Introduction

The Equation of State (EOS) of asymmetric nuclear matter can be written approximately as $E(\rho \mathrm{,}\delta )=E_0(\rho \mathrm{,}\delta =0)+S(\rho ){\delta }^2+O\left({\delta }^4\right)+\dots \mathrm{,}$ (1) where δ=(ρ_n-ρ_p)/(ρn+ρp), which is related to different areas of nuclear physics, such as the nuclear structure of ground-state nuclei, the dynamics of neutron-rich heavy ion collisions (HICs), the physics of neutron stars, etc. [1, 2]. To date, the isoscalar component of the EOS has been well-constrained by the kaon condensation and the flow data of HICs [3]. However, the symmetry energy, particularly the density dependence of the symmetry energy S(ρ), still has large uncertainties away from the normal density.

Heavy-ion collisions with neutron-rich nuclei in facilities, such as the Heavy Ion Research Facility in Lanzhou (HIRFL/Lanzhou), Facility for Rare Isotope Beams at Michigan States University (FRIB/MSU), the Radioactive Isotope Beam Factory (RIBF/RIKEN), will produce rare isotope beams to study the dynamical evolution of neutron-rich nuclear systems and obtain information on the density dependence of the symmetry energy by comparing the acquired data with the transport model simulations [4, 5]. Many theoretical and experimental studies have been conducted to constrain the density dependence of the symmetry energy via intermediate-energy HICs [2, 6-20]. In addition, Zheng et al. proposed a novel method for studying the symmetry energy [21], which may impose additional restriction on the constraints in the future. There are also a lot of studies that aim to constrain the density dependence of symmetry energy at the suprasaturation density by analyzing the properties of neutron stars, such as the mass-radius relationship and tidal deformability [23-34], subsequent to the publication of GW170817 events data [23, 24]. By analyzing the combination of the observables of the neutron skin of ²⁰⁸Pb, the collective flow, isospin diffusion, or π^-/π⁺ ratios of HICs, and properties of neutron stars, the relatively tight constraints on the symmetry energy were inferred from 0.3ρ₀ to 3ρ₀[14, 17, 12, 22]. In a recent review paper by Li et al.[35], the constraints on the symmetry energy were compiled and yielded the average values and their variances. For the slope of the symmetry energy, i.e. $L=3{\rho }_0\frac{\partial S(\rho )}{\partial \rho }{|}_{{\rho }_0}$, the value is 57.7± 19 MeV, and the symmetry energy at twice the normal density S(2ρ₀) is 51± 13 MeV.

However, there is tension around the symmetry energy constraints following the Lead Radius Experiment (PREX-II) reporting of a model-independent extraction of the neutron skin of ²⁰⁸Pb and Δ r_np=0.29± 0.07 fm [36]. Yue et al. [17] analyzed newly published neutron skin data with ground-state properties and giant monopole resonances of finite nuclei, the properties of neutron stars for PSR J0740+6620, the mass-radius of PSR J0030+045, and the neutron star tidal deformability extracted from the GW170817 event to obtain the constraints of symmetry energy in a large density region. Their calculations suggested that S(ρ₀)=34.5± 1.5 MeV and L=85.5 ± 22.2 MeV. The values obtained are consistent with the constraints of the symmetry energy from the SπRIT data [13] for Sn+Sn at 270A MeV, but are less than the values obtained for the analysis of PREX-II using a specific class of relativistic energy density functional [36], in which L=106 ± 37 MeV and S₀=38.7± 4.7 MeV. Their constraints on the symmetry energy also overlap with the symmetry energy extracted from the charge radius of ⁵⁴Ni, where 21 MeV < L < 88 MeV [37], and the combination of astrophysical data with PREX-II and chiral effective field theory, where $L{=53}_{-15}^{+14}$ MeV [38].

Currently, the big challenge in constraining the density dependence of the symmetry energy with HICs is to understand the model dependence and reduce the uncertainties of the symmetry energy constraints. The understanding of the model dependence has been achieved in the last 10 years via the transport model evaluation project, and significant progress has been made in the treatment of nucleon-nucleon collisions and the mean-field potential [39-43]. The reduction of the uncertainties of symmetry energy constraints is still underway, which can be realized by simultaneously describing multi-observables using transport models. The data for isospin-sensitive observables from intermediate-energy HICs, such as isoscaling [44], double neutron to proton ratio [45], isospin diffusion [46], rapidity dependence of the isobaric ratio [10], and neutron-excess to proton ratio [11], have been used for comparison with the predictions from different transport models [4, 5]. Among these transport models, the ImQMD model successfully describes the neutron-to-proton yield ratio (n/p ratio), isospin diffusion, and isospin transport ratio as a function of rapidity, and constraints on the symmetry energy at the subsaturation density have been achieved [5].

Except for the aforementioned isospin-sensitive observables, the neutron number divided by the proton number of IMFs was considered to be complementary information of the neutron-to-proton yield ratio and can be used to extract information related to the symmetry energy [6]. This can be explained by the multi-fragmentation mechanism where the copious production of IMFs and its isospin contents are closely related to the dynamic mechanism of the HICs [47-49].

The measurement of IMFs has been performed at the National Superconducting Conducting Cyclotron Laboratory at Michigan State University [50], but there are few models for the theoretical analysis of the isospin effects of IMFs and their use to constrain the symmetry energy. Thus, it is worth to check whether the ImQMD model can simultaneously describe the isospin properties of IMFs.

In this study, we focus on the properties of IMFs (Z=3-8) for the central collision (b=2 fm) of ^112,124Sn+^112,124Sn at E_beam=50 MeV/u using the ImQMD model coupled with the GEMINI model. Furthermore, we compared the calculated results to the data for learning the form of the symmetry energy and the reaction dynamics.

Theoretical model and symmetry energy

The version of ImQMD used in this study was the same as that used in Ref. [5], i.e., ImQMD05, in which the isospin-independent momentum-dependent interaction was utilized. To simulate low-intermediate energy HICs, the basic properties of the initial nuclei are important, and they are guaranteed by requiring that the binding energy and root-mean-square (rms) radius of the sampled nuclei fall within the range of the experimental data with specific uncertainties. For the standard initialization treatment, the initial nuclei remain stable within 200 fm/c, which is sufficient for Sn+Sn at 50 MeV/u because the reaction dynamics are completed within 100 fm/c [51]. Additional details regarding the initialization of the ImQMD model can be found in Refs. [51-55].

The nucleonic mean field potential is derived from the Skyrme-type energy density functional, and the energy density of the symmetry potential is as follows: $w_{\text{asy}}^{\text{pot}}=\frac{C_{\text{s}}}{2}{\left(\frac{\rho }{{\rho }_0}\right)}^{\gamma }{\delta }^2\rho \mathrm{.}$ (2) where ρ is the nucleon density. C_s is the symmetry potential parameter, C_s=35.2 MeV, and γ is a parameter related to the density dependence of the symmetry energy. Based on these interaction parameters, we can obtain the symmetry energy per nucleon for cold nuclear matter, as follows: $S(\rho )=\frac{1}{3}\frac{{\hslash }^2}{2m}{\rho }_0^{2/3}{\left(\frac{3{\pi }^2}{2}\frac{\rho }{{\rho }_0}\right)}^{2/3}+\frac{C_s}{2}{\left(\frac{\rho }{{\rho }_0}\right)}^{\gamma }\mathrm{,}$ (3) where m denotes the nucleon mass. The corresponding values of the symmetry energy S(ρ₀) and the slope of the symmetry energy $L=3{\rho }_0\frac{\partial S(\rho )}{\partial \rho }{|}_{{\rho }_0}$ at a normal density are listed in Table 1. For this particular parameterization, the symmetry energy increases with a decrease of γ at a subsaturation density, whereas the opposite is true for supranormal density.

The ImQMD simulations were terminated at 400 fm/c when the systems reached “dynamical freeze-out”. The primary fragments were recognized using the isospin-dependent minimum spanning tree method [51], and the excitation energy was calculated in the rest frame of the fragment by subtracting the ground state energy from the energy of the fragment in its rest frame. It should be noted that the fragments produced during the ImQMD calculations are primary, and they may have very neutron-rich or very neutron-poor “unphysical” fragments. The probability of generation of these “unphysical” primary fragments is tiny in ImQMD simulations, and their ground state energies are approximately obtained using the liquid-drop mass formula.

The sequential decay of the primary fragments was performed using the GEMINI code [56, 57], which describes the de-excitation process of the compound nucleus and facilitates binary decay, including light-particle evaporation and symmetric fission. The decay continues until the particle emission is energetically forbidden or impossible owing to competition with γ-ray emission.

Results and discussions

Before studying the isospin properties of the IMFs, we focused on the cluster formation mechanism of HICs. Figure 1 shows the time evolution of the multiplicities of emitted nucleons (top panels) and fragments (bottom panels), for the collisions of ¹²⁴Sn+¹²⁴Sn and ¹¹²Sn+¹¹²Sn. It is evident that the nucleons are emitted prior to IMFs with Z=3-8, and IMFs with Z=3-8 prior to fragments with Z=9-20. For example, for the reaction system ¹¹²Sn+¹¹²Sn, the emission of nucleons begins from t∼50 fm/c, but the emission of IMFs with Z=3-8 occurs at approximately t∼100 fm/c and Z=9-20 fragments at approximately t∼140 fm/c. Such a dynamic process results in a decrease in the fragment velocity with the fragment charge number. After t∼250 fm/c, the primary fragments are completely formed. Ideally, the system should be in “dynamical freeze-out,” and its multiplicities should not change with time. However, it is difficult for the QMD type Hamiltonian to bind the fragments for a long time, and the pseudo-emission of nucleons occurs at the late stage of the simulations of HICs. Therefore, the multiplicities of fragments with Z=3-8 or 9-20 decrease slightly, and the magnitude depends on the stiffness of the symmetry energy.

Fig. 1Full-screen View

(Color online) The time evolution of the multiplicities of nucleons and fragments with Z=3-8 and Z=9-20 for ¹²⁴Sn+¹²⁴Sn (panels (a) and (c)) and ¹¹²Sn+¹¹²Sn (panels (b) and (d)) for different symmetry energy cases.

To understand the influence of different symmetry potentials on the reaction dynamics, we investigated the time evolution of the multiplicities of nucleons and fragments for γ=0.5 (solid lines) and γ=2.0 (dashed lines). The results show that the multiplicities of the nucleons and the fragments depend on γ and the isospin asymmetry of the reaction system.

The multiplicities of emitted neutrons calculated with γ=0.5 are greater than those with γ=2.0, and the effect is the opposite for the multiplicities of protons. Consequently, it is expected that the neutron-to-proton yield ratio would be sensitive to the stiffness of the symmetry energy. Numerous studies have investigated for constraining the symmetry energy using n/p ratios [45, 10, 54, 58, 59, 39].

For the multiplicities of fragments with Z=3-8 (or 9-20), the calculations show that the multiplicities of IMFs obtained with γ=0.5 are smaller than those with γ=2.0. This is because a higher density is reached in the overlap region and more compressional energy is stored for the soft isospin asymmetric nuclear equation of the state (γ=0.5) compared to that of the stiff isospin asymmetric nuclear equation of state (γ=2.0) in the simulations. Thus, the reaction system simulated with γ=0.5 disintegrates into more light-charged fragments compared to the case of γ=2.0. Owing to the conservation of the nucleon number in the reaction system, the multiplicities for fragments with Z=3-8 (or 9-20) for the γ=0.5 case are smaller than those for γ=2.0.

Considering the reaction dynamics, it is expected that the size of the fragments and their kinetic energies should be correlated to the stiffness of the symmetry energy. In Fig. 2, the average center-of-mass kinetic energy per nucleon $\langle E_{\text{c}\text{.m}\text{.}}/A\rangle$ is plotted for the primary fragments as a function of their charge number Z for ¹²⁴Sn+¹²⁴Sn (left panel),and ¹¹²Sn+¹¹²Sn (right panel). The solid red lines represent the calculated results for γ=0.5, and the dashed red lines are the results obtained for γ=2.0. For both symmetry energies, the values of $\langle E_{\text{c}\text{.m}\text{.}}/A\rangle$ decrease with an increase in Z owing to the aforementioned reaction dynamics. It is interesting to note that $\langle E_{\text{c}\text{.m}\text{.}}/A\rangle$ as a function of Z is also sensitive to the density dependence of the symmetry energy, and the values of $\langle E_{\text{c}\text{.m}\text{.}}/A\rangle$ obtained for γ=0.5 are larger than those obtained for γ=2.0 when Z>3. This can be understood based on the reaction dynamics shown in Fig. 1, wherein the multiplicities of the IMFs are larger for γ=2.0 compared to the case of γ=0.5. If there is an energy equipartition about fragments, one can expect each IMF to carry less kinetic energy for the case of γ=2.0 compared to the case of γ=0.5. The difference between the values of $\langle E_{\text{c}\text{.m}\text{.}}/A\rangle$ obtained for γ=0.5 and γ=2.0 increase with an increase in Z, and the difference exceeds 10% at Z=8.

Fig. 2Full-screen View

(Color online) The average kinetic energy per nucleon as a function of the fragment charge number Z. The solid symbols represent γ=0.5 and the open circles represent γ=2.0. The red lines are the results for primary fragments, and the blue lines with symbols represent the cold fragments.

However, the measured fragments in the experiments cold, which indicate that the hot primary fragments should be deexcited. In this study, the decay of the primary fragments created in the ImQMD simulations was performed using the GEMINI code [56, 57] using their default parameter sets. The blue lines with symbols represent the results for the cold fragments. For Z=3 and 4, the values of $\langle E_{\text{c}\text{.m}\text{.}}\rangle /A$ are reduced by ∼7% compared to the results obtained for the primary fragments. This reduction is caused by the decay of primary fragments, which converts the fragments with higher mass or charge numbers to those with lower mass or charge numbers. For fragments with Z ≥ 6, the effects of decay on $\langle E_{\text{c}\text{.m}\text{.}}/A\rangle$ are weak, and the sensitivities of $\langle E_{\text{c}\text{.m}\text{.}}/A\rangle$ to the stiffness of the symmetry energy remain. Consequently, $\langle E_{\text{c}\text{.m}\text{.}}/A\rangle$ as a function of Z of the IMFs can also be used to understand the fragmentation mechanism and to probe the density dependence of the symmetry energy.

We will now discuss the isospin contents of the IMFs in more detail. The $\langle N\rangle /Z$ of the IMFs was investigated, which is defined as $\langle N\rangle /Z=\frac{{\displaystyle {\sum }_i{N}_i\times Y(N_i\mathrm{,}Z)}}{{\displaystyle {\sum }_{i}Y(N_i\mathrm{,}Z)}}/Z\mathrm{.}$ (4) where Y(Ni,Z) is the yield of the fragment with a neutron number Ni and charge number Z. The summation is for all isotopes at a given Z. As an example, Fig. 3 present the $\langle N\rangle /Z$ ratio of primary fragments (red lines) as a function of E_c.m./A for fragments with Z=3, 4, 5, 6, 7, 8. The left six panels are for ¹²⁴Sn+¹²⁴Sn and the right six panels are for the reaction system ¹¹²Sn+¹¹²Sn. In general, the slopes of the $\langle N\rangle /Z$ ratio as a function of E_c.m./A are negative, because the Coulomb energy accelerates the neutron-poor fragments to a higher kinetic energy region. These behavior occurs for all elements of the IMFs. Furthermore, the curves of the $\langle N\rangle /Z$ ratios as a function of kinetic energy E_c.m./A depend on the stiffness of the symmetry energy, and the results obtained for γ=0.5 and 2.0 cross over at a specific energy, E_cross. The values of E_cross increase from 4 MeV to approximately 10 MeV as Z increases from 3 to 8, owing to the Coulomb interaction.

Fig. 3Full-screen View

The $\langle N/Z\rangle$ ratios as functions of kinetic energy per nucleon for Z= 3, 4, 5, 6, 7, 8 for primary (red lines) and cold fragments (blue lines) obtained for γ=0.5 and 2.0. The left six panels are for ¹²⁴Sn+¹²⁴Sn, and the right panels are for ¹¹²Sn+¹¹²Sn.

Below the cross energy, the values of $\langle N\rangle /Z$ obtained for γ =0.5 are smaller than those obtained for γ =2.0, and turn over above E_cross. The predicted magnitude of the slope of the $\langle N\rangle /Z$ ratio as a function of E_c.m./A obtained for γ=0.5 is smaller compared to that for γ =2.0. This behavior can be understood based on the cluster formation mechanism. For the Z≤4 fragments with high kinetic energy, and the isospin effects were governed by the dynamics. Thus, the behavior of $\langle N\rangle /Z$ of the fragments in the high kinetic energy region is similar to the n/p ratios of the emitted nucleons. Consequently, one can expect the values of $\langle N\rangle /Z$ obtained for γ=0.5 to be larger than those obtained for γ=2.0.

Fragments with low kinetic energy or heavier fragments are formed at a later stage of the reaction, and their isospin content is determined by their free energy. The free energy of the primary fragments, f(A, δ, T), can be approximated as $\begin{array}{ll}f(A\mathrm{,}\delta \mathrm{,}T)=\hfill & -a_{\text{sym}}^v(\rho \mathrm{,}T)(N-Z{)}^2/A+a_{\text{sym}}^{\text{surf}}(\rho \mathrm{,}T){\delta }^2{A}^{2/3}\hfill \\ \hfill & -a_{\text{c}}(\rho \mathrm{,}T)Z(Z-1)/A^{1/3}\hfill \\ \hfill & +a_{\text{v}}(\rho \mathrm{,}T)A-a_{\text{s}}(\rho \mathrm{,}T)A^{2/3}\mathrm{.}\hfill \end{array}$ (5) where $a_{\text{sym}}^{\text{v}}$ and $a_{\text{sym}}^{\text{surf}}$ are the bulk-and surface-related symmetry energy coefficients, respectively. ac is a coefficient related to the Coulomb energy, and a_v and a_s are the volume and surface energy coefficients, respectively. If we assume that the fragments are at thermal equilibrium and stable, the isospin asymmetry of the primary fragment is determined by the minimum free energy of the cluster at temperature T, i.e., $\frac{\partial f(A\mathrm{,}\delta \mathrm{,}T)}{\partial \delta }=0$. Therefore,the isospin asymmetry of the primary fragments should be proportional to ${\delta }_{\text{frag}}=\frac{a_{\text{c}}(\rho \mathrm{,}T)A^{2/3}\left(A-1\right)}{a_{\text{c}}(\rho \mathrm{,}T)A^{5/3}+4a_{\text{sym}}^{\text{v}}(\rho \mathrm{,}T)A-4a_{\text{sym}}^{\text{surf}}{A}^{2/3}}\mathrm{.}$ (6) The Coulomb interaction increases the isospin asymmetry of the primary fragments and the symmetry energy term decreases the isospin asymmetry of the fragments. Since the fragments are formed during the expansion phase during which the system is at a subsaturation density, the values of the symmetry energy $a_{\text{sym}}^v(\rho \mathrm{,}T)$ obtained for γ=0.5 are larger than those obtained for γ=2.0, and it is expected that the $\langle N\rangle /Z$ values of the fragments obtained for γ=2.0 will be larger than those for γ=0.5, according to Eq. (6).

To investigate the secondary sequential decay effects, we present $\langle N\rangle /Z$ as a function of E_c.m./A as blue lines in Fig. 3. For fragments with Z≥5, the secondary decay reduces the sensitivity of $\langle N\rangle /Z$ to the stiffness of the symmetry energy because of abundant neutron evaporation. Specifically, the decay weakly influences the isospin contents of the fragments obtained for γ=0.5, and strongly suppresses the $\langle N\rangle /Z$ obtained for γ=2.0. This is because the E^*/A obtained for γ=0.5 is smaller than that obtained for γ=2.0. For γ=0.5, the values of E^*/A varied from 1.6 to 1.8 MeV. For γ=2.0, the values of E^*/A varied from 1.8 to 2.4 MeV. The hardly understanding point is that the sensitivity of $\langle N\rangle /Z$ to the stiffness of the symmetry energy is the same or is enhanced for Z=3 and 4 after the decay. This could be related to the “³He-⁴He puzzle”[60-62] and the decay of ⁸Be[50]. Thus, the cluster formation mechanism in transport models and the sequential decay model require further refinement.

To avoid the cluster formation flaw in the transport model, we calculated ${\displaystyle \sum }N/{\displaystyle \sum }Z$ for fragments with Z from 3 to 8. ${\displaystyle \sum }N$ and ${\displaystyle \sum }Z$ are defined as follows: ${\displaystyle \sum }N={\displaystyle \sum _{i\mathrm{,}j}}{N}_i\times Y(Z_j\mathrm{,}{N}_i);$ (7) ${\displaystyle \sum }Z={\displaystyle \sum _{i\mathrm{,}j}}{Z}_j\times Y(Z_j\mathrm{,}{N}_i)\mathrm{.}$ (8) The summation is over all IMFs for a given kinetic energy bin (1.5 MeV/nucleon) for all events. This is similar to the method proposed in Ref. [6] for studying the isospin effects and symmetry energy in HICs.

The data on the isospin observable ${\displaystyle \sum }N/{\displaystyle \sum }Z$ for the central collision of Sn+Sn were reported in Ref.[50]. It should be noted that the experimental central collisions are defined by the high multiplicity of charged particles, and the corresponding real impact parameter distribution is Gaussian [63-65]. For the central collision events selected in Refs.[50], the reduced impact parameter ranges from 0 to approximately 0.4 and peakes at approximately 0.2. Thus, we performed simulations at a representative value of the impact parameter, such as the mean of the impact parameter distribution, b=2 fm. We also selected fragments emitted at center-of-mass angles of 70° ≤ θ_c.m. ≤ 110°, which is the same as that in Ref.[50]. The calculated results are shown in Fig. 4 (a)-(d). Panels (a) and (c) show ¹²⁴Sn+¹²⁴Sn, and panels (b) and (d) show ¹¹²Sn+¹¹²Sn. The calculated results for ${\displaystyle \sum }N/{\displaystyle \sum }Z$ for the primary fragments (top panels) strongly depend on the density dependence of the symmetry, which is consistent with the results in Ref.[6]. However, ${\displaystyle \sum }N/{\displaystyle \sum }Z$ as a function of E_c.m./A obtained in the transverse direction is weakly sensitive to the isospin asymmetry of the system. This is because the IMFs preferentially form in the forward and backward regions owing to the IMFs formation mechanism. In panels (e) and (f), we present ${\displaystyle \sum }N/{\displaystyle \sum }Z$ as a function of E_c.m./A in the non-transverse direction region, i.e., θ_c.m.<70° and θ_c.m. > 110°, and ${\displaystyle \sum }N/{\displaystyle \sum }Z$ in the non-transverse direction not only depend on the stiffness of the symmetry energy, but also on the isospin asymmetry of the systems.

Fig. 4Full-screen View

${\displaystyle \sum }N/{\displaystyle \sum }Z$ as a function of the kinetic energy per nucleon for ¹²⁴Sn+¹²⁴Sn (left four panels) and for ¹¹²Sn+¹¹²Sn (right four panels). Panel (a), (b), (e) and (f) represent the results for the primary fragments, and (c), (d), (g) and (h) are the results for the cold fragments.

The results of ${\displaystyle \sum }N/{\displaystyle \sum }Z$ of the IMFs after decay are plotted in panels (c) and (d) for the transverse direction and in panels (g) and (h) for the non-transverse direction. The secondary sequential decay mainly influences the ${\displaystyle \sum }N/{\displaystyle \sum }Z$ ratios in the lower kinetic energy region because the fragments with lower kinetic energy mainly consist of neutron-rich isotopes. However, the ${\displaystyle \sum }N/{\displaystyle \sum }Z$ ratios still maintain a sensitivity to the density dependence of the symmetry energy after sequential decay. In the transverse direction, the calculations indicate that the sensitivity of ${\displaystyle \sum }N/{\displaystyle \sum }Z$ increases as E_c.m./A increases. We compared the calculated results in the transverse direction with the data (solid points in panels (c) and (d))[50], which includes the ⁸Be contribution, to understand the model prediction. It is evident that the calculations with γ=0.5 are favored, and this conclusion is consistent with the constraints on γ obtained with the n/p ratio, isospin diffusion, and isospin transport ratio as a function of rapidity[5, 54]. For ${\displaystyle \sum }N/{\displaystyle \sum }Z$ as a function of E_c.m./A in the non-transverse direction, our calculations show that the values of ${\displaystyle \sum }N/{\displaystyle \sum }Z$ are not only sensitive to the symmetry energy, but also the isospin asymmetry of the systems. However, the sensitivity of ${\displaystyle \sum }N/{\displaystyle \sum }Z$ in the non-transverse direction is weaker compared to that in the transverse direction.

Finally, we analyze the double ratios of ${\displaystyle \sum }N/{\displaystyle \sum }Z$ for the two systems A=¹²⁴Sn+¹²⁴Sn and B=¹¹²Sn+¹¹²Sn, i.e., $DR(N/Z)=({\displaystyle \sum }N/{\displaystyle \sum }Z{)}_A/({\displaystyle \sum }N/{\displaystyle \sum }Z{)}_B\mathrm{.}$ (9) The idea of this double ratio is similar to the double neutron-to-proton ratio, which has been used to isolate the isospin effects and constrain the density dependence of the symmetry energy [54, 6, 45]. To obtain good statistics of the results of DR(N/Z) in the transverse direction, we performed simulations with 100,000 events. In Fig. 5 (a), we present the DR(N/Z) ratios for the cold fragments in the transverse direction for an experimental case such that 70°≤ θ_c.m.≤ 110°. The shaded region with a solid pattern (dashed pattern) is the result obtained for γ=0.5 (γ=2.0). The DR(N/Z) ratios obtained for γ=2.0 are larger than those obtained for the γ=0.5 case. This can be observed from Fig. 3. These trends reflect the fact that the fragments originate from the composite system that survives the pre-equilibrium emission, as indicated in Ref.[6]. Compared to the data, the calculations for the case of γ=0.5 are favored. We also analyzed DR(N/Z) in the non-transverse direction, and the results are shown in panel (b). The calculations show that the sensitivity of DR(N/Z) to the stiffness of the symmetry energy is maintained in the non-transverse direction, and the DR(N/Z) in the non-transverse direction has good statistics because the IMFs are distributed in the forward and backward regions.

Fig. 5Full-screen View

Double N/Z ratios as a function of the kinetic energy per nucleon. The top panels are the results for the transverse direction, and the bottom panels are the results for non-transverse direction.

Summary and prospective

In summary, we investigated the isospin properties of intermediate mass fragments for the reaction ^112,124Sn+^112,124Sn at a beam energy of 50 MeV per nucleon. Our results indicated that the average kinetic energy per nucleon of the fragments $\langle E_{\text{c}\text{.m}\text{.}}/A\rangle$ as a function of Z is sensitive to the density dependence of the symmetry energy, and this sensitivity does not change significantly by secondary sequential decay. Furthermore, the $\langle N\rangle /Z$ ratio of the IMFs as a function of the kinetic energy per nucleon is also sensitive to the density dependence of the symmetry energy. To circumvent the difficulties associated with cluster formation in transport model calculations, we also investigated the ${\displaystyle \sum }N/{\displaystyle \sum }Z$ and DR(N/Z) ratios as a function of kinetic energy. The calculations revealed that both observables are sensitive to the density dependence of the symmetry energy, and the sensitivity was maintained even after secondary sequential decay. By comparing the calculations to the obtained data, the symmetry energy with γ=0.5 is favored. The results are consistent with the conclusions obtained by simultaneously describing the n/p ratios, isospin diffusion, and isospin transport ratio as a function of rapidity.

Nevertheless, the tight constraints on the symmetry energy still need more work. For example, the effects of the momentum-dependent symmetry potential should be well understood because the isospin-dependent momentum-dependent symmetry potential influences the cluster formation and the n/p ratio of the emitted nucleons [66, 67]. Thus, the curves of the average kinetic energy of the IMFs as a function of Z are modified. The ${\displaystyle \sum }N/{\displaystyle \sum }Z$ ratios and DR(N/Z) may also be influenced by the different signs associated with effective mass splitting. These points should be carefully investigated in the future. Our analysis and experimental studies[50] also suggest that a sequential decay model should be developed.