1. Introduction
The synthesis of unknown nuclides to expand the chart of nuclides is a prominent topic in nuclear physics research. The unknown nuclides primarily reside on the neutron-rich side of the chart. In recent years, there has been significant interest in the multinucleon transfer (MNT) reaction near the Coulomb potential barrier in heavy-ion reactions. Theoretical predictions and experimental results of the MNT reaction have shown that the MNT reaction is an effective way to produce neutron-rich nuclei in the heavy mass region [1–6]. Notably, Niwase et al synthesized the new neutron-rich nuclide 241U using the MNT reaction of 238U+198Pt at RIKEN in Japan in 2023, which bolstered confidence among researchers regarding the utilization of MNT reactions for producing neutron-rich nuclides within the heavy mass region [6]. Upgrades and constructions of big science facilities are underway worldwide to facilitate research on synthesizing neutron-rich nuclei in the medium and heavy mass range via the MNT reaction. In China, the High Intensity Heavy-ion Accelerator Facility (HIAF) is anticipated to commence operations by 2025, promising significant advancements in the synthesis and identification of new nuclides and superheavy elements, as well as their subsequent structural and property investigations [7]. However, the reaction products of MNT exhibit a wide distribution in mass, angle and kinetic energy, which presents challenges in collecting and separating desired reaction products. Therefore, there is an urgent need for theoretical models to carry out more mechanistic studies on the MNT reaction to provide crucial references and guidance for future experiments. The theoretical methods used to study the MNT reaction include the dinuclear system (DNS) model [8–15], the Langevin equation [16–18], the GRAZING model [19–21], the time-dependent Hartree–Fock (TDHF) model [22, 23], the improved quantum molecular dynamics (ImQMD) model [24–33], etc. For more information on the MNT reaction, see the review papers [34–39] and their references.
Most nuclei are deformed, which leads to diverse orientations between projectile and target during nuclear reactions. The orientation of these nuclei plays a crucial role in the heavy ion collision process, and it directly affects the reaction dynamics and the properties of the resulting fragments, such as cross sections, exit angle and total kinetic energy. In recent years, the orientation effect in MNT reactions has been intensively explored. In [22, 40–44], the authors have thoroughly investigated the initial nuclear orientation effect in MNT reactions using the TDHF model. The orientation effect on physical quantities such as composite system lifetime, total kinetic energy loss and interaction time are systematically analyzed in these studies. The energy dependence of the orientation effect has been explored based on the ImQMD model and the Langevin equation in [28] and [45], respectively. They found that the impact of the orientation effect gradually diminishes with increasing energy. In [18], Amano et al investigated the effect of nuclear orientations on the mass drift in heavy ion collisions using the Langevin equation and revealed that the nuclear orientation effects play a crucial role in the strong correlation between fragment mass and its emitting angle. Hind and Nishio et al analyzed the experimental data of fission fragments to study the influence of the orientation effect of the 238U target on the products, and the results showed that the collision of the target nucleus with tip orientation mainly produced quasifission fragments [46, 47]. In [48], a study based on the DNS model was conducted to explore the contribution of different components of driving potential in various orientations in the 86Kr+166Er reaction. The results show that the Coulomb potential is the main factor causing significant differences in the driving potential in different orientations. The effect of dynamic deformation on the potential energy surface and product distribution in MNT reactions has been investigated based on the DNS model in [8]. The results show that side–side collisions favor the production of neutron-rich nuclei in the 136Xe+208Pb and 136Xe+198Pt systems.
In the MNT reaction, a significant number of nucleons are transferred between the projectile and target within reaction time on the order of zs, and the orientation of the deformed nuclei significantly alters the reaction kinetics. Therefore, it is essential to use a dynamic model further to investigate the effect of orientation on the MNT reaction. The ImQMD model is an ideal tool for such studies. The model was developed by Ning Wang et al based on the QMD model and has been successfully applied in studies related to fusion reactions and MNT reactions in the low-energy region, as well as fragmentation reactions in the intermediate-energy region. More detailed information and applications of the ImQMD model can be found in the review [33]. 238U is a typically deformed nucleus, and the 238U+238U reaction is a more suitable reaction system to study the orientation effect of MNT reactions. Numerous studies have extensively investigated this reaction using the ImQMD model. In [26], Tian et al thoroughly discuss the formation and properties of the composite system, highlighting that its lifetime, shape, momentum distribution, and other characteristics strongly depend on the incident energy. In [27], the charge, mass, and excitation energy distributions of the products at 7.0 MeV/nucleon energy are analyzed by combining the ImQMD model with the statistical decay model HIVAP by Zhao et al, and the theoretical calculations are consistent with the experimental data on mass distribution. In [28], Zhao et al focus on investigating the influence of initial deformation and orientation effect on the composite system lifetime and the production probability of superheavy nuclides, with particular attention to the energy dependence of composite system lifetimes. Zhao et al further explore mechanisms behind transuranic nuclide production through this reaction at 7.0 MeV/nucleon energy; when combined with HIVAP calculation results, it successfully reproduces isotopic distribution experimental data [29]. Lastly, the production cross sections for primary and final fragments with charges Z = 70–120 are predicted in [30]. It showed that the reaction could produce about 60 unknown neutron-rich nuclides, and the outgoing angles of these nuclides are generally located in the range of θlab ≤ 60°.
In this paper, the ImQMD model will be employed to study the orientation effect of the initial nuclei in the 238U+238U reaction. Unlike [28], this paper specifically focuses on the influence of the orientation effect on the nuclei deformation at the contact moment, the composite system lifetime, the transfer rate, and the fragments production mechanism. The structure of this paper is organized as follows: section 2 introduces the ImQMD model and tests the stability of the deformed 238U; the study of the orientation effect will be carried out in section 3 ; finally, the conclusion is given in section 4 .
2. ImQMD model
2.1. Model introduction
The ImQMD model is an improved version of QMD, the nucleus–nucleus interaction potential in the model is adopted as the Skyrme interaction potential and the fermionic properties of the nuclei are improved using the phase space occupancy constraint method. Similar to the QMD model, each nucleon in the ImQMD model is represented by a Gaussian wave packet of coherent states. During the dynamic evolution process, the nucleon moves in a mean field produced by other nucleons and its evolution is based on Hamiltonian equations:
$\begin{eqnarray}\begin{array}{rcl}{\dot{{\boldsymbol{r}}}}_{i} & = & \displaystyle \frac{{\rm{\partial }}H}{{\rm{\partial }}{{\boldsymbol{p}}}_{i}},\\ {\dot{{\boldsymbol{p}}}}_{i} & = & -\displaystyle \frac{{\rm{\partial }}H}{{\rm{\partial }}{{\boldsymbol{r}}}_{i}},\end{array}\end{eqnarray}$
where ri and pi are the centers of the wavepackets of the nucleons in coordinate and momentum space. The Hamiltonian H is expressed as: $\begin{eqnarray}H=T+{U}_{\mathrm{Coul}\ }+{U}_{\mathrm{loc}},\end{eqnarray}$
where T = ${\sum }_{i}\tfrac{{{\boldsymbol{p}}}_{i}^{2}}{2m}$ denotes the kinetic energy. The Coulomb interaction potential energy is calculated by the following equation: $\begin{eqnarray}\begin{array}{rcl}{U}_{\mathrm{Coul}\ } & = & \displaystyle \frac{1}{2}\displaystyle \int {\rho }_{p}({\boldsymbol{r}})\displaystyle \frac{{e}^{2}}{\left|{\boldsymbol{r}}-{{\boldsymbol{r}}}^{{\prime} }\right|}{\rho }_{p}\left({{\boldsymbol{r}}}^{{\prime} }\right){\rm{d}}{\boldsymbol{r}}{\rm{d}}{{\boldsymbol{r}}}^{{\prime} }\\ & & -{e}^{2}\displaystyle \frac{3}{4}{\left(\displaystyle \frac{3}{\pi }\right)}^{1/3}\displaystyle \int {\rho }_{p}^{4/3}{\rm{d}}{\bf{r}},\end{array}\end{eqnarray}$
where the density of protons ρp(r) is expressed as: $\begin{eqnarray}{\rho }_{{\rm{p}}}({\boldsymbol{r}})=\displaystyle \sum _{i}\displaystyle \frac{1}{{\left(2\pi {\sigma }_{r}^{2}\right)}^{3/2}}\exp \left[-\displaystyle \frac{{\left({\boldsymbol{r}}-{{\boldsymbol{r}}}_{i}\right)}^{2}}{2{\sigma }_{r}^{2}}\right],\end{eqnarray}$
where σr is the wave packet width of the proton in coordinate space. The nuclear local interaction potential energy is obtained through integration over the energy density of Skyrme interaction potential: $\begin{eqnarray}{U}_{\mathrm{loc}}=\int {{ \mathcal H }}_{\mathrm{loc}}({\boldsymbol{r}}){\rm{d}}{\boldsymbol{r}},\end{eqnarray}$
where the Skyrme interaction potential energy density is given as: $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal H }}_{\mathrm{loc}}({\boldsymbol{r}}) & = & \displaystyle \frac{\alpha }{2}\displaystyle \frac{{\rho }^{2}}{{\rho }_{0}}+\displaystyle \frac{\beta }{\gamma +1}\displaystyle \frac{{\rho }^{\gamma +1}}{{\rho }_{0}^{\gamma }}+\displaystyle \frac{{g}_{0}}{2}{\left({\rm{\nabla }}\rho \right)}^{2}\\ & & +{g}_{\tau }\displaystyle \frac{{\rho }^{\eta +1}}{{\rho }_{0}^{\eta }}+\displaystyle \frac{{C}_{S}}{2{\rho }_{0}}\left[{\rho }^{2}-{\kappa }_{S}{\left({\rm{\nabla }}\rho \right)}^{2}\right]{\delta }^{2}.\end{array}\end{eqnarray}$
Here ρ0 is the saturation density, $\delta =\left({\rho }_{{\rm{n}}}-{\rho }_{{\rm{p}}}\right)/\left({\rho }_{{\rm{n}}}+{\rho }_{{\rm{p}}}\right)$ is the isospin asymmetry, and ρ = ρn + ρp is the nucleon density. The neutron density ρn takes the same form as the proton density ρp. The parameters are taken as IQ2 parameters, as shown in table 1. The IQ2 parameters have been successfully applied to describe fusion reactions [49] and MNT reactions [32, 50] in heavy ion collisions.Table 1. IQ2 model parameters. |
| α | β | γ | g0 | gτ | η | Cs | κs | ρ0 |
|---|---|---|---|---|---|---|---|---|
| (MeV) | (MeV) | (MeV fm2) | (MeV) | (MeV) | (fm2) | (fm3) | ||
| −365 | 303 | 7/6 | 7.0 | 12.5 | 2/3 | 32.0 | 0.08 | 0.165 |
In this work, the z-axis and x-axis are set as the beam direction and the impact parameter direction respectively. The initial center-of-mass distance between the projectile and the target is 30 fm. The collision parameter ranges from 1 to bmax = RP+RT fm, where RP and RT denote the radii of the projectile and target, respectively. This work does not include reactions with collision parameters exceeding bmax because most of them are elastic scattering. This paper aims to analyze the influence of the orientation effect on the composite system and the properties of the outgoing fragments in the 238U+238U reaction, the collision process for each collision parameter is simulated 1000 times, which is appropriate for the research work in this paper. For each event, the collision process is simulated up to t = 2000 fm/c with a step size of t = 1 fm/c.
2.2. Stability test for initial nuclei
The nuclei can have arbitrary orientations when they collide, as shown in figure 1(a), where two long ellipsoidal colliding nuclei are used as an example. A unique distribution exists for the orientation configuration of the projectile and target, which should be considered in the calculations to describe various nuclear reactions adequately. However, this approach remains intricate, and the actual calculations consider only a few unique orientation configurations, such as tip–tip, side–side, tip–side, and side–tip collisions. To visually depict the tip and side orientations more effectively, figure 1(b) presents a schematic diagram of the projectile-target orientation configurations exemplifying a tip–side collision.
Figure 1. (a) The schematic diagram of the orientation configurations of two prolately deformed colliding nuclei. (b) The schematic diagram of tip-side collision. |
The projectile and target can be initialized based on the hard-sphere or Fermi density distribution. In principle, the Fermi distribution better describes the diffuseness of nuclear density and provides a random initial orientation for sampling, which is closer to the actual nuclear reaction process. However, during the sampling process, due to the role of the surface diffuseness of the nucleus, the density of the tail region of the Fermi distribution is minimal, and the calculated Fermi momentum is also relatively small. The influence of wave packet width poses significant challenges for sampling when Fermi momentum approaches zero. Therefore, calculations usually neglect initial nucleus deformation and orientation effects by initializing with a hard sphere density distribution. Figure 2(a) shows the density distribution of 238U of the initial moment based on the hard-sphere distribution without considering deformation and orientation effects, the black, red, and green curves represent the density distributions of nucleons, neutrons, and protons, respectively. Since the nucleon is represented by a Gaussian wave packet in the ImQMD model, the wave packet effect of the nucleon compensates for the diffuseness effect, and the surface diffuseness of the nucleus at the initial moment obtained by the hard sphere distribution sampling is reasonable.
Figure 2. Density distribution of 238U at the initial time. The black for nucleons, the red for neutrons, and the green for protons. (a) sphere, (b) tip and (c) side. |
238U is a typical deformation nucleus with the quadrupole deformation parameter β2 of 0.215 [51], and it is essential to consider the effects of initial nucleus deformation and orientation in the simulation process. In this study, we employ the density distribution of deformed hard spheres to initialize the ellipsoidal nucleus. Assuming that the three axes of the ellipsoidal nucleus coincide with the Cartesian coordinate system's x, y, and z axes, the nucleons are uniformly extracted in an ellipsoid with radius R(θ, φ) = R0[1 + α0Y20(θ, φ) + α2Y22(θ, φ) +α2Y2−2(θ, φ)]. Here, ${\alpha }_{0}={\beta }_{2}\cos \gamma $, ${\alpha }_{2}=1/\sqrt{2}{\beta }_{2}\sin \gamma $. When γ = 0, the ellipsoid is a rotational ellipsoid whose symmetry axis is the z-axis, corresponding to the tip orientation of 238U. Similarly, when γ = 2π/3, it is also a rotational ellipsoid, but the symmetry axis is the x-axis, corresponding to the side orientation of 238U. Figures 2(b) and (c) show the density distributions of the 238U projectile at the initial moment for the tip orientation (where the symmetry axis of 238U nucleus coincides with the collision axis) and the side orientation (where the symmetry axis of 238U nucleus is perpendicular to the collision axis), respectively. On the one hand, the saturation values of the density distributions inside the nucleus are reasonable and similar for both orientations, and the density distributions for both orientations reasonably describe the surface diffuseness of the nucleus at the initial moment. On the other hand, compared to side orientation (figure 2(c)), tip orientation (figure 2(b)) displays a wider density distribution that approaches zero at z = 11 fm. In contrast, side orientation approaches zero near 10 fm, which aligns with our physical understanding of an ellipsoid shape. Overall, the initial deformation nuclei sampled from the deformed hard sphere distribution are reasonable.
After sampling, it is imperative to check the stability of the selected initial projectile and target based on the ground-state properties of the nuclei. Figure 3 shows the time evolution of binding energy, root mean square radius, and deformation physical quantity β of the ground state 238U in different orientations. The first, second, and third columns correspond to the sphere, tip, and side orientations, respectively. It can be seen that the binding energy and root mean square radius are stabilized up to 1000 fm/c for all orientations, satisfying the requirements for nuclear stability in MNT reactions. To evaluate the stability of initial nucleus deformation, a physical quantity β = LZ/LX is defined as an approximate measure of nuclear deformation. Here, LZ and LX represent the lengths parallel to the z-axis and x-axis when the nuclear density reaches half of the saturation density (0.16 fm−3), respectively, as illustrated in figure 4. Analysis in conjunction with figure 1(b), the ratio LZ/LX provides a good approximation for the ratio of the semi-axis length of the symmetry axis with the other two semi-axes length of the rotating ellipsoid, or its reciprocal. Therefore, the closer the β value is to 1, the smaller the axisymmetric deformation of the nucleus. It can be seen that each orientation's β maintains a certain degree of stability under IQ2 parameters until 600 fm/c especially. Although less stable than binding energy and root mean square radius, the stability suffices for reactions involving heavy nuclei such as 238U+238U where interaction between projectile and target occurs early. The definition of β draws on the work in [28], which investigated the superheavy nuclei production probability for different collision orientations based on the ImQMD model and focused on the energy dependence of the orientation effect. This paper focuses on studying the influence of the orientation effect on the deformation of the nucleus at the contact moment, the composite system lifetime, the transfer rate and the fragments production mechanism.
Figure 3. The time evolution of (a)–(c) the binding energy, (d)–(f) the root-mean-square radius, and (g)–(i) the deformation variable β of initial 238U nuclei. Columns 1 through 3 represent the hard sphere, tip orientation and side orientation, respectively. |
Figure 4. Contour plot of the density distribution for the initial nuclei 238U (the density unit is fm−3). |
3. Results and discussion
3.1. Effect of nucleus orientation on nuclear deformation at the contact moment
This section aims to statistically analyze the deformation physical quantity β values of the projectile at the contact moment for all events to investigate the influence of the orientation effect on nuclear deformation. Figure 5 illustrates the probability distribution of β value for the projectile at the contact moment in the 238U+238U reaction with a collision parameter b = 1 fm. The arrows indicate the initial β value, which is the statistically averaged value obtained from all sampling events, and (a)–(e) denote the cases of the sphere–sphere, tip–tip, tip–side, side–side, and side–tip collisions, respectively. The most available β values of the projectile for sphere-sphere, tip-tip, tip-side, side-side, and side-tip collisions are 0.9, 1.1, 1.05, 0.75, and 0.75, respectively, while the initial β values in the sphere, tip orientation, and side orientation are 1.0, 1.1 and 0.9, respectively. This result indicates that the projectile retains its initial orientation characteristics to some extent at the contact moment. Meanwhile, the most available β value is lower than its initial value. This phenomenon is particularly evident in figure 6, which shows statistical average values of the β. The average values of β at the contact moment are 0.78, 0.78, 0.91, 1.04 and 1.06 in side–side, side–tip, sphere–sphere, tip–side, and tip–tip collisions, respectively; all of these values are lower than their respective initial values. This trend is mainly due to the Coulomb repulsion that causes the nuclei to suffer compression in the collision direction from the initial time to the contact moment. Additionally, upon analyzing figures 5(b) and (d), it can be found that the probability distributions of β values show a similar shape, as well as the most available β values and their corresponding locations are very similar. In these cases, the corresponding projectiles are in the tip orientation, and the targets are in the tip and side orientations. Similarly, in figures 5(c) and (e), the shapes of the probability distributions of β, the most available values, and their locations are also very close. At this time, the corresponding projectile is in side orientation, and the target nuclei are in the side and tip orientations, respectively. These results indicate that the β value of the projectile at the contact moment is less affected by the target orientation and vice versa. It can also be seen from figure 6 that statistical mean values of the β are almost the same when the projectile is both in the tip orientation (side orientation).
Figure 5. The probability distribution of β value for the projectile at the contact moment in the 238U+238U reaction with a collision parameter b = 1 fm, the initial value is indicated by the arrows. (a)–(e) Represent sphere–sphere, tip–tip, side––side, tip–side, and side–tip collisions, respectively. |
Figure 6. Statistical average values of the deformation physical quantity β of the projectile at the contact moment in side–side, side–tip, sphere–sphere, tip–side, and tip–tip collisions. |
3.2. Effect of nucleus orientation on the composite system lifetime
The time interval from the projectile and the target contact (i.e. the formation of the neck) to their subsequent separation (i.e. the disappearance of the neck) is referred to as the composite system lifetime, also known as neck lifetime. This parameter significantly influences nucleon exchange and energy dissipation between the projectile and target, making it a crucial physical quantity in understanding MNT reaction dynamics. Figure 7 illustrates the probability distribution of the composite system lifetime for the 238U+238U reaction at collision parameters b of 1, 5, and 8 fm, respectively. The black, red, blue, and green curves represent the sphere–sphere, tip–tip, side–side, and tip–side collisions. Since the properties and dynamics of tip–side and side–tip collisions are similar in the 238U+238U reaction, only the result of tip–side collision will be discussed later. The same colors and symbols will continue to be used in figures 8 to 10. In figure 7, columns one to three show results for collision parameters b = 1, 5, and 8 fm, respectively. As the collision parameter increases, the peak of the probability distribution gradually shifts towards the left, indicating a steady reduction in composite system lifetime. A larger collision parameter implies more peripheral collisions, leading to shorter lifetimes for composite systems. For b = 1 fm, a clear distinction can be observed among four different types of collisions based on probability distribution patterns. The tip–tip collision exhibits a narrow peak with relatively concentrated probabilities. While side–side collisions exhibit broader distributions with generally shorter lifetimes compared to tip–tip collisions, indicating a higher occurrence of events involving the transfer of a few nucleons in this orientation configuration within 100–350 fm/c. The probability distribution for tip–side collisions falls between the two cases above. At b = 5 fm, all four collision cases display broader probability distributions, and their differences are somewhat reduced, accompanied by a decrease in the peak value compared to that at b = 1 fm, suggesting an increased diversity of reaction mechanisms under this collision parameter. At b = 8 fm, the trend of probability distributions for all four collisions is nearly the same except for slightly higher peak values observed in tip–tip collisions. This phenomenon is consistent with the conclusion in [42] that the composite system lifetime of MNT reactions is less dependent on the orientation effect in peripheral collisions.
Figure 7. Probability distribution of the composite system lifetime in four typical collisions for 238U+238U reaction. |
Figure 8. The impact parameter dependence of the composite system lifetime in four typical collisions for 238U+238U reaction. |
The dependence of the composite system lifetime average value in the 238U+238U reaction on collision parameters is illustrated in figure 8. Consistent with the analysis of the probability distribution for composite system lifetimes, peripheral MNT reactions show a lesser impact from the orientation effect on the composite system lifetime. For collision parameter b ≤ 5 fm, the tip–tip collisions have the longest lifetimes, the side–side collisions have the shortest lifetimes, and the composite system lifetime with different collision orientations ranges between 300–700 fm/c. In reference [43], Beck et al investigated how the composite system lifetime varies with the incident energy in the 238U+238U center collision using the TDHF model. Their findings revealed that the lifetime of the composite system at the incident energy Ec.m. = 833 MeV is about 2 zs, i.e. 600 fm/c, which is consistent with the results presented in this paper. Furthermore, they observed no discernible trend in the composite system lifetime for all collision orientations at different incident energies. However, it was noted that at lower energies, the composite system for tip–tip collisions has the longest lifetime, and the composite system for side–side collisions has the shortest lifetime, which is consistent with the results of this paper.
3.3. Effect of nucleus orientation on transfer rate
To study the influence of the orientation effect on nucleon transfer, we define a physical parameter
$\begin{eqnarray*}{\eta }^{(i)}=\displaystyle \frac{{N}_{{\rm{P}}}^{(i)}-{N}_{{\rm{T}}}^{(i)}}{{N}_{{\rm{P}}}^{(i)}+{N}_{{\rm{T}}}^{(i)}},\end{eqnarray*}$
to quantify the degree of nucleon exchange between projectile and target, here the index i indicates the projectile-like fragments (PLF) or the target-like fragments (TLF), while ${N}_{{\rm{P}}}^{(i)}$ and ${N}_{{\rm{T}}}^{(i)}$ are the numbers of nucleons in ith reaction fragments originated from the projectile and target nuclei, respectively. The projectile and the target studied in this work are the same, both 238U. When ${N}_{{\rm{P}}}^{(i)}={N}_{{\rm{T}}}^{(i)}$, the fragments contain an equal number of nucleons from both the projectile and target nuclei, resulting in the transfer rate η(i) of zero. A positive transfer rate indicates a greater contribution from the projectile nucleus, while a negative transfer rate suggests a greater contribution from the target nucleus. The transfer rate η(i) reaches its maximum value of 1 when all nucleons in the fragment originate from the projectile, with ${N}_{{\rm{P}}}^{(i)}={A}_{{\rm{P}}}$ (the projectile's mass number) and ${N}_{{\rm{T}}}^{(i)}=0$. Conversely, the transfer rate η(i) is at its minimum value of −1 when all nucleons in the fragment are derived from the target, with ${N}_{{\rm{P}}}^{(i)}=0$ and ${N}_{{\rm{T}}}^{(i)}={A}_{{\rm{T}}}$ (the target's mass number). For the PLF, the closer the transfer rate is to 1, the fewer nucleons are transferred; similarly, for the TLF, the closer the transfer rate is to -1, the fewer nucleons are transferred. In a single event, there may be no obvious pattern between the transfer rates of PLF and TLF. However, for a reaction system with the same projectile and target, after statistical averaging over all events, the transfer rates η of PLF and TLF should be equal in magnitude but opposite in sign because the situation of nucleon transfer between the projectile and target is the same under nuclear–nuclear interaction. Figure 9 presents the transfer rate η versus collision parameters for the 238U+238U reaction under four different typical collision cases. Solid and hollow symbols correspond to PLF and TLF, respectively. Due to identical projectiles and targets, near symmetry in transfer rates exists between PLF and TLF as mentioned above, although slight asymmetries may arise due to individual nucleon emissions. This paper analyzes the transfer rate η for PLF as an illustrative example. As collision parameters increase, so does the transfer rate η, indicating that more nucleons originate from the projectile while fewer are transferred. In different collision orientation configurations, the difference of η decreases gradually with the increase of the collision parameter. The difference in transfer rates is most pronounced for the collision parameter b = 1 fm, with tip–tip collisions transferring the highest number of nucleons and side–side collisions transferring the fewest, with other collisions falling in between. The average composite system lifetimes (see figure 8) can explain this phenomenon. For small collision parameters, tip–tip collisions have longer lifetimes, allowing for more nucleon transfers; conversely, side–side collisions have shorter lifetimes, resulting in fewer transferred nucleons. When b ≥ 8 fm, the slight differences in lifetimes do not cause significant changes in the transfer rates, which are already very close for each collision case.
Figure 9. The impact parameter dependence of the transfer rate η in four typical collisions for 238U+238U reaction. |
3.4. Effect of nucleus orientation on fragments production mechanism
This section analyzes the influence of the orientation effect on fragment production mechanisms. All reaction mechanisms are now categorized into three main types: transfer reactions, elastic and inelastic scattering, and other reactions, the latter including processes such as fragmentation and ternary breakup. Figure 10 illustrates the dependence of the probability of these reaction types on the collision parameters for different collision orientations in the 238U+238U reaction. In figure 10(a), solid and hollow symbols denote transfer reactions and other reaction types, respectively, while in figure 10(b), solid symbols denote elastic and inelastic scattering. As shown in figure 10, the transfer reaction mechanism predominates when the collision parameter is small and is accompanied by a few other reactions, which may originate from the ternary breakup events due to the high density of the neck. As the collision parameter increases, there is a rapid decrease in the proportion of transfer reactions and an increase in elastic and inelastic scattering events; meanwhile, the proportion of other reactions becomes nearly zero. As shown in figure 10(a), when 1 ≤ b ≤ 6 fm, the probability of transfer reactions changes at a slower rate with increasing collision parameters. For b > 6 fm, there is a sharp decline in probability for transfer reactions with increasing collision parameters while the number of elastic and inelastic scattering events rapidly rises due to competition between these two mechanisms. It is worth pointing out that within the range of 10 ≤ b ≤ 13 fm, the transfer probability for side–side collisions is significantly higher compared to other cases, mainly because the larger nucleus radius perpendicular to the collision direction in the case of side–side orientations increases the probability of transfer of a few nucleons. Sekizawa et al investigated the orientation effect of the MNT reaction in the 64Ni+238U system in [42], revealing the variation of the total kinetic energy loss (TKEL) with the collision parameter for specific orientations. It concluded that the TKEL of fragments is slightly higher for side-oriented 238U targets than tip-oriented ones with larger collision parameters. This result implies that the side orientation in peripheral collisions is more likely to transfer nucleons, which is consistent with the conclusion that the transfer probability of side–side collisions in the 238U+238U reaction is significantly higher than that of other collisions at 10 ≤ b ≤ 13 fm in this paper. The composite system lifetime for these events ranges from 100–300 fm/c.
Figure 10. The impact parameter dependence of the probability of transfer reaction, elastic and inelastic scattering, and other reaction types in four typical collision orientations for 238U+238U reaction. (a) Solid and hollow symbols denote transfer reactions and other reaction types, respectively, and (b) solid symbols denote elastic and inelastic scattering. |
4. Conclusions
In this study, the orientation effect in the MNT reactions of 238U+238U at Ec.m. = 833 MeV is investigated within the framework of the ImQMD model. The initial deformation nuclei are extracted using the deformed hard sphere density distribution, and the influence of the orientation effect on the nuclei deformation at the contact moment, the composite system lifetime, the transfer rate, and the fragments production mechanism are investigated. It is found that the projectile and the target at the contact moment retain their respective orientation characteristics to a certain extent, and the value of the deformation physical quantity β of the projectile (target) at the contact moment is less affected by the target (projectile). In addition, the composite system lifetime and transfer rate exhibit less dependence on the orientation effect in peripheral collisions. In contrast, the influence of the orientation effect becomes more prominent in more central collisions with the collision parameter b ≤ 6 fm. By analyzing the relationship between the probability distributions of different reaction types with the collision parameter, it is observed that when b ≤ 6 fm, the transfer mechanism is predominant, and the transfer probability is less affected by the collision parameter. However, for b > 6 fm, the probability of the transfer reaction decreases significantly with the increase of the collision parameter. Notably, within the 10 ≤b ≤ 13 fm, the side–side orientation configuration produces more products through the transfer mechanism than other orientation configurations. This phenomenon can be attributed primarily to the larger nucleus radius perpendicular to the collision direction in the case of side–side orientations, which increases the probability of transfer of a few nucleons.