−0.059−0.0620.0170.0176.2786.0916.2046.1381972.399
    272Hs0.2110.2170.213−0.030−0.0560.0060.0196.2826.0876.2056.1341983.376
    274Hs0.1900.1940.191−0.039−0.0410.0060.0106.2966.0936.2166.1391994.504
    276Hs0.1740.1790.176−0.048−0.0280.0070.0076.3166.1006.2336.1472004.934
    DD-ME2
    264Hs0.2600.2670.263−0.001−0.061−0.0120.0146.1786.0736.1366.1211928.426
    266Hs0.2610.2690.264−0.023−0.066−0.0010.0196.2006.0866.1546.1331942.991
    268Hs0.2590.2690.263−0.042−0.0680.0110.0216.2206.0976.1716.1441957.270
    270Hs0.2520.2640.257−0.058−0.0600.0170.0176.2366.1056.1846.1521971.027
    272Hs0.2130.2220.216−0.032−0.0540.0070.0126.2406.1016.1856.1481981.994
    274Hs0.1960.2040.199−0.039−0.0410.0070.0106.2566.1076.1986.1541993.391
    276Hs0.1780.1860.181−0.048−0.0270.0080.0076.2716.1136.2106.1602004.657
    NL3*
    264Hs0.2650.2710.2670.004−0.060−0.0140.0136.2606.0796.1866.1261931.827
    266Hs0.2630.2700.266−0.017−0.065−0.0040.0176.2846.0906.2066.1371945.933
    268Hs0.2620.2690.265−0.036−0.0670.0070.0186.3076.1016.2256.1481959.582
    270Hs0.2560.2650.260−0.054−0.0610.0150.0176.3266.1096.2406.1561972.574
    272Hs0.2330.2410.236−0.045−0.0510.0100.0126.3446.1156.2546.1621984.027
    274Hs0.2040.2110.207−0.039−0.0400.0070.0096.3576.1186.2646.1641995.465
    276Hs0.1850.1930.188−0.047−0.0280.0080.0066.3766.1256.2796.1712006.648
    DownLoad: CSV
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    Table 2

    Table 2.  Ground state properties, including quadrupole deformation parameters, of neutrons and protons (β2,n and β2,p), deformation parameters β2,β4,β6,β8 and β10, mass radius Rt, radii of protons and neutrons (Rp and Rn), charge radius Rc, and binding energies EB of N=162 isotones using MDC-RMF with five effective interactions: PC-PK1, PK1, PKDD, DD-ME2, and NL3*.
    β2,nβ2,pβ2β4β6β8β10Rn/fmRp/fmRt/fmRc/fmEB/MeV
    PC-PK1
    266Rf0.2610.2740.266−0.039−0.0600.0050.0136.3036.0836.2186.1291953.531
    268Sg0.2600.2740.266−0.048−0.0630.0090.0156.3046.1026.2256.1491961.448
    270Hs0.2570.2690.261−0.057−0.0610.0120.0156.3066.1206.2326.1671967.408
    272Ds0.2490.2580.253−0.061−0.0550.0150.0126.3086.1396.2406.1861971.338
    PK1
    266Rf0.2560.2650.260−0.038−0.0620.0070.0176.2966.0536.2026.1001957.699
    268Sg0.2570.2680.261−0.046−0.0670.0100.0186.2996.0756.2116.1221966.814
    270Hs0.2450.2510.248−0.044−0.0620.0100.0166.2976.0916.2166.1381973.766
    272Ds0.2250.2280.226−0.037−0.0540.0070.0126.2926.1046.2176.1511979.041
    PKDD
    266Rf0.2560.2660.260−0.043−0.0630.0090.0186.2726.0506.1866.0971955.230
    268Sg0.2590.2720.264−0.051−0.0700.0130.0206.2776.0736.1976.1201965.167
    270Hs0.2520.2610.256−0.059−0.0620.0170.0176.2786.0916.2046.1381972.399
    272Ds0.2410.2450.242−0.060−0.0560.0170.0156.2786.1076.2106.1541977.775
    DD-ME2
    266Rf0.2550.2680.260−0.040−0.0610.0080.0176.2246.0626.1616.1091955.711
    268Sg0.2570.2710.262−0.048−0.0650.0120.0196.2316.0866.1746.1321964.571
    270Hs0.2520.2640.257−0.058−0.0600.0170.0176.2366.1056.1846.1521971.027
    272Ds0.2420.2490.245−0.060−0.0540.0170.0156.2416.1236.1936.1701975.320
    NL3*
    266Rf0.2610.2720.266−0.038−0.0620.0070.0176.3256.0706.2266.1161956.503
    268Sg0.2610.2730.266−0.047−0.0660.0110.0186.3256.0916.2336.1371965.582
    270Hs0.2560.2650.260−0.054−0.0610.0150.0176.3266.1096.2406.1561972.574
    272Ds0.2480.2530.250−0.058−0.0550.0160.0146.3266.1276.2476.1741977.689
    DownLoad: CSV
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    To determine the dimension of the deformation space when investigating the ground states of SHN using MDC-CDFTs, we calculate the binding energies of Hs isotopes and isotones with N=162 in a different deformation space {βλ;λ=0,2,,λmax} with λmax being the maximum order of deformation parameters, meaning all deformation parameters βλβλmax are considered self-consistently, while other deformation parameters are constrained to zero. In Fig. 1, the binding energies of N=162 isotones with five different effective interactions are plotted as a function of λmax. For convenience, we take 270Hs with the effective interaction PC-PK1 as an example to discuss the influence of each order of deformation on the binding energy in detail. When constraining 270Hs to be spherical, i.e., in the deformation space {βλ;λ=0}, the resulting binding energy is 1956.39 MeV, which is close to the prediction from the relativistic continuum Hartree-Bogoliubov theory, which is 1952.65 MeV [76], but not close to the value given in AME2020 (marked by a black square in Fig. 1). After considering the quadrupole deformation β2, the binding energy of 270Hs changes considerably (by approximately 8.43 MeV) and approaches that from AME2020. This result indicates the importance of the quadrupole deformation. The influence of the hexadecapole deformation β4 on the binding energy is smaller, with a change of only 0.68 MeV. Considering β6, the energy change is approximately 1.87 MeV, which is larger than that of β4, and EB approaches the value provided in AME2020. Including β8 and β10 does not affect the binding energy, which converges well at {βλ;λ=0,2,,10}. From these results, we can conclude that to obtain a proper description of 270Hs, one should consider the β6 deformation at least with respect to the binding energy. Calculated binding energy versus λmax with other density functionals are also displayed in Fig. 1, and although the binding energies with five effective interactions differ from each other, the overall trend of EB changing with λmax are similar. The binding energy of 270Hs is predominantly changed by β2, then β6 and β4. The influence of β8 and β10 can be ignored. The RECs for 270Hs with PC-PK1 are 2.27, 2.03, 2.29, 2.31, and 2.31 MeV in deformation spaces {βλ;λ=0,,λmax}, with λmax=2,4,6,8, and 10, respectively. The values of RECs change slightly in different deformation spaces and only minimally influence the trends of binding energies with respect to λmax.

    Figure 1

    Figure 1.  (color online) Binding energies of N=162 isotones for PC-PK1, DD-ME2, PKDD, NL3*, and PK1 as a function of λmax. The black square and black circle represent the values from AME2020 and FRMD(2012), respectively.

    To check whether the above conclusion is valid for other SHN, we perform similar calculations for even-even isotones with N=162 and Hs isotopes, with the results also presented in Figs. 1 and 2. From these figures, we find that the binding energies of these nuclei are significantly changed by β2. The influence of β4 and β6 should also be considered, and the contribution to the total energy from β6 is larger than that from β4. For Hs isotopes, with decreasing neutron number, the value of β2 increases substantially and the differences of the total energies between the spherical case and in the ground states increase, which can be observed in Fig. 2. For isotones with N=162, the value of β2 only minimally changes with proton number. For these nuclei, the relationship between binding energies and βλmax are the same as for 270Hs.

    Figure 2

    Figure 2.  (color online) Binding energies of Hs isotopes for PC-PK1, DD-ME2, PKDD, NL3*, and PK1 as a function of λmax. The black square and black circle represent the values from AME2020 and FRMD(2012), respectively.

    It is well known that shell structure is particularly important for SHN and very sensitive to the deformation of the nucleus [28]. Using 270Hs as an example, we explore how the deformations influence the shell gaps at Z=108 and N=162 by investigating the structure of single-particle levels (SPLs) in different deformation spaces. In Fig. 3, we illustrate the SPLs for protons and neutrons of 270Hs versus λmax, calculated with PC-PK1. When λmax=10, i.e., for the ground state, the energy gaps at Z=108 and N=162 are approximately 1.34 MeV and 1.85 MeV, which are considerably large for such a heavy nucleus [77] and result in deformed shells.

    Figure 3

    Figure 3.  (color online) Single proton and neutron levels of 270Hs from PC-PK1 calculations with different λmax. In the spherical case (λmax=0), each level is labelled by |nlj. When λmax0, each level is labelled by the projection Ω of total angular momentum on the symmetry axis and the parity π. Single-particle levels with positive and negative parities are presented by red and black lines, respectively.

    In the spherical limit, λmax=0, each single particle state is labelled by |nlj, where n, l, and j denote the radial quantum number, orbital angular momentum, and total angular momentum, respectively. It is obvious that there are no shell gaps at Z=108 and N=162. After including β2, a spherical orbital |nlj with a degeneracy of 2j+1 splits into (2j+1)/2 levels, with each one represented by Ωπ with the projection Ω of total angular momentum on the symmetry axis and the parity π. It is found that due to quadrupole correlations, the shell gaps at Z=108 and N=162 appear, with values of 0.66 MeV and 1.17 MeV, respectively. When including β4 into the deformation space, the order of SPLs around the two gaps changes and the shell gaps at Z=108 (up to approximately 1.28 MeV) and N=162 (up to approximately 1.56 MeV) increase considerably. The impact of β6 on the shell gap at Z=108 is smaller, only 0.02 MeV. Although for neutrons, the shell gap at N=162 increases by approximately 0.26 MeV. The inclusion of β8 and β10 produces only extremely minimal changes in the shell gaps and the order of SPLs. From this, one can conclude that β2 plays a vital role in the formation of the shell closures Z=108 and N=162, which are further enhanced by β4. The influence of β6 is relatively small and the effects of β8 and β10 can be considered negligible. However, there remains a question: where do the Y60 correlations come from? By checking the SPLs, we find that two proton levels 1/2+ originating from the spherical orbitals 3s1/2 and 1i13/2 are very close to each other and the mixing of these two spherical orbitals in the deformed SPLs results in Y60 correlations. For neutrons, these correlations originate from the mixing of the spherical orbitals 3p3/2 and 1j15/2 in the levels 3/2, close to the neutron Fermi energy.

    In this study, we investigate the ground state properties of SHN around 270Hs in multidimensional deformation spaces using the MDC-RMF model with five density functionals. The influence of higher-order deformation parameters on the ground state of nuclei near 270Hs are investigated, including the binding energies and SPLs. We show that the binding energies of deformed SHN around 270Hs are significantly affected by higher-order deformations. In particular, the influence of β6 on binding energy is larger than that of β4. For the doubly magic nucleus 270Hs, the deformed shell gaps at Z=108 and N=162 are mainly determined by quarupole correlations and are enhanced by the inclusion of β4. In conclusion, at least the β6 degree of freedom should be considered in the investigation of SHN using CDFTs. It is also very interesting to investigate how the higher-order deformations influence other properties of SHN, such as the moment of inertia and energy spectra, by using density functional theories. Additionally, we would like to mention that the calculations performed in this study can also be performed using the deformation relativistic Hartree-Bogoliubov (DRHBc) theory [78-84], in which the scalar potential and densities are expanded in terms of the Legendre polynomials, though the DRHBc theory is much more time consuming than the MDC-RMF model. Very recently, the influence of higher-order deformations on possible bound nuclei beyond the drip line has also been investigated in the transfermium region from No (Z=102) to Ds (Z=110) using the DRHBc theory [85], and the development of a nuclear mass table using the DRHBc theory is in progress [86-89].

    We thank Bing-Nan Lu, Yu-Ting Rong, and Kun Wang for helpful discussions. The results described in this paper were obtained on the High-performance Computing Cluster of ITP-CAS and the ScGrid of the Supercomputing Center, Computer Network Information Center of the Chinese Academy of Sciences.

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3. Chen, B., Dong, J., Wang, Y. et al. Robustness of N=152 and Z=108 shell closures in superheavy mass region[J]. Chinese Physics C, 2025, 49(1): 011001. doi: 10.1088/1674-1137/ad8d4b
4. Moscato, P., Grebogi, R. Approximating the nuclear binding energy using analytic continued fractions[J]. Scientific Reports, 2024, 14(1): 11559. doi: 10.1038/s41598-024-61389-5
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6. Rong, Y.-T., Wu, X.-Y., Lu, B.-N. et al. Structures of 96Zr and 96Ru with covariant density functional theory | [96Zr和96Ru结构的协变密度泛函理论研究][J]. Scientia Sinica: Physica, Mechanica et Astronomica, 2024, 54(9): 292010. doi: 10.1360/SSPMA-2024-0059
7. Zheng, R.-Y., Sun, X.-X., Shen, G.-F. et al. Evolution of N = 20, 28, 50 shell closures in the 20 ≤ Z ≤ 30 region in deformed relativistic Hartree-Bogoliubov theory in continuum[J]. Chinese Physics C, 2024, 48(1): 014107. doi: 10.1088/1674-1137/ad0bf2
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9. Sharma, H., Jain, S., Kumar, R. et al. Optimum orientation of compact and elongated hexadecapole deformed actinide targets: Application to synthesizing superheavy nuclei[J]. Physical Review C, 2023, 108(4): 044613. doi: 10.1103/PhysRevC.108.044613
10. Zhang, K.Y., Yang, S.Q., An, J.L. et al. Missed prediction of the neutron halo in 37Mg[J]. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 2023. doi: 10.1016/j.physletb.2023.138112
11. Liu, L.-M., Xu, J., Peng, G.-X. Measuring deformed neutron skin with free spectator nucleons in relativistic heavy-ion collisions[J]. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 2023. doi: 10.1016/j.physletb.2023.137701
12. Deng, X.-Q., Zhou, S.-G. Examination of promising reactions with Am 241 and Cm 244 targets for the synthesis of new superheavy elements within the dinuclear system model with a dynamical potential energy surface[J]. Physical Review C, 2023, 107(1): 014616. doi: 10.1103/PhysRevC.107.014616
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14. An, R., Jiang, X., Cao, L.-G. et al. Evolution of nuclear charge radii in copper and indium isotopes[J]. Chinese Physics C, 2022, 46(6): 064101. doi: 10.1088/1674-1137/ac501a

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Xiao-Qian Wang, Xiang-Xiang Sun and Shan-Gui Zhou. Microscopic study of higher-order deformation effects on the ground states of superheavy nuclei around 270Hs[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac3904
Xiao-Qian Wang, Xiang-Xiang Sun and Shan-Gui Zhou. Microscopic study of higher-order deformation effects on the ground states of superheavy nuclei around 270Hs[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac3904 shu
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Microscopic study of higher-order deformation effects on the ground states of superheavy nuclei around 270Hs

    Corresponding author: Xiang-Xiang Sun, sunxiangxiang@ucas.ac.cn
  • 1. CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, China
  • 2. School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
  • 3. School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China

Abstract: We investigate the effects of higher-order deformations βλ (λ=4,6,8, and 10) on the ground state properties of superheavy nuclei (SHN) near the doubly magic deformed nucleus 270Hs using the multidimensionally-constrained relativistic mean-field (MDC-RMF) model with five effective interactions: PC-PK1, PK1, NL3*, DD-ME2, and PKDD. The doubly magic properties of 270Hs include large energy gaps at N=162 and Z=108 in the single-particle spectra. By investigating the binding energies and single-particle levels of 270Hs in the multidimensional deformation space, we find that, among these higher-order deformations, the deformation β6 has the greatest impact on the binding energy and influences the shell gaps considerably. Similar conclusions hold for other SHN near 270Hs. Our calculations demonstrate that the deformation β6 must be considered when studying SHN using MDC-RMF.

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    I.   INTRODUCTION
    • One of the challenges in modern nuclear physics is exploring the mass and charge limits of atomic nuclei [1-8]. The prediction of the existence of an “island of stability” of superheavy nuclei (SHN) was made in the 1960s [9-14]. Currently, only the elements with Z118 have been synthesized [15-17]. Various predictions of the center of the “island of stability” have been made [9-14, 18-20], but the position of this island is not well established. In contrast to the “island of stability,” the existence of a “shallow” of SHN has been well established theoretically and experimentally. This “shallow” connects the continent of stable nuclei to the “island of stability.” The center of this shallow is predicted to be around Z=108 and N=162 and consists of deformed SHN [21-24]. 270108Hs162 is a doubly magic deformed nucleus [25, 26] and offers a prototype for exploring the structure of SHN.

      There are currently two kinds of theoretical approaches for studying the structures and properties of SHN, the macroscopic-microscopic method (MMM) and the microscopic method. Generally, the surface of a nucleus is parameterized as [27]

      R(θ,φ)=R0[1+β00+λ=1λμ=λβλμYλμ(θ,φ)],

      (1)

      where βλμ is the deformation parameter and R0 is the radius of a sphere with the same volume. However, there is an interesting consideration: how large should the dimension of the deformation space be when studying deformed SHN? In 1991, Patyk and Sobiczewski investigated the ground state properties of the heaviest even-even nuclei with proton numbers Z=90–114 and neutron numbers N=136–168 using the MMM and found that the β6 degree of freedom is important for binding energies and the formation of deformed shells [23, 28]. Additionally, β6 has a considerable influence on the moments of inertia [29, 30] and high-K isomers [31, 32]. The microscopic description of the structure of SHN can be achieved using density functional theories, with few studies investigating the influence of β6 on the binding energy and shell structure of SHN to date.

      Covariant density functional theory (CDFT) is one of the most successful self-consistent approaches and has been used to describe ground and excited states of nuclei throughout the nuclear chart [33-41]. To investigate the ground state properties, potential energy surfaces (PESs), and fission barriers of heavy nuclei and SHN, multidimensionally-constrained (MDC) CDFTs have been developed [41-44]. MDC-CDFTs have been applied to investigate hypernuclei [45-48], the fission barriers and the PESs of actinide nuclei [42, 43, 49], the ground state properties and PESs of 270Hs [50], the nonaxial octupole Y32 correlations in N=150 isotones [51] and Zr isotopes [52], octupole correlations in MχD of 78Br [53] and Ba isotopes [54], etc. In MDC-CDFTs, reflection and axial symmetry are both broken, and the shape degrees of freedom βλμ, where µ is an even number, are self-consistently included, such as β20,β22,β30,β32,β40,β42, and β44. Either the Bardeen-Cooper-Schrieffer (BCS) approach or the Bogoliubov transformation have been implemented to consider the pairing effects. With two different approaches to treat pairing correlations, there are two types of MDC-CDFTs: when using the BCS approach, the MDC relativistic mean-field (RMF) model is used, and for the Bogoliubov transformation, the MDC relativistic Hartree-Bogoliubov (RHB) theory is employed.

      In this study, we use the MDC-RMF model to investigate the ground state properties of SHN around the doubly magic deformed nucleus 270Hs and focus on the influence of the higher-order deformations. This paper is organized as follows. The MDC-CDFTs is introduced in Sec. II. In Sec. III, the results and discussions are presented. Finally, we summarize the study in Sec. IV.

    II.   THEORETICAL FRAMEWORK
    • In the CDFT, nucleons interact with each other through the exchange of mesons and photons or point-coupling interactions. To obtain accurate saturation properties of nuclear matter, the non-linear coupling terms or the density dependence of the coupling constants are introduced. Subsequently, there are four kinds of covariant density functionals: either meson exchange (ME) or point-coupling (PC) combined with the non-linear (NL) or density dependent (DD) couplings. In this study, both the ME and PC density functionals are used. The main formulae of the MDC-CDFTs can be found in Refs. [41, 43, 50, 52]. For convenience, we only introduce the MDC-RMF with the NL-PC effective interactions briefly.

      The NL-PC Lagrangian is

      L=ˉψ(iγμμM)ψLlinLnlLderLCou,

      (2)

      where the linear, nonlinear, derivative couplings, and the Coulomb terms respectively are

      Llin=12αSρ2S+12αVρ2V+12αTSρ2TS+12αTVρTV2,

      (3)

      Lnl=13βSρ3S+14γSρ4S+14γV[ρ2V]2,

      (4)

      Lder=12δS[νρS]2+12δV[νρV]2+12δTS[νρTS]2+12δTV[νρTV]2,

      (5)

      LCou=14FμνFμν+e1τ32A0ρV,

      (6)

      where M represents the nucleon mass, e is the unit charge, and αS, αV, αTS, αTV, βS, γS, γV, δS, δV, δTS, and δTV are coupling constants. The isoscalar density ρS, isovector density ρTS, the time-like components of isoscalar current ρV, and the time-like components of isovector currents ρTV are defined as

      ρS=ˉψψ,ρTS=ˉψτψ,ρV=ˉψγ0ψ,ρTV=ˉψτγ0ψ.

      (7)

      The single particle wave function ψk(r) with energy ϵk of a nucleon is obtained by solving the Dirac equation

      ˆhψk(r)=ϵkψk(r),

      (8)

      with the Dirac Hamiltonian

      ˆh=αp+β[M+S(r)]+V(r),

      (9)

      where the scalar potential S(r) and vector potential V(r) are

      S=αSρS+αTSρTSτ+βSρ2S+γSρ3S+δSΔρS+δTSΔρTSτ,V=αVρV+αTVρTVτ+γVρ2VρV+δVΔρV+δTVΔρTVτ+e1τ32A0.

      (10)

      In the MDC-CDFTs, the wave functions are expanded using the axially deformed harmonic oscillator (ADHO) basis [55, 56], which is obtained by solving the Schrödinger equation

      [22M2+VB(z,ρ)]Φα(rσ)=EαΦα(rσ),

      (11)

      where r=(z,ρ), where ρ=x2+y2, and

      VB(z,ρ)=12M(ω2ρρ2+ω2zz2),

      (12)

      is the ADHO potential with the oscillator frequency, represented by ωρ (ωz), perpendicular to (along) the z axis. More detailed formulae on the applications of ADHO in MDC-RMF can be found in Refs. [41, 43, 50, 52].

      After obtaining the ADHO basis, the single-particle wave functions can be expanded using the basis

      ψi(rσ)=(αfαiΦα(rσ)αgαiΦα(rσ)),

      (13)

      where α denotes a set of quantum numbers of the ADHO basis function, α{nz,nρ,ml,ms}, and fαi and gαi are the expansion coefficients. For the truncation of the ADHO basis, we follow Refs. [55, 57]. Finally, the wave functions are obtained by self-consistent iterations.

      270Hs is an axially deformed nucleus in the ground state [50, 58, 59]. Reflection-asymmetric deformations normally occur only for ultra-neutron-rich nuclei with N182 in the SHN region [59]. Thus, we only consider the axially symmetric deformations βλ, where λ is an even number, in this study. To investigate the influence of each shape degree of freedom on the bulk properties of SHN, constraint calculations on mass multipole moments are performed [27]. In MDC-CDFTs, a modified linear-constraint method is implemented [42, 43] and the Routhian reads

      E=ERMF+λ12CλQλ.

      (14)

      After the nth iteration, the variable C(n+1)λ is determined by

      C(n+1)λ=C(n)λ+kλ(β(n)λβλ),

      (15)

      where C(n)λ is the value of the nth iteration, kλ is a constant, and βλ is the desired value of the deformation parameter.

      The intrinsic multipole moments are calculated as

      Qλ,τ=d3rρτ(r)rλYλ0(Ω),

      (16)

      where τ represents the nucleon (the neutron or the proton), and ρτ is the corresponding vector density. The deformation parameter βλ,τ is given by

      βλ,τ=4π3NτRλQλ,τ,

      (17)

      where R=r0A1/3, with r0=1.2 fm, and Nτ represents the corresponding particle's number, A, N, or Z.

    III.   RESULTS AND DISCUSSIONS
    • To investigate the influence of higher-order deformations on the ground state properties of SHN, the doubly magic deformed nucleus 270Hs, even-even Hs isotopes from 264Hs to 276Hs, and N=162 isotones from 266Rf to 272Ds are analyzed. When investigating the ground state properties and the influence of higher-order deformations, the ADHO basis with Nf=20 shells is adopted, leading to an accuracy of 0.1 MeV in total energy of 270Hs [50] in the MDC-RMF caclulations. In the particle-particle channel, a separable pairing force is adopted. Here, the strength and effective range of this force are taken to be the same as those in Ref. [50]: G=1.1G0, where G0=728 MeV fm3, and a=0.644 fm. The effective interactions PC-PK1 [60], PK1 [61], PKDD [61], DD-ME2 [62], and NL3* [63] are employed in the particle-hole channel.

      The ground state properties, including deformation parameters βλ (λ=2,4,6,8, and 10), radii, and binding energies of even-even Hs isotopes with the above-mentioned five effective interactions are given in Table 1, and even-even isotones with N=162 are listed in Table 2. The binding energies of one nucleus with five effective interactions differ from each other, e.g., the largest binding energy of 270Hs is 1973.77 MeV with PK1 and the smallest is 1967.41 MeV with PC-PK1. Such results are relatively close to the empirical values in AME2020, EB=1969.65 MeV [64-66], and are also comparable to the predictions of other models, such as MMM, with EB=1969.20 MeV [23], the Skyrme Hartree-Fock Bogoliubov mass model (HFB-24), with EB=1968.45 MeV [67], the Weizsäcker-Skyrme (WS) mass formula WS4, with EB=1970.27 MeV [68], the finite range droplet model (FRDM(2012)), with EB=1971.48 MeV [58], and several RMF calculations [69-73]. For other nuclei, similar conclusions can also be drawn. From these two tables, it is clear that all considered nuclei are deformed in the MDC-RMF calculations with five effective interactions. This is consistent with the results obtained from MMM calculations [23, 58, 59] and other global studies [74, 75]. Additionally, it has been shown that the inclusion of the rotational energy correction (REC) can improve the description of binding energies with PC-PK1 [60]. In this study, after considering RECs in PC-PK1 calculations, the binding energy of 270Hs changes from 1967.45 to 1969.76 MeV, which is closer to the value provided in AME2020.

      β2,nβ2,pβ2β4β6β8β10Rn/fmRp/fmRt/fmRc/fmEB/MeV
      PC-PK1
      264Hs0.2700.2800.274−0.002−0.060−0.0130.0116.2456.0906.1826.1381924.415
      266Hs0.2660.2760.271−0.021−0.063−0.0040.0146.2676.1016.2006.1481939.205
      268Hs0.2620.2730.266−0.040−0.0630.0040.0156.2886.1116.2176.1581953.554
      270Hs0.2570.2690.261−0.057−0.0610.0120.0156.3066.1206.2326.1671967.408
      272Hs0.2450.2580.250−0.060−0.0490.0130.0106.3306.1316.2526.1781979.303
      274Hs0.2160.2280.221−0.053−0.0380.0090.0066.3446.1356.2636.1821990.951
      276Hs0.1880.1980.192−0.049−0.0270.0070.0036.3576.1396.2736.1852002.778
      PK1
      264Hs0.2530.2580.2550.006−0.058−0.0160.0116.2286.0586.1596.1051934.074
      266Hs0.2530.2580.255−0.014−0.065−0.0060.0166.2536.0706.1796.1181947.952
      268Hs0.2560.2610.258−0.034−0.0700.0050.0196.2786.0846.2016.1311961.285
      270Hs02450.2510.248−0.044−0.0620.0100.0166.2976.0916.2166.1381973.766
      272Hs0.2110.2160.213−0.029−0.0530.0050.0106.3056.0906.2216.1371985.924
      274Hs0.1940.1980.195−0.038−0.0400.0060.0106.3226.0976.2346.1441997.412
      276Hs0.1780.1820.180−0.047−0.0280.0070.0076.3426.1056.2506.1512008.356
      PKDD
      264Hs0.2500.2550.2520.001−0.060−0.0150.0116.2076.0536.1456.1011932.544
      266Hs0.2530.2580.255−0.020−0.066−0.0040.0166.2336.0676.1666.1151946.294
      268Hs0.2580.2640.260−0.041−0.0720.0090.0216.2596.0826.1886.1291959.686
      270Hs0.2520.2610.256−0.059−0.0620.0170.0176.2786.0916.2046.1381972.399
      272Hs0.2110.2170.213−0.030−0.0560.0060.0196.2826.0876.2056.1341983.376
      274Hs0.1900.1940.191−0.039−0.0410.0060.0106.2966.0936.2166.1391994.504
      276Hs0.1740.1790.176−0.048−0.0280.0070.0076.3166.1006.2336.1472004.934
      DD-ME2
      264Hs0.2600.2670.263−0.001−0.061−0.0120.0146.1786.0736.1366.1211928.426
      266Hs0.2610.2690.264−0.023−0.066−0.0010.0196.2006.0866.1546.1331942.991
      268Hs0.2590.2690.263−0.042−0.0680.0110.0216.2206.0976.1716.1441957.270
      270Hs0.2520.2640.257−0.058−0.0600.0170.0176.2366.1056.1846.1521971.027
      272Hs0.2130.2220.216−0.032−0.0540.0070.0126.2406.1016.1856.1481981.994
      274Hs0.1960.2040.199−0.039−0.0410.0070.0106.2566.1076.1986.1541993.391
      276Hs0.1780.1860.181−0.048−0.0270.0080.0076.2716.1136.2106.1602004.657
      NL3*
      264Hs0.2650.2710.2670.004−0.060−0.0140.0136.2606.0796.1866.1261931.827
      266Hs0.2630.2700.266−0.017−0.065−0.0040.0176.2846.0906.2066.1371945.933
      268Hs0.2620.2690.265−0.036−0.0670.0070.0186.3076.1016.2256.1481959.582
      270Hs0.2560.2650.260−0.054−0.0610.0150.0176.3266.1096.2406.1561972.574
      272Hs0.2330.2410.236−0.045−0.0510.0100.0126.3446.1156.2546.1621984.027
      274Hs0.2040.2110.207−0.039−0.0400.0070.0096.3576.1186.2646.1641995.465
      276Hs0.1850.1930.188−0.047−0.0280.0080.0066.3766.1256.2796.1712006.648

      Table 1.  Ground state properties, including quadrupole deformation parameters, of neutrons and protons (β2,n and β2,p), deformation parameters β2, β4, β6, β8, and β10, mass radius Rt, radii of protons and neutrons (Rp and Rn), charge radius Rc, and binding energy EB of Hs isotopes using MDC-RMF with five effective interactions: PC-PK1, PK1, PKDD, DD-ME2, and NL3*.

      β2,nβ2,pβ2β4β6β8β10Rn/fmRp/fmRt/fmRc/fmEB/MeV
      PC-PK1
      266Rf0.2610.2740.266−0.039−0.0600.0050.0136.3036.0836.2186.1291953.531
      268Sg0.2600.2740.266−0.048−0.0630.0090.0156.3046.1026.2256.1491961.448
      270Hs0.2570.2690.261−0.057−0.0610.0120.0156.3066.1206.2326.1671967.408
      272Ds0.2490.2580.253−0.061−0.0550.0150.0126.3086.1396.2406.1861971.338
      PK1
      266Rf0.2560.2650.260−0.038−0.0620.0070.0176.2966.0536.2026.1001957.699
      268Sg0.2570.2680.261−0.046−0.0670.0100.0186.2996.0756.2116.1221966.814
      270Hs0.2450.2510.248−0.044−0.0620.0100.0166.2976.0916.2166.1381973.766
      272Ds0.2250.2280.226−0.037−0.0540.0070.0126.2926.1046.2176.1511979.041
      PKDD
      266Rf0.2560.2660.260−0.043−0.0630.0090.0186.2726.0506.1866.0971955.230
      268Sg0.2590.2720.264−0.051−0.0700.0130.0206.2776.0736.1976.1201965.167
      270Hs0.2520.2610.256−0.059−0.0620.0170.0176.2786.0916.2046.1381972.399
      272Ds0.2410.2450.242−0.060−0.0560.0170.0156.2786.1076.2106.1541977.775
      DD-ME2
      266Rf0.2550.2680.260−0.040−0.0610.0080.0176.2246.0626.1616.1091955.711
      268Sg0.2570.2710.262−0.048−0.0650.0120.0196.2316.0866.1746.1321964.571
      270Hs0.2520.2640.257−0.058−0.0600.0170.0176.2366.1056.1846.1521971.027
      272Ds0.2420.2490.245−0.060−0.0540.0170.0156.2416.1236.1936.1701975.320
      NL3*
      266Rf0.2610.2720.266−0.038−0.0620.0070.0176.3256.0706.2266.1161956.503
      268Sg0.2610.2730.266−0.047−0.0660.0110.0186.3256.0916.2336.1371965.582
      270Hs0.2560.2650.260−0.054−0.0610.0150.0176.3266.1096.2406.1561972.574
      272Ds0.2480.2530.250−0.058−0.0550.0160.0146.3266.1276.2476.1741977.689

      Table 2.  Ground state properties, including quadrupole deformation parameters, of neutrons and protons (β2,n and β2,p), deformation parameters β2,β4,β6,β8 and β10, mass radius Rt, radii of protons and neutrons (Rp and Rn), charge radius Rc, and binding energies EB of N=162 isotones using MDC-RMF with five effective interactions: PC-PK1, PK1, PKDD, DD-ME2, and NL3*.

      To determine the dimension of the deformation space when investigating the ground states of SHN using MDC-CDFTs, we calculate the binding energies of Hs isotopes and isotones with N=162 in a different deformation space {βλ;λ=0,2,,λmax} with λmax being the maximum order of deformation parameters, meaning all deformation parameters βλβλmax are considered self-consistently, while other deformation parameters are constrained to zero. In Fig. 1, the binding energies of N=162 isotones with five different effective interactions are plotted as a function of λmax. For convenience, we take 270Hs with the effective interaction PC-PK1 as an example to discuss the influence of each order of deformation on the binding energy in detail. When constraining 270Hs to be spherical, i.e., in the deformation space {βλ;λ=0}, the resulting binding energy is 1956.39 MeV, which is close to the prediction from the relativistic continuum Hartree-Bogoliubov theory, which is 1952.65 MeV [76], but not close to the value given in AME2020 (marked by a black square in Fig. 1). After considering the quadrupole deformation β2, the binding energy of 270Hs changes considerably (by approximately 8.43 MeV) and approaches that from AME2020. This result indicates the importance of the quadrupole deformation. The influence of the hexadecapole deformation β4 on the binding energy is smaller, with a change of only 0.68 MeV. Considering β6, the energy change is approximately 1.87 MeV, which is larger than that of β4, and EB approaches the value provided in AME2020. Including β8 and β10 does not affect the binding energy, which converges well at {βλ;λ=0,2,,10}. From these results, we can conclude that to obtain a proper description of 270Hs, one should consider the β6 deformation at least with respect to the binding energy. Calculated binding energy versus λmax with other density functionals are also displayed in Fig. 1, and although the binding energies with five effective interactions differ from each other, the overall trend of EB changing with λmax are similar. The binding energy of 270Hs is predominantly changed by β2, then β6 and β4. The influence of β8 and β10 can be ignored. The RECs for 270Hs with PC-PK1 are 2.27, 2.03, 2.29, 2.31, and 2.31 MeV in deformation spaces {βλ;λ=0,,λmax}, with λmax=2,4,6,8, and 10, respectively. The values of RECs change slightly in different deformation spaces and only minimally influence the trends of binding energies with respect to λmax.

      Figure 1.  (color online) Binding energies of N=162 isotones for PC-PK1, DD-ME2, PKDD, NL3*, and PK1 as a function of λmax. The black square and black circle represent the values from AME2020 and FRMD(2012), respectively.

      To check whether the above conclusion is valid for other SHN, we perform similar calculations for even-even isotones with N=162 and Hs isotopes, with the results also presented in Figs. 1 and 2. From these figures, we find that the binding energies of these nuclei are significantly changed by β2. The influence of β4 and β6 should also be considered, and the contribution to the total energy from β6 is larger than that from β4. For Hs isotopes, with decreasing neutron number, the value of β2 increases substantially and the differences of the total energies between the spherical case and in the ground states increase, which can be observed in Fig. 2. For isotones with N=162, the value of β2 only minimally changes with proton number. For these nuclei, the relationship between binding energies and βλmax are the same as for 270Hs.

      Figure 2.  (color online) Binding energies of Hs isotopes for PC-PK1, DD-ME2, PKDD, NL3*, and PK1 as a function of λmax. The black square and black circle represent the values from AME2020 and FRMD(2012), respectively.

      It is well known that shell structure is particularly important for SHN and very sensitive to the deformation of the nucleus [28]. Using 270Hs as an example, we explore how the deformations influence the shell gaps at Z=108 and N=162 by investigating the structure of single-particle levels (SPLs) in different deformation spaces. In Fig. 3, we illustrate the SPLs for protons and neutrons of 270Hs versus λmax, calculated with PC-PK1. When λmax=10, i.e., for the ground state, the energy gaps at Z=108 and N=162 are approximately 1.34 MeV and 1.85 MeV, which are considerably large for such a heavy nucleus [77] and result in deformed shells.

      Figure 3.  (color online) Single proton and neutron levels of 270Hs from PC-PK1 calculations with different λmax. In the spherical case (λmax=0), each level is labelled by |nlj. When λmax0, each level is labelled by the projection Ω of total angular momentum on the symmetry axis and the parity π. Single-particle levels with positive and negative parities are presented by red and black lines, respectively.

      In the spherical limit, λmax=0, each single particle state is labelled by |nlj, where n, l, and j denote the radial quantum number, orbital angular momentum, and total angular momentum, respectively. It is obvious that there are no shell gaps at Z=108 and N=162. After including β2, a spherical orbital |nlj with a degeneracy of 2j+1 splits into (2j+1)/2 levels, with each one represented by Ωπ with the projection Ω of total angular momentum on the symmetry axis and the parity π. It is found that due to quadrupole correlations, the shell gaps at Z=108 and N=162 appear, with values of 0.66 MeV and 1.17 MeV, respectively. When including β4 into the deformation space, the order of SPLs around the two gaps changes and the shell gaps at Z=108 (up to approximately 1.28 MeV) and N=162 (up to approximately 1.56 MeV) increase considerably. The impact of β6 on the shell gap at Z=108 is smaller, only 0.02 MeV. Although for neutrons, the shell gap at N=162 increases by approximately 0.26 MeV. The inclusion of β8 and β10 produces only extremely minimal changes in the shell gaps and the order of SPLs. From this, one can conclude that β2 plays a vital role in the formation of the shell closures Z=108 and N=162, which are further enhanced by β4. The influence of β6 is relatively small and the effects of β8 and β10 can be considered negligible. However, there remains a question: where do the Y60 correlations come from? By checking the SPLs, we find that two proton levels 1/2+ originating from the spherical orbitals 3s1/2 and 1i13/2 are very close to each other and the mixing of these two spherical orbitals in the deformed SPLs results in Y60 correlations. For neutrons, these correlations originate from the mixing of the spherical orbitals 3p3/2 and 1j15/2 in the levels 3/2, close to the neutron Fermi energy.

    IV.   SUMMARY
    • In this study, we investigate the ground state properties of SHN around 270Hs in multidimensional deformation spaces using the MDC-RMF model with five density functionals. The influence of higher-order deformation parameters on the ground state of nuclei near 270Hs are investigated, including the binding energies and SPLs. We show that the binding energies of deformed SHN around 270Hs are significantly affected by higher-order deformations. In particular, the influence of β6 on binding energy is larger than that of β4. For the doubly magic nucleus 270Hs, the deformed shell gaps at Z=108 and N=162 are mainly determined by quarupole correlations and are enhanced by the inclusion of β4. In conclusion, at least the β6 degree of freedom should be considered in the investigation of SHN using CDFTs. It is also very interesting to investigate how the higher-order deformations influence other properties of SHN, such as the moment of inertia and energy spectra, by using density functional theories. Additionally, we would like to mention that the calculations performed in this study can also be performed using the deformation relativistic Hartree-Bogoliubov (DRHBc) theory [78-84], in which the scalar potential and densities are expanded in terms of the Legendre polynomials, though the DRHBc theory is much more time consuming than the MDC-RMF model. Very recently, the influence of higher-order deformations on possible bound nuclei beyond the drip line has also been investigated in the transfermium region from No (Z=102) to Ds (Z=110) using the DRHBc theory [85], and the development of a nuclear mass table using the DRHBc theory is in progress [86-89].

    ACKNOWLEDGMENTS
    • We thank Bing-Nan Lu, Yu-Ting Rong, and Kun Wang for helpful discussions. The results described in this paper were obtained on the High-performance Computing Cluster of ITP-CAS and the ScGrid of the Supercomputing Center, Computer Network Information Center of the Chinese Academy of Sciences.

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