Introduction

Between 1969 and 1995, approximately 75 new neutron-rich nuclides were discovered via multinucleon transfer (MNT) reactions [1]; however, no new transuranium or superheavy nuclides were detected using MNT reactions. The constraints of projectile fragmentation (PF) and fusion-evaporation (FE) reactions have rekindled interest in MNT for generating exotic nuclei, particularly those in the N=126 and superheavy regions [2-5].

Zagrebaev et al. first proposed that the reaction ¹³⁶Xe + ²⁰⁸Pb at energies near the Coulomb barrier was suitable for creating new neutron-rich isotopes near the neutron shell closure at N = 126 [6]. More than 50 unknown nuclei may be produced in such a reaction with cross-sections of not less than 1 μb. Building on this, Dubna conducted MNT reaction experiments in 2012 with ¹³⁶Xe + ²⁰⁸Pb [7], which resulted in the transfers of up to 16 nucleons from Xe to Pb. Experimental results from the ¹³⁶Xe + ¹⁹⁸Pt reaction obtained at Grand Accélérateur National d’Ions Lourds (GANIL) in 2015 revealed that the cross-sections for producing N = 126 nuclei in MNT reactions significantly exceed those from PF [8]. Most recently, Dubna reported the transfer of approximately 27 nucleons from the projectile to the target nucleus in the ¹³⁶Xe + ²³⁸U reaction [9]. MNT reactions can also be used to synthesize transuranium nuclei. In 2018, experiments conducted at Texas A&M University using the ²³⁸U + ²³²Th reaction compared the energies and half-lives of alpha emitters with known and predicted values, revealing the production of new elements with atomic numbers up to 116 [10]. A breakthrough in the production of new nuclides using the MNT process was the reaction ⁴⁸Ca + ²⁴⁸Cm conducted at GSI [11]. Five new neutron-deficient isotopes, including ²¹⁶U, ²¹⁹Np, ²²³Am, ²²⁹Am and ²³³Bk were discovered. In 2023, Niwase et al. synthesized the new neutron-rich isotope ²⁴¹U via the MNT reaction ²³⁸U + ¹⁹⁸Pt at the KEK Isotope Separation System (KISS) facility [12].

The MNT reactions involve quasi-elastic (QE), deep inelastic (DI), and quasi-fission (QF) processes. The ⁴⁰Ar + ²³²Th reaction is a representative experiment illustrating the dynamic mechanism of the MNT reactions[13]. Subsequently, Wilczyński effectively explained this experiment based on friction theory and scattering into negative angles [14].

The theoretical frameworks employed in investigating the MNT reactions include both phenomenological and microscopic models. Phenomenological approaches include the Langevin equations [15-22], GRAZING model [23-25], and dinuclear system (DNS) model [26-43]. Microscopic approaches include the improved quantum molecular dynamics model (ImQMD) [44-50] and time-dependent Hartree-Fock approach (TDHF) [51-55]. These theoretical models of heavy-ion collisions have guided the production of new exotic nuclei [56-64].

To comprehensively analyze heavy-ion collisions, Zagrebaev and Greiner applied the Langevin equations to MNT reactions, covering the entire process from the entrance stage to the formation of the fused system, including the DI, QF, and fission processes. They used the evolution of mass asymmetry to describe nucleon transfer [16, 17] and subsequently introduced charge asymmetry as a degree of freedom to account for the equilibration of the neutron-to-proton ratio [18, 19]. Subsequent studies further developed the Langevin model [65-68]. As a phenomenological framework, the current Langevin approach considers only microscopic effects by incorporating the shell correction energy into the driving potential, regardless of other phenomena. Therefore, to improve the Langevin approach for self-consistently simulating MNT processes including microscopic mechanisms, a Langevin equation model was developed by simplifying the current framework and was employed to simulate the dynamics of MNT reactions in the initial stage. Calculations of the Wilczyński plot were used to verify that the simplified model adequately described energy dissipation and angular distributions, thereby demonstrating that fluctuation-dissipation in the friction mechanism primarily affects energy dissipation during MNT reactions, particularly for the DI and QF processes.

The remainder of this article is organized as follows: Sect. 2 introduces the Langevin equations, while Sect. 3 presents the results and discussion. Finally, a summay of this work is presented in Sect. 4.

Theoretical model

In this study, we employed a two-center parameterization to describe the shape of the nuclear system [69], as shown in Fig. 1. We used three collective coordinates as follows: Z₀/R_cn, where z₀ is the distance between the centers of two oscillator potentials, made dimensionless by the radius of the spherical compound nucleus R_CN; δ, representing the deformation of the fragment; and ηA, the mass asymmetry of the colliding nuclei, defined as ${\eta }_A=\left(A_1-A_2\right)/\left(A_1+A_2\right)$. A₁ and A₂ represent the mass numbers of the target and projectile, respectively, and indicate the mass numbers of the left and right components of the system. We assumed that each fragment underwent identical deformation, denoted by $\delta ={\delta }_1={\delta }_2$, defined by ${\delta }_i=\left(3{\beta }_i-3\right)/\left(1+2{\beta }_i\right)$ with ${\beta }_i=a_i/b_i$. Here, a and b represent the half-lengths of the axes of the ellipse in the z₀-direction and orthogonal to z₀, respectively.

Fig. 1Full-screen View

The profile function ρ(z) of two-center parameterization

The collective coordinates $q_i=｛r\left(i\mathrm{.}e\mathrm{.},z_0/R_{cn}\right)\mathrm{,}{\eta }_{\text{A}}\mathrm{,}\delta ｝$ and their conjugate momenta pi are driven by the general form of the Langevin equations: $\begin{array}{l}\frac{\text{d}{q}_i}{\text{d}t}={\left(m^{-1}\right)}_{ij}{p}_j\mathrm{,}\\ \frac{\text{d}{p}_i}{\text{d}t}=-\frac{\partial V}{\partial q_i}-\frac{1}{2}{p}_j\frac{\partial {\left(m^{-1}\right)}_{jk}}{\partial q_i}{p}_k-{\gamma }_{ij}{\left(m^{-1}\right)}_{jk}{p}_k+g_{ij}{\Gamma }_j\left(t\right)\mathrm{,}\end{array}$ (1) where the shape-dependent transport coefficients mij and γij correspond to the inertia and friction tensors, respectively. ${\Gamma }_i\left(t\right)$ are normalized random variables with a Gaussian distribution, and gij are the random force amplitudes determined from the fluctuation dissipation theorem, $g_{ij}{g}_{jk}={\gamma }_{ik}T$. The nuclear temperature T is defined as $T=\sqrt{E^{*}/a}$, where a, the level density parameter is assigned a constant value of $A_{\text{cn}}/10{\text{MeV}}^{-1}$. The excitation energy $E^{*}$ is given by $E^{*}=E_{\text{tot}}-V-E_{\text{kin}}$, where the total, potential, and kinetic energies are represented by $E_{\text{tot}}\left(E_{\text{c}\mathrm{.}\text{m}\mathrm{.}}\right)$, V, and $E_{\text{kin}}=p_j{\left(m^{-1}\right)}_{jk}{p}_k/2$, respectively.

The inertia tensor components mij associated with the collective coordinates are calculated using the Werner-Wheeler approach for incompressible irrotational flow [70, 71]. Using the wall-plus-window approach of one-body dissipation, the friction tensor γij describes the dissipation of necked-in nuclear shapes featuring two fragments connected by a well-pronounced neck [71-74]. The potential energy V includes both the rotational energy V_rot and potential energy V_FRLDM obtained within the finite-range liquid-drop model (FRLDM) replacing the surface energy of the liquid-drop model with a Yukawa-plus-exponential potential nuclear energy [75-78]. The formulae and parameters in our model are consistent with Eqs. (29-34) in Ref. [78]. Consequently, the potential energy utilized is adiabatic and provides the nuclear mass for a uniform charge distribution (${\eta }_Z={\eta }_A$).

The evolution of the angle θ, which indicates the relative orientation of the nuclei, and the relative angular momentum $\mathcal{l}$ are governed by the following equations: $\begin{array}{l}\frac{\text{d}\theta }{\text{d}t}=\frac{\mathcal{l}}{{\Im }_{12}}\mathrm{,}\\ \frac{\text{d}\mathcal{l}}{\text{d}t}=-\frac{\partial V}{\partial \theta }-{\gamma }_{\tan }\frac{\mathcal{l}}{{\Im }_{12}}{R}_{\text{c}\mathrm{.}\text{m}\mathrm{.}}^2+R_{\text{c}\mathrm{.}\text{m}\mathrm{.}}\sqrt{{\gamma }_{\tan }T}{\Gamma }_{\tan }\left(t\right)\mathrm{.}\end{array}$ (2) Here, ${\Im }_{12}$ is the inertia moment of the relative motion; R_c.m. is the distance between the mass centers of the nuclei; and ${\gamma }_{\tan }$ is the tangential friction force of the colliding nuclei. ${\gamma }_{\tan }$ is assumed to vary with the radial friction ${\gamma }_{\tan }={\gamma }_{\tan }^0{\gamma }_{\text{rr}}$, where ${\gamma }_{\tan }^0$ is an adjustable parameter.

Two-center parameterization employs ϵ to describe the neck of a mononucleus system. To reduce the computational complexity, we assume that ϵ evolves over time instead of being included as a collective coordinate. The value of ϵ shifts from 1 in the entrance channel to 0.35 in the exit channel [79]. Therefore, the potential energy also undergoes a transition from entrance to exit as follows: $\begin{array}{l}V\left(q;\tau \right)=V\left(q\mathrm{,}ϵ=1\right)f_{ϵ}\left(\tau \right)+V\left(q\mathrm{,}ϵ=0.35\right)\left[1-f_{ϵ}\left(\tau \right)\right]\mathrm{,}\\ f\left(\tau \right)=\exp \left(-\tau /{\tau }_{ϵ}\right)\mathrm{,}\end{array}$ (3) where τ represents the relaxation time of the neck and ${\tau }_{ϵ}$ is the parameter whose suggested value is 10^-21 s [20].

Notably, if nucleon transfer is considered under the evolution of mass asymmetry ηA driven by Eq. (1), the friction γr and ${\gamma }_{{\eta }_A}$ are zero before the two separated nuclei come into contact and form mononucleus, indicating the absence of nucleon transfer. However, this does not correspond to reality, particularly when simulating large-impact parameters and grazing processes, where it diverges from experimental results. Consequently, to describe nucleon transfer before the contact point, this study employs the inertialess-reduced Langevin equation [16], which is derived from the corresponding master equation for the distribution function [80]. Because of the different equations used to calculate nucleon transfer at various stages, their transition requires further consideration.

Initially, Zagrebaev and Greiner utilized inertialess-reduced Langevin equations for mass and charge asymmetries to comprehensively handle the nucleon transfer process without considering the momentum and kinetic energy of mass asymmetry [16-18]. Karpov and Saiko further developed the model by solving the full Langevin equations, which also incorporated mass and charge fluctuations [20-22]. Referring to these studies, the approach employed in this study employs two different equations for the separated and mononucleus stages. In addition, our model is simplified from the aforementioned studies as follows: In selecting the collective coordinates, identical deformations were assumed for both nuclei; the intrinsic rotation of the nuclei was neglected; the potential energy was described using the FRLDM, without considering shell correction energies and the initial sudden approximation; and the use of phenomenological friction in the entrance channel was eliminated to reduce the number of model parameters.

Results and discussion

3.1

Wilczyński plot for ⁴⁰Ar + ²³²Th system

Trajectory calculations start at r = 5, which for the ⁴⁰Ar + ²³²Th system is equivalent to a distance of 40 fm between the centers of mass of the projectile and target nuclei. In addition, the excitation energies of the initial conditions were set to zero. At the scission point, the nuclei were separated, resulting in the formation of primary fragments, which marked the termination of the calculation. Five hundred events were simulated for each impact parameter.

Wilczyński proposed constructing a contour map of the double differential cross-section on the energy and scattering angle plane [14], known as the Wilczyński plot for studying the dynamics of nucleus-nucleus collisions. This allows the analysis of the relationship between the dissipation of total kinetic energy (TKE) and rotational degrees of freedom from the various quantities involved in the MNT reaction. We calculated the Wilczyński plot for ⁴⁰Ar + ²³²Th system shown in Fig. 2 at E_lab = 388 MeV (E_c.m. = 331 MeV).

Fig. 2Full-screen View

(Color online) Calculated Wilczyński plot for the ⁴⁰Ar + ²³²Th collision at E_lab = 388 MeV

Although the friction model has effectively conformed to the peaks observed in the Wilczyński plot [81], it fails to describe the statistical fluctuations. Peaks resembling those predicted by the friction model are obtained when the fluctuation component is turned off. By comparing this curve with the experimental data, we can adjust the model parameters ${\gamma }_{\text{tan}}\mathrm{,}{\tau }_{ϵ}$. In the simulation, the energy dissipation and angular evolution of the DI and QF processes were consistent with the experimental results. In addition, we observed trajectories with negative deflection angles. For instance, Fig. 3 shows one such trajectory for angular momentum $L=100\hslash$ with a deflection angle of ${\theta }_{\text{c}\mathrm{.}\text{m}\text{.}}=-27$.

Fig. 3Full-screen View

(Color online) Trajectory for the ⁴⁰Ar + ²³²Th collision at E_lab = 388 MeV: (a) Trajectory and its projections of the evolution of energy (V+E_kin), elongation, and mass of heavier nucleus; (b) Profile of the trajectory at the indicated time

Because ${\gamma }_{\text{tan}}^0$ is a model parameter that determines the rate of angular momentum dissipation, it can manifest in the relationship between the calculated TKE and scattering angle. Therefore, prior to simulation, ${\gamma }_{\text{tan}}^0$ must be adjusted based on the the peak of experimental Wilczyński plot. The value of ${\gamma }_{\text{tan}}^0$ is set to 0.05 in Fig. 2. However, experimental data for the Wilczyński plot is unavailable for all systems. For such systems, the adjustment of ${\gamma }_{\text{tan}}^0$ can be guided by the Sommerfeld-like parameter ${\eta }^{\prime }=\frac{Z_1{Z}_2\sqrt{\mu }}{\hslash \sqrt{2\left(E_{\text{c}\mathrm{.}\text{m}\text{.}}-V_{\text{B}}\right)}}$ [82], where μ and V_B are the reduced mass and Coulomb barrier, respectively. For systems with η’ less than 200, such as the ⁴⁰Ar + ²³²Th reaction with η’ = 120, the peak of the Wilczyński plot shifted towards negative angles as the total kinetic energy loss (TKEL) increased. However, when η’ increased to 400, the peak tended to stabilize in the direction of the constant scattering angle. The peak progressively extended towards larger scattering angles with further increase in η’.

3.2

Calculations for ¹³⁶Xe + ²³⁸U, ²⁰⁹Bi systems

For further comparison with the experimental data, we simulated the ¹³⁶Xe + ²³⁸U reaction at E_lab = 1110 MeV (E_c.m. = 706.4 MeV). The average TKE and total excitation energy $E_{\text{total}}^{*}$ for the fragments, calculated as a function of the primary fragment mass, are presented in Fig. 4, along with a comparison with experimental results [9].

Fig. 4Full-screen View

(Color online) Average TKE and $E_{\text{total}}^{*}$ calculated as a function of the primary fragment mass. The empty symbols correspond to the experimental results [9]

The calculated energy dissipation for large mass transfers is consistent with the experimental results, indicating that the simulation of the average TKE for fragment masses below 230 is reasonable. However, Langevin calculations for target-like fragments (TLFs) significantly underestimate the dissipated energy in the reaction, particularly in cases involving fewer nucleon transfers because of the simplified descriptions near the contact point. The results (Fig. 4) include only trajectories where the two nuclei come into contact. Trajectories involving grazing processes, in which the two nuclei did not directly touch under large collision parameters, were excluded.

We simulated the ¹³⁶Xe + ²⁰⁹Bi reaction for E_lab = 1130 MeV (E_c.m. = 684 MeV). In Fig. 5, the center-of-mass angular distribution $\text{d}\sigma /\text{d}\text{Ω}$ of the differential cross section is plotted versus θ_c.m. for events corresponding to TKE between 300 and 650 MeV. The angular distribution is consistent with the experimental values, except for the region near θ_gr+10°. The grazing angle θ_gr for this reaction is 54. This discrepancy can also be attributed to the description of the model before the contact point. In particular, for heavy systems, the continuous use of FRLDM as the potential energy is unsuitable for the approach stage. Therefore, the model requires further refinement to describe the separated stage. ${\gamma }_{\text{tan}}^0$ was set to 0.11 for the ¹³⁶Xe + ²³⁸U and ¹³⁶Xe + ²⁰⁹Bi reactions.

Fig. 5Full-screen View

(Color online) Center-of-mass angular distribution of the damped cross section for events with TKE within the indicated range in the ¹³⁶Xe + ²⁰⁹Bi reaction at E_lab = 1130 MeV. The experimental data are from [83]. Histogram is the calculation

The interaction time, τ_int, is a fundamental characteristic of nuclear reactions, which represents the time from when the two nuclei contact each other until they subsequently separate into fragments, as illustrated in Fig.6 for the studied reactions. In addition, the distributions of the total simulation time t_tot, starting from t = 0 at r = 5, are presented. These include trajectories with grazing collisions under large impact parameters. Longer interaction times within these distributions correspond to the most dissipative collisions, such as the QF process.

Fig. 6Full-screen View

(Color online) Reaction time distributions for the(a) ⁴⁰Ar + ²³²Th collision at E_lab = 388 MeV, (b) ¹³⁶Xe + ²³⁸U reaction at E_lab = 1.11 GeV, and (c) ¹³⁶Xe + ²⁰⁹Bi reaction at E_lab = 1130 MeV

Summary

In this study, a Langevin equation model was developed to simplify previous research by reducing the number of adjustable model parameters and streamlining the physical processes considered. This model was employed to investigate multinucleon transfer processes, enabling the analysis of double differential cross sections across energy and scattering angles. The classical ⁴⁰Ar + ²³²Th reaction was simulated at E_lab = 388 MeV, and the calculated Wilczyński plot was displayed. For comparisons with experimental results, the ¹³⁶Xe + ²³⁸U reaction was simulated at E_lab = 1110 MeV, alongside the ¹³⁶Xe + ²⁰⁹Bi reaction at E_lab = 1130 MeV, with subsequent calculations of TKE-mass distributions and angular distributions, respectively. Finally, the interaction time distributions of the reactions were calculated. These results indicate that the simplified Langevin equation model effectively describes the energy dissipation in MNT reactions.