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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
Z⩽118 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 aroundZ=108 andN=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=1λ∑μ=−λβ∗λμYλμ(θ,φ)],
(1) where
βλμ is the deformation parameter andR0 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 numbersZ=90 –114 and neutron numbersN=136 –168 using the MMM and found that theβ6 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 ofβ6 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 octupoleY32 correlations inN=150 isotones [51] and Zr isotopes [52], octupole correlations in MχD of78Br [53] and Ba isotopes [54], etc. In MDC-CDFTs, reflection and axial symmetry are both broken, and the shape degrees of freedomβλμ , where µ is an even number, are self-consistently included, such asβ20,β22, β30,β32,β40,β42 , andβ44 . 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(iγμ∂μ−M)ψ−Llin−Lnl−Lder−LCou,
(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+e1−τ32A0ρV,
(6) where M represents the nucleon mass, e is the unit charge, and
αS ,αV ,αTS ,αTV ,βS ,γS ,γV ,δS ,δV ,δTS , andδTV are coupling constants. The isoscalar densityρS , isovector densityρTS , the time-like components of isoscalar currentρV , and the time-like components of isovector currentsρTV are defined asρS=ˉψψ,ρTS=ˉψτψ,ρV=ˉψγ0ψ,ρTV=ˉψτγ0ψ.
(7) The single particle wave function
ψk(r) with energyϵk of a nucleon is obtained by solving the Dirac equationˆhψk(r)=ϵkψk(r),
(8) with the Dirac Hamiltonian
ˆh=α⋅p+β[M+S(r)]+V(r),
(9) where the scalar potential
S(r) and vector potentialV(r) areS=αSρS+αTSρTS⋅τ+βSρ2S+γSρ3S+δSΔρS+δTSΔρTS⋅τ,V=αVρV+αTVρTV⋅τ+γVρ2VρV+δVΔρV+δTVΔρTV⋅τ+e1−τ32A0.
(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
[−ℏ22M∇2+VB(z,ρ)]Φα(rσ)=EαΦα(rσ),
(11) where
r=(z,ρ) , whereρ=√x2+y2 , andVB(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} , andfαi andgαi 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 withN⩾182 in the SHN region [59]. Thus, we only consider the axially symmetric deformationsβλ , 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 readsE′=ERMF+∑λ12CλQλ.
(14) After the nth iteration, the variable
C(n+1)λ is determined byC(n+1)λ=C(n)λ+kλ(β(n)λ−βλ),
(15) where
C(n)λ is the value of the nth iteration,kλ is a constant, andβλ 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 parameterβλ,τ is given byβλ,τ=4π3NτRλQλ,τ,
(17) where
R=r0A1/3 , withr0=1.2 fm, andNτ 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 from264Hs to276Hs , andN=162 isotones from266Rf to272Ds are analyzed. When investigating the ground state properties and the influence of higher-order deformations, the ADHO basis withNf=20 shells is adopted, leading to an accuracy of 0.1 MeV in total energy of270Hs [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 , whereG0=728 MeV fm3, anda=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 , and10 ), 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 withN=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 of270Hs 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, withEB=1969.20 MeV [23], the Skyrme Hartree-Fock Bogoliubov mass model (HFB-24), withEB=1968.45 MeV [67], the Weizsäcker-Skyrme (WS) mass formula WS4, withEB=1970.27 MeV [68], the finite range droplet model (FRDM(2012)), withEB=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 of270Hs changes from 1967.45 to 1969.76 MeV, which is closer to the value provided in AME2020.β2,n β2,p β2 β4 β6 β8 β10 Rn /fmRp /fmRt /fmRc /fmEB /MeVPC-PK1 264Hs 0.270 0.280 0.274 −0.002 −0.060 −0.013 0.011 6.245 6.090 6.182 6.138 1924.415 266Hs 0.266 0.276 0.271 −0.021 −0.063 −0.004 0.014 6.267 6.101 6.200 6.148 1939.205 268Hs 0.262 0.273 0.266 −0.040 −0.063 0.004 0.015 6.288 6.111 6.217 6.158 1953.554 270Hs 0.257 0.269 0.261 −0.057 −0.061 0.012 0.015 6.306 6.120 6.232 6.167 1967.408 272Hs 0.245 0.258 0.250 −0.060 −0.049 0.013 0.010 6.330 6.131 6.252 6.178 1979.303 274Hs 0.216 0.228 0.221 −0.053 −0.038 0.009 0.006 6.344 6.135 6.263 6.182 1990.951 276Hs 0.188 0.198 0.192 −0.049 −0.027 0.007 0.003 6.357 6.139 6.273 6.185 2002.778 PK1 264Hs 0.253 0.258 0.255 0.006 −0.058 −0.016 0.011 6.228 6.058 6.159 6.105 1934.074 266Hs 0.253 0.258 0.255 −0.014 −0.065 −0.006 0.016 6.253 6.070 6.179 6.118 1947.952 268Hs 0.256 0.261 0.258 −0.034 −0.070 0.005 0.019 6.278 6.084 6.201 6.131 1961.285 270Hs 0245 0.251 0.248 −0.044 −0.062 0.010 0.016 6.297 6.091 6.216 6.138 1973.766 272Hs 0.211 0.216 0.213 −0.029 −0.053 0.005 0.010 6.305 6.090 6.221 6.137 1985.924 274Hs 0.194 0.198 0.195 −0.038 −0.040 0.006 0.010 6.322 6.097 6.234 6.144 1997.412 276Hs 0.178 0.182 0.180 −0.047 −0.028 0.007 0.007 6.342 6.105 6.250 6.151 2008.356 PKDD 264Hs 0.250 0.255 0.252 0.001 −0.060 −0.015 0.011 6.207 6.053 6.145 6.101 1932.544 266Hs 0.253 0.258 0.255 −0.020 −0.066 −0.004 0.016 6.233 6.067 6.166 6.115 1946.294 268Hs 0.258 0.264 0.260 −0.041 −0.072 0.009 0.021 6.259 6.082 6.188 6.129 1959.686 270Hs 0.252 0.261 0.256 −0.059 −0.062 0.017 0.017 6.278 6.091 6.204 6.138 1972.399 272Hs 0.211 0.217 0.213 −0.030 −0.056 0.006 0.019 6.282 6.087 6.205 6.134 1983.376 274Hs 0.190 0.194 0.191 −0.039 −0.041 0.006 0.010 6.296 6.093 6.216 6.139 1994.504 276Hs 0.174 0.179 0.176 −0.048 −0.028 0.007 0.007 6.316 6.100 6.233 6.147 2004.934 DD-ME2 264Hs 0.260 0.267 0.263 −0.001 −0.061 −0.012 0.014 6.178 6.073 6.136 6.121 1928.426 266Hs 0.261 0.269 0.264 −0.023 −0.066 −0.001 0.019 6.200 6.086 6.154 6.133 1942.991 268Hs 0.259 0.269 0.263 −0.042 −0.068 0.011 0.021 6.220 6.097 6.171 6.144 1957.270 270Hs 0.252 0.264 0.257 −0.058 −0.060 0.017 0.017 6.236 6.105 6.184 6.152 1971.027 272Hs 0.213 0.222 0.216 −0.032 −0.054 0.007 0.012 6.240 6.101 6.185 6.148 1981.994 274Hs 0.196 0.204 0.199 −0.039 −0.041 0.007 0.010 6.256 6.107 6.198 6.154 1993.391 276Hs 0.178 0.186 0.181 −0.048 −0.027 0.008 0.007 6.271 6.113 6.210 6.160 2004.657 NL3* 264Hs 0.265 0.271 0.267 0.004 −0.060 −0.014 0.013 6.260 6.079 6.186 6.126 1931.827 266Hs 0.263 0.270 0.266 −0.017 −0.065 −0.004 0.017 6.284 6.090 6.206 6.137 1945.933 268Hs 0.262 0.269 0.265 −0.036 −0.067 0.007 0.018 6.307 6.101 6.225 6.148 1959.582 270Hs 0.256 0.265 0.260 −0.054 −0.061 0.015 0.017 6.326 6.109 6.240 6.156 1972.574 272Hs 0.233 0.241 0.236 −0.045 −0.051 0.010 0.012 6.344 6.115 6.254 6.162 1984.027 274Hs 0.204 0.211 0.207 −0.039 −0.040 0.007 0.009 6.357 6.118 6.264 6.164 1995.465 276Hs 0.185 0.193 0.188 −0.047 −0.028 0.008 0.006 6.376 6.125 6.279 6.171 2006.648 Table 1. Ground state properties, including quadrupole deformation parameters, of neutrons and protons (
β2,n andβ2,p ), deformation parametersβ2 ,β4 ,β6 ,β8 , andβ10 , mass radiusRt , radii of protons and neutrons (Rp andRn ), charge radiusRc , and binding energyEB of Hs isotopes using MDC-RMF with five effective interactions: PC-PK1, PK1, PKDD, DD-ME2, and NL3*.β2,n β2,p β2 β4 β6 β8 β10 Rn /fmRp /fmRt /fmRc /fmEB /MeVPC-PK1 266Rf 0.261 0.274 0.266 −0.039 −0.060 0.005 0.013 6.303 6.083 6.218 6.129 1953.531 268Sg 0.260 0.274 0.266 −0.048 −0.063 0.009 0.015 6.304 6.102 6.225 6.149 1961.448 270Hs 0.257 0.269 0.261 −0.057 −0.061 0.012 0.015 6.306 6.120 6.232 6.167 1967.408 272Ds 0.249 0.258 0.253 −0.061 −0.055 0.015 0.012 6.308 6.139 6.240 6.186 1971.338 PK1 266Rf 0.256 0.265 0.260 −0.038 −0.062 0.007 0.017 6.296 6.053 6.202 6.100 1957.699 268Sg 0.257 0.268 0.261 −0.046 −0.067 0.010 0.018 6.299 6.075 6.211 6.122 1966.814 270Hs 0.245 0.251 0.248 −0.044 −0.062 0.010 0.016 6.297 6.091 6.216 6.138 1973.766 272Ds 0.225 0.228 0.226 −0.037 −0.054 0.007 0.012 6.292 6.104 6.217 6.151 1979.041 PKDD 266Rf 0.256 0.266 0.260 −0.043 −0.063 0.009 0.018 6.272 6.050 6.186 6.097 1955.230 268Sg 0.259 0.272 0.264 −0.051 −0.070 0.013 0.020 6.277 6.073 6.197 6.120 1965.167 270Hs 0.252 0.261 0.256 −0.059 −0.062 0.017 0.017 6.278 6.091 6.204 6.138 1972.399 272Ds 0.241 0.245 0.242 −0.060 −0.056 0.017 0.015 6.278 6.107 6.210 6.154 1977.775 DD-ME2 266Rf 0.255 0.268 0.260 −0.040 −0.061 0.008 0.017 6.224 6.062 6.161 6.109 1955.711 268Sg 0.257 0.271 0.262 −0.048 −0.065 0.012 0.019 6.231 6.086 6.174 6.132 1964.571 270Hs 0.252 0.264 0.257 −0.058 −0.060 0.017 0.017 6.236 6.105 6.184 6.152 1971.027 272Ds 0.242 0.249 0.245 −0.060 −0.054 0.017 0.015 6.241 6.123 6.193 6.170 1975.320 NL3* 266Rf 0.261 0.272 0.266 −0.038 −0.062 0.007 0.017 6.325 6.070 6.226 6.116 1956.503 268Sg 0.261 0.273 0.266 −0.047 −0.066 0.011 0.018 6.325 6.091 6.233 6.137 1965.582 270Hs 0.256 0.265 0.260 −0.054 −0.061 0.015 0.017 6.326 6.109 6.240 6.156 1972.574 272Ds 0.248 0.253 0.250 −0.058 −0.055 0.016 0.014 6.326 6.127 6.247 6.174 1977.689 Table 2. Ground state properties, including quadrupole deformation parameters, of neutrons and protons (
β2,n andβ2,p ), deformation parametersβ2,β4,β6,β8 andβ10 , mass radiusRt , radii of protons and neutrons (Rp andRn ), charge radiusRc , and binding energiesEB ofN=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} withλmax being the maximum order of deformation parameters, meaning all deformation parametersβλ⩽βλmax are considered self-consistently, while other deformation parameters are constrained to zero. In Fig. 1, the binding energies ofN=162 isotones with five different effective interactions are plotted as a function ofλmax . For convenience, we take270Hs 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 constraining270Hs 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 deformationβ2 , the binding energy of270Hs 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 deformationβ4 on the binding energy is smaller, with a change of only 0.68 MeV. Consideringβ6 , the energy change is approximately 1.87 MeV, which is larger than that ofβ4 , andEB approaches the value provided in AME2020. Includingβ8 andβ10 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 of270Hs , one should consider theβ6 deformation at least with respect to the binding energy. Calculated binding energy versusλmax 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 ofEB changing withλmax are similar. The binding energy of270Hs is predominantly changed byβ2 , thenβ6 andβ4 . The influence ofβ8 andβ10 can be ignored. The RECs for270Hs with PC-PK1 are 2.27, 2.03, 2.29, 2.31, and 2.31 MeV in deformation spaces{βλ;λ=0,⋯,λmax} , withλmax=2,4,6,8 , and10 , respectively. The values of RECs change slightly in different deformation spaces and only minimally influence the trends of binding energies with respect toλmax .Figure 1. (color online) Binding energies of
N=162 isotones for PC-PK1, DD-ME2, PKDD, NL3*, and PK1 as a function ofλmax . 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 byβ2 . The influence ofβ4 andβ6 should also be considered, and the contribution to the total energy fromβ6 is larger than that fromβ4 . For Hs isotopes, with decreasing neutron number, the value ofβ2 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 withN=162 , the value ofβ2 only minimally changes with proton number. For these nuclei, the relationship between binding energies andβλmax are the same as for270Hs .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 atZ=108 andN=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 of270Hs versusλmax , calculated with PC-PK1. Whenλmax=10 , i.e., for the ground state, the energy gaps atZ=108 andN=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 differentλmax . In the spherical case (λmax=0 ), each level is labelled by|nlj⟩ . Whenλmax≠0 , 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 atZ=108 andN=162 . After includingβ2 , a spherical orbital|nlj⟩ with a degeneracy of2j+1 splits into(2j+1)/2 levels, with each one represented byΩπ 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 atZ=108 andN=162 appear, with values of 0.66 MeV and 1.17 MeV, respectively. When includingβ4 into the deformation space, the order of SPLs around the two gaps changes and the shell gaps atZ=108 (up to approximately 1.28 MeV) andN=162 (up to approximately 1.56 MeV) increase considerably. The impact ofβ6 on the shell gap atZ=108 is smaller, only 0.02 MeV. Although for neutrons, the shell gap atN=162 increases by approximately 0.26 MeV. The inclusion ofβ8 andβ10 produces only extremely minimal changes in the shell gaps and the order of SPLs. From this, one can conclude thatβ2 plays a vital role in the formation of the shell closuresZ=108 andN=162 , which are further enhanced byβ4 . The influence ofβ6 is relatively small and the effects ofβ8 andβ10 can be considered negligible. However, there remains a question: where do theY60 correlations come from? By checking the SPLs, we find that two proton levels1/2+ originating from the spherical orbitals3s1/2 and1i13/2 are very close to each other and the mixing of these two spherical orbitals in the deformed SPLs results inY60 correlations. For neutrons, these correlations originate from the mixing of the spherical orbitals 3p3/2 and 1j15/2 in the levels3/2− , 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 near270Hs are investigated, including the binding energies and SPLs. We show that the binding energies of deformed SHN around270Hs are significantly affected by higher-order deformations. In particular, the influence ofβ6 on binding energy is larger than that ofβ4 . For the doubly magic nucleus270Hs , the deformed shell gaps atZ=108 andN=162 are mainly determined by quarupole correlations and are enhanced by the inclusion ofβ4 . In conclusion, at least theβ6 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.
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |
PC-PK1 | ||||||||||||
![]() | 0.270 | 0.280 | 0.274 | −0.002 | −0.060 | −0.013 | 0.011 | 6.245 | 6.090 | 6.182 | 6.138 | 1924.415 |
![]() | 0.266 | 0.276 | 0.271 | −0.021 | −0.063 | −0.004 | 0.014 | 6.267 | 6.101 | 6.200 | 6.148 | 1939.205 |
![]() | 0.262 | 0.273 | 0.266 | −0.040 | −0.063 | 0.004 | 0.015 | 6.288 | 6.111 | 6.217 | 6.158 | 1953.554 |
![]() | 0.257 | 0.269 | 0.261 | −0.057 | −0.061 | 0.012 | 0.015 | 6.306 | 6.120 | 6.232 | 6.167 | 1967.408 |
![]() | 0.245 | 0.258 | 0.250 | −0.060 | −0.049 | 0.013 | 0.010 | 6.330 | 6.131 | 6.252 | 6.178 | 1979.303 |
![]() | 0.216 | 0.228 | 0.221 | −0.053 | −0.038 | 0.009 | 0.006 | 6.344 | 6.135 | 6.263 | 6.182 | 1990.951 |
![]() | 0.188 | 0.198 | 0.192 | −0.049 | −0.027 | 0.007 | 0.003 | 6.357 | 6.139 | 6.273 | 6.185 | 2002.778 |
PK1 | ||||||||||||
![]() | 0.253 | 0.258 | 0.255 | 0.006 | −0.058 | −0.016 | 0.011 | 6.228 | 6.058 | 6.159 | 6.105 | 1934.074 |
![]() | 0.253 | 0.258 | 0.255 | −0.014 | −0.065 | −0.006 | 0.016 | 6.253 | 6.070 | 6.179 | 6.118 | 1947.952 |
![]() | 0.256 | 0.261 | 0.258 | −0.034 | −0.070 | 0.005 | 0.019 | 6.278 | 6.084 | 6.201 | 6.131 | 1961.285 |
![]() | 0245 | 0.251 | 0.248 | −0.044 | −0.062 | 0.010 | 0.016 | 6.297 | 6.091 | 6.216 | 6.138 | 1973.766 |
![]() | 0.211 | 0.216 | 0.213 | −0.029 | −0.053 | 0.005 | 0.010 | 6.305 | 6.090 | 6.221 | 6.137 | 1985.924 |
![]() | 0.194 | 0.198 | 0.195 | −0.038 | −0.040 | 0.006 | 0.010 | 6.322 | 6.097 | 6.234 | 6.144 | 1997.412 |
![]() | 0.178 | 0.182 | 0.180 | −0.047 | −0.028 | 0.007 | 0.007 | 6.342 | 6.105 | 6.250 | 6.151 | 2008.356 |
PKDD | ||||||||||||
![]() | 0.250 | 0.255 | 0.252 | 0.001 | −0.060 | −0.015 | 0.011 | 6.207 | 6.053 | 6.145 | 6.101 | 1932.544 |
![]() | 0.253 | 0.258 | 0.255 | −0.020 | −0.066 | −0.004 | 0.016 | 6.233 | 6.067 | 6.166 | 6.115 | 1946.294 |
![]() | 0.258 | 0.264 | 0.260 | −0.041 | −0.072 | 0.009 | 0.021 | 6.259 | 6.082 | 6.188 | 6.129 | 1959.686 |
![]() | 0.252 | 0.261 | 0.256 | −0.059 | −0.062 | 0.017 | 0.017 | 6.278 | 6.091 | 6.204 | 6.138 | 1972.399 |
![]() | 0.211 | 0.217 | 0.213 | −0.030 | −0.056 | 0.006 | 0.019 | 6.282 | 6.087 | 6.205 | 6.134 | 1983.376 |
![]() | 0.190 | 0.194 | 0.191 | −0.039 | −0.041 | 0.006 | 0.010 | 6.296 | 6.093 | 6.216 | 6.139 | 1994.504 |
![]() | 0.174 | 0.179 | 0.176 | −0.048 | −0.028 | 0.007 | 0.007 | 6.316 | 6.100 | 6.233 | 6.147 | 2004.934 |
DD-ME2 | ||||||||||||
![]() | 0.260 | 0.267 | 0.263 | −0.001 | −0.061 | −0.012 | 0.014 | 6.178 | 6.073 | 6.136 | 6.121 | 1928.426 |
![]() | 0.261 | 0.269 | 0.264 | −0.023 | −0.066 | −0.001 | 0.019 | 6.200 | 6.086 | 6.154 | 6.133 | 1942.991 |
![]() | 0.259 | 0.269 | 0.263 | −0.042 | −0.068 | 0.011 | 0.021 | 6.220 | 6.097 | 6.171 | 6.144 | 1957.270 |
![]() | 0.252 | 0.264 | 0.257 | −0.058 | −0.060 | 0.017 | 0.017 | 6.236 | 6.105 | 6.184 | 6.152 | 1971.027 |
![]() | 0.213 | 0.222 | 0.216 | −0.032 | −0.054 | 0.007 | 0.012 | 6.240 | 6.101 | 6.185 | 6.148 | 1981.994 |
![]() | 0.196 | 0.204 | 0.199 | −0.039 | −0.041 | 0.007 | 0.010 | 6.256 | 6.107 | 6.198 | 6.154 | 1993.391 |
![]() | 0.178 | 0.186 | 0.181 | −0.048 | −0.027 | 0.008 | 0.007 | 6.271 | 6.113 | 6.210 | 6.160 | 2004.657 |
NL3* | ||||||||||||
![]() | 0.265 | 0.271 | 0.267 | 0.004 | −0.060 | −0.014 | 0.013 | 6.260 | 6.079 | 6.186 | 6.126 | 1931.827 |
![]() | 0.263 | 0.270 | 0.266 | −0.017 | −0.065 | −0.004 | 0.017 | 6.284 | 6.090 | 6.206 | 6.137 | 1945.933 |
![]() | 0.262 | 0.269 | 0.265 | −0.036 | −0.067 | 0.007 | 0.018 | 6.307 | 6.101 | 6.225 | 6.148 | 1959.582 |
![]() | 0.256 | 0.265 | 0.260 | −0.054 | −0.061 | 0.015 | 0.017 | 6.326 | 6.109 | 6.240 | 6.156 | 1972.574 |
![]() | 0.233 | 0.241 | 0.236 | −0.045 | −0.051 | 0.010 | 0.012 | 6.344 | 6.115 | 6.254 | 6.162 | 1984.027 |
![]() | 0.204 | 0.211 | 0.207 | −0.039 | −0.040 | 0.007 | 0.009 | 6.357 | 6.118 | 6.264 | 6.164 | 1995.465 |
![]() | 0.185 | 0.193 | 0.188 | −0.047 | −0.028 | 0.008 | 0.006 | 6.376 | 6.125 | 6.279 | 6.171 | 2006.648 |