1　Introduction

Over the past decades, the synthesis of superheavy nuclei (SHN) has attracted much attention. SHN synthesis has been achieved in experiments via massive fusion reactions. The seventh period in the periodic table was filled with the superheavy element tennessine (Ts) using the Dubna gas-filled recoil separator (DGFRS) at the Flerov Laboratory of Nuclear Reactions in Dubna, Russia [1]. The existence of superheavy elements (SHE) was predicted in the late 1960s by the macroscopic-microscopic theory of the atomic nucleus [2]. The synthesis of SHN is associated with testing the shell model beyond the doubly magic nucleus ²⁰⁸Pb, searching for the "island of stability,” exploring the limit of the mass of atomic nucleus, and providing a strong Coulomb field such as quantum electrodynamics (QED) in a super-strong electric field [3]. The superheavy nucleus (SHN) (Z≥106) exists due to the strong binding shell effect against Coulomb repulsion. Therefore, the position of shell closure is particularly significant for the properties of SHN, such as half-lives of the α decay chain and spontaneous fission, formation probability, etc.; theoretical models predicted the shell closures at Z=114 and N=184 [4, 5]. Attempts to synthesize elements beyond Og (Z=118) were performed using different systems, for example, ⁶⁴Ni+²³⁸U [6], ⁵⁸Fe+²⁴⁴Pu [7], ⁵⁴Cr+²⁴⁸Cm [8, 9], and ⁵⁰Ti+²⁴⁹Cf [10]. The mass and angle distributions of fission fragments were measured [11]. Systematic analysis of different reactions are required for preferentially producing new SHNs in experiments.

The history of the synthesis of SHN goes back 40 years when multi-nucleon transfer reactions in collisions of two actinide nuclei were conducted [12, 13]. However, the yields of the heavy fragments in strongly damped collisions were found to decrease rapidly with increasing atomic numbers and were unable to produce SHN because of the significantly low cross section. Combinations with a doubly magic nucleus or nearly magic nucleus are usually chosen owing to the larger reaction Q values. Reactions with ²⁰⁸Pb or ²⁰⁹ Bi-based targets were first proposed by Oganessian et al. [14, 15]. The SHEs from Bh to Cn were synthesized in the cold fusion reactions at GSI (Darmstadt, Germany) with the heavy-ion accelerator UNILAC and the SHIP separator [16, 17]. Experiments on the synthesis of element Nh (Z=113) in the ⁷⁰Zn+²⁰⁹Bi reaction were performed successfully at RIKEN (Tokyo, Japan) [18]. However, creating superheavy isotopes beyond Nh in cold fusion reactions is challenging because of the significantly low cross section (σ<0.1 pb). Superheavy elements from Fl (Z=114) to Og (Z=118) were synthesized at the Flerov Laboratory of Nuclear Reactions (FLNR) in Dubna (Russia) with the double magic nuclide ⁴⁸Ca bombarding actinide nuclei [19-22]; in this experiment, more neutron-rich SHN was produced and identified by the subsequent α-decay chain. The decay properties of ²⁷¹Ds in the cold fusion reaction of ⁶⁴Ni+²⁰⁸Pb ${\to }^{271}$ Ds+n were identified using a gas-filled recoil separator at the Institute of Modern Physics (IMP) in Lanzhou [23]. Constructing new facilities worldwide, such as RIBF (RIKEN, Japan), SPIRAL2 (GANIL in Caen, France), FRIB (MSU, USA), and HIAF (IMP, China), and using significantly neutron-rich radioactive beams, we can potentially create SHNs on the "island of stability" soon.

The formation dynamics of SHN in massive fusion and multinucleon transfer reactions are complicated, and are associated with the coupling of several degrees of freedom, such as radial elongation, mass or charge asymmetry, shape configuration, relative motion energy, etc. Several macroscopic models were developed to describe the fusion hindrance in massive systems, for example, the macroscopic dynamical model [24], fusion-by-diffusion (FBD) model [25, 26], dynamical models based on Langevin-type equations [27-29], dinuclear system (DNS) model [30-34], etc. Recently, the time-dependent Hartree-Fock (TDHF) method was also applied to investigate the quasifission and fusion-fission dynamics in the reactions of ⁴⁸Ca+^239,244Pu [35]. Modifications of macroscopic models are required for self-consistent and reasonable explanation of fusion dynamics in massive systems. The production cross sections of SHEs Z=119 and 120 were estimated within the multidimensional Langevin-type equations [36] and DNS models [37-42] for different reaction systems. A systematic study on SHN production beyond oganesson (Z=118) is needed to predict the optimal projectile-target combinations and reaction mechanisms.

In this work, stochastic diffusion in the nucleon transfer process is applied to the DNS model via the Monte Carlo procedure. A systematic analysis of the production of new superheavy elements was performed. The remainder of this paper is organized as follows. In Sect. 2 a brief description of the DNS model is presented. A comparison with the available data and predictions of new elements Z=119-122 are discussed in Sect. 3. A summary is provided in Sect. 4.

2　Model description

We apply DNS model to the quasi-fission and fusion dynamics, multinucleon transfer reactions, and deep inelastic collisions We assume that the dissipation of the relative motion and rotation of the colliding system into the internal degrees of freedom are at the touching configuration. The DNS system evolves along two main degrees of freedom to form a compound nucleus: the radial motion via the decay of DNS and the nucleon transfer via the mass asymmetry η=(A₁-A₂)/(A₁+A₂) [43-45]. In accordance with the temporal sequence, the system undergoes capture 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 evaporating γ-rays, neutrons, light charged particles, and binary fission. The production cross section of the superheavy residue is estimated by summing partial waves with angular momentum J at the incident center of mass energy E_c.m. as

Here, T(E_c.m.,J) is the penetration probability and is given by a Gaussian-type barrier distribution. The fusion probability P_CN is described by the DNS model, considering the competition between the quasi-fission and fission of the heavy fragment. The survival probability W_sur was calculated using a statistical approach [46-49].

In the DNS model, the time evolution of the distribution probability P(Z₁,N₁,E₁, t) for fragment 1 with proton number Z₁, neutron number N₁, and excitation energy E₁ are described by the following master equation:

Here $W_{Z_1\mathrm{,}{N}_1;Z_1^{\text{'}}\mathrm{,}{N}_1}$ ($W_{Z_1\mathrm{,}{N}_1;Z_1\mathrm{,}{N}_1^{\text{'}}}$) is the mean transition probability from the channel(Z₁,N₁,E₁) to ($Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}{E}_1^{\text{'}}$), ［or (Z₁,N₁,E₁) to ($Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{E}_1^{\text{'}}$)］; d_Z1,Z1 denotes the microscopic dimension corresponding to the macroscopic state (Z₁,N₁,E₁).The sum is taken over all possible proton and neutron numbers that fragment Z₁’ and N₁’ may take; however, only one nucleon transfer is considered in the model with the relations Z₁’=Z₁±1 and N₁’=N₁±1. The excitation energy E₁ is determined by the dissipation energy from the relative motion and the potential energy surface of the DNS. The quasi-fission rate Λ^qf and fission rate Λ^fis are given by the one-dimensional Kramers formula [50]. The motion of nucleons in the interacting potential is governed by the single-particle Hamiltonian [32], which is influenced by the local excitation energy of the DNS.

The potential energy surface (PES) of the DNS is given by

The DNS fragments satisfy the relation Z₁+Z₂=Z and N₁+N₂=N, where Z and N are the proton and neutron numbers of the composite system, respectively. The symbol α denotes the quantities of Z₁, N₁, Z₂, N₂, J, R, β₁, β₂, θ₁, θ₂; B(Zi,Ni) (i=1, 2) and B(Z,N) are the negative binding energies of the fragment (Zi,Ni) and the compound nucleus (Z,N), respectively; $V_{\text{rot}}^{\text{CN}}$ is the rotational energy of the compound nucleus; βi represents the quadrupole deformations of the two fragments and is taken as the ground-state values; θi denotes the polar angles between the collision orientations and symmetry axes of the deformed nuclei. The collision direction is sampled using the Monte Carlo method with ${\theta }_i=\frac{\pi }{2}{\xi }_i$, where ξi is a random number. The angles θi (i=1, 2) of binary DNS fragments are stochastically sampled; the angles differ in the nucleon transfer process. The interaction potential between fragments (Z1,N1) and (Z2,N2) includes the nuclear, Coulomb, and centrifugal parts

where μ is the reduced mass of the two DNS fragments. The nuclear potential was calculated using the double-folding method based on the Skyrme interaction force without considering the momentum and spin dependence [51]. The Coulomb potential was obtained using Wong’s formula [52]. The distance R between the centers of two fragments was chosen as the minimal position of the interaction pocket. The minimal path in the valley of the PES is called the driving potential and is dependent on mass asymmetry. Fig. 1 shows the driving potentials for the tip-tip and waist-waist collisions; it also show the average value of random orientations in the reaction of ⁴⁸Ca + ²³⁸U. The tip-tip orientation indicates that the interaction potential is the minimaum when the polar angle θ corresponds to the inner fusion barrier of B_fus= 9.9 MeV; the waist-waist case is the maximal potential for the DNS fragments with B_fus= 4.8 MeV. In all orientations, the driving potential exhibits a symmetric structure. The driving potential with random collisions is close to the average values of the tip-tip and waist-waist collisions. The inner fusion barrier is related to the collision orientation; the waist-waist collisions undergo a high barrier to form a compound nucleus. However, the bump structure toward a decrease in mass asymmetry hinders the quasifission process.

Fig. 1Full-screen View

(Color online) The driving potentials in the reaction of ⁴⁸Ca+²³⁸U with the waist-waist, tip-tip, random collisions and fixed angles, respectively.

In order to form a compound nucleus (CN) overcoming the internal fusion barrier, the DNS must have sufficient local excitation. The formation probability of the compound nucleus at Coulomb barrier B and angular momentum J is given by the summation of the BG point

Here, the interaction time τ_int(E_c.m.,J,B) is obtained using the deflection function method [53]. The excitation energy E₁ of the DNS fragment (Z₁, N₁) is related to the collision orientation θ₁ and θ₂; N_t is the total event for the Monte Carlo integration with ${\theta }_i=\frac{\pi }{2}\xi$. The fusion probability is calculated using a Gaussian distribution f(B) as

The collision orientation influences the PES of the DNS because of the different interaction potential with the stochastic angle. Consequently, the formation probability of CN is related to the orientation of the two DNS fragments. Figure 2 shows the dependence of the fusion probability on excitation energy at different orientations in the nucleon transfer process in the reaction of ⁴⁸Ca+²³⁸U. The fusion probability increases with the excitation energy of the compound nucleus. The waist-waist case leads to a high fusion probability owing to the lower inner fusion barrier and a small peak towards the symmetric diffusion (quasifission path). The statistical error is included for random collisions, in which the fusion probability lies between fixed orientations. Usually, the tip-tip orientation is chosen in the calculation of the fusion probability, which is considered to be a probable nucleon transfer [32].

Fig. 2Full-screen View

(Color online) A comparison of the fusion probability a function of excitation energy in the reaction of ⁴⁸Ca+²³⁸U at different collision orientations.

Once the compound nucleus is formed by cascade nucleon transfer, de-excitation occurs by emitting γ-rays, particles (n, p, d, α, etc.), and binary fission. The survival probability of heavy nuclei after evaporating particles is crucial for assessing cross sections, which is usually calculated using a statistical process. The probability in the channel for evaporating the x-th neutron, y-th proton, and zth-α is expressed as [46]

Here, $E_{\text{CN}}^{*}$, J, and ${\Gamma }_{\text{tot}}$ are the excitation energy, spin of the compound nucleus, and sum of partial widths of particle evaporation, respectively. The excitation energy Es^* before evaporating the sth particles is given by:

with the initial value $E_1^{*}=E_{\text{CN}}^{*}$ and s=i+j+k. The nuclear temperature Ti is given by $E_i^{*}=a{T}_i^2-T_i$, where a is the level density parameter. The widths of the neutron evaporation and fission are calculated using the Weisskopf evaporation theory. The fission barrier is evaluated from the macroscopic liquid drop model and shell correction energy and is given as

where the macroscopic part $B_{\text{f}}^{\text{LD}}$ is calculated using the liquid drop model. The microcosmic shell correction energy is calculated using the Strutinsky method obtained from Ref. [54]. The damping energy E_D is associated with the level density and mass number of the compound nucleus; the shell correction and excitation energy dependence are considered in the calculation [46].

3　Results and discussion

The production rate of SHNs in massive fusion reactions is significantly low owing to the fusion hindrance in heavy systems, which enables the quasifission process in binary collisions. The evaporation residue (ER) excitation functions in different channels favor experimental measurements with optimal projectile-target combinations and suitable beam energies. The reaction dynamics in the competition of quasifission and fusion-fission reactions, level density, separation energy of evaporated particles, and fission barrier of the compound nucleus influence the ER cross sections. Figure 3 shows the ER excitation functions in the reaction of ⁴⁸Ca+²³⁸U; the results are compared with the data from Dubna [55]. The ER cross sections for producing SHN significantly depend on the orientations of both DNS fragments in the nucleon transfer process. The tip-tip collisions have lower cross sections that the waist-waist orientations and approximately lead to a two-order reduction because of the higher inner fusion barrier for merging the compound nucleus. The cross sections with the stochastic selection of the collision angle of two DNS fragments lie between the tip-tip and waist-waist orientation. The results with tip-tip collisions are consistent with the available data; this results were chosen in the calculation. The formation probabilities of compound nuclei in the fusion reactions are mainly determined by the inner fusion barrier and quasifission barrier (the height of the potential pocket), which correspond to 9.9 MeV (4.8 MeV) and 2.43 MeV (2.41 MeV) in the tip-tip (waist-waist) collisions. The lower inner fusion barrier and higher quasifission barrier are favorable for compound nucleus formation. The maximal yield of SHN ²⁸³Cn with 3 pb is positioned in the 3n channel via tip-tip collisions at an excitation energy of 35 MeV. The SHN was still far from the neutron shell closure (N=184). A new reaction mechanism is expected to create a neutron-rich SHN. Pure neutron channels are the dominant decay modes for surviving SHN. The cross sections with mixed channels of protons and α were reduced by two orders of magnitude with the 3 particle channels, for example, p2n and α2n. The charged particle channels and isospin diffusion are important for producing proton-rich actinide nuclides close to the drip line in the fusion-evaporation reactions and multinucleon transfer dynamics [56, 57].

Fig. 3Full-screen View

(Color online) Calculated ER cross sections with different channels of (2-5)n, 1p(2-4)n and 1α(1-3)n and compared with the experimental data in the reaction of ⁴⁸Ca+²³⁸U [55].

Superheavy elements from Fl (Z=114) to Og (Z=118) were successfully synthesized with ⁴⁸ Ca-induced reactions on actinide targets. This manifests a strong shell effect in the production and decay chains. Figure 4 shows the ER excitation functions for producing superheavy elements 114-118 with ²⁴⁴Pu, ²⁴⁸Cm, ²⁴⁹Bk, and ²⁴⁹Cf as the targets. Different evaporation channels are distinguished by colored lines and compared with experimental data [55, 58]. The 3n and 4n channels are the dominant decay modes for SHN production. The 2n channel is pronounced with an increase in the charge number of the SHN. However, the cross sections in the 4n and 5n channels decrease with increasing SHN. Unlike the cold fusion reactions, the maximal cross section in the ⁴⁸Ca induced reactions weakly depends on the mass of ER nucleus, e.g., the value of 8 pb for ²⁸⁸Fl and 0.6 pb for ²⁹⁴Og. The construction of the target material for the synthesis of new SHNs is a challenge. The hot fusion reactions also provide a possible way to create new elements in the eighth period with projectiles ⁴⁵Sc, ⁵⁰Ti, ⁵⁴Cr, ⁵⁸Fe, ⁶⁴Ni, etc. In addition to the reaction dynamics in the formation of SHN in hot fusion reactions, nuclear structure effects, such as shell effect, neutron separation energy, odd-even effect, microscopic state of compound nucleus (isomeric state), etc., are important in the evaluation of the production cross section.

Fig. 4Full-screen View

(Color online) ER excitation functions with pure neutron channels and compared with the experimental data in the reactions of ⁴⁸Ca+²⁴⁴Pu, ²⁴⁸Cm, ²⁴⁹Bk, ²⁴⁹Cf [55, 58].

Attempts to synthesize superheavy elements 119 and 120 were made at different laboratories worldwide, for example, FLNR, GSI, GANIL, etc. However, no decay chains were observed in these experiments. Theoretical predictions with various models were also made to produce the element 119 with a number of systems, for example, ²⁴⁹Cf (⁴⁵Sc,xn) ^294-x119, ²⁵¹Cf (⁴⁵Sc,xn) ^296-x119, ²⁴⁷Bk (⁵⁰Ti,xn) ^297-x119, ²⁴⁹Bk (⁵⁰Ti,xn) ^299-x119, ²⁵⁴Es(⁴⁸Ca,xn)^302-x119. Some results for the optimal ER cross sections in different reactions leading to the formation of Z=119 are listed in Table 1 . The optimal system of ²⁵²Es (⁴⁴Ca,3n) ²⁹³119 is possible with a larger cross section of 4 bps by the DNS model [41]. Further confirmation of the reliability of the calculation through different models is required. However, constructing the target material ²⁵²Es in experiments is significantly difficult. The reaction of ⁴⁵Sc+^249,251Cf is also feasible for synthesizing a new element with a cross section above 0.1 pb. Difference of one order magnitude for producing the element 119 in the reaction ⁵⁰Ti+²⁴⁹Bk exists in the model predictions, for example, 0.03 pb in the 4n channel at the excitation energy of 36 MeV by the FBD model [61], 0.57 pb at 41 MeV by the diffusion model with Langevin-type equations [62], 0.11 pb in Ref. [40], and 0.013 pb in our calculation by the DNS model. The 3n and 4n channels are optimal ways to create the new element, as shown in Fig. 5. The isotopic dependence of the ER cross sections is weak. A larger mass asymmetry in the bombarding system was favorable for producing SHN. The synthesis of element 119 in laboratories is possible in future experiments with high-intensity accelerators worldwide. Reliable predictions in theories are helpful for experimental management.

Fig. 5Full-screen View

The evaporation residue cross sections with channels of (2-5)n of element Z=119 in collisions by different reactions.

The synthesis of superheavy element 120 is particularly important for understanding the shell structure in the domain of SHNs. The strong shell effect enhances the fission barrier and α decay half-life, which is favorable for producing and surviving SHN. A possible experiment is planned at HIAF with high-intensity beams. Figure 6 shows a systematic comparison for producing SHN of Z=120 with actinide nuclide-based reactions; different panels represent different reaction systems to create the superheavy element 120 with different neutron richness. The compound nuclei formed by different combinations were close to the neutron shell closure (N=184). Channels 2, 3, and 4n were available for SHN production with excitation energies of approximately 25, 30, and 40 MeV, respectively. The maximal ER cross section depends on the isotopic projectiles. The 3n channel of the ^44,45Sc + ²⁵²Es reaction is favorable for synthesizing the new SHN owing to the large mass asymmetry in the entrance system. A smaller neutron separation energy is available for cooling the compound nuclei formed during the fusion reactions. In Table 2 , the production cross sections of Z=120 with the optimal channels and possible combinations are compared. Differences in model predictions exist in the calculations, for example, the production of ²⁹⁵120 in collisions of ⁵⁰Ti on ²⁴⁹Cf of 0.006 pb and 0.046 pb by the FBD model [61] and multidimensional Langevin-type equations [59], respectively. Calculations support the 3n channel in collisions of ⁵⁰Ti on californium isotopes is available for synthesizing the element 120 with the cross section above 0.1 pb at the excitation energy of approximately 35 MeV. The 2n channel in the reactions of ^49,51V+²⁴⁹Bk can also create a new element. The position of the maximal cross section is mainly determined by the odd-even effect of neutron evaporation and the energy dependence of the fusion probability. Note that the proton shell closure Z=120 was predicted with the relativistic mean-field model by including the isospin dependence of the spin-orbit interaction and the effective mass [65]. The shell correction energy calculated by the macroscopic-microscopic model is used in the calculation, and the Z=114 proton shell closure is given by the approach [54]. The production cross section of element 120 is enhanced with shell correction by the relativistic mean-field model.

Fig. 6Full-screen View

(Color online) The same as in Fig. 5; for the production of element Z=120

As an extension of the model prediction, we analyzed the formation of superheavy elements 121 and 122 in massive fusion reactions. Figure 7 shows the ER excitation functions in the reactions of ⁵⁴Cr+^247,249Bk and ^58,64Fe+²⁴³Am; in panels (a) and (b), the stable nuclide ⁵⁴Cr and isotopic target nuclei are selected. The (2–4)n channels in the ⁵⁴Cr+²⁴⁹Bk reaction are favorable with significantly low cross sections at the level of 1 fb; The influence of stable and radioactive nuclides on SHN production is compared in panels (c) and (d). The different structure of (2–5)n channels is caused by the neutron separation energy in the cascade evaporation, that is, the smaller separation energy for the compound nucleus ³⁰¹121, resulting in a larger 2n channel probability. The production cross section of element 122 is significantly low in the fusion reactions, as shown in Fig. 8. The neutron separation energy and fission barrier are sensitive to the survival of the SHN. The polar angle distribution, mass, total kinetic energy spectra, and excitation energy dependence of fission fragments from the SHNs are useful for extracting the fission barrier and shell evolution. Further experiments are required in the future. The systems of ⁵⁴Cr+ ^249,251Cf and ^58,64Fe+²⁴⁸Cm were chosen to synthesize element 122, as shown in Fig. 8. The cross section below 0.1 fb is out of the limit of the experimental measurement. The new reaction mechanism is expected for creating the neutron-rich SHN and new element, i.e., the multinucleon transfer reaction, incomplete fusion with radioactive nuclide etc.

Fig. 7Full-screen View

(Color online) The evaporation residue cross sections with channels of (2-5)n of element Z=121 in collisions by different combinations.

Fig. 8Full-screen View

(Color online) The same as in Fig. 5; for the production of element Z=122

4　Conclusion

Within the framework of the DNS model, SHN formation in the fusion-evaporation reactions was thoroughly investigated. Stochastic collision orientations in the nucleon transfer process were implemented into the model via the Monte Carlo approach. The calculated results are consistent with the experimental data from Dubna. The maximal cross sections of the evaporation residues appear in the (2–5)n evaporation channels. The yields in the 1pxn, 1α xn, and 1p1α xn evaporation channels are significantly lower than those of pure neutron evaporation. The reactions of ²⁴⁹Cf (⁴⁵Sc,xn)^294-x119, ²⁵¹Cf(⁴⁵Sc,xn)^296-x119, ²⁴⁷Bk(⁵⁰Ti,xn) ^297-x119, and ²⁴⁹Bk(⁵⁰Ti,xn)^299-x119 were investigated to synthesize the new element 119. We conclude that the maximum cross sections are close to 1 pb in the 3n evaporation channel for the systems and weakly depend on the isotopic target nucleus. The synthesis of the element Z=120 was investigated using a series of isotopic projectile nuclei bombarding actinide targets. The optimal combination is the reaction of ⁴⁴Sc+²⁵²Es in the 3n channel with a cross section of 3 pb. The production of superheavy elements 121 and 122 was obtained at a level below 1 fb in the massive fusion reactions. A new reaction mechanism still needs to be explored for the production of new elements. Therefore, the synthesis of new SHNs in experiments provides a good theoretical basis for selecting collision combinations.