# Characteristics of Fragments Produced by Ternary Fission in Intermediate Energy

Authors: Xian Li; Chengqian Wang; Yaohui Xu
Journal: International Journal of Photoenergy
 (2022)

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## Abstract

The characteristics of fragments produced in the ternary fissions have been researched by using the improved quantum molecular dynamics model in heavy197Au+197Au at 15 MeV/nucleon. The physical quantities of Z, n/z, θ, ptrans/plong about the nuclear fragments have strong dependencies on the impact parameters, and their curves are systematically drawn. By the study of these theoretical results above, we have concluded that there is a cut-off point at b =6 fm, which divides the impact parameters into two regions: the central or semi-peripheral and peripheral regions, and the ternary breakup mechanisms should be very different at these two impact parameter regions. Further research has showed that the regional property of n/z is related to the isospin symmetry of nuclear products, and the conclusion reflects the properties of nuclear fragments may be an observable probe into the nucleon interactions during heavy ion collisions.


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## Body

## 1. Introduction

In the high energy area of nuclear reaction, where the incident energy of each nucleon in some nuclear system is more than 1GeV, multifragmentation is universal, and the interaction between nucleons is dominated by two-body collisions. When the incident energy of each nucleon drops below 10 MeV, the reaction system will process as follows: fusion, fusion-fission, or deep-inelastic scattering, and the nucleons involved in the reaction move mainly under the mean field [1]. However, for the intermediate energy, the incident energy range from 10 MeV to several hundreds MeV. There exists a new fission mechanism in this region: ternary fission, and it is a new reaction mechanism and is not the same as multifragmentation or fusion-fission and deep-inelastic. In recent 20-30 years, a large number of studies have been conducted on the multiple fragmentation of heavy ions in the intermediate energy zone, and many experiments have confirmed the ternary fission in the intermediate energy and heavy nuclear collisions [2, 3]. This fission type of the ternary reaction is seen as a transition mode of heavy nuclear reaction from one-body energy dissipation in the low-energy collision into two-body energy dissipation on the high-energy reaction [4–7]. Therefore we need to study the fission mechanism of the ternary fission in order to clarify the transition conditions of the dynamical and energy dissipation mechanisms between the one-body and the two-body, and then reveal the microscopic mechanisms of nucleons interaction in the heavy and intermediate energy ion collisions system.In the early, the heavy system197Au+197Au at energy of 15 MeV/nucleon have been conducted at the laboratory of Laboratori Nazionali del Sud (LNS) in Catania, and detected by the multidetector array arranged in 4π geometry. Their findings suggest that there exists a kind of ternary fission in which the reaction system undergoes twice sequential break-ups and separates into three comparable fragments, and be called the fast ternary fission [8–10]. In deeper study, they have found the three fragments of a ternary fission in the space are arranged almost in a line, and concluded that the breakup mechanism of the ternary reaction should belong to deep inelastic scattering [11, 12]. There are many theoretical dynamical models in order to research the fission mechanism of nucleus-nucleus on heavy ion collision. For example, a simple deterministic dynamical model of Blocki [13] and the numerical simulations for both extreme situations: one-body energy dissipation and two-body energy dissipation, were performed [14, 15], and also “QMD-CHIMERA code” of Lukasik into the heavy reaction system of 197Au+197Au [16]. But the microcosmic mechanism of nucleon and nucleon forces in the ternary reaction caused by heavy ion collisions is still unclear, which is also a concern of people. Further study of this problem may reveal the mechanism of the microscopic motion of the dissipative transition between one-body and two-body. Therefore, it is necessary to apply a suitable theoretical dynamical model to carefully analyze the mechanism of ternary reaction in the heavy nuclear system.In this work, we have used the improved quantum molecular dynamics (ImQMD) model to simulate the reaction process of the197Au+197Au nuclear reaction system at the energy of 15 MeV/nucleon, and selected the ternary fissions on the same condition with the experiments [8–12]. The properties of fission fragments are carefully researched, such as the proton number about nuclear fragment (Z), the ratio of neutrons to protons (n/z), the angle distribution (θ), and the ratio of transverse momentum to longitudinal momentum (ptrans/plong) and so on.


## 2. Theoretical Model

In this paper, we have chosed a microscopic transport model: ImQMD model to simulate the heavy nuclear collision process. And It is based on the QMD model, which is mentioned by Aichelin [17, 18], and mainly applied to the research of the higher energy heavy-ion collisions. But in the tens of MeV energy region, the QMD model is not suitable. With the addition of some improvements, such as the surface symmetry energy part, the number of phases occupied and width of the wave packet, it has been renamed the ImQMD model and applied to heavy ion collisions in the low and intermediate energy region [19–21].The introduction about the ImQMD model is described in detail in our previous paper [22] .The evolution of heavy ion collision system is constrained by the canonical equation:
(1)r⟶̇i=∂H∂p⟶i,p⟶̇i=−∂H∂r⟶i.H is the Hamiltonian of the collision system, and is made up of the kinetic energy T, the local interaction potential Uloc and Coulomb interaction potential Ucoul terms. So it is writing as
(2)H=T+Uloc+Ucoul.The formulas of these three terms above are calculated as follows:(3)T=∑ipi22mi,(4)Uloc=∫Hlocr⟶dr⟶=∫α2ρ2ρ0+βγ+1ργ+1ρ0γ+g02∇ρ2+gτρη+1ρ0η+Cs2ρ0ρ2−κs∇ρ2δ2dr⟶(5)Ucoul=12∫ρpr⟶e2r⟶−r⟶′ρpr⟶′dr⟶dr⟶′−e2343π1/3∫ρpr⟶4/3dr⟶The term ofδ=ρn−ρp/ρn+ρp is the isospin asymmetry, and ρ,ρn,ρp are separately the nucleon, neutron and proton densities. In the fomula (4), the first and second parts are two-body and three-body component, the third and fourth terms are part of the surface and momentum-dependent, the last term is about the symmetry energy.The set of parameters appeared in the fomulas of (4) and (5) are called IQ2 parameters and shown in Table 1 [20]. The IQ2 parameters are connected with the interaction potential energy functional, and they are obtained from the Skyrme interaction of SkM and SLy series. By comparing the properties of some ground state nuclei, such as root mean square radius, binding energy, the cross section of reaction system with experimental data, finally, the researchers have found that the IQ2 parameters were appropriate for describing heavy nuclear reactions in the incident energy of tens of MeV regions [19]. The softness or hardness parameters of the symmetry energy in the equation (4) indeed have an effect on the final nuclear products, and the softness parameters are adopted in the IQ2 parameters of ImQMD model.Table 1
The IQ2 parameters [22].
αβγg0gτηCSκSρ0MeVMeVMeVfm2MeVMeVfm2fm‐3-3563037/67.012.52/332.00.080.165


## 3. Results and Discussions

We have selected 20 projectile nuclei and 20 target nuclei by using the ImQMD model with IQ2 parameters, and their binding energies maintain near7.92MeV/nucleon and the root-mean-square radii keep around 5.35fm within 6000 fm/c, respectively. We have calculated more than 200,000 different collisions, through using random numbers to produce a Euler angle and then the projectile or target nuclei would rotate around their centers of mass with the angle. For the small impact parameter area of 0-3 fm, we have simulated about 5000 events at each impact parameter, almost 15000 reactions at each medium impact parameter of 4-6 fm, and more than 20000 collisions for every peripheral impact parameter of 7-12 fm.With the same constraint as experiment research [10], we have selected a kind of the ternary collision on the following conditions:
(6)Aprojectile+Atarget−70≤A1+A2+A3≤Aprojectile+Atarget,(7)∑i=13p⟶longi>0.8p,0∑i=13p⟶transi<0.04p.0The symbols of A1, A2, and A3 in the formula (6) respect the mass numbers of the nuclear fragments produced in a ternary fission, separately. In the formula (6), the total mass number of the reaction minus 70 is because about up to 70 mass units to be missed for the evaporation of undetected nucleons, for example α particles. The longitudinal and transversal momenta of the collision system should be satisfied as the the formula (7), and p0 is the momentum of some 197Au projectile in incident beam.In our earlier study, we have simulated the mass distributions and angular distributions of fragments in the ternary fission of the197Au+197Au reaction system at 15 A MeV by the ImQMD model, and the results have shown that the theoretical results are consistent with experimental data very well [22–24]. The facts confirm that the ImQMD is suitable way to research the heavy nuclear reaction in intermediate energy area.Figure1 shows the correlation between the proton numbers of nuclear fragments in ternary fission and impact parameter. A tripartite reaction will produces three nuclear fragments, and the biggest, the middle and the smallest fragments are recorded as A1, A2, and A3. The mean values about mass numbers of the ternary fission nuclear products are dependence on the impact parameter. The values in the black-squares, red-circles and blue-triangles lines, respectively, represent the mean mass number of A1, A2, and A3 at each impact parameter. From Figure 1, we can find that the black-squares, red-circles and blue-triangles lines nearly remain the same. But after the impact parameter arriving 6 fm, we can see that the proton numbers about the fragments of A1 and A2 are growing sharply with the impact parameter increasing. Meanwhile, the smallest fragment in the ternary fission is quickly falling with the rising impact parameter. The results indicate that in the impact parameters region from 0 to 6 fm, the ternary reactions seem to be the same fission mechanism, and the other part of the impact parameters has another fission mechanism.Figure 1
The correlation between the proton numbers of nuclear fragments in ternary fission and the impact parameter.In Figure2, it is shown that the correlation between θ and the impact parameter. The angle of θ is defined as the scattering direction deviating from the direction of the incident beam. The black-square, red-dot and blue-triangle lines express separately the angle of θ about the maximum, medium and minimum fragments at each impact parameter. Before the impact parameter is equal to 6 fm, these three color lines distribute in the angle range from 22° to 30°. We also find that the black-square, red-dot lines nearly almost overlap indicating that the medium and minimum fragments are close in space. At the same time the vertical distance between two curves of the red-dot and blue-triangle lines is remained around 5°.So we can infer that the smallest nuclear fragment has strong correlation with the medium mass nuclear fragments and weak link with the largest nuclear fragment. Moreover all the three color lines have downward trends until the impact parameter reaching 6 fm. After b =6 fm, the curves of the red-dot and blue-triangle lines begin to rise, and the values of the red-dot distribute in the interval from 28° to 38°.But the black-square line is still falling until b =9 fm, and the distribution range is from 24° to 18°. And the slope of the descending part in the black-square curve is greater than those of the red-dot and blue-triangle curves in the region b =4-7 fm. This result means for the larger impact parameter the smallest nuclear fragment exists in a small value and narrow angle range, and disparts away from the other two bigger nuclear fragments. From the characteristics of the angle distributions, we can conclude that the microcosmic mechanisms of the ternary reaction are different in different impact parameter areas. In the region b <6 fm, the fission model belongs to the sequential ternary fission [22], and the second fission produced by one of primary fragments forms the smallest and medium mass fragments, so their angle distributions are close to each other. When b >6 fm, the fission model pertains to direct ternary fission [22] and the smallest fragment is from the neck of the nuclear system and its angle distribution stays away from those of large and medium fragments.Figure 2
Correlation betweenθ and the impact parameter. The black-square, red-dot and blue-triangle lines express separately the angle of θ about the maximum, medium and minimum fragments in the ternary reaction.Figure3 displays the dependence of the ratio of ptrans and plong on the impact parameter in 197Au+197Au system at 15 MeV/nucleon. The physical quantities of ptrans and plong are separately the transverse and longitudinal momentums of nuclear reaction system. The longitudinal momentum (plong) of the reaction system is along the incident beam, and the transverse momentum (ptrans) is perpendicular to the direction of the beam. In Figure 3, it is shown that black-square curve is falling down with the rising impact parameter, and the value of ptrans/plong approximately changes from 0.024 to 0.008. Based on this result, we can infer that the the transverse momentum has a small percentage of the total momentum of the system. But the percentage of the transverse momentum in the small impact parameter scope, such as b =0-4 fm, is approximate three times for that of the transverse momentum in the great impact parameter range, such as b =9-12 fm. From the characteristics of the black-square line in Figure 3, we can conclude that the quantity of the transverse momentum transferred from the incident beam of the system is reduced with the impact parameter increasing. Moreover, we have compared the slopes of a curve in small with that in great impact parameter ranges, and observed that the impact parameter 6 fm is cut-off point. Before this point the curve descends flat, and after that point it declines precipitously. It seems the mechanisms of the ternary reactions at small and great impact parameter ranges are imparity.Figure 3
Correlation between the ratio of ptrans and plong and the impact parameter.Normally stable nuclei have the same number of protons and neutrons, and the ratio of (n/z) will tend to stabilize at 1.0, as shown by the beta stability line. When the nucleus become larger, both the numbers of protons and neutrons in the nucleus will increase, and the number of neutrons has risen more than the number of protons, for providing the attractive force to balance the Coulomb repulsion between protons. Then the ratio of (n/z) of larger nucleus will be greater than 1.0, and which will lead to the growth of the binding energy of the nucleus and become unstable.Figure4 displays the ratio of neutrons to protons (n/z) about the fragments in the ternary fission of 197Au+197Au system at 15 MeV/nucleon is depending on the impact parameters. In this figure, the black-squares, red-circles and blue-triangles lines, respectively, represent the ratios of n/z for the biggest, the middle and the smallest fragments at the impact parameters from 0 to 12 fm. As the biggest fragment A1, the values of blue-triangles line are maximum ranged from 1.45 to 1.48, then the values of red-circles line on behalf of the middle fragment become smaller about between 1.42 and 1.45, the last the values of the black-squares line representing the smallest fragment in the ternary reaction are the lowest varying from 1.38 to 1.42. The results of Figure 4 show that the n/z ratio is strongly dependent on the collision parameters. As the similar characteristics with the Figures 1-3, it is special when the parameter is 6 fm. We can find that the slopes of the curve are getting higher after that point, and also the blue-triangles line has a lower slope than the red-circles line. Especially the slope of the black-squares line shows grows up from b =4 fm, reaches a maximum at b =8 fm and then falls down until b =11 fm, then it goes up again.Figure 4
The dependence of the ratio of neutrons to protons on the impact parameter in197Au+197Au system at 15 MeV/nucleon.In our previous paper [22], we have also found the curves describing the properties of the ternary reactions with a inflection point at b =6 fm. Add to that the result of Figures 1-4, we assume that the fission mechanisms of ternary reaction are different between b =0-6 fm and b =7-12 fm. When the impact parameter is less than 6 fm the fission mechanism belongs to deep inelastic scattering, there are a lot of nucleons and energy transfer between target and projectile nuclei; and in the range of the impact parameter is greater than 6 fm, it belongs to semi-peripheral or peripheral collisions, there is a few exchange on nucleons and energy in reaction system. Under these two microscopic mechanisms, the nucleus-nucleus interaction force of the nuclear reaction system should be significantly different.


## 4. Summary

The characteristics of fragments in the ternary fission of the heavy197Au+197Au system at the incident energy of 15 MeV/nucleon have been researched through the ImQMD model. Firstly, we have found their proton numbers depending on the impact parameter in Figure 1, and observed that the impact parameter b =6 fm was a special point. Before this point the curves of A1, A2 and A3 were horizontal line, and after that point the proton numbers about the fragments of A1 and A2 increased sharply with the impact parameter rising. In order to deep research, the angular distributions for A1, A2 and A3 of ternary fissions were analyzed. It was found that the distributions of the angle θ about the fragments A1, A2 and A3 were strongly dependent on the impact parameters. The angle distribution curves about A2 and A3 appeared flat around 30° and slow down at b =4-7 fm. The value of the curve about A1 distributed in a narrow area and the average value was near 24°. Also the impact parameter b =6 fm presented obvious different. After this point, both the distributions of θ about A1 and A2 were rising and that of A3 was falling. In order to analyze the reason for the change above, we have investigated the dependance of ptrans/plong on the impact parameter and found the ratio of ptrans/plong reduced with the impact parameter increasing. Finally, in Figure 4 the red-circle line representing the ratio of n/z about the fragment A1 became rising quickly after b =6 fm, and the blue-triangle line on behalf of the ratio of n/z about the fragment A2 was also going up but slowly. Meanwhile the black-square line began to fall down for the mass number of the third fragment got smaller. These results have indicated this breakup mechanism of ternary fissions on central and semi-central collision is different with that on the peripheral collision.At last, we have summarized the correlation between the impact parameter and the quantities of Z,n/z, θ and ptrans/plong about the fragments produced by ternary fission. The results were shown that the ternary fission system of 197Au+197Au at 15 MeV/nucleon had different mechanisms at central and peripheral impact parameter regions, and the cut-off point is at the impact parameter b =6 fm. Under different fission mechanisms, the interaction forces of the nucleons in the nuclear reaction system are different, for example, the symmetry energy term, which corresponds to the isospin effect of the nucleus. Studying the characteristics of nuclear products, such as the ratio of n/z may be a possible way to analyze the interaction forces of the nucleons. In the next step, we will study the isospin effect of the trifission products under the same impact parameter, and further analyze how much the symmetry energy is affected by the isospin asymmetry of the nucleus.



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