1 Introduction

In recent years, the nuclear physics community has conducted extensive research on heavy-ion collisions near the Coulomb barrier. In particular, the production, identification, and structural characterization of neutron-rich exotic nuclei far from the stability line have garnered significant attention from both nuclear physicists and nuclear astrophysicists. These studies hold critical theoretical value for topics such as the synthesis of superheavy elements, the nuclear symmetry energy, and the rapid neutron-capture process ($r$-process). Deep inelastic collisions (DIC) can involve substantial nucleon transfer near the barrier energy. The primary products typically possess low excitation energies and can de-excite to the ground state by emitting only a few nucleons. Consequently, DIC serves as one of the most important pathways for producing neutron-rich exotic nuclei far from the stability line. Although nuclear physicists observed the phenomenon of deep inelastic collisions as early as the 1960s, it was not until the 1970s that the importance of this reaction mechanism was gradually recognized. Since then, various theoretical frameworks have been developed for its study, including the GRAZING model, multi-dimensional Langevin dynamical models, the Dinuclear System (DNS) model, and the Improved Quantum Molecular Dynamics (ImQMD) model.

The primary characteristics of deep inelastic collisions include the formation of a quasi-rigidly rotating dinuclear system by the projectile and target nuclei, where nucleon transfer is driven by the neutron-proton ($N/Z$) asymmetry. The relative kinetic energy of the projectile and target dissipates rapidly, and a large amount of angular momentum is transferred into the intrinsic spin of the reaction products, eventually leading the system to split into two fragments. Due to the intense dissipation of the total kinetic energy, which exhibits strongly damped characteristics, this process is also referred to as a strongly damped collision. Research indicates that projectile-like fragments (PLF) resulting from the transfer of a few nucleons typically originate from shorter interaction times and are mostly emitted near the grazing angle. In contrast, large-scale nucleon transfer often corresponds to longer contact times, during which the projectile may undergo significant deflection. Consequently, the maximum of the angular distribution for multinucleon transfer reaction products shifts toward smaller angles as the number of exchanged nucleons increases. This suggests that the probability of producing exotic products via deep inelastic collisions may increase as the emission angle approaches zero degrees. However, definitive experimental verification of this phenomenon is currently lacking.

Despite the widespread interest in measuring and identifying multinucleon transfer reaction products near zero degrees, experiments face the challenge of effectively separating low-energy beams. Furthermore, the broad momentum distribution of deep inelastic collision products makes it difficult for existing experimental setups to accurately measure cross-sections near zero degrees. Existing measurement data can be found in \cite{Ref_1, Ref_2, Ref_3, Ref_4, Ref_5, Ref_6, Ref_7, Ref_8, Ref_9, Ref_10, Ref_11}. The Super Separator Spectrometer ($S^3$) at GANIL is a high-precision spectrometer dedicated to the separation and analysis of atomic and molecular ions. This spectrometer is capable of producing and utilizing low-energy heavy-ion beams with energies of $5-15$ MeV/u and high intensities. It possesses the capability to filter exotic products of deep inelastic collisions in the zero-degree direction, which can be used to construct secondary beams of low-energy exotic nuclei. Therefore, predicting the reaction cross-sections of exotic products from deep inelastic collisions near zero degrees is of vital importance. To explore the characteristics of neutron-rich exotic products in the deep inelastic collision process, Stefan et al. conducted nuclear reaction experiments on the $^{136}\text{Xe} + ^{198}\text{Pt}$ system. The angular acceptance of the GANIL laboratory is approximately $\pm 4^\circ$.

Abstract

Using the Improved Quantum Molecular Dynamics (ImQMD) model coupled with the GEMINI++ statistical decay model, we investigate the deep inelastic collision (DIC) processes of the $^{40}\text{Ar} + ^{208}\text{Pb}$ system at an incident energy of $E_{\text{lab}} = 290$ MeV/u. By analyzing the total kinetic energy (TKE) and mass distributions of the reaction products, we clearly demonstrate that projectile-like fragments (PLFs) primarily originate from the deep inelastic collision mechanism.

In this study, we calculated the differential cross-sections, emission angles, and the contact time between the projectile and target for various PLF isotopes, such as Carbon ($^{13,14,15,16}\text{C}$). The results indicate that the differential cross-sections for neutron-rich PLFs peak as the emission angle approaches zero degrees. Further analysis reveals a correlation where a higher neutron-to-proton ratio ($N/Z$) in the PLFs corresponds to a longer contact time between the projectile and target, as well as a smaller emission angle.

The underlying dynamical mechanism can be described as follows: after the initial contact between the projectile and the target nucleus, a dinuclear system with a neck structure is formed. This system rotates around the center of mass for a duration exceeding $100$ fm/c, covering a rotation angle of approximately $180$ degrees. During this prolonged rotational contact, a significant amount of nucleon transfer occurs between the projectile and the target. Due to the positive $Q_{gg}$ values for the transfer of $1p+2n$, $1p+3n$, and $1p+4n$ from the target nucleus—specifically $Q_{gg}$ values of $5.805$ MeV—this is conducive to the occurrence of such reactions, as the differential cross-sections are relatively large \cite{21, 22, 23}. The findings of this study provide a critical reference for conducting low-energy nuclear physics experiments and extracting secondary beams of exotic nuclei using the next generation of zero-degree spectrometers.

Keywords: ImQMD model; Deep inelastic collisions; GEMINI++ model; Neutron-rich projectile-like fragments; Differential cross section.

2 Model Description

Over the past few decades, various transport models have achieved significant success in describing nuclear reactions, with the Quantum Molecular Dynamics (QMD) model being one of the most prominent. The Improved Quantum Molecular Dynamics (ImQMD) model incorporates several key enhancements: (1) the introduction of surface energy and surface symmetry energy terms into the potential energy density functional; (2) the implementation of system-size-dependent wave-packet widths; and (3) the use of a phase-space occupancy constraint method to improve the description of nuclear Fermi characteristics.

In the ImQMD model, the wave function of an individual nucleon is represented as a Gaussian wave packet:
$$\psi_i(\mathbf{r}) = \frac{1}{(2\pi\sigma_r^2)^{3/4}} \exp\left[-\frac{(\mathbf{r}-\mathbf{r}_i)^2}{4\sigma_r^2} + \frac{i}{\hbar} \mathbf{p}_i \cdot \mathbf{r}\right]$$

The wave packet centers in coordinate and momentum space are $\mathbf{r}_i$ and $\mathbf{p}_i$, respectively. We adopt a system-size-dependent wave packet width $L = 0.09A^{1/3} + 0.88$. The evolution is governed by Hamilton's equations:
$$\begin{aligned} \dot{\mathbf{r}}_i &= \frac{\partial H}{\partial \mathbf{p}_i} \ \dot{\mathbf{p}}_i &= -\frac{\partial H}{\partial \mathbf{r}_i} \end{aligned}$$

The Hamiltonian $H = T + U_{coul} + U_{loc}$ consists of kinetic energy, Coulomb energy, and local nuclear interaction potential energy. The local potential energy is given by:
$$V_{loc}(\mathbf{r}) = \int v_{eff}(\mathbf{r}, \mathbf{r'}) \rho(\mathbf{r'}) d\mathbf{r'}$$
The nuclear interaction potential includes two-body interactions, nonlinear density-dependent terms, surface energy, symmetry energy, and momentum-dependent interactions. The interaction potential parameters are listed in [TABLE:7]. For two-body collisions, the center-of-mass energy is:
$$\sqrt{s} = \sqrt{(E_1 + E_2)^2 - (\mathbf{p}_1 + \mathbf{p}_2)^2}$$
The center-of-mass velocity is $\boldsymbol{\beta} = (\mathbf{p}_1 + \mathbf{p}_2) / (E_1 + E_2)$. In the two-body center-of-mass system, the momentum of the first particle is:
$$\mathbf{p} = \gamma\left( \frac{\mathbf{p}_1 \cdot \boldsymbol{\beta}}{\beta} - \beta E_1 \right) \frac{\boldsymbol{\beta}}{\beta} + \left(\mathbf{p}_1 - \frac{\mathbf{p}_1 \cdot \boldsymbol{\beta}}{\beta} \frac{\boldsymbol{\beta}}{\beta}\right)$$

3 Results and Discussion

To accurately simulate the production of neutron-rich projectile fragments, we first use the ImQMD model to sample initial nuclei and evolve them for $t = 200$ fm/c to ensure stability. We simulated $10^5$ reaction events at $E_{\text{lab}} = 20$ MeV/u. The ImQMD model evolves the system for $t = 1000$ fm/c, followed by the GEMINI++ statistical decay model for de-excitation.

3.1 Optimal Impact Parameter and Reaction Mechanism

In low-energy heavy-ion reactions, fusion occurs at small impact parameters, while deep inelastic collisions dominate at larger ones. [FIGURE:1] shows the probability of two-fragment events ($A \ge 20$). Below $b = 5$ fm, fusion dominates. We selected $b_{min} = 5$ fm and $b_{max} = r_P + r_T + 0.5$ fm.

[FIGURE:1] also shows the fragment mass distribution. The primary products (red squares) show three peaks: projectile-like fragments ($A \approx 20$), target-like fragments ($A \approx 236$), and the compound nucleus ($A \approx 250$). After GEMINI++ de-excitation (black triangles), the PLF distribution remains stable, indicating low excitation energy. [FIGURE:2] and [FIGURE:3] show the total kinetic energy (TKE) distribution. The TKE is significantly lower than the incident energy, exhibiting the strong damping characteristic of deep inelastic collisions.

3.2 Differential Cross-sections of Projectile-like Fragments

We calculated differential cross-sections for isotopes like $^{16}\text{C}$. The total cross-section is:
$$\sigma = 2\pi \int_{b_{\min}}^{b_{\max}} b P(b) db$$
[FIGURE:N] compares ImQMD+GEMINI++ (red line) with NNCLE (black line) and DIT+GEMINI++ (blue dashed line) against experimental data (red points). The ImQMD model shows that neutron-rich PLF cross-sections peak near zero degrees and decrease toward the grazing angle. For high $N/Z$ products like $^{16}\text{C}$, ImQMD provides high agreement with experimental data without the "second peak" at the grazing angle seen in other models.

[FIGURE:N] illustrates the evolution at $b = 10$ fm. At $t = 160$ fm/c, a dinuclear system forms and rotates for approximately $480$ fm/c through $110$ degrees before fragmenting. During this time, significant nucleon transfer occurs. Positive $Q_{gg}$ values for $1p+2n$ ($5.805$ MeV), $1p+3n$, and $1p+4n$ channels facilitate the production of neutron-rich fragments.

3.3 Contact Time and Emission Angle

[FIGURE:1] shows a negative correlation between the $N/Z$ ratio of PLFs and their emission angles. As the $N/Z$ ratio becomes more extreme, the average emission angle decreases. [FIGURE:2] shows that PLFs with extreme $N/Z$ ratios are associated with longer contact times (e.g., $10^{-21}$ s for $^{24}\text{O}$). Prolonged contact enhances rotation, leading to emission at smaller angles. This "zero-degree advantage" provides a theoretical basis for extracting exotic nuclei. Predicted cross-sections for $^{18,19,20}\text{N}$ and $^{24,25,26}\text{N}$ near zero degrees are presented in [TABLE:1].

4 Conclusion

Using the ImQMD model coupled with GEMINI++, we investigated DIC processes in the $^{64}\text{Ni} + ^{208}\text{Pb}$ system at $E_{\text{lab}} = 5.9$ MeV/u. The TKE distribution confirms that neutron-rich PLFs originate from the deep inelastic mechanism with maximum cross-sections at zero degrees. The calculated cross-sections for $^{65,66,67,68}\text{Ni}$ agree well with experimental data. We found that extreme $N/Z$ ratios correspond to longer contact times and smaller emission angles. This research supports the strategy of using zero-degree spectrometers to extract secondary exotic nuclear beams.