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], and ²⁸⁶Mc [8] – in the last year has brought the total count to 3,327 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 3,039 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 8,000 to 10,000 unknown bound isotopes in the nuclei chart [10-12]. This suggests that at least over 5,000 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]; and ^223,229Am, 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 Accëlërateur National d’Ions Lourds (GANIL) in France, and Argonne National Laboratory (ANL) in the USA, actively working towards 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 of ^132,136Xe + 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 number A=208A=208 in proximity to the double magic nucleus ²⁰⁸Pb. These projectiles were directed to bombard targets ²³²Th and ²⁴⁸Cm 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 Sec. 2, we provide a concise overview of the DNS model. Sec. 3 presents the calculation results and subsequent discussion. Finally, a summary and conclusions drawn from our study are detailed in Sec. 4.

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: $\begin{array}{c}{\sigma }_{\text{tr}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_{\text{c}\text{.m}\text{.}}\right)={\displaystyle \sum _{J=0}^{J_{\text{max}}}{\sigma }_{\text{cap}}}\left(E_{\text{c}\mathrm{.}\text{m}\text{.}}\mathrm{,}J\right){\displaystyle \int }f\left(B\right)\\ \times P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{J}_1\mathrm{,}B\right)\times W_{\text{sur}}\left(E_1\mathrm{,}{J}_1\mathrm{,}s\right)\text{d}B\mathrm{.}\end{array}$ (1) ${\sigma }_{\text{cap}}\left(E_{\text{c}\text{.m}\text{.}}\mathrm{,}J\right)$ denotes the cross-section of the DNS formation derived by the Hill–Wheeler formula with a barrier distribution function[43]. $W_{\text{sur}}\left(E_1\mathrm{,}{J}_1\mathrm{,}s\right)$ denotes the survival probability of the fragment formation during the MNT process. Furthermore, s denotes the decay channels for fragments $\left(Z_1\mathrm{,}{N}_1\right)$ such as neutrons, protons, deuteron, alpha rays, and gamma rays. E_c.m. denotes the incident energy at the center of the mass frame. The highest angular momentum J_max was calculated for the colliding system in the grazing configuration. The angular momentum J was considered at the initial collision configuration before dissipation. E₁ and J₁ denote the excitation energy and angular momentum of the fragment with proton number Z₁ and neutron number N₁, respectively, in DNS, respectively. Additionally, $P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{J}_1\mathrm{,}B\right)$ is the formation probability of fragments $\left(Z_1\mathrm{,}{N}_1\right)$. For the barrier distribution function, we adopted an asymmetric Gaussian[44] form. $f\left(B\right)=\frac{1}{N}\exp \left[-{\left(\frac{B-B_{\text{m}}}{\Delta }\right)}^2\right]$ (2) The quantities $\Delta$ and B_m were evaluated using $\Delta$ =(B_t+B_s)/2 and B_m=(B_t+B_s)/2. Furthermore, B_t and B_s represent the height of the Coulomb barrier and minimum point of deformation under tip-tip collision, respectively. The normalization constant satisfies ${\displaystyle \int }f\left(B\right)\text{d}B=1$.

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, $P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\right)$, represents the proton number Z₁, neutron number N₁, and excitation energy E₁ for DNS fragment 1, which is described by the following master equation: $\begin{array}{l}\frac{\text{d}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}\beta \mathrm{,}t\right)}{\text{d}t}\\ ={\displaystyle \sum _{Z_1^{\text{'}}}{W}_{Z_1\mathrm{,}{N}_1\mathrm{,}\beta ;{Z^{\prime }}_1\mathrm{,}{N}_1\mathrm{,}{\beta }^{\text{'}}}}\left(t\right)[d_{Z_1\mathrm{,}{N}_1}P\left({Z^{\prime }}_1\mathrm{,}{N}_1\mathrm{,}{E^{\prime }}_1\mathrm{,}{\beta }^{\text{'}}\mathrm{,}t\right)\\ -d_{{Z^{\prime }}_1\mathrm{,}{N}_1}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}\beta \mathrm{,}t\right)]\\ +{\displaystyle \sum _{{N^{\prime }}_1}{W}_{Z_1\mathrm{,}{N}_1\mathrm{,}\beta ;Z_1\mathrm{,}{N^{\prime }}_1\mathrm{,}{\beta }^{\text{'}}}}\left(t\right)[d_{Z_1\mathrm{,}{N}_1}P\left(Z_1\mathrm{,}{N^{\prime }}_1\mathrm{,}{E^{\prime }}_1\mathrm{,}{\beta }^{\text{'}}\mathrm{,}t\right)\\ -d_{Z_1\mathrm{,}{N^{\prime }}_1}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}\beta \mathrm{,}t\right)]\\ -\left[{\Lambda }_{A_1\mathrm{,}{E}_1\mathrm{,}t}^{\text{qf}}\left(\Theta \right)+{\Lambda }_{A_1\mathrm{,}{E}_1\mathrm{,}t}^{\text{fis}}\left(\Theta \right)\right]P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}t\right)\end{array}$ (3) Specifically, $W_{Z_1\mathrm{,}{N}_1\mathrm{,}\beta ;Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}\beta }$ ($W_{Z_1\mathrm{,}{N}_1\mathrm{,}\beta \mathrm{,};Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}\beta }$) is the mean transition probability from channel($Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}\beta$) to ($Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}{E}_1^{\text{'}}\mathrm{,}\beta$), [or ($Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}\beta$) to ($Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{E}_1^{\text{'}}\mathrm{,}\beta$)], and $d_{Z_1\mathrm{,}{Z}_1}$ denotes the microscopic dimension corresponding to the macroscopic state ($Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1$). The sum is considered over all possible proton and neutron numbers that fragment ($Z_1^{\text{'}}$, $N_1^{\text{'}}$) may take; however, only one nucleon transfer is considered in the model, with the relations $Z_1^{\text{'}}=Z_1\pm 1$ and $N_1^{\text{'}}=N_1\pm 1$. The quasi-fission width Λ _qf and fission width ${\Lambda }^{\text{f}is}$ were calculated using Kramers’ formula [45, 46].

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]. $\begin{array}{c}{W}_{Z_1\mathrm{,}{N}_1\mathrm{,}\beta ;Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}{\beta }^{\text{'}}}=\frac{{\tau }_{\text{mem}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}\beta \mathrm{,}{E}_1;Z_1^{\text{'}}\mathrm{,}{N}_1{\beta }^{\text{'}}\mathrm{,}{E}_1^{\text{'}}\right)}{d_{Z_1\mathrm{,}{N}_1}{d}_{Z_1^{\text{'}}\mathrm{,}{N}_1}{\hslash }^2}\\ \times {\displaystyle \sum _{i{i}^{\prime }}{\left|\langle Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}{E}_1^{\text{'}}\mathrm{,}{i}^{\text{'}}\left|V\right|Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}i\rangle \right|}^2}\mathrm{.}\end{array}$ (4) The neutron transition coefficient has the same formula. The relaxation time is calculated using the deflection function method [49]. The memory time τ_mem and V interaction elements can be found in [47].

The motion of the nucleons in the interacting potential is governed by a single-particle Hamiltonian[41, 50]: $H\left(t\right)=H_0\left(t\right)+V\left(t\right)$ (5) with $H_0\left(t\right)={\displaystyle \sum _K{\displaystyle \sum _{{\nu }_K}{\epsilon }_{{\nu }_K}}}\left(t\right){\alpha }_{{\nu }_K}^{+}\left(t\right){\alpha }_{{\nu }_K}\left(t\right)$ (6) $\begin{array}{c}V\left(t\right)={\displaystyle \sum _{K\mathrm{,}{K}^{\text{'}}}{\displaystyle \sum _{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}{u}_{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}}}{\alpha }_{{\alpha }_K}^{+}\left(t\right){\alpha }_{{\beta }_K}\left(t\right)\\ ={\displaystyle \sum _{K\mathrm{,}{K}^{\prime }}{V}_{K\mathrm{,}{K}^{\prime }}}\left(t\right)\end{array}$ (7) Here, indices K and K’ (K,K’ = 1, 2) denote fragments 1 and 2, respectively. Furthermore, quantities ε_{ν K} and $u_{{\alpha }_K,{\beta }_{K^{\prime }}}$ 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: $\begin{array}{c}{u}_{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}=\\ U_{K\mathrm{,}{K}^{\prime }}\left(t\right)\left\{\exp \left[-\frac{1}{2}{\left(\frac{{\epsilon }_{{\alpha }_K}\left(t\right)-{\epsilon }_{{\beta }_K}\left(t\right)}{{\Delta }_{K\mathrm{,}{K}^{\prime }}\left(t\right)}\right)}^2\right]-{\delta }_{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}\right\}.\end{array}$ (8) Detailed calculations of these parameters and the mean transition probabilities are described in Ref. [41, 50]. $\Delta {\epsilon }_K=\sqrt{\frac{4{\epsilon }_K^{*}}{g_K}}\mathrm{,}{\epsilon }_K^{*}={\epsilon }^{*}\frac{A_K}{A}\mathrm{,}{g}_K=A_K/\mathrm{12,}$ (9) where ε^* denotes the local excitation energy of the DNS. The microscopic dimensions of fragment ($Z_K\mathrm{,}{N}_K$) were evaluated using the valence states NK = $g_K\Delta {\epsilon }_K$ and valence nucleons mK = N_K/2 (K=1,2) as: $d\left(m_1\mathrm{,}{m}_2\right)=\left(\begin{array}{c}{N}_1\\ m_1\end{array}\right)\left(\begin{array}{c}{N}_2\\ m_2\end{array}\right).$ (10) The local excitation energy E₁ was 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] ${\epsilon }^{\ast }\left(t\right)=E^{\text{diss}}\left(t\right)-\left(U\left(\left\{a\right\}\right)-U\left(\left\{{\alpha }_{\text{EN}}\right\}\right)\right)\mathrm{.}$ (11) The quantities of the entrance channel, denoted as α_EN, encompass proton and neutron numbers, quadrupole deformation parameters, and orientation angles. Specifically, they are represented as Z_P, N_P, Z_T, N_T, R, β_P, β_T, θ_P, and θ_T for the projectile and target nuclei by symbols P and T, respectively. The interaction time τ_int is 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: $\begin{array}{c}{U}_{\text{dr}}\left(t\right)=Q_{\text{gg}}+V_{\text{C}}\left(Z_1\mathrm{,}{N}_1;{\beta }_1\mathrm{,}{Z}_2\mathrm{,}{N}_2\mathrm{,}{\beta }_2\mathrm{,}t\right)\\ +V_{\text{N}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{\beta }_1;Z_2\mathrm{,}{N}_2\mathrm{,}{\beta }_2\mathrm{,}t\right)+V_{\text{def}}\left(t\right)\end{array}$ (12) with $V_{\text{def}}\left(t\right)=\frac{1}{2}{C}_1{\left({\beta }_1-{{\beta }^{\prime }}_T\left(t\right)\right)}^2+\frac{1}{2}{C}_2{\left({\beta }_2-{{\beta }^{\prime }}_{\text{P}}\left(t\right)\right)}^2$ (13) $C_i=\left(\lambda -1\right)\left(\lambda +2\right)R_{\text{N}}^2\delta -\frac{3}{2\pi }\frac{Z^2{e}^2}{R_{\text{N}}\left(2\lambda +1\right)}\mathrm{.}$ (14) where Q_gg, derived from the negative binding energies of fragments $(Z_{\text{i}}\mathrm{,}{N}_{\text{i}})$, was calculated using the liquid drop model plus shell correction[11]. θi denotes the angle between the collision orientations and symmetry axes of the deformed nuclei. Furthermore, V_C and V_N were 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, V_def(t) denotes the deformation energy of DNS at reaction time t. The evolutions of quadrupole deformations of the projectile- and target-like fragments occurs from the initial configuration as $\begin{array}{l}{{\beta }^{\prime }}_{\text{T}}\left(t\right)={\beta }_{\text{T}}\exp \left(-t/{\tau }_{\beta }\right)+{\beta }_1\left[1-\exp \left(-t/{\tau }_{\beta }\right)\right]\mathrm{,}\\ {{\beta }^{\prime }}_{\text{P}}\left(t\right)={\beta }_{\text{P}}\exp \left(-t/{\tau }_{\beta }\right)+{\beta }_2\left[1-\exp \left(-t/{\tau }_{\beta }\right)\right]\end{array}$ (15) where the deformation relaxation was ${\tau }_{\beta }=4\times {10}^{-21}s$.

The total kinetic energy (TKE-mass) of the primary fragment was evaluated using the following expression: $\text{TKE}=E_{\text{c}\text{.m}\text{.}}+Q_{\text{gg}}-E^{\text{diss}}\mathrm{,}$ (16) where $Q_{\text{gg}}=M_{\text{P}}+M_{\text{T}}-M_{\text{PLF}}-M_{\text{TLF}}$ and E_c.m. denote the incident energy in the center-of-mass frame. Masses M_P, M_T, M_PLF, and M_TLF correspond to the projectile, target, projectile-like fragment, and target-like fragment, respectively.

The survival probability $W_{\text{sur}}\left(E_1\mathrm{,}{J}_1\mathrm{,}s\right)$ 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 evaporating γ-rays and light particles (including neutrons, protons, and α particles) in competition with fission[44]. The probability in the channel for evaporating the x-th neutron, y-th proton, and z- alpha particle can be expressed as follows: $\begin{array}{l}{W}_{\text{sur}}\left(E_{\text{1}}^{*}\mathrm{,}x\mathrm{,}y\mathrm{,}z\mathrm{,}J\right)=P\left(E_1^{*}\mathrm{,}x\mathrm{,}y\mathrm{,}z\mathrm{,}J\right)\\ \times {\displaystyle \prod _{i=1}^x\frac{{\Gamma }_{\text{n}}\left(E_i^{*}\mathrm{,}J\right)}{{\Gamma }_{\text{tot}}\left(E_i^{*}\mathrm{,}J\right)}}{\displaystyle \prod _{j=1}^y\frac{{\Gamma }_{\text{p}}\left(E_j^{*}\mathrm{,}J\right)}{{\Gamma }_{\text{tot}}\left(E_j^{*}\mathrm{,}J\right)}}{\displaystyle \prod _{k=1}^z\frac{{\Gamma }_{\alpha }\left(E_k^{*}\mathrm{,}J\right)}{{\Gamma }_{\text{tot}}\left(E_k^{*}\mathrm{,}J\right)}}\mathrm{.}\end{array}$ (17) where $E_1^{*}$ and J denote the excitation energies evaluated from the mass tables in Refs. [11] and spin of the excited nucleus, respectively. The total width ${\Gamma }_{\text{t}ot}$ is the sum of the partial widths of the particle evaporation, γ emission, and fission. The excitation energy $E_s^{*}$ before evaporating the sth particle can be evaluated as follows: $E_{s+1}^{*}=E_s^{*}-B_i^{\text{n}}-B_j^{\text{p}}-B_k^{\alpha }-2T_s,$ (18) where the initial conditions are $E_1^{*}$ and s=i+j+k. Furthermore, $B_i^{\text{n}}$, $B_j^{\text{p}}$, and $B_k^{\alpha }$ are the separation energies of the ith neutron, jth proton, kth alpha, respectively. The nuclear temperature Ti is obtained by $E_{\text{i}}^{*}=a{T}_{\text{i}}^{\text{2}}-T_{\text{i}}$, where a denotes the level-density parameter. The fission and particle decay widths were calculated using the Weisskopf evaporation theory and Bohr–Wheeler formula, respectively. The realization probability $P\left(E_1^{*}\mathrm{,}x\mathrm{,}y\mathrm{,}z\mathrm{,}J\right)$ was calculated using Jackson’s formula[58].

Results and Discussion.

We calculated the production cross-sections of actinide isotope chains with atomic number Z = 93-100 in the collisions of ^132,136Xe + ²⁴⁸Cm at incident energy E_lab = 699-790 MeV, as shown in Fig. 1. Compared with the available experimental data for ^129,132,136Xe + ²⁴⁸Cm, which are represented by a solid red star, solid blue circle, and solid black squares with error bars, respectively, our calculation of ¹³⁶Xe + ²⁴⁸Cm, marked by solid black lines, and ¹³²Xe + 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 projectile ^129,132,136Xe isotopes induced reactions with the target ²⁴⁸Cm 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 projectile ¹³² Xe-induced reactions tend to shift to the proton-rich region when compared to the experimental results. Based on the data presented in Figs. 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 free parameters both for calculating the primary fragment cross-section and for estimating the survival probability against fission.

Fig. 1Full-screen View

(Color online) Calculation and experiment results of production cross-sections of actinide isotopic chains with Z = 93–100 in reactions of ^129,132,136Xe + ²⁴⁸Cm at E_lab = 699-790 MeV. The available experimental data are considered from [59, 60], marked by solid black square for ¹³⁶Xe induced reactions, solid red star for ¹³²Xe induced reactions, solid blue circle for ¹²⁹Xe induced 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 with A=208 bombarding targets ²⁴⁸Cm and ²³²Th at incident energy E_c.m.=1.1× V_B. 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. 2(a), interaction potential V_CN of ²⁰⁸Pt + ²⁴⁸Cm, ²⁰⁸Hg + ²⁴⁸Cm, ²⁰⁸Pb + ²⁴⁸Cm, ²⁰⁸Po + ²⁴⁸Cm, and ²⁰⁸Rn + ²⁴⁸Cm reactions were marked by solid black, dashed red, dash-dot blue, dash-dot-dot green, and short dashed olive lines, respectively. The tendencies of V_CN distributions for these collisions were similar. A larger Coulomb potential results in a larger interaction potential V_CN. Specifically, V_CN increases exponentially with decreasing distance R decreasing 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. 2(b), 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^-21 s, as shown in Fig. 2(c).

Fig. 2Full-screen View

(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 ²⁰⁸Hg, 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 E_c.m. = 1.1 × V_B, which decreases exponentially as angular momentum increases. In panel(c), for a given angular momentum $L=50\hslash$ 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 Eqs.(12). These are composed of the Coulomb potential, binding energy, and nuclear potential. They are calculated using the Wong formula, the liquid-drop model with shell correction, and the double folding method, respectively, as cited in [42]. The driving potential of projectiles ²⁰⁸Hg, ²⁰⁸Pb, and ²⁰⁸Po on targets ²⁴⁸Cm and ²³²Th during tip-tip collision at a fixed distance is plotted as a function of mass asymmetry, denoted as η. Here, $\eta =\left(A_{\text{T}}-A_{\text{P}}\right)/\left(A_{\text{T}}+A_{\text{P}}\right)$. This is illustrated in Fig. 3(a)(e), represented by solid black, dashed red, and dash-dot blue lines, respectively.

Fig. 3Full-screen View

(Color online) Potential energy surface and driver potentials of projectiles ²⁰⁸Hg, ²⁰⁸Pb, and ²⁰⁸Po induced reactions with targets ²⁴⁸Cm and ²³²Th 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) and (e), respectively. Potential energy surfaces for these collisions are shown in panels (b), (c), (d), (f), (g), (h), respectively. Open stars denote projectile–target injection points. These solid black lines represent valley trajectories on the two-dimensional potential energy surface.

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 at η = 0.2, 0 to derive the potentials of the target ²⁴⁸Cm-based reactions. One pocket in the driving potentials for the target ²³²Th-based reactions appears at η = 0.2. The neutron subshell number N=162 can potentially play a crucial role in pocket formation. For projectiles, such as ²⁰⁸Po, which are distant from the β- 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 Figs. 4, with driving potential trajectories represented as solid grey 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 projectiles ²⁰⁸Hg, ²⁰⁸Pb, and ²⁰⁸Po with targets ²⁴⁸Cm and ²³²Th at the incident energy E_c.m. = 1.1 × V_B exhibited 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. 4Full-screen View

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

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 projectiles ²⁰⁸Hg, ²⁰⁸Pb, and ²⁰⁸Po with target ²⁴⁸Cm at E_c.m. = 1.1 × V_B, 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 numbers Z = 104-116) exceeding 10 picobarns. The neutron subshell N=162 may be particularly influential, especially in the ²⁰⁸Po + ²⁴⁸Cm collision.

Fig. 5Full-screen View

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

Secondary production cross-sections for actinide target-like 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 projectiles ²⁰⁸Pt, ²⁰⁸Hg, ²⁰⁸Pb, ²⁰⁸Po, ²⁰⁸Rn, and ²⁰⁸Ra bombarding on targets ²⁴⁸Cm at E_c.m. = 1.1 × V_B. The line representations for these projectiles in Fig. 6 are as follows: solid black for ²⁰⁸Pt, dashed red for ²⁰⁸Hg, dash-dot green for ²⁰⁸Pb, dashed-dot-dot blue for ²⁰⁸Po, and short dashed olive for ²⁰⁸Rn. Observations indicate that collisions characterized by a smaller Coulomb force lean towards 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 reactions ²⁰⁸Pt+²⁴⁸Cm, ²⁰⁸Hg+²⁴⁸Cm, ²⁰⁸Pb+²⁴⁸Cm, ²⁰⁸Po+²⁴⁸Cm, ²⁰⁸Rn+²⁴⁸Cm, and ²⁰⁸Ra+²⁴⁸Cm. These predictions are detailed in Table 1. For the new neutron-rich 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. The ²⁰⁸Rn+²⁴⁸Cm reactions result in the largest cross-sections for new neutron-deficient actinide isotopes. The open circles represent the new neutron-rich actinide nuclides.

Fig. 6Full-screen View

(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 ²⁴⁸Cm at E_c.m. = 1.1 × V_B. 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.

Figure 7 shows secondary production cross-sections of all the formed fragments in collisions of ²⁰⁸Os+²⁴⁸Cm, ²⁰⁸Pt+²⁴⁸Cm, ²⁰⁸Hg+²⁴⁸Cm, ²⁰⁸Pb+²⁴⁸Cm, ²⁰⁸Po+²⁴⁸Cm, ²⁰⁸Rn+²⁴⁸Cm, ²⁰⁸Ra+²⁴⁸Cm, and primary production cross-sections of ²⁰⁸Pb+²⁴⁸Cm at the incident energy E_c.m. = 1.1 × V_B as N-Z panel. 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. 7Full-screen View

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

Conclusion

Using the DNS model framework, we systematically calculated the production cross-sections of MNT fragments in reactions involving projectiles such as ²⁰⁸Os, ²⁰⁸Pt, ²⁰⁸Hg, ²⁰⁸Pb,²⁰⁸Po, ²⁰⁸Rn, ²⁰⁸Ra, and ^132,136Xe colliding with targets ²³²Th and ²⁴⁸Cm around Coulomb barrier energies. To investigate the isospin diffusion on the formation of actinide products during the MNT process, the same number of projectiles with A=208 were selected. Our calculation for ^132,136Xe + 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 subshell N=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 the N/Z ratio 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 numbers Z=104-110.