Peng?Hui Chen · Chang Geng · Zu?Xing Yang · Xiang?Hua Zeng,3 · Zhao?Qing Feng

Abstract We systematically calculated the multinucleon transfer reactions of 208Os, 208Pt, 208Hg, 208Pb, 208Po, 208Rn, 208Ra, and 132,136 Xe when bombarded on 232 Th and 248 Cm at Coulomb barrier energies within the dinuclear system model.These results are in good agreement with the available experimental data.The influence of Coulomb and shell effects on actinide production in these reactions has been rigorously studied.We calculated and analyzed the potential energy surface (PES) and total kinetic energy (TKE) mass distributions for the reactions involving 208Hg, 208Pb, and 208 Po with 248 Cm and 232Th.The PES and TKE spectra shed light on the fragment formation mechanisms in multinucleon transfer reactions, with clear indications of isospin and shell effects.The production cross sections for multinucleon transfer products show a strong dependence on isobar projectiles with a mass number A=208.Isobar projectiles with high N/Z ratios are advantageous for generating neutron-rich target-like fragments.Conversely, products induced by isobar projectiles with larger charge numbers tend to shift toward proton-rich regions.The intertwining of the Coulomb potential and shell effect is evident in the production cross sections of actinide isotopes.Drawing from reactions induced by radioactive projectiles, we anticipate the discovery of several new actinide isotopes near the nuclear drip lines, extending our reach into the superheavy nuclei domain.

Keywords Dinuclear system model · Isobaric collisions · Multinucleon transfer reactions · Neutron-rich actinides

1 Introduction

To date, the synthesis of 11 isotopes–149Lu [1],207Th [2],251,264Lr [3, 4],166Pm,168Sm,170Eu,172Gd [5],204Ac [6],39Na [7], and286Mc [8]–in the last year, has brought the total count to 3327 known nuclei in the nuclide chart.This comprises 288 natural nuclides (254 stable isotopes with lifespans longer than the Earth’s age and 34 unstable nuclides)and 3039 species of nuclei synthesized in global laboratories.These syntheses utilize techniques such as fusion-evaporation (FE), multinucleon transfer (MNT), deep-inelastic reactions (DIR), projectile fragmentation (PF), spallation,fission (SF), neutron capture (NC), and thermonuclear tests(TT) [9].However, various theoretical models predict the existence of an additional 8000 to 10,000 unknown bound isotopes in the nuclei chart [10–12].This suggests that at least over 5000 nuclides remain to be discovered by nuclear experimentalists, particularly in the realms of nuclear drip lines and the stability islands of superheavy nuclei.

In recent years, laboratories across the globe have made significant advancements in nuclear synthesis.From an experimental standpoint, several new nuclear species have been produced:207Th,235Cm,214U,222Np,211Pa,280Ds [2,13–15], and others via fusion-evaporation (FE) reactions;110Zr,121Tc,129Pd, and more through projectile fragmentation (PF) [16]; and223,229Am,233Bk [17], among others,using multinucleon transfer (MNT) reactions.This study has garnered significant attention from leading research facilities, including the Lanzhou Heavy Ion Research Facility (HIRFL) in China, Joint Institute for Nuclear Research (JINR) in Russia, Helmholtz Centre for Heavy Ion Research (GSI) in Germany, Grand Acc?l?rateur National d’Ions Lourds (GANIL) in France, and Argonne National Laboratory (ANL) in the USA, actively working toward the synthesis of new nuclides in regions near the drip lines and within the superheavy region.

In efforts to elucidate the damped collision mechanism and predict the synthesis cross sections of target nuclides,theorists have crafted intricate yet pragmatic models to represent the multinucleon transfer reactions occurring at incident energies close to the Coulomb barrier.Notable among these are the GRAZING model [18–20], the dinuclear system (DNS) model [21–30], and a dynamical model anchored in the Langevin equations [31, 32].There are also microscopic methods that account for the degrees of freedom inherent to nucleons.Examples include the time-dependent Hartree (TDHF) approach [33–35] and the improved quantum molecular dynamics model (ImQMD)[36, 37].Although each model possesses its distinctive attributes, they are all capable of faithfully replicating available experimental data.The DNS model, in particular,offers a comprehensive view, considering factors, such as the shell effect, dynamic deformations, fission, quasi-fission, deep-inelastic mechanisms, and the odd-even effect.Furthermore, it stands out for its exceptional computational efficiency.

In this study, we compared the calculated cross sections of target-like fragments resulting from MNT reactions of132,136Xe +248Cm at incident energies proximate to the Coulomb barriers with the available experimental data,using the DNS model as a basis.To delve into the interplay between the Coulomb force and shell effect in the MNT process, we selected isobaric projectiles with a mass numberA=208 in proximity to the double magic nucleus208Pb.These projectiles were directed to bombard targets232Th and248Cm at energies consistent with the Coulomb barrier.Our analysis further explores the production cross sections of yet-to-be-identified actinide isotopes in isobaric collisions.The structure of this study is as follows:In Sect.2, we provide a concise overview of the DNS model.Section 3 presents the calculation results and subsequent discussion.Finally, a summary and conclusions drawn from our study are detailed in Sect.4.

2 Model description

Initially, Volkov introduced a concept to depict deep-inelastic heavy-ion collisions [38].Later on, Adamian incorporated a quasi-fission component into the massive fusion process [39, 40].Subsequent enhancements, including modifications to the relative motion energy and angular momentum of the colliding nuclei, in tandem with nucleon transfer,were developed within the DNS framework by the Lanzhou Group [41].The production cross sections of superheavy nuclei (SHN), quasi-fission, and fusion–fission dynamics have been comprehensively examined within the dynamical DNS model.The dynamic evolution of the colliding system involves sequential processes: overcoming the Coulomb barrier to form the DNS, adjusting the relative motion energy,angular momentum, mass, charge asymmetry, among other factors, within the potential energy surface, and finally,the de-excitation of primary fragments [42].The production cross section of the MNT fragments was determined as follows:

σcap(Ec.m.,J) denotes the cross section of the DNS formation derived by the Hill–Wheeler formula with a barrier distribution function [43].Wsur(E1,J1,s) denotes the survival probability of the fragment formation during the MNT process.Furthermore, s denotes the decay channels for fragments(Z1,N1) such as neutrons, protons, deuteron, alpha rays, and gamma rays.Ec.m.denotes the incident energy at the centerof-mass frame.The highest angular momentumJmaxwas calculated for the colliding system in the grazing configuration.The angular momentumJwas considered at the initial collision configuration before dissipation.E1andJ1denote the excitation energy and angular momentum of the fragment with proton numberZ1and neutron numberN1, respectively,in DNS, respectively.Additionally,P(Z1,N1,J1,B) is the formation probability of fragments (Z1,N1).For the barrier distribution function, we adopted an asymmetric Gaussian[44] form.

The quantities △ andBmwere evaluated using △=(Bt+Bs)/2 andBm=(Bt+Bs)/2.Furthermore,BtandBsrepresent the height of the Coulomb barrier and minimum point of deformation under tip–tip collision, respectively.The normalization constant satisfies ∫f(B)dB=1.

In the DNS model, the solution for nucleon transfer and relative motion involves a set of microscopic derivations, and the master equations distinguish between protons and neutrons.The fragment distribution probability,P(Z1,N1,E1) ,represents the proton numberZ1, neutron numberN1, and excitation energyE1for DNS fragment 1, which is described by the following master equation:

The excited DNS creates a valence space where the valence nucleons are symmetrically distributed around the Fermi surface.Only particles in states within this valence space participate actively in nucleon transfer.The local excitation energy and nucleon transfer influence the transition probability.These are microscopically determined from the interaction potential within the valence space, as detailed in [47, 48].

The neutron transition coefficient has the same formula.The relaxation time is calculated using the deflection function method [49].The memory timeτmemandVinteraction elements can be found in [47].

The motion of the nucleons in the interacting potential is governed by a single-particle Hamiltonian [41, 50]:

with

Here, indicesKandK′(K,K′= 1, 2) denote fragments 1 and 2, respectively.Furthermore, quantitiesεvKanduαK,βK′represent the single-particle energy and interaction matrix elements, respectively.Single-particle states are defined with respect to the centers of the interacting nuclei and are assumed to be orthogonalized in the overlap region.Thus,the annihilation and creation operators depend on the time.The single-particle matrix elements are parameterized as follows:

whereε?denotes the local excitation energy of the DNS.The microscopic dimensions of fragment (ZK,NK) were evaluated using the valence statesNK=gKΔεKand valence nucleonsmK=NK∕2 (K=1,2 ) as:

The local excitation energyE1was derived from the dissipation energy coupled to the potential energy surface (PES) of the relative motion of the DNS.The excitation energy in the equilibrium stage was determined by dividing the fragments by their mass.The angular momentum of the main fragment was determined by the moment of inertia.The local excitation energy was evaluated using [47, 48]

The quantities of the entrance channel, denoted asαEN,encompass proton and neutron numbers, quadrupole deformation parameters, and orientation angles.Specifically, they are represented asZP,NP,ZT,NT,R,βP,βT,θP, andθTfor the projectile and target nuclei by symbols P and T, respectively.The interaction timeτintis obtained using the deflection function method [51].The energy dissipated in the DNS exhibits an exponential increase.The potential energy surface (PES) of the DNS can be evaluated as follows:

whereQgg, derived from the negative binding energies of fragments (Zi,Ni) , was calculated using the liquid-drop model plus shell correction [11].θidenotes the angle between the collision orientations and symmetry axes of the deformed nuclei.Furthermore,VCandVNwere calculated using Wong’s formula [52] and the double-folding potential [53], respectively.The quadrupole deformations of the ground- state nuclei were obtained from Ref.[54].Additionally,Vdef(t) denotes the deformation energy of DNS at reaction timet.The evolutions of quadrupole deformations of the projectile- and target-like fragments occur from the initial configuration as

where the deformation relaxation wasτβ=4×10-21s.

The total kinetic energy (TKE-mass) of the primary fragment was evaluated using the following expression:

whereQgg=MP+MT-MPLF-MTLFandEc.m.denote the incident energy in the center-of-mass frame.MassesMP,MT,MPLF, andMTLFcorrespond to the projectile, target, projectile-like fragment, and target-like fragment, respectively.

The survival probabilityWsur(E1,J1,s) is particularly important in the evaluation of the cross section, which is usually calculated using a statistical model.The physical process for understanding excited nuclei is straightforward.However, the magnitude of the survival probability is strongly dependent on the ingredients in the statistical model, such as the level-density parameter [55], separation energy [54], shell correction [54], and fission barrier[56, 57], and others.The excited fragments were cooled by evaporatingγ-rays and light particles (including neutrons, protons, andαparticles) in competition with fission[44].The probability in the channel for evaporating thexth neutron,yth proton, andz-alpha particle can be expressed as follows:

3 Results and discussion

We calculated the production cross sections of actinide isotope chains with atomic numberZ= 93–100 in the collisions of132,136Xe +248Cm at incident energyElab= 699–790 MeV,as shown in Fig.1.Compared with the available experimental data for129,132,136Xe +248Cm, which are represented by a solid red star, solid blue circle, and solid black squares with error bars, respectively, our calculation of136Xe +248Cm,marked by solid black lines, and132Xe +248Cm, marked by dashed red lines, could reproduce the tendency of the cross-sectional distribution of actinide isotopic chains.From experimental data [59, 60], it was determined that projectile129,132,136Xe isotopes-induced reactions with the target248Cm to provide actinide products that have a large overlap distribution area in the neutron-rich region.This was not clearly distinguishable as expected.From our calculation in terms of the deep-inelastic mechanism, relative proton-rich projectile132Xe-induced reactions tend to shift to the proton-rich region when compared to the experimental results.Based on the data presented in Fig.1, target-like fragments have production cross sections of magnitude levels from 100 millibarns to 10 nanobarns.When significantly distanced from the target, the formation cross section of products below the target declines more gradually than that of trans-target products.This suggests that quasi-fission plays a more dominant role in these collisions.It is worth noting that our calculations have limitations: They rely on a model with freeparameters both for calculating the primary fragment cross section and for estimating the survival probability against fission.

Fig.1 (Color online) Calculation and experiment results of production cross sections of actinide isotopic chains with Z = 93–100 in reactions of 129,132,136 Xe + 248 Cm at Elab =699–790 MeV.The available experimental data are considered from [59, 60], marked by solid black square for 136 Xe-induced reactions, solid red star for 132Xe-induced reactions,solid blue circle for 129Xeinduced reactions.Our calculations for 136Xe-induced reactions were shown by solid black lines,132Xe-induced reactions shown by dashed red lines

To investigate the competition between the Coulomb repulsive potential and shell effect in MNT reactions, we calculated the reactions of isobaric projectiles withA=208 bombarding targets248Cm and232Th at incident energyEc.m.=1.1×VB.The calculation details of these collisions are presented below.The interaction potential between the colliding partners was combined with the Coulomb and nuclear potentials.In Fig.2a, interaction potentialVCNof208Pt +248Cm,208Hg +248Cm,208Pb +248Cm,208Po +248Cm, and208Rn +248Cm reactions was marked by solid black, dashed red, dash-dot blue, dash-dot-dot green, and short dashed olive lines, respectively.The tendencies ofVCNdistributions for these collisions were similar.A larger Coulomb potential results in a larger interaction potentialVCN.Specifically,VCNincreases exponentially with decreasing distanceRdecreasing in the attraction region of the nuclear force, where it increases slowly.Nucleon transfer occurred in the touch configuration.Based on the deflection function,the sticking times of the colliding partners are calculated for all impact parameters [49], as shown in Fig.2b, which decreases exponentially as angular momentum increases.In these collisions, a relatively large Coulomb potential resulted in a longer sticking time with a fixed impact parameter.During the sticking time, the kinetic energy dissipates into the composite system to heat with the internal excitation energy, which increases exponentially with reaction time and reaches equilibrium at approximately 2 ×10-21s, as shown in Fig.2c.

Fig.2 (Color online) In panel a, solid black, red, blue, green, olive lines indicate the interaction potential of the tip–tip collisions as a function of surface distance in reactions induced by projectiles 208 Hg, 208Pb, 208Po, 208Pt, and 208Rn, respectively, with target 248Cm;Panel b shows distributions of reaction time to the angular momentum of collisions for these five reaction systems at incident energy Ec.m. =1.1×VB , which decreases exponentially as angular momentum increases.In panel c, for a given angular momentum L=50?across these five collision systems, the internal excitation energies are shown to increase exponentially with reaction time

After the colliding partners are captured, the dissipated kinetic energy, combined with the angular momentum in the DNS, allows them to diffuse along the potential energy surface (PES).This is followed by nucleon rearrangement between the colliding entities, determined by solving a set of master equations.The PES and driving potential were derived using Eq.(12).These are composed of the Coulomb potential, binding energy, and nuclear potential.They are calculated using the Wong formula, the liquiddrop model with shell correction, and the double-folding method, respectively, as cited in [42].The driving potential of projectiles208Hg,208Pb, and208Po on targets248Cm and232Th during tip–tip collision at a fixed distance is plotted as a function of mass asymmetry, denoted asη.Here,η=(AT-AP)∕(AT+AP).This is illustrated in Fig.3a, e,represented by solid black, dashed red, and dash-dot blue lines, respectively.

Open circles and stars represent the projectile–target injection points.Panels (a) and (e) show that the tendencies of the driving potential trajectories for these collisions are similar.Two pockets appeared atη=0.2,0 to derive the potentials of the target248Cm-based reactions.One pocket in the driving potentials for the target232Th-based reactions appears atη=0.2.The neutron subshell numberN=162 can potentially play a crucial role in pocket formation.For projectiles, such as208Po, which are distant from theβ- stable line, their initial points of interaction lie significantly away from their respective driving potential trajectories.As diffusion starts, they quickly converge toward the driving potential path.Generally, using the PES, one can broadly predict the spectral distribution trend across each isotope chain.

By solving a set of master equations, we derived the production probabilities of primary fragments with their respective excitation energies.These equations categorized the fragments based on mass number and kinetic energy as delineated in Eq.(16).These distributions are illustrated in Fig.4, with driving potential trajectories represented as solid gray lines.From Fig.4, we observe two prominent peaks in the high kinetic regions, situated around the projectile–target injection points.Additionally, cross sections tend to concentrate in the valleys of the driving potential trajectories.Reactions involving projectiles208Hg,208Pb, and208Po with targets248Cm and232Th at the incident energyEc.m.=1.1×VBexhibited TKE-mass distributions that were both symmetric and expansive.The TKE-mass distribution spans broadly within the kinetic range of 500–800 MeV and mass range of 160–280 MeV,suggesting a potential transfer of more than 30 nucleons.

Fig.3 (Color online) Potential energy surface and driver potentials of projectiles 208Hg-, 208Pb-, and 208Po-induced reactions with targets 248 Cm and 232 Th at tip–tip collisions are listed in Fig.3.Specifically,208Hg-, 208Pb-, and 208Po-induced reactions were represented by solid black, dashed red, and dash-dot blue lines in panels a, e, respectively.Potential energy surfaces for these collisions are shown in panels b–d, f–h, respectively.Open stars denote projectile–target injection points.These solid black lines represent valley trajectories on the two-dimensional potential energy surface

Utilizing the statistical evaporation program, we calculated the survival probability of the excited primary fragments,which in turn determined the production cross sections of the secondary fragments.The production cross sections of primary and secondary fragments, delineated by mass and charge numbers in the collisions from projectiles208Hg,208Pb, and208Po with target248Cm atEc.m.=1.1×VB, are depicted in Fig.5 panels (a) through (f).The solid blue and dashed red lines represent secondary and primary fragments, respectively, while regions of superheavy nuclei are highlighted with rectangular shading.Our findings show that primary fragments span a vast charge spectrum, even reaching the superheavy regions.Conversely, secondary fragment production was significantly dampened by de-excitation.This reduction is attributed to the fact that highly excited primary trans-target fragments, having minimal fission barriers, are prone to undergoing fission.Our predictions indicate cross sections for superheavy nuclei(with atomic numbersZ= 104–116) exceeding 10 picobarns.The neutron subshellN=162 may be particularly influential,especially in the208Po +248Cm collision.

Fig.4 (Color online) Panels a–f display the calculated TKE-mass distribution of primary reaction products from head-on collisions involving projectiles 208Hg, 208Pb, and 208 Po with targets 248 Cm and 232 Th at Ec.m. =1.1×VB.Driving potential trajectories are also incorporated

Fig.5 (Color online) The calculated primary and secondary yields mass and charge distribution for 208Hg-, 208Pb-, and 208 Po-induced reactions with targets 248 Cm at Ec.m. =1.1×VB were listed in panels a–f, respectively.Dashed red and solid blue lines represented primary and secondary yields.The superheavy region ( Z >104) is shown by a rectangular shadow

Fig.6 (Color online) Predicted isotopic distribution cross sections for target-like fragments with atomic numbers Z = 89–103 resulting from the collisions of projectiles Pt, Hg, Pb, Po, and Rn with mass number A=208 bombarding on target 248 Cm at Ec.m. =1.1×VB.The patterns correspond to the following projectiles: solid black for Pt, dashed red for Hg, dash-dot green for Pb, dashed-dot-dot blue for Po, and short dashed olive for Rn.Predicted new actinide isotopes are highlighted with open circles

Secondary production cross sections for actinide targetlike fragments, including isotopes of Actinium, Thorium,Protactinium, Uranium, Neptunium, Plutonium, Americium, Curium, Berkelium, Californium, Einsteinium, Fermium, Mendelevium, Nobelium, and Lawrencium, have been calculated.These calculations pertain to collisions involving projectiles208Pt,208Hg,208Pb,208Po,208Rn, and208Ra bombarding on targets248Cm atEc.m.=1.1×VB.The line representations for these projectiles in Fig.6 are as follows: solid black for208Pt, dashed red for208Hg, dashdot green for208Pb, dashed-dot-dot blue for208Po, and short dashed olive for208Rn.Observations indicate that collisions characterized by a smaller Coulomb force lean toward the neutron-rich region, whereas those with a more substantial Coulomb force gravitate to neutron-deficient areas.Numerous previously unidentified actinide isotopes are predicted from reactions208Pt +248Cm,208Hg +248Cm,208Pb +248Cm,208Po +248Cm,208Rn +248Cm, and208Ra +248Cm.These predictions are detailed in Table 1.For the new neutronrich actinide isotopes,208Pt +248Cm reactions produce the largest cross sections.However,208Pt remains unclear.Notably, unknown actinide products are highly dependent on the Coulomb potential.The208Rn +248Cm reactions result in the largest cross sections for new neutron-deficient actinide isotopes.The open circles represent the new neutron-rich actinide nuclides.

Figure 7 shows secondary production cross sections of all the formed fragments in collisions of208Os +248Cm,208Pt +248Cm,208Hg +248Cm,208Pb +248Cm,208Po +248Cm,208Rn+248Cm,208Ra +248Cm, and primary production cross sections of208Pb +248Cm at the incident energyEc.m.=1.1×VBasN-Zpanel.Panels (g) and (h) clearly show the de-excitation effects.Panels (a), (b), (c), (d), (e), (f), and (h) show that many new isotopes are predicted, including neutron-rich and neutron-deficient isotopes and even superheavy nuclei.The projectile–target injection points and all existing isotopes in the nuclide chart are represented by solid black triangles and open squares, respectively.

Fig.7 (Color online) The production cross sections of all secondary fragments formed in the collisions involving 208 Pt +248Cm, 208 Hg + 248Cm, 208 Pb +248Cm, 208 Po + 248 Cm 208 Rn +248Cm, and 208 Ra + 248 Cm at the incident energy Ec.m. =1.1×VB along with the primary fragments from 208 Pb + 248 Cm are listed in N-Z panels.Open stars denote projectile–target injection points

Table 1 Calculated cross sections of unknown actinide isotopes with Z = 89–103 in the reactions of projectiles 208Pt,- 208Hg-, and 208Pb-induced MNT reactions with target 248 Cm at incident energy Ec.m. =1.1×VB

4 Conclusion

Using the DNS model framework, we systematically calculated the production cross sections of MNT fragments in reactions involving projectiles such as208Os,208Pt,208Hg,208Pb,208Po,208Rn,208Ra, and132,136Xe colliding with targets232Th and248Cm around Coulomb barrier energies.To investigate the isospin diffusion on the formation of actinide products during the MNT process, the same number of projectiles withA=208 was selected.Our calculation for132,136Xe +248Cm is consistent with the available experimental data.The sticking time for these colliding systems, inferred from deflection functions, was significantly influenced by the Coulomb force, especially at smaller impact parameters.Furthermore, PES and TKE of these reactions, which can contribute to predicting the tendency of cross-sectional diffusion, are discussed.A relatively large cross section from TKE appears around the pockets in PES, where the neutron subshellN=162 is evident.The de-excitation process strongly depresses the primary cross section of actinide isotopes up to four magnitude levels.The production cross section of the new actinides is highly dependent on theN/Zratio of the isobaric projectile.It was determined that the Coulomb force coupled with the shell effect plays a crucial role in the production of cross sections of actinides products in MNT reactions.These five colliding systems predicted a wide array of previously unknown heavy isotopes, with accessible cross-sectional values even for superheavy nuclei within the charge numbersZ= 104–110.

Author contributions All authors contributed to the study conception and design.Material preparation, data collection, and analysis were performed by Chang Geng and Peng-Hui Chen.The first draft of the manuscript was written by Peng-Hui Chen and Chang Geng, and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.

Data availability The data that support the findings of this study are openly available in Science Data Bank at https:// doi.org/ 10.57760/ scien cedb.j00186.00249 and https:// cstr.cn/ 31253.11.scien cedb.j00186.00249.

Declarations

Conflict of interest The authors declare that they have no competing interests.