Preamble

Production of superheavy nuclei in the V and Cr induced reactions with the Skyrme energy-density functional approach Xiao-Jun Chen , Zi-Han Wang , Yu-Cui Gao Ya-Ling Zhang , Niu Wan , Ming-Hui Huang , and Zhao-Qing Feng 1 School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China State Key Laboratory of Heavy Ion Science and Technology, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China (Dated: November 29, 2025) Within the framework of the dinuclear system model, the synthesis of superheavy nuclei in the V and Cr induced reactions have been systematically investigated. The nucleus- nucleus potential and potential energy surface is calculated by the well-known Skyrme energy- density functional theory with the parameters of SKM, SKM*, Ska, Z and SLy4, respectively, which correspond to the different nuclear equation of state. The available experimental data from Dubna, GSI, RIKEN and Berkeley are nicely reproduced. It is found that the incompressibility modulus influences the fusion-evaporation residue excitation functions and production cross sections of superheavy nuclei. The Ca induced reactions, i.e., Cm have the larger cross sections for synthesizing livermorium in comparison with the systems of Pu and U. The production cross sections of new elements with Z=119 and 120 in the reactions of Cm and Cm are thoroughly analyzed and strongly depend on Skyrme parameters.

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Skyrme energy density functional, potential energy surface, DNS model I. INTRODUCTION In past several decades, the synthesis of superheavy nuclei (SHN) attracted much attention and has been ob- tained much progress in experiments up to the elemen- t oganesson (Z=118). The competition of complete fu- sion and quasifission dynamics enables a very complicat- ed process in the formation of SHN. The multidimen- sional quantum tunneling, collective excitation, dynami- cal deformation, nucleon or cluster transfer etc, influence the fusion dynamics to form the compound nucleus by overcoming the Coulomb barrier [ ]. Up to now, the nu- clear fusion reactions have been extensively investigated, in particular on the topics of weakly bound nuclei in- duced reactions [ ], nuclear fusion at deep sub-barrier energies for astrophysical interests [ ] and synthesis of superheavy nuclei (SHN) [ ]. Roughly, one third nu- clides on the nuclear chart were synthesized in laborato- ries via the fusion reactions [ ]. The hunting of super- heavy nuclei (SHN) in the nature or synthesizing SHN in laboratories, in particular around the ’island of sta- bility’ predicted theoretically, is the topical issue in past and nowadays. The cold-fusion reactions with Pb or Bi based targets were firstly proposed by Oganessian et al. [ ]. The superheavy elements (SHEs) from Bh to Cn were successfully synthesized in the cold-fusion reac- tions at GSI (Darmstadt, Germany) with the heavy-ion accelerator UNILAC and the SHIP separator [ ]. Ex- periments on the synthesis of element Nh (Z=113) in the Bi reaction have been performed successfully at RIKEN (Tokyo, Japan) [ ]. The SHEs from Fl (Z=114) to Og (Z=118) have been synthesized at the Flerov Lab- oratory of Nuclear Reactions (FLNR) at Dubna (Russi- a) with the double- magic nuclide Ca bombarding ac- tinide nuclei [ ]. With constructing the new facilities in the world such as RIBF (RIKEN, Japan) [ ], SPI- RAL2 (GANIL in Caen, France) [ ], FRIB (MSU, USA) ], HIAF (IMP, China) [ ], the SHNs on the ’island of stability’ by using the neutron-rich radioactive beams induced fusion reactions or via the multinucleon trans- fer (MNT) reactions might be possible in experiments.

Accurate estimation of SHN production cross sections is particularly important for the synthesis in experiments, i.e., the optimal combination of reaction system, beam energy, evaporation channel etc.

The fusion dynamics is strongly governed by the nucleus-nucleus (NN) potential.

The significant quan- tity is the Coulomb barrier, which is determined by the attractive nuclear and repulsive Coulomb potential- Usually, the NN potential is used to calculate the fusion cross-section with a quantum penetration or the coupled-channel approach. The fusion probability rapid- ly decreases with reducing the incident energy below the Coulomb barrier.

An empirical formula was proposed by Bass for estimating the Coulomb barrier and fusion cross section [ ]. In the light and medium reaction sys- tems, the compound nucleus is formed after overcoming the Coulomb barrier. However, the quasifission mecha- nism may take place in the heavy projectile-target com- binations, in which the disintegration of colliding sys- tem after a few of nucleon transfer hinders the com- pound nucleus formation.

The neck dynamics, shape evolution, collective excitation and nucleon transfer in- fluences the NN potential. It has been known that the NN potential in the fusion reactions is associated with the shape evolution and beam energy by the dynamical

models, e.g., time-dependent Hartree-Fock (TDHF) ap- proach [ ], quantum molecular dynamics (QMD) model ]. There are mainly two sorts of NN potential, name- ly, the phenomenological potentials such as the Woods- Saxon potential [ ], proximity potential [ ], po- tentials (Yukawa-plus-exponential, DDM3Y, Migdal) via the double-folding method [ ], and the adiabat- ic potential [ ]. It is also possible to construct the NN potential within the energy-density functional approach based on the effective nucleon-nucleon interaction, i.e., the Skyrme force [ ], the finite-range Gogny interac- tion [ ] etc. The advantage of the energy-density func- tional approach enables a unified description of nuclear structure, nuclear dynamics and nuclear matter based on the effective nucleon-nucleon interaction.

In this work, the NN potential is calculated within the Skyrme energy-density functional. The potential energy surface is obtained with the approach and the production of SHN is discussed by implementing into the dinuclear system (DNS) model.

The article is organized as fol- lows. In Section 2 we give a brief description of Skyrme energy-density functional theory and the potential energy surface in the DNS model. The fusion-evaporation exci- tation functions on SHN production in the V and Cr induced reactions on actinide nuclei are to be systematically investigated In Section 3. A summa- ry and perspective on the NN potential are presented in Section 4.

II. BRIEF DESCRIPTION OF THE MODEL The nucleus-nucleus potential is a basic quantity for describing the nuclear dynamics in the low-energy heavy- ion collisions. The interaction potential in binary colli- sions depend on the collision orientation and is composed of nuclear and Coulomb contribution as follows [ Here the Coulomb potential is calculated by the well Wong’s formula [ denotes the symbols being the projectile or target nu- cleus and the relative momentum of inertia. The represent the nuclear radii, quadrupole deformations and polar angles between the beam direction and the sym- metry axes of deformed nuclei, respectively. the center-of-mass distance of projectile and target nu- clides. Shown in Fig. 1 [FIGURE:1] is the definition of the quantities and the integration variables . The de- formation effect is included in the nuclear and Coulomb potentials, which results in the orientation dependence of the Coulomb barrier and influences the quasifission dy- namics in the massive fusion reactions. The multiple in- tegral with the energy-density functional by the Skyrme force is performed in the spherical coordinate system ( The nuclear potential is calculated by the Skyrme energy-density functional as [ are the binding energies con- tributed from the nucleon-nucleon force of colliding sys- tem, projectile and target nuclei by the relation of drdθdϕ , respectively. energy-density functional )] is derived from the Skyrme force as (see in detail in Appendix) 6 (1 + 2 (1 + )) (3) with the spin-exchange operator

ˆ P σ = 1

2(1 + ˆ

The zero-range effective forces between nucleons in the nuclear environment provide the energy-density function- al and are available for the ground-state properties of fi- nite nuclei and nuclear matter at saturation density [ 34 ].

The energies of colliding system, projectile and target nuclei are calculated by

respectively. The density profiles of proton and neutron distributions for projectile and target nuclides are taken to be frozen of the Woods-Saxon form as 1 + exp[ with the diffuseness coefficients being the values of 0.55-0.65 fm. The projectile radii with the quadrupole deformation is given by

R P = R 0 P

76 + 0 . Similarly, the neutron and proton density distributions of target nucle- us are obtained.

Within the help of the well-known extended Thomas- Fermi (ETF) approximation, the kinetic energy term is obtained up to the second order extension. The energy- density is expressed as [

ε [ ρ p ( r ) , ρ n ( r )] = ¯ h 2

with the kinetic energy term

τ i ( r ) =3

f i = 1 + 2 m ¯ h 2 (3 t 1 + 5 t 2 16 + t 2 x 2 4 ) ρ i ( r ) . (12)

Here the local density the kinetic energy density are satisfied in the calculation. The potential part in the energy-density functional is given by

ν sk ( r ) = t 0 2 [(1 + 1

[(1 + 1 (1 + 1 (1 + 1 (1 + 1 (1 + 1 2)](( The parameters , the density- dependent stiffness and the spin-orbit strength listed in Table 1 [TABLE:1].

The six sets of Skyrme parameters SkM, SkM ], Z [ ], SLy4 [ ], are taken in the cal- culation. The binding energy, root-mean-square radii of finite nuclei around the magic numbers and nuclear mat- ter properties at saturation density are self-consistently described with the forces.

The nucleus-nucleus (NN) potential is of importance in the description of nuclear dynamics, i.e., the cay, low-energy heavy-ion collisions, quasi-fission process, fusion-evaporation reactions etc [ Consequent- ly, the quasifission yields, fusion-fission products, fusion cross section, isotopic distribution, angular and kinetic energy spectra in the deep inelastic collisions or multinu- cleon transfer reactions, are influenced by the NN poten- tial. The deformation, collective excitation, shape evolu- tion, initial orientation etc influence the nucleus-nucleus potential [ ]. In the realistic nuclear reaction, the density profile varies with the evolution time of collid- ing system and leads to the complicated NN potential.

Two typical approximations are usually used in the re- action models, i.e., sudden approximation and adiabatic approach. In the calculation, the sudden approximation with the frozen nuclear density is used in the potential energy surface and the estimation of SHN production.

As a typical reaction system, the Ca induced fusion re- actions on the actinide nuclides were chosen for succuss- fully synthesizing the SHN with Z=112-118 at Dubna.

We select the Skyrme forces SKM, SKM , Ska, Z and SLy4 corresponding to the different incompressible mod- ulus of nuclear matter at the normal density. Shown in Am and Cm, respectively. The d- ifference of the NN potentials from the Skyrme forces is obvious and caused from the different compression mod- ulus of nuclear matter. It is assumed that the DNS is formed at the bottom of potential pocket. The broader and deeper pocket is available for the compound nucleus formation against the quasifission reactions.

The formation of superheavy nuclei in massive fusion reaction is complicated and associated with the nucleon transfer, shape evolution, neck formation, relative mo- tion energy and angular momentum dissipation. In the dinuclear system model, the density profiles of colliding nuclei are taken to be frozen and the neck dynamics is not taken into account. The nucleon transfer is coupled to the relative degrees of freedom via a set of master e- quations by the potential energy surface (PES). The PES is given by ).The ) are the negative binding energies of fragment compound nucleus ), respectively.

The symbol denotes the quantities of . The represent the quadrupole deformations of two DNS fragments at ground state. denote the angles between the collision orientations and the symmetry axes of deformed nuclei. The nucleus-nucleus potential between fragments ( ) and ( ) includes the nuclear and Coulomb interaction.

In the calculation, the distance between the centers of the two fragments is chosen to be the value at the touching configuration, in which the DNS is assumed to be formed. Shown in Fig. 2 [FIGURE:2] is the driving potential as a function of mass asymmetry ) in the reactions of

TABLE I. Parameters of the Skyrme forces used in the calculation.

Parameter (MeV fm (MeV fm (MeV fm (MeV fm (MeV fm (MeV) Cm, respectively. The Businaro-Gallone (B.G.) point is marked at the maximal position of driving potential.

The difference of driving potentials with the Skyrme forces SKM, SKM , Ska, Z and SLy4 is caused from the NN potential. The fusion probability is related to the inner fusion barrier, which is defined the difference of driving potential at the B.G. point and the one at incident position. The low inner fusion barrier is favorable for the compound nucleus formation.

The DNS model has been applied to the the quasi- fission and fusion dynamics, multinucleon transfer reac- tions and deep inelastic collisions, in which the dissi- pation of relative motion and rotation of colliding sys- tem into the internal degrees of freedom is assumed at the touching configuration. The DNS system evolves a- long two main degrees of freedom to form a compound nucleus, namely, the radial motion via the decay of DNS and the nucleon transfer via the mass asymme- ) or the charge asymmetry ]. In accordance with the temporal sequence, the system undergoes the cap- ture by overcoming the Coulomb barrier, the competition of quasi-fission and complete fusion by cascade nucleon transfer, and the formation of cold residue nuclide by e- vaporating -rays, neutrons, light charged particles and binary fission. The production cross section of the su- perheavy residue is estimated by the sum of partial wave with the angular momentum at incident center of mass energy

σ ER ( E c.m. ) = π ¯ h 2

J =0 (2 J + 1) T ( E c.m. , J )

Here, ) is the penetration probability and given by the Hill-Wheeler formula and a Gaussian-type barri- er distribution [ ]. The distribution function is taken as the Gaussian form with the normalization constant satisfying the unity re- lation The quantities and ∆are evaluated by 2 and ∆= ( respectively. The are the Coulomb barrier at waist-to-waist orientation and the minimum barrier by varying the quadrupole deformation of the colliding partners. The fusion probability is described by the DNS model and taking into account the competition of Comparison of the nucleus-nucleus potentials with the Skyrme forces SKM, SKM , Ska, Z and SLy4 in collisions of Am (left panel) and Cm (right panel), respectively.

the quasi-fission and fission of the heavy fragment [ ], in which the nucleon transfer is described by solving a set of microscopically derived master equations by distinguish- ing protons and neutrons. The survival probability is calculated with the Weisskopf statistical theory [ in which the decay of compound nucleus formed in the fusion reaction is cooled by evaporating rays and light particles including neutrons, protons, in competition with binary fission.

III. RESULTS AND DISCUSSION The nucleus-nucleus potential is of significance in the low-energy heavy-ion collisions, i.e., the fusion cross section, quasifission dynamics, fusion-fission reaction, fusion-evaporation for synthesizing the heavy or super- heavy nuclei. A repulsive core with the incompressibility modulus of K=228 MeV is introduced into the Michigan- 3-Yukawa-Reid effective potential for describing the fu- sion cross sections at energies far below the Coulomb barrier in the reaction of ]. The ener- gy density function method paves a bridge between the nucleon-nucleon effective force and nuclear matter equa- tion of state, i.e., the Skyrme effective interaction, rel- ativistic invariant theory based on the meson-exchange concept, chiral effective theory etc. The effective Skyrme force is expected for a unified description of the nucle- ar ground-state properties, nuclear reactions and nucle- ar matter. The potential energy surface influences the nuclear dynamics in the fusion-evaporation and fusion- fission reactions. Consequently, the fusion cross sections are correlated to the incompressibility modulus and ef- fective nucleon-nucleon interaction. Shown in Fig. 3 [FIGURE:3] is the fusion-evaporation residue excitation functions in the reaction of Am for producing moscovium with the Skyrme parameters SKM, SKM, Ska, Z, SLy4 and the available data from Dubna [ ]. It is obvious that the maximal cross section of 20 pb at the excitation of 35 MeV in the 3n evaporation channel from the experi- mental measurements. The production cross sections of SHNs with the different Skyrme forces are related to the density in the overlap region and nuclear matter proper- ties. Overall, the fusion-evaporation excitation functions with the parameter SLy4 (K=230 MeV) are consistent with the available data. The uncertainties of the Skyrme forces on the SHN production cross sections are collected and compared in Fig. 3 (f). It is demonstrated that the maximal cross sections in the 2n-5n evaporation chan- nels are similar for different Skyrme forces. The fusion- evaporation residue excitation functions with the Skyrme parameters SKM, SKM, Ska, Z and SLy4 in the reaction Cm are calculated and compared with the available data from Dubna [ ], GSI [ ] and RIKEN ]. It is obvious that the maximal cross section in the 4n channel with 5 pb at =40 MeV. The uncertainties of production cross sections are huge for different Skyrme forces owing to the larger overlap density in comparison with the reaction of Am. The lower cross sec- tion for producing livermorium than the ones of moscovi- um is mainly caused from the reduction of fission barrier of compound nucleus, which manifests the dominance of shell effect in the formation of SHN.

Recently, new superheavy elements beyond oganesson attracted attention in experiments and theories.

Ca induced reactions meet some obstacles owing to the management of target material. The combination of V and Cr on actinide nuclei paves the way for synthe- sizing the new elements 119 and 120. In comparison with Ca induced reactions, the production cross section of SHN is usually reduced by the high inner fusion barrier and Q-value with the V and Cr induced reac- tions. Shown in Fig. 5 [FIGURE:5] is the fusion-evaporation residue excitation functions in the reaction of U. It is The driving potential as a function of mass asymmetry in the reactions of (a) Am and (b) respectively.

Comparison of the fusion-evaporation residue excitation functions in the reaction of Am for producing moscovium with the Skyrme parameters SKM, SKM, Ska, Z, SLy4 and the available data from Dubna [ The fusion-evaporation residue excitation functions with the Skyrme parameters SKM, SKM, Ska, Z and SLy4 in the reaction of Cm for producing livermorium and compared with the available data from Refs [ ], GSI data from Ref. [ ] and RIKEN data from Ref. [

obvious that the 5n channel has the cross section below 0.01 pb and the distribution structure is different with the Skyrme parameters SKM, SKM*, SKa, Z and SLy4.

Roughly, the 2n-4n channels with the maximal cross sec- tions of 0.1-1 pb are obtained with the Skyrme forces SKM, SKM*, SKa and Z. However, the parameter SLy4 predicts the larger cross sections for flerovium produc- tion. The difference of Skyrme forces for calculating the fusion-evaporation excitation functions is mainly caused from the NN potential and potential pocket (width and height). The narrower potential pocket enables the lower fusion probability and results in the reduction of the SHN cross sections. The attempt of the Ti beams on was performed at Berkeley [ ]. Comparison of the calcu- lated results and the experimental data is shown in Fig.

6. Basically, the 4n channel is available for producing liv-

ermorium and consistent with the data with the Skyrme forces SKM, SKM* and Z. The parameter Ska predicted the unexpected very-low cross sections. The uncertain- ties of SHN production cross section are huge with the magnitude of 2-5 order, in particular with decreasing the excitation energy.

V induced reactions attracted much attention for synthesizing the new SHN beyond Ca. More mass asymmetric system is available for enhancing the fusion probability. As a test for the new reaction mechanism, the system of Am is feasible for producing o- ganesson in experiments. The fusion-evaporation excita- tion functions with the Skyrme parameters SKM, SKM, SKa, Z and SLy4 are calculated as shown in Fig. 7 [FIGURE:7]. The cross sections in 2-4n are possible for measurement in experiments. However, for synthesizing the new element 119, the production cross section below 1 fb is found with the Skyrme forces SKM, SKM, SKa and Z as shown in In recent years, the high-intensity Cr beams are pro- vided at the accelerators in the world, i.e., SHE factory in Dubna, SHE special accelerator (CAFe II) and gas filled recoil separator spectrometer (SHANS2). Shown in the DNS model with the Skyrme energy density func- tional by SKM, SKM , SKa, Z and SLy4 in the reaction U and compared with the Dubna data [ The production cross section of Lv in the 4n channel is nicely reproduced with the Skyrme forces SKM, SKM* and Z. The 4n channel at the excitation energy of 41 MeV is the optical way for synthesizing SHN and the excitation energy goes up in comparison with the one of 38 MeV in the reaction of Cm for producing livermorium.

The larger binding energy of double magic nucleus en- larges the Q-value and is favorable for the SHN formation owing to the survival probability. Accurate estimation of the fusion-evaporation excitation function induced by Cr on actinide nuclei is helpful for synthesizing new element in laboratories.

We calculated the oganesson production in the Pu reaction as shown in Fig.

10. The 2-4n evaporation channels are dominant for

the SHN formation with the cross sections within the region of 1-10 pb. Measurements on the system is be- ing expected for checking the prediction and extending the synthesis of new SHN with the Cr beams. element synthesis attracted much attention in experi- ments, e.g., ]. The fusion-evaporation excitation functions for synthesizing the new SHEs with Z=119 and Am and are calculated as shown in Fig. 11 [FIGURE:11] and in Fig. 12 [FIGURE:12], respec- tively. It is obvious that the production cross sections of 119 and 120 are below 1 fb with the prediction of Skyrme forces SKM, SKM , SKa and Z, except for SLy4. The depth and width of the potential pocket of NN potential are correlated with the nuclear matter properties of EOS and the diffusion magnitude of proton and neutron den- sity profiles. It has been manifested the mass tables of nuclides also influence the PES and separation energies, consequently the SHN production [ IV. CONCLUSIONS In summary, the nucleus-nucleus potential is calculat- ed with the Skyrme energy density functional, which is associated with the incompressible modulus of nuclear matter with the Skyrme forces SKM, SKM , Ska, Z and SLy4. The fusion-evaporation excitation functions in the V and Cr induced reactions on actinide nuclei are systematically investigated for producing SHN within the DNS model. It is demonstrated that the dis- crepancy of fusion-evaporation cross sections with the d- ifferent Skyrme forces comes from the stiffness of nuclear incompressibility modulus and from the inner fusion bar- rier. The available data from Dubna, GSI and RIKEN and the fusion-evaporation residue excitation function- s in the reactions of Am and Cm are nicely reproduced by SKM, SKM and Z. The extension cross sections for new SHE production with Z=119 and 120 are 1 fb in the Am reaction and below 0.1 fb in collisions of Cm, respectively.

Acknowledgements This work was supported by the the National Natural Science Foundation of Chi- na Projects (Grant Nos 12575132, W2412040, 12175072, 12311540139) National search and Development Program of China (Grant No. 2024YFE0110400).

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The fusion-evaporation residue excitation functions calculated by the DNS model with the Skyrme parameters SKM, SKM, SKa, Z and SLy4 in the reaction of Production of SHN livermorium with the Skyrme energy density functional by SKM, SKM, Ska, Z and SLy4 in the reaction of Pu and compared with the recent experimental data from Berkeley [

The same as shown in Fig. 5, but for the reaction of The fusion-evaporation excitation functions with the Skyrme parameters SKM, SKM*, SKa, Z and SLy4 in the reaction of Cm for producing SHE with Z=119.

Production of SHN livermorium calculated by the DNS model with the Skyrme energy density functional by SKM, SKM*, SKa, Z and SLy4 in the reaction of U and compared with the recent experimental data from Dubna [ The same as shown in Fig. 5, but for the reaction of

The fusion-evaporation excitation functions for synthesizing the new SHE with Z=119 in the reaction of The same as shown in Fig. 11, but for producing the new SHE with Z=120 in the reaction of

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