Qing-Zhen Chai(柴清禎),Wei-Juan Zhao(趙維娟),and Hua-Lei Wang(王華磊)

School of Physics and Engineering,Zhengzhou University,Zhengzhou 450001,China

Abstract The first(namely,inner)fission barriers for even-A N=152 nuclei have been studied systematically in the framework of macroscopic-microscopic model by means of potential energy surface(PES)calculations in the threedimensional(β2,γ,β4)deformation space.Their collective properties,such as ground-state deformations,are compared with previous calculations and available observations,showing a consistent trend.In addition,it has been found that the microscopic shell correction energy plays an important role on surviving fission in these N=152 deformed shell nuclei.The inclusion of non-axial symmetric degree of freedom γ will pull the fission barrier down more significantly with respect to the calculation involving in hexadecapole deformation β4.Furthermore,the calculated Woods-Saxon(WS)single particle levels indicate that the large microscopic shell correction energies due to low level densities may be responsible for such a reduction on the inner fission barrier.

Key words:fission barriers,potential energy surface calculations,ground-state deformations,shell correction energy,single particle levels

1 Introduction

The region of superheavy elements(SHE),characterized by the extreme values of proton number Z,is one of the nuclear landscape and an arena of active experimental and theoretical studies.[1?2]Contrary to other regions of the nuclear chart,the SHE are stabilized only by quantum shell effects.In general,the stability of SHE is defined crudely by the fission barrier,such as its shape and size.The height of the first fission barrier,Bf,which is usually defined as the difference between the energies of the corresponding saddle in the potential energy surface(PES)and the ground state,is one of most important quantities.It dominates the survival probability of SHE synthesized in heavy-ion reactions and has a marked impact on the spontaneous fission half-lives.[3]

Indeed,it has been a long standing problem to accurately describe the fission phenomenon since it was interpreted,for the first time,by the barrier penetration about 80 years ago.[4]Fission is a process whereby a complex quantum system(nucleus)goes from an equilibrated shape to a highly deformed shape that it finally splits into two lighter fragments.[5]Up to date,considerable effort has been made to understand the fission problem in both theory and experiment.Experimentally,the empirical values Bfcorresponding to the lowest state have been estimated from modelling analyses of the available experimental data for the neutron-induced fission cross sections.[5]All known empirical values in even-even heavy nuclei are only extended to the actinide nuclei,e.g.,the known heaviest even-even nucleus is252Cf.Certainly,the theoretical investigation of fission barriers used widely should reproduce these empirical values firstly.References[1,6]have been reviewed these theoretical frameworks,including the macroscopic-microscopic(MM)models,[7?11]the nonrelativistic energy density functionals based zero-range Skyrme and finite-range Gogny interactions,[12?14]the extended Thomas-Fermi plus Strutinsky methods[15?16]and the covariant density functional theory(CDFT).[17?19]Among these,the MM approaches usually have very high descriptive power as well as simplicity of calculation.Accordingly,in the present work,the multidimensional PES calculation will be based on the framework of MM model.

Furthermore,as is well known,the transuranium nuclei,which are produced artificially in heavy-ion induced nuclear fusion reactions,are the gateway to the so-called SHE.In addition,the deformed shell at N=152 is experimentally known for a long time.The analysis of singleparticle spectra of heavy nuclei in multidimensional deformation space has reproduced the appearance of the strong,experimentally-known shell at N=152 and explained that it is mainly due to the quadrupole deformation β2,though importantly enhanced by the high-order deformation.[20?22]The even-even N=152 isotones consist of 7 nuclei experimentally,ranging from246Pu to258Sg.The half-life of these isotones are 10.84 d(246Pu),3.48×105y(248Cm),13.08 y(250Cf),25.39 h(252Fm),51 s(254No),6.4 ms(256Rf)and 2.9 ms(258Sg),respectively.

Motivated by the previous PES studies in superheavy nuclei,[23?24]and transuranium nuclei,[25]we wonder the influence of different deformation parameters on the first fission barrier in these N=152 nuclei,where the second fission barrier,even the third one,is beyond the scope of the present work for the deformation space(β2,γ,β4).Furthermore,we have extended the calculation to the very neutron-rich heavy nucleus244U and neutron-de ficient nucleus260Hs in order to assess the current status of theoretical model and provide some useful informations for future experimental confirmation.Thus,we will investigate systematically the even-A N=152 isotones with 92≤Z≤108 in present work,focusing on the specific effects of non-axial quadrupole deformation γ and hexadecapole deformation β4on the ground-state and saddle points(which determine the first fission barrier)in the PES.

The article is organized as follows.In Sec.2 we present a brief description of the theoretical formalism used to obtain the main ingredient of the present study,i.e.,the PES calculations.Section 3 is devoted to the numerical calculated results and discussions on these N=152 isotones.Finally a brief summary is given in Sec.4.

2 Model and Method

The pairing-deformation self-consistent PES calculation,[26?27]which is an approximation of the Hartree-Fock mothod,is based on the MM models presented in this work.It has been used extensively in the description of the overall systematics of nuclear properties,such as nuclear ground-state masses,shapes,and fission barriers.The basic idea in the MM approach is that the total potential energy of a deformed nucleus can be decomposed in two parts,

where Emacis the macroscopic bulk-energy term,being a smooth function of Z,N and deformation,and Emicis the microscopic quantum correction calculated from a phenomenological(non-self-consistent)single-particle potential well.In the following,we will briefly outline the unified procedure and simultaneously provide the necessary references.

First,the macroscopic energy is obtained from the standard liquid-drop(LD)model.[28]Since our primary attention is just on the PES and the difference between the points(e.g.,the minimum and saddle point)on it,the nuclear potential energy relative to the energy of a spherical LD is adopted,which can be written as[29]

where the relative surface and Coulomb energies Bsand Bcare only functions of nuclear shape.The spherical surface energy E(0)sand the fissility parameter χ are Z-and N-dependent.The surface energy tends to hold the nucleus together,and the Coulomb energy tends to pull it apart.

Then the microscopic correction part,which arises because of the nonuniform distribution of single-particle levels,mainly contains a shell correction and a pairing correction.The shell correction energy is calculated by Strutinsky method,[30]δEshell=Esp?and the pairing correction energy is obtained by Lipkin-Nogami(LN)method,[31]δEpair=ELN?Esp.The strutinsky smoothingis performed with a sixth-order Laguerre polynomial and a smoothing range γ =1.20~ω0,where~ω0=41/A1/3MeV.The single-particle energies Espare calculated from a phenomenological Woods-Saxon(WS)potential with the set of universal parameters.[32]During the diagonalization process of the WS Hamiltonian,deformed harmonic oscillator states with the principal quantum number N≤12 and 14 have been used as a basis for protons and neutrons,respectively.The pairing corrections originate from the short-range interaction of correlated pairs of nucleons moving in time-reversed orbits.The approximately particle-number-conserving LN method using here avoids the spurious pairing phase transition for large spacings between the single-particle levels at the Fermi surface encountered in the traditional Bardeen-Cooper-Schrieffer(BCS)calculation.In the paring windows,dozens of single-particle levels,the respective states(e.g.half of the particle number Z or N)just below and above the Fermi energy,are included empirically for both protons and neutrons.

Consequently,the PES of a given nuclear system is obtained in the multi-dimensional deformation space(β2,γ,β4)and the nuclear equilibrium deformation is determined by minimizing the PES.Note that nuclear shape is defined by the standard parametrization in which it is expanded in spherical harmonics Yλμ(θ,?).[32]There is a fundamental limitation in λ,because the range of the individual“bumps” on the nuclear surface decreases with increasing λ and obviously should not be smaller than a nucleon diameter.A limiting value of λ

3 Results and Discussions

Table 1 shows the values R4/2,[34]P-factor,[35]and the calculated ground-state properties in these N=152 isotones,which are confronted with experiments and/or other accepted theories.The well deformed axially symmetric rotor are indicated roughly by the values of R4/2and the P-factor(P=NpNn/(Np+Nn)≥4),demonstrating the strong collectivity among these nuclei.It also suggested that ifis about(40–50)keV in heavy mass region,the state cannot be of any other nature than rotational.[36?37]As expected,the calculated deformations β2are the main contributions in the nuclear shape parametrization.They are all located in the range of 0.2～0.3,which is the typical range of the well deformed nuclei.Our results are in agreement with several other results are based on the fold-Yukawa(FY)single-particle potential and the finite-range droplet model(FRDM),[38]the Hartree-Fock-BCS(HFBCS),[39]and the extended Thomas-Fermi plus Strutinsky integral(ETFSI)methods,[40]although they are a little underestimation with experimental values[41]in248Cm and250Cf.This is further analysed by Dudek et al.[42]where a corrected formula is suggested to modify the shape inconsistency.The nature of shape inconsistencies may arise from that the charge distributions are more deformed than the mass distributions in the Woods-Saxon calculations.

Table 1 The values of R4/2,[34]P-factor,[35]the excitation energies of the first 2+state E( ),and the calculated results for ground-state equilibrium deformation parameters β2,β4for even-A N=152 isotones,together with the FY+FRDM(FF),[38]HFBCS,[39]and ETFSI[40]calculations and existing experimental values(Exp.)[41]for comparison.

Table 1 The values of R4/2,[34]P-factor,[35]the excitation energies of the first 2+state E( ),and the calculated results for ground-state equilibrium deformation parameters β2,β4for even-A N=152 isotones,together with the FY+FRDM(FF),[38]HFBCS,[39]and ETFSI[40]calculations and existing experimental values(Exp.)[41]for comparison.

aE )and adopted β2-values for these nuclei are taken from Ref.[41].The uncertainties are less than 1.0 keV and 0.015 for E( and ,respectively.bThe calculated ground-state|γ|values of these isotones are always less than 2?.

Nuclei R4/2 P E(2+1)a/keV β2 β4 TRSb FF HFBCS ETFSI Exp. TRS FF HFBCS ETFSI 244U–7.22 – 0.2300.2500.250.25–0.0410.038 0 0.05 246Pu3.3088.21 46.7 0.2330.2500.250.26–0.0400.038 0 0.04 248Cm 3.313 9.10 43.4 0.236 0.250 0.28 0.26 0.286 0.036 0.039 ?0.02 0.04 250Cf 3.321 9.90 42.7 0.240 0.250 0.28 0.26 0.298 0.030 0.027 ?0.02 0.04 252Fm – 9.10 46.6 0.2410.2500.29 0.26 – 0.0210.027?0.02 0.02 254No 3.290 8.21 44.2 0.240 0.251 0.26 0.26 – 0.013 0.015 ?0.01 0.02 256Rf 3.360 7.22 44.0 0.239 0.252 0.27 0.27 – 0.004 0.002 ?0.02 0 258Sg – 6.12 – 0.240 0.252 0.27 0.25 – ?0.008 ?0.036 ?0.03 ?0.02 260Hs – 4.88 – 0.239 0.253 0.24 0.25 – ?0.015 ?0.023 ?0.03 ?0.02

Fig.1 (Color online)(a)Comparison of calcualted potential energy curves of the total energy(black),the microscopic shell correction(red),and the macroscopic liquid drop energy(blue)for even-even N=152 isotones as a function of quadruple deformation β2.The γ and β4deformation are set to zero in order to address the effect of β2parameter alone.(b)The calculated total barrier and its macroscopic and microscopic contributions at saddle point of total energy curves in(a).

As mentioned above,it is the quantum shell effects that make the SHE stabilize.As shown in Fig.1,there is no doubt that the microscopic shell correction energy plays an important role with the addition of proton numbers,i.e.,extending to the region of SHE region.For convenience of description,the potential energy curves have been calculated just with the main component deformation β2.The presence of ground-state minimum is mainly ascribed to the microscopic shell correction energy,not the macroscopic liquid drop energy.In general,the height of fission barrier becomes lower close to the drip-line nuclei than long-lived nuclei such as256Rf and258Sg.However,according to the β-stability line given by the empirical formula Z=A/(1.98+0.0155A2/3),[43]the most β-stable nucleus is248Cm.It is interesting to identify the competition of the β decay and spontaneous fission in these nuclei.Note that the macroscopic liquid drop energy in260Hs decreases gently with increasing quadrupole deformation β2.Thus,it can be concluded that it survives completely due to microscopic shell

Further,while there is no evidence for pure hexadecupole excitation in spectra yet,the known important role for the ground-state shape of heavy nulcei of hexadecupole deformation has been presented over several decades.[33]Figure 2 shows the calculated potential energy curves against the quadrupole deformation β2with and without the inclusion of hexadecupole deformation β4.In the lighter N=152 isotones,the inclusion of β4could drive the minima of potential energy curves down a little.Besides,it also generates a markedly reduction on the saddle point in heavier ones.Especially in260Hs,the reduction of the fission barrier height is more than 1 MeV.Since a 1-MeV difference in the fission barrier could result in several orders of magnitude difference in the fission half-life,the influence of β4degree of freedom on fission barrier should not be neglectful in these nuclei.

Fig.2 Comparison of calcualted potential energy curves with(magenta)and without(black)hexadecapole deformation β4for even-even N=152 isotones as a function of quadruple deformation β2.Both potential energy curves have been obtained with γ =0?.

Moreover,to investigate the effect of non-axial quadrupole deformation γ on fission barrier in these N=152 isotones,we systematically displayed the calculated potential energy curves as the function of β2as well.Up to date,the non-axial quadrupole deformation γ has been manifested itself by the wobbling motion,signature inversion and chiral doublets in many nuclei.[44?46]For example,it is suggested that the triaxial minima are about 0.5 MeV shallower than axial minima in N=76 isotones.[47]In the present work,it can be found that the inclusion of the triaxial deformation would pull the fission barrier down signi ficantly as well,exhibiting in Fig.3. Indeed,this is consistent with the other MM model calculations that the inner fission barrier is usually lowered when the triaxial deformation is allowed in the actinide region.[48]For example,the fission barrier in256Rf obtained with triaxial PES calculations is about 6 MeV whereas the value given by axial PES calculations has been predicted close to 9 MeV.

Fig.3 (Color online)Comparison of calcualted potential energy curves with(green)and without(black)nonaxialquadrupole deformation γ for even-even N=152 isotones as a function of quadruple deformation β2.Both potential energy curves have been obtained with β4=0.

Fig.4 (Color online)(a)Calculated potential energy curves against γ at saddle point of β2in246Pu,248Cm,and250Cf.(b)Calculated potential energy curves against β2in248Cm.Each calculated deformation space is with the inclusion of(β2)(black),(β2,β4)(magenta),(β2,γ)(green),(β2,γ,β4)(blue)degrees of freedom,respectively.For the clearness,all the curves have subtracted to the minimum energy of PES.The experimental fission barrier denotes by sphere scatter

To exemplify the effects of β4and γ on fission barrier,taking the248Cm nucleus as an example,we have depicted the calculated potential energy curves for this nucleus in Fig.4.Based on the energy curves in Fig.4(a),one can see the potential energy is relatively flat near the minima with respect to the triaxial deformation γ.But for a crude evaluation,the influence of γ may be more significant in a lower β2saddle point to some extent.Furthermore,it is worth noticing that the hexadecapole deformation β4pulls down the minimum of PES while the triaxial deformation γ pushes up the saddle point of PES remarkably in Fig.4(b).The in fl uence of γ is the larger of the two,and they are all important to the reproduction of experimental fission barrier.In addition,it has been pointed out that if only a few shape degrees of freedom are constrained,the spurious saddle points may be obtained inevitably.[10]In the present deformation space(β2,γ,β4),it seems more suitable than other selected deformation spaces,considering it is sufficient to describe the experiment value in248Cm.

Fig.5 (Color online)The calculated Woods-Saxon single-particle levels near the Fermi surface of248Cm for protons(a)and for neutrons(b).The positive(negative)parity levels are denoted by the solid(dashed)lines.

Fig.6 (Color online)Similar to Fig.5,but as a function of β4.The blue dotted line denotes the equilibrium deformation β4=0.036.

As mentioned earlier,the microscopic part of total energy arises due to fluctuations in the actual distribution of single-particle levels in the nucleus relative to a smooth distribution of levels.[29]Therefore,in order to further analyze their different effects on fission barrier,we represent the calculated Woods-Saxon single particle levels near the Fermi surface of248Cm in Fig.5.For convenience,it is intelligible to use origin deformation parameters with β4=0 and γ =0.Apparently,the deformed shell gaps can be clearly seen near the Fermi surface,together with the asymptotic quantum numbers ?[Nnzml].The present single-particle structures are consistent with calculated single-particle spectra for doubly magic nuclei270Hs and298Fl by Sobiczewski et al.[37]Since the total single-particle correction are determined by the shell correction,which would be enhanced at the low level density near the Fermi surface,[29]one can expect the shell correction is larger at the ground state than at the saddle point.Furthermore,the similar mechanism implying by single-particle structure would be emerging in the β4and γ degrees of freedom.The β4and γ deformation degrees of freedom will simultaneously affect the energies at both ground state and saddle point though they may have somewhat different impacts on them.As illustrated in Figs.6 and 7,the single-particle level density near the Fermi surface is lower at the inclusion of β4(with γ,i.e.,triaxial saddle)than without β4(without γ,i.e.,axial saddle).For the effect of γ deformation on inner fission barrier in248Cm,a low level density in TS could result in larger shell correction energy(negative value)than the ones in AS.Then the total energy of TS will be lower than the total energy of AS.On the other hand,as shown in Fig.3,the equilibrium deformation γ equals zero,i.e.,the γ deformation do not influence the total energy of the first minimum in248Cm.Obviously,according to the Bf=Esaddle?Eminimum,the height of fission barrier of TS would be lower than that of AS.On the contrary,the effect of β4on the inner fission barrier may not impact on the saddle point but on the minimum of248Cm as depicted in Fig.2.Then the total energy of the ground state with the inclusion of β4deformation should be lower than the ones without β4.Thus,the effect of β4deformation on the inner fission barrier has strengthened a little the height of fission barrier.However,as shown in Fig.2,the inclusion of β4deformation may reduce the inner fission barriers in the heavier N=152 isotones as well,where such an oscillating e ff ect of β4deformation on the inner fission barrier in N=152 isotones is an interesting foundation.

Fig.7 (Color online)Calculated proton(a)and neutron(b)single-particle levels near the Fermi surface for 248Cm at three typical deformation grid points,i.e.,GS(ground state),AS(axial saddle)and TS(triaxial saddle)points.The red lines indicate the Fermi energy levels.

4 Summary

In conclusion,systematic investigation of the first fission barrier in even-A N=152 isotones with 92≤Z≤108 has been preformed using multidimensional PES calculations in the deformation space(β2,γ,β4).The calculated ground-state deformations are basically in agreement with previous study and existing experimental data.And the collective properties revealing with the value of R4/2,P-factor andare also consisted with present work.Furthermore,the spectacular reproduction of the fission barrier by our PES calculations toward interesting physics.The effect of hexadecupole deformation β4and non-axial quadrupole deformation γ on the first fission barrier is inequable but both unnegligible.In addition,it is found that the microscopic shell correction energy due to the nonuiform distribution of calculated WS single-particle levels should be responsible for the evolution of fission barrier.Low level density would result in the large shell correction energy,which corresponds to the reduction of the total potential energy.Therefore,the effects of various deformation can be understood by Bf=Esaddle?Eminimum.