Introduction

The cluster structure in an atomic nucleus is a spatially located subsystem consisting of strongly related nucleons with much greater internal binding energy than external nucleons, which can be treated as a whole without considering its internal structure [1]. In 1968, Ikeda proposed that nuclear cluster states tend to occur in excited states near cluster threshold energy [2]. In some weakly bound nuclei, the cluster structure is more obvious, and the cluster structure is also ubiquitous in light nuclei; for example, the configuration of ⁶Li is composed of a α particle and a deuteron, the 2α structure for ⁸Be, 3α for ¹²C, and 5 α for ²⁰Ne [3, 4]. The most convenient method to study the cluster structure inside a nucleus is to separate the cluster via pick-up or stripping reactions. A theoretical explanation of the pre-equilibrium reaction was initially developed using the exciton model. Semi-classical theories have been less successful in explaining the angular distribution of emitted particles. The Boltzmann master equation theory was mainly used to calculate the energy spectra of particles emitted during nucleon-induced reactions and heavy-ion reactions. Details can be found in the review paper [5]. However, the emission of preequilibrium clusters in transfer reactions around the Coulomb barrier is also an important physical problem. The emission of preequilibrium clusters is a complex process that is related not only to the cluster structure of the collision system but also to the dynamics of the reaction. In the treatment of nuclear structures, the cluster state is the overlap of the single-particle wave functions. In the nuclear reaction, the formation of a preequilibrium particle is different from the cluster emitted during the de-excitation of a composite nucleus, and a pre-equilibrium cluster is formed before the formation of the compound nucleus. Its emission continues until the formation of the composite nucleus, and the pre-equilibrium cluster may be emitted from any fragment during the reaction. Cluster emission provides important information for the study of single-particle states or multiparticle correlation of nuclei and is a powerful tool for nuclear spectroscopy [5]. Moreover, cluster emission in intermediate- and high-energy heavy-ion collisions is also important for investigating nuclear fragmentation and little-bang nucleosynthesis [6-8].

Since multinucleon transfer (MNT) reactions and deep inelastic heavy-ion collisions were proposed in the 1970s [9-11], a large number of experiments have been carried out to measure the double differential cross-sections, angular distributions, and energy distributions of different reaction systems. However, it is worth noting that there has been relatively little experimental and theoretical research on pre-equilibrium cluster emissions in transfer reactions. In the 1980s, scientists at RIKEN in Japan and IMP in China measured the pre-equilibrium cluster emission of the transfer reactions of ¹⁴N+¹⁵⁹Tb, ¹⁶⁹Tm, ¹⁸¹Ta, ¹⁹⁷Au, ²⁰⁹Bi [12] and ¹²C+²⁰⁹Bi [13, 14], respectively. The angular distributions, kinetic energy spectra, and total production cross-sections of the emitted particles were measured experimentally. As we all know, since the concept of superheavy stable island was proposed in the 1960s, the synthesis of superheavy nuclei has become an important frontier in the field of nuclear physics. In the past few decades, 15 types of superheavy elements Z=104~118 [15] have been artificially synthesized by hot fusion or cold fusion reactions. However, owing to the limitations of projectile-target materials and experimental conditions, it is difficult for the fusion evaporation reaction to reach the next period of the periodic table. With MNT reactions, many nuclei can be generated depending on the transfer channels, and with the development of separation and detection technology, the MNT reaction may be the most promising method for synthesizing unknown superheavy elements. This mechanism has been applied to the production of heavy and super-heavy isotopes [16, 17].

The study of pre-equilibrium cluster emission in the MNT reaction is not only of great significance for understanding the cluster structure of the collision system but also for exploring the formation mechanism of the cluster, the kinetic information of the reaction process, and the nuclear astrophysical process. The High Intensity Accelerator Facility (HIAF) built in Huizhou, China has a large energy range and a wide range of particle beams [18], which provides a good experimental platform for the study of nuclear cluster structures and cluster emission.

In this study, we systematically investigated the preequilibrium cluster emission in transfer reactions. The remainder of this paper is organized as follows. In Sect. 2, we provide a brief description of the DNS model for describing preequilibrium cluster production. In Sect. 3, the production cross sections, kinetic energy spectra, and angular distributions of the preequilibrium clusters are analyzed and discussed. A summary and perspective are presented in Sect. 4.

Brief description of the model

The dinuclear system (DNS) model was first proposed by Volkov [19] to describe deep inelastic heavy-ion collisions. For the first time, Adamian et al. applied the DNS model to fusion evaporation reactions in competition with the quasi-fission process to study the synthesis of superheavy nuclei [20-22]. The Lanzhou nuclear physics group has further developed the DNS model [23-26]; for example, they introduced the barrier distribution function method in the capture process, considered the effects of quasi-fission and fission in the fusion stage, used statistical evaporation theory, and the Bohr-Wheeler formula to calculate the survival probability of superheavy nuclei. The DNS model has been widely used to study the production cross-section, quasi-fission, fusion dynamics, etc., in the synthesis of superheavy nuclei based on fusion evaporation (FE) reactions and multinucleon transfer (MNT) reactions [27-30].

Using the DNS model, we calculated the temporal evolution, kinetic energy spectra, and angular distributions of the pre-equilibrium clusters in the transfer reactions with incident energy near the Coulomb barrier. Compared to our previous work [31], we introduced the transfer of clusters in the master equation of the DNS model, and the Coulomb force was considered in the pre-equilibrium cluster emission process. A pre-equilibrium particle is formed before the formation of the compound nucleus, and its emission continues until the composite nucleus is formed. The cross section of pre-equilibrium particle emission (ν= n, p, d, t, ³He, α, ^6,7Li and ^8,9Be) is defined as $\begin{array}{c}{\sigma }_{\nu }\left(E_k\mathrm{,}\theta \mathrm{,}t\right)={\displaystyle \sum _{J=0}^{J_{\text{max}}}}{\displaystyle \sum _{Z_1=Z_{\nu }}^{Z_{\text{max}}}}{\displaystyle \sum _{N_1=N_{\nu }}^{N_{\text{max}}}}\pi {\overline{\lambda }}^2\left(2J+1\right){\displaystyle \int }f(B)\\ \times T\left(E_{\text{c}\text{.m}\text{.}}\mathrm{,}J\mathrm{,}B\right)P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\left(E_{\text{c}\text{.m}\text{.}}\mathrm{,}J\right)\mathrm{,}t\mathrm{,}B\right)\\ \times P_{\nu }\left(Z_{\nu }\mathrm{,}{N}_{\nu }\mathrm{,}{E}_k\right)\text{d}B\mathrm{.}\end{array}$ (1) Here, Ek and θ are the kinetic energy and emission angle of the particles when they are ejected from projectile-like or target-like fragments, respectively, and t is the time of the reaction process. The reduced de Broglie wavelength $\overline{\lambda }=\hslash /\sqrt{2\mu E_{\text{c}\text{.m}\text{.}}}$, and $P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\left(E_{\text{c}\text{.m}\text{.}}\mathrm{,}J\right)\mathrm{,}t\mathrm{,}B\right)$ denotes the realization probability of the DNS fragment (Z₁,N₁). $P_{\nu }\left(Z_{\nu }\mathrm{,}{N}_{\nu }\mathrm{,}{E}_k\right)$ is the emission probability of the pre-equilibrium particles. E₁ is the excitation energy for fragment (Z₁,N₁), which is associated with the center-of-mass energy E_c.m. and the incident angular momentum J. The maximal angular momentum J_max is taken as the grazing collision of the two colliding nuclei. The DNS fragments (Z₁,N₁) range from light (Zν,Nν) to the composite system $\left(Z_{\text{max}}\mathrm{,}{N}_{\text{max}}\right)$, where $Z_{\text{max}}=Z_{\text{T}}+Z_{\text{P}}$ and $N_{\text{max}}=N_{\text{T}}+N_{\text{P}}$ are the total number of protons and neutrons, respectively.

2.2

The nucleon and cluster transfer dynamics

In the nucleon transfer process, the distribution probability of DNS fragments is obtained by numerically solving a set of master equations [34]. Fragment (Z₁,N₁) has the proton number of Z₁, the neutron number of N₁, the internal excitation energy of E₁, and the quadrupole deformation β₁, and the time evolution equation of its distribution probability can be described as $\begin{array}{l}\frac{\text{d}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}{\beta }_1\mathrm{,}B\mathrm{,}t\right)}{\text{d}t}\\ ={\displaystyle \sum _{Z_1^{\text{'}}=Z_1\pm 1}}{W}_{Z_1\mathrm{,}{N}_1\mathrm{,}{\beta }_1;Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}{\beta }_1^{\text{'}}}^{\text{p}}\left(t\right)\\ \times \left[d_{Z_1\mathrm{,}{N}_1}P\left(Z_1^{\text{'}}\mathrm{,}{N}_1\mathrm{,}{E}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}\mathrm{,}B\mathrm{,}t\right)-d_{Z_1^{\text{'}}\mathrm{,}{N}_1}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}{\beta }_1\mathrm{,}B\mathrm{,}t\right)\right]\\ +{\displaystyle \sum _{N_1^{\text{'}}=N_1\pm 1}}{W}_{Z_1\mathrm{,}{N}_1\mathrm{,}{\beta }_1;Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}}^{\text{n}}\left(t\right)\\ \times \left[d_{Z_1\mathrm{,}{N}_1}P\left(Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{E}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}\mathrm{,}B\mathrm{,}t\right)-d_{Z_1\mathrm{,}{N}_1^{\text{'}}}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}{\beta }_1\mathrm{,}B\mathrm{,}t\right)\right]\\ +{\displaystyle \sum _{Z_1^{\text{'}}=Z_1\pm \mathrm{1,}{N}_1^{\text{'}}=N_1\pm 1}}{W}_{Z_1\mathrm{,}{N}_1\mathrm{,}{\beta }_1;Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}}^{\text{d}}\left(t\right)\\ \times \left[d_{Z_1\mathrm{,}{N}_1}P\left(Z_1^{\text{'}}\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{E}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}\mathrm{,}B\mathrm{,}t\right)-d_{Z_1^{\text{'}}\mathrm{,}{N}_1^{\text{'}}}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}{\beta }_1\mathrm{,}B\mathrm{,}t\right)\right]\\ +{\displaystyle \sum _{Z_1^{\text{'}}=Z_1\pm \mathrm{1,}{N}_1^{\text{'}}=N_1\pm 2}}{W}_{Z_1\mathrm{,}{N}_1\mathrm{,}{\beta }_1;Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}}^{\text{t}}\left(t\right)\\ \times \left[d_{Z_1\mathrm{,}{N}_1}P\left(Z_1^{\text{'}}\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{E}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}\mathrm{,}B\mathrm{,}t\right)-d_{Z_1^{\text{'}}\mathrm{,}{N}_1^{\text{'}}}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}{\beta }_1\mathrm{,}B\mathrm{,}t\right)\right]\\ +{\displaystyle \sum _{Z_1^{\text{'}}=Z_1\pm \mathrm{2,}{N}_1^{\text{'}}=N_1\pm 1}}{W}_{Z_1\mathrm{,}{N}_1\mathrm{,}{\beta }_1;Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}}^{{}^3\text{He}}\left(t\right)\\ \times \left[d_{Z_1\mathrm{,}{N}_1}P\left(Z_1^{\text{'}}\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{E}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}\mathrm{,}B\mathrm{,}t\right)-d_{Z_1^{\text{'}}\mathrm{,}{N}_1^{\text{'}}}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}{\beta }_1\mathrm{,}B\mathrm{,}t\right)\right]\\ +{\displaystyle \sum _{Z_1^{\text{'}}=Z_1\pm \mathrm{2,}{N}_1^{\text{'}}=N_1\pm 2}}{W}_{Z_1\mathrm{,}{N}_1\mathrm{,}{\beta }_1;Z_1\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}}^{\alpha }\left(t\right)\\ \times \left[d_{Z_1\mathrm{,}{N}_1}P\left(Z_1^{\text{'}}\mathrm{,}{N}_1^{\text{'}}\mathrm{,}{E}_1^{\text{'}}\mathrm{,}{\beta }_1^{\text{'}}\mathrm{,}B\mathrm{,}t\right)-d_{Z_1^{\text{'}}\mathrm{,}{N}_1^{\text{'}}}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}{\beta }_1\mathrm{,}B\mathrm{,}t\right)\right]\mathrm{.}\end{array}$ (6) In this equation, $W_{Z_1\mathrm{,}{N}_1\mathrm{,}{\beta }_1;{Z^{\prime }}_1\mathrm{,}{N^{\prime }}_1\mathrm{,}{{\beta }^{\prime }}_1}$ is the mean transition probability from channels (Z₁,N₁,E₁,β₁) to $\left({Z^{\prime }}_1\mathrm{,}{N^{\prime }}_1\mathrm{,}{E^{\prime }}_1\mathrm{,}{{\beta }^{\prime }}_1\right)$. The quantities $d_{Z_1\mathrm{,}{N}_1}$ indicate the microscopic dimensions corresponding to the macroscopic state (Z₁,N₁,E₁,β₁). In this process, the transfer of nucleons or clusters satisfies the relationships ${Z^{\prime }}_1=Z_1\pm Z_{\nu }$ and ${N^{\prime }}_1=N_1\pm N_{\nu }$, which represent the transfer of a neutron, proton, deuteron, tritium, ³He, α. Note that we ignored the quasi-fission of DNS and fission of heavy fragments in the dissipation process. The initial probabilities of the projectile and target nuclei are set as $P\left(Z_{\text{proj}}\mathrm{,}{N}_{\text{proj}}\mathrm{,}{E}_1=\mathrm{0,}t=0\right)=P\left(Z_{\text{targ}}\mathrm{,}{N}_{\text{targ}}\mathrm{,}{E}_1=\mathrm{0,}t=0\right)=0.5$. The nucleon transfer process satisfies the unitary condition, ${\displaystyle {\sum }_{Z_1\mathrm{,}{N}_1}}P\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}t\right)=1$.

Similar to the cascade transfer of nucleons [25], the transfer of clusters is also described by the single-particle Hamiltonian, $H(t)=H_0\left(t\right)+V(t)\mathrm{.}$ (7) 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, annihilation and creation operators depend on the reaction time. The total single particle energy is $H_0\left(t\right)={\displaystyle \sum _K}{\displaystyle \sum _{{\nu }_K}}{\epsilon }_{{\nu }_K}\left(t\right){\alpha }_{{\nu }_K}^{+}\left(t\right){\alpha }_{{\nu }_K}\left(t\right)\mathrm{.}$ (8) The interaction potential is $\begin{array}{c}V(t)={\displaystyle \sum _{K\mathrm{,}{K}^{\text{'}}}}{\displaystyle \sum _{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}}{u}_{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}\left(t\right){\alpha }_{{\alpha }_K}^{+}\left(t\right){\alpha }_{{\beta }_{K^{\prime }}}\left(t\right)\\ ={\displaystyle \sum _{K\mathrm{,}{K}^{\prime }}}{V}_{K\mathrm{,}{K}^{\prime }}\left(t\right)\mathrm{.}\end{array}$ (9) The quantity ${\epsilon }_{\nu K}$ represents the single-particle energies, and $u_{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}$ are the interaction matrix elements parameterized in the following form: $\begin{array}{l}{u}_{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}=U_{K\mathrm{,}{K}^{\prime }}\left(t\right)\\ \times \left\{\exp \left[-\frac{1}{2}{\left(\frac{{\epsilon }_{{\alpha }_K}\left(t\right)-{\epsilon }_{{\beta }_K}\left(t\right)}{{\text{Δ}}_{K\mathrm{,}{K}^{\prime }}\left(t\right)}\right)}^2\right]-{\delta }_{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}\right\}\mathrm{.}\end{array}$ (10) Here, the calculations of $U_{K\mathrm{,}{K}^{\prime }}\left(t\right)$ and ${\delta }_{{\alpha }_K\mathrm{,}{\beta }_{K^{\prime }}}\left(t\right)$ are described in Ref. [35].

In the relaxation process of relative motion, the DNS is excited by the dissipation of the relative kinetic energy and angular momentum. The excited DNS opens a valence space in which valence nucleons have a symmetrical distribution around the Fermi surface. Only the particles in states within the valence space are actively excited and undergo transfer. The average of these quantities was calculated in the valence space as follows: $\text{Δ}{\epsilon }_K=\sqrt{\frac{4{\epsilon }_K^{*}}{g_K}}\mathrm{,}{\epsilon }_K^{*}={\epsilon }^{*}\frac{A_K}{A}\mathrm{,}{g}_K=A_K/12.$ (11) Here, ${\epsilon }^{*}$ is the local excitation energy of the DNS fragments, which provides the excitation energy for the mean transition probability. The number of valence states in the valence space is $N_K=g_K\text{Δ}{\epsilon }_K$, gK is the single-particle density around the Fermi surface. The number of valence nucleons was $m_K=N_{K/2}$. The microscopic dimension for the fragment ($Z_K\mathrm{,}{N}_K$) is evaluated by $d\left(m_1\mathrm{,}{m}_2\right)=\left(\begin{array}{c}{N}_1\\ m_1\end{array}\right)\left(\begin{array}{c}{N}_2\\ m_2\end{array}\right).$ (12) The mean transition probability is related with the local excitation energy and the transfer of nucleons or clusters, and it can be microscopically derived from the interaction potential in valence space as $\begin{array}{l}{W}_{Z_1\mathrm{,}{N}_1;{Z^{\prime }}_1\mathrm{,}{N^{\prime }}_1}^{\nu }=G_{\nu }\frac{{\tau }_{\text{mem}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1;{Z^{\prime }}_1\mathrm{,}{N^{\prime }}_1\mathrm{,}{E^{\prime }}_1\right)}{d_{Z_1\mathrm{,}{N}_1}{d}_{Z_1^{\text{'}}\mathrm{,}{N^{\prime }}_1}{\hslash }^2}\\ \times {\displaystyle \sum _{i{i}^{\prime }}}{\left|\langle {Z^{\prime }}_1\mathrm{,}{N^{\prime }}_1\mathrm{,}{E^{\prime }}_1\mathrm{,}{i}^{\text{'}}\left|V\right|Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\mathrm{,}\text{ and }i\rangle \right|}^2\mathrm{.}\end{array}$ (13) $G_{\nu }$ represents the spin-isospin statistical factors, and we use the winger density approach to identify the particle types [36, 37], that is, $G_{\nu }=\mathrm{1,1,3}/\mathrm{8,1}/\mathrm{12,1}/\mathrm{12,1}/96$ for neutrons, protons, deuterons, tritium, ³He, α, respectively. The cluster transition probability is related to the cluster formation probability with cluster structure, cluster potential, cluster binding energy, Mott effect, etc. Further improvements are needed in future work.

The memory time is connected with the internal excitation energy [38], ${\tau }_{\text{mem}}\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1;{Z^{\prime }}_1\mathrm{,}{N}_1\mathrm{,}{E^{\prime }}_1\right)={\left[\frac{2\pi {\hslash }^2}{{\displaystyle {\sum }_{K{K}^{\prime }}\langle V_{KK}{V}_{K{K}^{\prime }}^{*}\rangle }}\right]}^{1/2}\mathrm{,}$ (14) $\begin{array}{l}\langle V_{KK}{V}_{K{K}^{\prime }}^{*}\rangle =\frac{1}{4}{U}_{K{K}^{\prime }}^2{g}_K{g^{\prime }}_K{\text{Δ}}_{K{K}^{\prime }}\text{Δ}{\epsilon }_K\text{Δ}{{\epsilon }^{\prime }}_K\\ \times {\left[{\text{Δ}}_{K{K}^{\prime }}^2+\frac{1}{6}\left({\left(\text{Δ}{\epsilon }_K\right)}^2+{\left(\text{Δ}{{\epsilon }^{\prime }}_K\right)}^2\right)\right]}^{-1/2}\mathrm{.}\end{array}$ (15) The interaction matrix elements are calculated by $\begin{array}{c}{\displaystyle \sum _{i{i}^{\prime }}}{\left|V_{i{i}^{\prime }}\right|}^2={\omega }_{11}\left(i_1\mathrm{,}{i^{\prime }}_1\right)+{\omega }_{22}\left(i_1\mathrm{,}{i^{\prime }}_1\right)\\ +{\omega }_{12}\left(i_1\mathrm{,}{i^{\prime }}_1\right)+{\omega }_{21}\left(i_1\mathrm{,}{i^{\prime }}_1\right)\mathrm{,}\end{array}$ (16) in which ${\omega }_{K{K}^{\prime }}\left(i\mathrm{,}{i^{\prime }}_1\right)=d_{Z_1\mathrm{,}{N}_1}\langle V_{K{K}^{\prime }}\mathrm{,}{V}_{K{K}^{\prime }}^{*}\rangle \mathrm{,}$ (17) with the states $i\left(Z_1\mathrm{,}{N}_1\mathrm{,}{E}_1\right)$ and $i^{\prime }\left({Z^{\prime }}_1\mathrm{,}{N^{\prime }}_1\mathrm{,}{E^{\prime }}_1\right)$.

In the relaxation process of relative motion, the DNS is excited by the dissipation of the relative kinetic energy. The local excitation energy is determined by the dissipation energy from the relative motion and the potential energy surface of the DNS [26, 28], ${\epsilon }^{\ast }\left(t\right)=E_{\text{diss}}\left(t\right)-\left(U(\{\alpha \})-U\left(\left\{{\alpha }_{\text{EN}}\right\}\right)\right)\mathrm{,}$ (18) where ${\alpha }_{\text{EN}}=Z_{\text{P}}\mathrm{,}{N}_{\text{P}}\mathrm{,}{Z}_{\text{T}}\mathrm{,}{N}_{\text{T}}\mathrm{,}J\mathrm{,}R\mathrm{,}{\beta }_{\text{P}}\mathrm{,}{\beta }_{\text{T}}\mathrm{,}{\theta }_{\text{P}}\mathrm{,}{\theta }_{\text{T}}$ for the projectile-target system. The excitation energy of DNS fragment(Z₁,N₁) is $E_1={\epsilon }^{*}\left(t={\tau }_{\text{int}}\right)A_1/A$. ${\tau }_{\text{int}}$ denotes the interaction time, which is associated with the reaction system and relative angular momentum, and can be obtained using the deflection function [39]. The energy dissipated into the DNS is $E_{\text{diss}}\left(t\right)=E_{\text{c}\text{.m}\text{.}}-B-\frac{\langle J(t)\rangle (\langle J(t)\rangle +1){\hslash }^2}{2{\zeta }_{\text{rel}}}-\langle E_{\text{rad}}\left(J\mathrm{,}t\right)\rangle \mathrm{,}$ (19) and the radial energy is $\langle E_{\text{rad}}\left(J\mathrm{,}t\right)\rangle =E_{\text{rad}}\left(J\mathrm{,0}\right)\exp \left(-t/{\tau }_r\right)\mathrm{,}$ (20) where the relaxation time of the radial motion is τr = 5 × 10^-22 s, and the initial radial energy is $E_{\text{rad}}\left(J\mathrm{,0}\right)=E_{\text{c}\text{.m}\text{.}}-B-J_i\left(J_i+1\right){\hslash }^2/\left(2{\zeta }_{\text{rel}}\right)$. The dissipation of the relative angular momentum is described by $\langle J(t)\rangle =J_{\text{st}}+\left(J_i-J_{\text{st}}\right)\exp \left(-t/{\tau }_J\right)\mathrm{.}$ (21) The angular momentum at the sticking limit is $J_{\text{st}}=J_i{\zeta }_{\text{rel}}/{\zeta }_{\text{tot}}$ and the relaxation time ${\tau }_J=15\times {10}^{-22}$ s. ζ_rel and ζ_tot are the relative and total moments of inertia of DNS, respectively. The initial angular momentum is set to $J_i=J$ in Eq. (1). The relaxation times of the radial kinetic energy and angular momentum are associated with the friction coefficients in binary collisions. The values in this work were obtained from an empirical analysis of deeply inelastic heavy-ion collisions [39, 40].

The potential energy surface (PES) of the DNS is evaluated as $\begin{array}{c}U(\{\alpha \})=B\left(Z_1\mathrm{,}{N}_1\right)+B\left(Z_2\mathrm{,}{N}_2\right)\\ -B(Z\mathrm{,}N)+V(\{\alpha \})\mathrm{,}\end{array}$ (22) with the relationship of $Z_1+Z_1=Z$ and $N_1+N_1=N$ [41, 42]. The symbol {α} denotes the quantities $Z_1\mathrm{,}{N}_1\mathrm{,}{Z}_2\mathrm{,}{N}_2\mathrm{,}J\mathrm{,}R\mathrm{,}{\beta }_1\mathrm{,}{\beta }_2\mathrm{,}{\theta }_1\mathrm{,}{\theta }_2$. In the calculation, the distance R between the centers of the two fragments was chosen to be the value at the touching configuration, in which DNS is assumed to be formed, and $R=r_0\times \left(A_{\text{P}}^{1/3}+A_{\text{T}}^{1/3}\right)$ with $r_0=1.2\sim 1.3$ fm. $B\left(Z_i\mathrm{,}{N}_i\right)(i=\mathrm{1,2})$ and $B(Z\mathrm{,}N)$ are the negative binding energies of fragment (Zi,Ni) and compound nucleus (Z,N), respectively. βi represents the quadrupole deformations of the two fragments in the ground state, and ${\theta }_i\left(i=\mathrm{1,2}\right)$ denotes the angles between the collision orientations and symmetry axes of the deformed nuclei. The interaction potential between fragment (Z₁,N₁) and (Z₂,N₂) is derived from $V(\{\alpha \})=V_{\text{C}}\left(\left\{\alpha \right\}\right)+V_{\text{N}}\left(\left\{\alpha \right\}\right)+V_{\text{def}}\left(t\right)\mathrm{,}$ (23) where V_C is the Coulomb potential using the Wong formula [43], V_N is the nucleus-nucleus potential using the double folding potential [20], and $V_{\text{def}}\left(t\right)$ denotes the deformation energy of the DNS at reaction time t, $V_{\text{def}}\left(t\right)=\frac{1}{2}{C}_1{\left({\beta }_1-{{\beta }^{\prime }}_{\text{T}}\left(t\right)\right)}^2+\frac{1}{2}{C}_2{\left({\beta }_2-{{\beta }^{\prime }}_{\text{P}}\left(t\right)\right)}^2\mathrm{.}$ (24) The quantity $C_i\left(i=\mathrm{1,2}\right)$ denotes the stiffness parameters of the nuclear surface, which are calculated using the liquid-drop model [44]. The detailed calculations of $V_{\text{def}}\left(t\right)$ can be obtained from Ref. [45] and the references therein. Figure 1 shows the PES in the collisions of ¹⁴N+²⁰⁹Bi and ⁶⁴Ni+¹⁹⁸Pt. The zigzag lines are the driving potentials, which are estimated by the minimal PES values during the process of transferring nucleons. The incident point is denoted by the star symbol.

Fig. 1Full-screen View

(Color online) The potential energy surfaces in the reactions of (a) ¹⁴N+²⁰⁹Bi and (b) ⁶⁴Ni+¹⁹⁸Pt, respectively

Results and discussion

The pre-equilibrium cluster emission in the transfer reaction is very complicated and is not only related to the structure of the collision system, for example, the pre-formation factor, but also to the dynamic evolution of the reaction process, that is, the dissipation of relative motion and the coupling of internal degrees of freedom of the reaction system. The emission of pre-equilibriumrium cluster is a non-equilibrium process of time and space evolution, which is a powerful probe for deeply investigating the MNT reaction dynamics.

The temporal evolution of the emission probability of n, p, d, t, ³He and α from the transfer reactions of ¹⁴N + ¹⁵⁹Tb, ¹⁶⁹Tm, ¹⁸¹Ta and ¹⁹⁷Au at E_lab=115 MeV is shown in Fig. 2 and Fig. 3, respectively. The formation of the compound nucleus is of the order of a few hundred zeptoseconds, while the reaction time of the preequilibrium process is approximately several zeptoseconds. It can be seen from the figures that the emission of the pre-equilibrium cluster continues until the formation of the composite nucleus. At the beginning of the reaction, the emission probability of the preequilibrium cluster increased rapidly, reaching a maximum value at approximately 20×10^-22~40×10^-22 s, and then remained stable or decreased gradually. The emission probabilities of α and hydrogen isotopes are comparable, and the yields are approximately 3~4 orders of magnitude lower than that of neutrons but much larger than that of ³He. The local excitation energy of the DNS fragment increases with time, and the emitted clusters can remove part of the energy, which is conducive to the formation of a compound nucleus with a lower excitation energy. The total emission cross sections of the preequilibrium clusters can be obtained by counting the temporal evolution of the cluster yields. The total emission cross sections of the different preequilibrium particles are shown in Table 1. It is obvious that the α yields are comparable with the proton emission

Fig. 2Full-screen View

Temporal evolution of the preequilibrium cluster emission in the reactions of (a) ¹⁴N + ¹⁵⁹Tb and (b) ¹⁴N + ¹⁶⁹Tm at the beam energy of 115 MeV

Fig. 3Full-screen View

The same as in Fig. 2, but for the collisions of ¹⁴N on ¹⁸¹Ta and ¹⁹⁷Au, respectively

Figure 4 shows the kinetic energy spectra of the light nuclei produced in the transfer reactions ¹⁴N + ¹⁵⁹Tb, ¹⁶⁹Tm, ¹⁸¹Ta at E_lab =115 MeV. It can be seen from the figure that the kinetic energy spectra of the different reactions show a similar shape, presenting the Boltzmann distribution. The emission of neutrons was the most important factor. Compared with the previous work [31], we introduced Coulomb barrier correction in this study. Hydrogen isotopes have similar emission probabilities, owing to the same amount of charge, and the peak of their kinetic energy spectrum is approximately 10 MeV. Because α and ³He are more charged, the kinetic energy spectra move in the direction of greater energy (i.e., to the right in the picture). In addition, we can see from Fig. 4 that the emission probability of α is approximately three to five orders of magnitude higher than ³He, because the former has a lower separation energy and is more easily emitted from the DNS fragments. The calculation results above are consistent with experimental data [12, 14].

Fig. 4Full-screen View

(Color online) Kinetic energy spectra of the light nuclei produced in the reactions of (a) ¹⁴N+¹⁵⁹Tb, (b) ¹⁶⁹Tm and (c) ¹⁸¹Ta at E_lab=115 MeV, respectively

In Fig. 5, we show the kinetic energy spectra of the pre-equilibrium clusters (n, p, d, t, ³He, α, ^6,7Li, ^8,9Be) in the transfer reactions induced by ¹²C and ¹⁴N to the same target nucleus ²⁰⁹Bi. The kinetic energy spectra of these pre-equilibrium particles in the transfer reactions show the nuclear structure effect and the dynamic characteristics of the nuclear reaction. The available experimental data for the α emission from the HIRFL for the massive transfer reaction of ¹²C+²⁰⁹Bi [14] and from RIKEN for ¹⁴N+²⁰⁹Bi [12] are well reproduced with the DNS model. The excitation energy of DNS fragments, transition probability, binding energy, and separation energy of the transferred nucleons (clusters) affect the kinetic energy spectra. The emission cross-section of the preequilibrium cluster is mainly related to its formation probability and emission probability. In our calculation, it is assumed that the clusters already exist in the DNS; therefore, the emission cross-sections of different clusters are mainly determined by the emission probabilities. The higher the charge of the emitted particle, the higher the Coulomb barrier. However, the larger the separation energy of the cluster, the smaller the decay width and the lower the emission probability. The kinetic energy spectra of the clusters are strongly related to the Coulomb barriers and excitation energies of the composite system.

Fig. 5Full-screen View

Kinetic energy spectra of the preequilibrium clusters produced in collisions of (a) ¹²C+²⁰⁹Bi at E_lab=73 MeV and (b) ¹⁴N+²⁰⁹Bi at E_lab=115 MeV, respectively. The available experimental data (α particle) are shown for comparison from HIRFL for the reaction of ¹²C+²⁰⁹Bi [14] (left panel) and from RIKEN for ¹⁴N+²⁰⁹Bi [12](right panel), respectively

The emission of the pre-equilibrium cluster is not only related to the reaction system, but also to the incident energy. Figure 6 shows a comparison of the time evolution and kinetic energy distribution of the transfer reaction, ⁵⁸Ni+¹⁹⁸Pt, at incident energies of 220 and 240 MeV. The left part of this figure shows the temporal evolution, and the right part shows the kinetic energy spectra of the pre-equilibrium particles. The kinetic energy of the pre-equilibrium cluster is mainly determined by the local excitation energy of the projectile-like and target-like fragments, and a high local excitation energy is beneficial to cluster emission. We can see that the emission probability at E_c.m. = 240 MeV is about 2 to 3 orders of magnitude higher than at E_c.m. = 220 MeV, indicating that the emission probability of the pre-equilibrium clusters increased with the incident energy. In Fig. 7, we compare the transfer reactions of bombarding the target nucleus ¹⁹⁸Pt with heavier isotopes of Ni at E_c.m. = 240 MeV, on the left is the kinetic energy spectra of the pre-equilibrium clusters emitted in the ⁶⁴Ni+¹⁹⁸Pt reaction, and on the right is the reaction of ⁷²Ni+¹⁹⁸Pt. Compared with the reaction system of ⁶⁴Ni+¹⁹⁸Pt, the reaction induced by ⁷²Ni seems to be more likely to emit neutrons, but the former is more likely to emit protons. In both reaction systems, the peak of the kinetic energy spectra of the proton isotopes was approximately 9 MeV, while the peak of α kinetic energy spectra was approximately 17 MeV.

Fig. 6Full-screen View

(a) The temporal evolution and (b) kinetic energy spectra of the preequilibrium clusters produced in collisions of ⁵⁸Ni+¹⁹⁸Pt at E_c.m.=220 MeV and E_c.m.=240 MeV, respectively

Fig. 7Full-screen View

Kinetic energy spectra of the preequilibrium clusters produced in collisions of ^64,72Ni+¹⁹⁸Pt at E_c.m.=240 MeV

The particles emitted in the pre-equilibrium process and from the compound nucleus have different kinetic energies and angular distributions [51]. Direct particles primarily emit in the same direction as the incident particles and have similar energy to each other, while the particles of the compound process emit in all directions, in equal amounts forward and backward. Pre-equilibrium particles tend to be emitted forward and are generally more energetic than those from the composite nucleus. Shown in Fig. 8 is the angular distributions of the emitted pre-equilibrium clusters in ^58,64,72Ni+¹⁹⁸Pt at E_c.m.=240 MeV. The angular distribution was different for different reaction systems. For the same reaction system, the shapes of the angular distributions of different clusters are quite similar because different clusters may evaporate from the same excited DNS fragment. It can also be seen from the figure that the angular distributions of the pre-equilibrium particles are anisotropic, and their shapes show characteristics similar to of those the angular distributions of fragments in multinucleon transfer reactions [39, 50]. In the program calculation, we ignored the values for which the output was less than 1E-10. Under the three reaction systems, the angular distributions of the particles increased rapidly when the angle of the center of mass system was approximately 26 and reached a maximum value between 32 and 36. There is a window of 32–160 for the pre-equilibrium neutron emission. The study of the angular distribution of pre-equilibrium clusters in the transfer reaction is of great significance to the study of the angular distribution of the primary fragments in the MNT reaction and is helpful for the management of experimental measurements.

Fig. 8Full-screen View

Angular distributions of preequilibrium clusters in the reactions of ^58,64,72Ni+¹⁹⁸Pt at E_c.m.=240 MeV

Conclusion

In summary, within the framework of the DNS model, we investigated the emission mechanism of the preequilibrium clusters in the massive transfer reactions near the Coulomb barrier energies, that is, the temporal evolution, kinetic energy spectra, and angular distributions of n, p, d, t, ³He, α, ^6,7Li, ^8,9Be in collisions of ¹²C+²⁰⁹Bi, ¹⁴N+¹⁵⁹Tb, ¹⁶⁹Tm, ¹⁸¹Ta, ¹⁹⁷Au, ²⁰⁹Bi, and ^58,64,72Ni+¹⁹⁸Pt. Cluster transfer and dynamic deformation are coupled to the relative dissipation of the angular momentum and motion energy in the DNS model. The emission of preequilibrium clusters strongly depends on the incident energy, separation energy, and Coulomb barrier from the primordial DNS fragments. The yields of hydrogen isotopes and α production have similar magnitudes but are more probable than those of heavy particles. The kinetic energy spectra manifest the difference of charged particles, that is, more kinetic energy for the α emission than the ones of protons. The pre-equilibrium clusters follow the MNT fragment emission on the angular distributions and are related to the correlation of the nucleons. The reaction mechanism is helpful for investigating the cluster structure of atomic nuclei and MNT fragment formation, that is, the yields, shell effect, and emission dynamics, which are being planned for future experiments at HIAF in Huizhou.