Search for True Ternary Fission in Reaction ⁴⁰Ar + ²⁰⁸Pb

Abstract

True ternary fission, the fission of a nucleus into three fragments of nearly equal mass, is an elusive and poorly known process influenced by shell effects. An increase in the probability of this process with respect to binary fission, which is very low in spontaneous and neutron-induced fission, has been envisaged. Heavy-ion-induced reactions are adopted due to the possibility of an increase in the fissility parameter and the excitation energy of the compound nuclei. Nuclei with mass number around A = 250, accessible in heavy-ion-induced reactions, are favorable and should be investigated. It is still debated if the process takes place in a single step, direct ternary fission, or in a two step, sequential ternary fission. The purpose of this work is to define experimental conditions and observables that allow the disentangling of the products from the direct and sequential ternary fission, as well as from the usual most probable binary fission. This step is essential for gaining insights into the ternary fission dynamics and the binary to ternary fission competition. The method proposed here is for simulating the kinematics of the ternary and binary fission processes to compute the energy distributions and angular correlations of direct and sequential ternary fission products, as well as those of binary fission. The reaction taken as a benchmark is ⁴⁰Ar + ²⁰⁸Pb at 230 MeV and is supposed to form the ²⁴⁸Fm* compound nucleus. The simulation results have been filtered by considering the response function of a multi-coincidence detection system virtually constructed using the Geant4 simulation toolkit. The simulations support the possibility of separating the products of different multimodal fission decays with the proposed setup that consequently represents an effective tool to obtain insights into ternary fission from the observables selected.

1. Introduction

The process of induced nuclear fission, namely the division of a nucleus into two lighter nuclei, the so-called binary fission (BF), was first proposed by Meitner and Frisch [1] as an attempt to interpret the data from O. Hahn and F. Strassmann [2,3] who obtained bombarding Uranium with neutrons. The qualitative explanation of the induced fission process was provided within the framework of the Liquid Drop Model (LDM), which was introduced a few years earlier by Bohr [4]. In 1939, Bohr and Wheeler formalized Meitner’s and Frisch’s intuition in their seminal paper [5].

Fission can also take place spontaneously, as was later discovered by G.N. Flerov and K.A. Petrzhak in 1941. In the case of nuclear reactions induced by heavy ions, projectile and target may fuse first to form a compound nucleus (CN), which later on may split into lighter nuclei, the so-called fission fragments, if energetically allowed.

Shortly after the discovery of binary fission, the breakup of heavy nuclei into three fragments, ternary fission (TF), was still theoretically predicted on the base of the LDM [6,7]. LDM calculations involving only the initial and final states indicate that by increasing the CN charge, TF becomes indeed energetically possible. In particular, the energy release is higher than that from BF if the number of fission fragments of fairly equal mass is bigger than 2 [8,9]. Experimental evidence of TF, consisting of the evidence of long-range alpha particles emitted in connection with spontaneous fission, was first presented by Alvarez and collaborators [10]. In the same period, ternary and quaternary fission, consisting of the emission of much heavier light fragments with mass up to 32 atomic mass units, was identified by San-Tsiang, Zah-Wei, Chastel, and Vigneron [11,12].

Many experimental works have been devoted to the study of spontaneous and neutron-induced fission, i.e., in nuclei, at low excitation energy. In these studies, it has emerged that mostly alpha particles are emitted as a ternary particle (TP) in a plane perpendicular to the fragment separation axis, with some degree of distortion toward the light fragment direction due to the Coulomb focusing of the heavier fragment [13,14]. Furthermore, the average alpha particle energy is nearly constant, around 16 MeV [15], but disagreement has been reported for the TF/BF ratios, e.g., in the spontaneous fission of ²⁵²Cf $\approx 0.1\%$ [16] and $0.24\pm 0.02\%$ [17] were obtained in different measurements.

Still, further efforts are necessary to consolidate the present knowledge about ternary fission. By using heavy-ion-induced reactions, heavier nuclei with higher fissility parameters and excitation energies become accessible. This change of the entrance channel conditions largely affects not only the TF but also other competing processes accompanied by alpha-particle emission, for instance, the quasifission and pre-theramalization stages of the formation of the compound nucleus [18,19,20]. The analysis of double-differential $\alpha$ spectra, measured in coincidence with two fission fragments from very heavy composite systems, evidenced alpha multiplicities much larger than those expected from extrapolation of the TF data [18]. These observations support the view that the alpha particles appear from nuclear matter fluctuations occurring in the neck of the fissioning system or from the CN before the scission and cannot be used to probe the potential energy surface (PES) at the scission configuration [21,22].

The breakup of a nucleus into three fragments of nearly equal mass $(A_1\simeq A_2\simeq A_3)$ is known as true ternary fission (TTF). This process may occur following one of two different paths, which is to be distinguished experimentally: (i) In one step, known as direct ternary fission (DTF) [23]; (ii) In two subsequent binary breakups, known as sequential ternary fission (STF) [24]. In both cases, three fragments emerge, but the kinematics is expected to be different. In STF, the first step is an asymmetrical binary fission (AsymBF). It produces one light and one heavy fragment. If the heavy fragment excitation energy is sufficient to overcome its fission barrier it, in turn, splits into two fragments within a time scale of ${10}^{-20}$ s [23]. The mixing of STF and DTF events is one of the major issues in the experimental observation of TTF.

Experimental evidence of the TTF has been claimed since the 60s [25]. A review of early experimental results can be found in ref. [23]. The production of ternary fragments, such as F, Ne, Na, Mg, Al, and Si isotopes, up to A = 35, has been identified by Gönnenwein et al. [26] in neutron-induced fission on ²⁴²Am target. However, TTF probability turned out to be very low. Typical values are those observed in the thermal neutron induced fission of ²³⁵U, where a TTF event occurs every $6.7(\pm 3.0){10}^6$ BF [27]. The low excitation energy seems to be a hindering factor of the TTF probability that, in addition, decreases exponentially with the increase in ternary fragment mass [28,29]. Because of this, TTF is still a poorly known process and its occurrence in superheavy nuclei (SHN) is not experimentally confirmed [21,30,31,32]. Yet, it is worth mentioning that the search of TF was mainly performed in spontaneous and neutron-induced fission, which gives access to fissioning nuclei with a neutron to proton ratios nearly constant and with atomic numbers in the narrow region Z = 90–98 [33,34,35,36,37,38,39,40].

The conditions favoring the breakup in more than two fragments were suggested by Swiatecki [8]. Within the framework of the liquid drop model, the probability of true ternary fission increases rapidly as ${\mathrm{Z}}^2/\mathrm{A}$ increases [8]. Many models have been proposed, including shell effects in the calculation of the PES. According to Diehl et al. [9] TTF is hindered by a double saddle path where the second saddle is higher in energy. With the increase in the charge of the nucleus, the second saddle tends to vanish and eventually disappear, making TTF accessible. Theoretical studies on ²⁵²Cf [41,42], and the giant nuclear system ²³⁸U + ²³⁸U [29] also indicate similar shell effects for TTF. Minima for tripartitions, which is composed by magic nuclei, appear in the PES proposed in ref. [43,44]. These indications are supported by recent observations of TTF in the spontaneous fission of ²⁵²Cf and in neutron-induced fission of ²³⁵U [43,45,46,47,48]. In both cases, the observation of TTF-producing nuclei with masses close to the magic ¹³²Sn, ⁷⁰Ni, and ⁴⁸Ca isotopes was claimed [45,46].

Further experimental results can be found in [25,49,50]. Particularly interesting are the systematics on ⁴⁰Ar-induced reactions [49,51] where one could expect a TTF/BF ratio up to 3% [51] and a cross section of about 0.5 mb for TTF, which has a ternary fragment with $A>23$.

By using heavy-ion-induced reactions, it would be possible to test the above conditions that are supposed to ignite TTF by changing the neutron to proton ratio of the fissioning nucleus and its excitation energy. However, any experiment devoted to the detection of the TTF has to deal with the overlap of DTF and STF fragments. Therefore, an analysis of the possible detector configurations has to be planned carefully.

This article aims to establish the guidelines to disentangle the products of DTF and STF mechanisms by analyzing the angular correlations and energies expected for TTF fragments to be identified in mass and charge. The reaction ⁴⁰Ar + ²⁰⁸Pb at the beam energy of 230 MeV, producing ²⁴⁸Fm* (${\mathrm{Z}}^2/\mathrm{A}$ = 40.3), is chosen as a case study to search for TTF. The beam energy is chosen so to keep the excitation energy low enough to be able to be sensitive to shell effects. The same reaction was exploited by Price et al. [52] by using mica and glass as a solid material to detect heavy ion tracks and rebuild the kinematics. They claim a value of $4\times {10}^{-3}$ for the ratio TTF/BF. The present availability of modern devices well suited for measuring the kinetic energies and performing the (A,Z) identification of heavy fragments over a wide mass range [53,54,55,56,57] motivates this study.

The article is organized as follows. In Section 2.1 and Section 2.2, we briefly highlight, for both DTF and STF processes, the kinematics equations and the resulting angular and energy distributions built on conservation laws. In Section 2.3, the asymBF and symmetric BF (symBF) events are discussed. A possible apparatus and computed observables for STF, DTF, and BF decays, by considering the events collected in 1 week of beam time, are described in Section 2.4 and Section 3, respectively. Finally, remarks and conclusions are given in Section 4.

2. Materials and Methods

2.1. Kinematics of DTF and STF Decay Mechanisms

Ternary fission fragments originated during DTF and STF decays are characterized by different kinematics given the single or dual step nature. Among all the possibilities trajectories, only those belonging to the same plane have been taken into account. A schematic view of both processes is shown in Figure 1.

Hence, the velocity vector of each fragment is described by 2 variables: the absolute value of velocity and the polar angle ($\theta$). Known a priori are the projectile (${\mathrm{A}}_{\mathrm{p}}$, $Z_p$) nucleus, the target (${\mathrm{A}}_{\mathrm{t}}$, $Z_t$) nucleus, and the energy of the projectile that impinges on a fixed target ($E_p$). For a detailed description of the evolution of the reaction, the mass and the atomic numbers of the fragment have to be fixed by assigning mass values $M(A_i,Z_i)$. From now on, we consider known the masses of the fragments, which we will indicate in synthetic form, not making explicit reference to the dependence on the atomic number and on the mass number, namely $M(A_i,Z_i)=M_i$ with $i=1,2,3$. $A_{23}$ is the mass number of the intermediate fragment in STF. The actual mass values are taken from [58] and the Q values of the process involved in DTF and STF can be computed.

By taking into account the energy and momentum conservation laws, we deduced the equations describing the DTF and STF events for fixed tripartitions. In all calculations, we assume that the fragments are cold, meaning that the excitation energy of the system is fully converted into kinetic energy of the fragments. We made this assumption at each separation stage by neglecting the amount of energy dissipated by particle evaporation and/or transformed in deformation of the fragments.

Assuming that all fragment velocity vectors belong to the same reaction plane, nine variables (mass, energy, and angle of each fragment) describe the kinematics of the exit channel. By fixing the mass tripartition ($M_1$, $M_2$, $M_3$), the kinetic energy and the emission angle of fragment 1 ($E_1$ and ${\theta }_1$) and the emission angle of fragment 2 (${\theta }_2$), and by considering the energy and momentum conservation laws, the remaining three quantities, namely the kinetic energy of fragment 2 ($E_2$) and the kinetic energy and the emission angle of the fragment 3 ($E_3$ and ${\theta }_3$), are calculated. The angles are given with respect to the beam direction and are positive in the anticlockwise direction.

2.1.1. One-Step Decay: Direct Ternary Fission

The kinematic plot for DTF is shown in the top panel of Figure 1. The tripartition occurs suddenly. Energy conservation for three body decay allows having a velocity of the third fragment in the following way:

$$\begin{array}{c}\hfill v_3={\left[{\displaystyle \frac{2(E_p+Q)-M_1{v}_1^2-M_2{v}_2^2}{M_3}}\right]}^{1/2}.\end{array}$$

(1)

The Q-value in the above equation is defined as

$$\begin{array}{ccc}& & Q=M_p+M_t-\sum _{i=1}^3{M}_i,\hfill \end{array}$$

where $M_p$, $M_t$, and $M_i$ are the masses of the projectile, target and of the i-th product nucleus, respectively.

By considering momentum conservation law, the angle of the third fragment and the velocity of the second fragment can be expressed, respectively, as follows:

$$\begin{array}{ccc}& & {\theta }_3=co{s}^{-1}\left[{\displaystyle \frac{M_p{v}_p-M_1{v}_{1}cos\left({\theta }_1\right)-M_2{v}_{2}cos\left({\theta }_2\right)}{\sqrt{M_3\{2(E_p+Q)-M_1{v}_1^2-M_2{v}_2^2\}}}}\right]\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & v_2=-{\displaystyle \frac{M_1{v}_{1}sin\left({\theta }_1\right)+M_3{v}_{3}sin\left({\theta }_3\right)}{M_{2}sin\left({\theta }_2\right)}}\hfill \end{array}$$

(3)

So, replacing the values of $v_3$ and ${\theta }_3$, the solutions of $v_2$ are as follows:

$$\begin{array}{c}\hfill v_2={\displaystyle \frac{-\beta \pm \sqrt{{\beta }^2-4\alpha \gamma }}{2\alpha }},\end{array}$$

(4)

where

$$\begin{array}{ccc}& & \alpha =M_2(M_2+M_3)\hfill \\ & & \beta =2M_2\{M_1{v}_{1}cos({\theta }_1-{\theta }_2)-\sqrt{2E_p{M}_p}cos\left({\theta }_2\right)\}\hfill \\ & & \gamma =M_1{v}_1(M_1{v}_1+M_3(v_1-2(E_p+Q))-\sqrt{8E_p{M}_p}cos\left({\theta }_1\right))+2M_p{E}_p\hfill \end{array}$$

Now, if we substitute the value of $v_2$ found in term of known parameters in the $v_3$ and ${\theta }_3$ expressions we obtain the three equations for $E_2$, $E_3$ and ${\theta }_3$ as function of the three input parameters ($E_1$, ${\theta }_1$ and ${\theta }_2$) for each ternary fragmentation.

2.1.2. Two-Step Decay: Sequential Ternary Fission

In the first step, the compound nucleus of mass number $A_{CN}$ breaks into two fragments: ($A_{23}$ and $A_1$). The Total Kinetic Energy (TKE, the sum of the kinetic energy of the two fragments ($A_{23}$ and $A_1$)) is calculated considering no excitation of the fragments. The velocity plot for STF is shown in the bottom panel of Figure 1. At each decay step, the fragments are emitted on the opposite sides in the center of mass of the fissioning nucleus. The velocities of the fragments in the lab frame are obtained by summing up the cm velocities with the proper velocity of the source in the cm frame, namely ${\overrightarrow{v}}_{cm}$ in the first step and ${\overrightarrow{v}}_{23,cm}$ in the second one.

We now analyze the kinematics of this two-step TF. Before the collision, the center-of-mass velocity is given by the following:

$$\begin{array}{c}\hfill v_{cm}={\displaystyle \frac{M_p{v}_p}{M_p+M_t}}={\displaystyle \frac{\sqrt{2E_p{M}_p}}{M_p+M_t}}.\end{array}$$

By using the momentum conservation law along the axis identified by the beam vector and the orthogonal one, we have the following:

$$\begin{array}{ccc}& & M_1{v}_{1}cos\left({\theta }_1\right)+M_{23}{v}_{23}cos\left({\theta }_{23}\right)=(M_1+M_{23})v_{cm}\hfill \\ & & M_1{v}_{1}sin\left({\theta }_1\right)+M_{23}{v}_{23}sin\left({\theta }_{23}\right)=0\hfill \end{array}$$

From this, we obtain the following:

$$\begin{array}{ccc}& & {\theta }_{23}={cot}^{-1}\left[cot\left({\theta }_1\right)-{\displaystyle \frac{(M_1+M_{23})v_{cm}}{M_1{v}_{1}sin\left({\theta }_1\right)}}\right]\hfill \\ & & v_{23}=-{\displaystyle \frac{M_1{v}_{1}sin\left({\theta }_1\right)}{M_{23}sin\left({\theta }_{23}\right)}}\hfill \end{array}$$

By recursively knowing the initial condition of the second fissioning nucleus ($A_{23}\to A_2+A_3$) and by using both the energy and momentum conservation laws, we obtain the following:

$$\begin{array}{ccc}& & M_2{v}_{2}cos\left({\theta }_2\right)+M_3{v}_{3}cos\left({\theta }_3\right)=M_{23}{v}_{23}cos\left({\theta }_{23}\right)\hfill \\ & & M_2{v}_{2}sin\left({\theta }_2\right)+M_3{v}_{3}sin\left({\theta }_3\right)=M_{23}{v}_{23}sin\left({\theta }_{23}\right)\hfill \\ & & {\displaystyle \frac{1}{2}}{M}_2{v}_2^2+{\displaystyle \frac{1}{2}}{M}_3{v}_3^2={\displaystyle \frac{1}{2}}{M}_{23}{v}_{23}^2+Q_2\hfill \end{array}$$

Thus, the angle and the velocity of the third fragment and the velocity of the second fragment are as follows:

$$\begin{array}{ccc}& & cos{\theta }_3={\displaystyle \frac{1}{M_3{V}_3}}\left[M_{23}{v}_{23}cos\left({\theta }_{23}\right)-M_2{v}_{2}cos\left({\theta }_2\right)\right]\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & v_3={\left[{\displaystyle \frac{M_{23}{v}_{23}^2+2Q_2-M_2{v}_2^2}{M_3}}\right]}^{{\displaystyle \frac{1}{2}}}\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & v_2={\displaystyle \frac{-\beta \pm \sqrt{{\beta }^2-4\alpha \gamma }}{2\alpha }}\hfill \end{array}$$

(7)

where

$$\begin{array}{ccc}& & \alpha =M_2(M_2+M_3)\hfill \\ & & \beta =-2M_2{M}_{23}{v}_{23}cos({\theta }_2-{\theta }_{23})\hfill \\ & & \gamma =M_{23}^2{v}_{23}^2-M_3{M}_{23}{v}_{23}^2-2M_3{Q}_2.\hfill \end{array}$$

Then, by substituting the value of $v_2$ in the expression of $v_3$ and ${\theta }_3$, the three equations for $E_2$, $E_3$, and ${\theta }_3$ as functions of the three input parameters ($E_1$, ${\theta }_1$ and ${\theta }_2$) for each ternary fragmentation are obtained.

2.2. DTF and STF Differences in Energy and Angular Distributions

The identification of the experimental signatures of the TTF mechanisms requires the selection of experimental conditions allowing to distinguish between DTF and STF on an event-by-event basis. We have simulated mass, charge, kinetic energy, and angular distributions of the fragments originating from these two mechanisms and searched for the experimental conditions for collecting valuable data. We consider the experimental feasibility of a measurement where the velocities of the three fragments are co-planar because angular momentum conservation arguments imposes this constraint (see discussions in [59,60,61]).

The ⁴⁰Ar + ²⁰⁸Pb at the ${\mathrm{E}}_{\mathrm{lab}}$ = 230 MeV reaction has been selected as it is addressed as a favorable pathway for TTF in theoretical works [52]. Furthermore, most of the features of the binary fission of this reaction were already well established in previous studies [62,63,64,65,66]. For instance, recently, ejectile production has been studied in mass, charge, and energy distributions and cross sections of the different transfer channels, leading to the production of neutron-rich heavy isotopes [62,63]. Moreover, the spontaneous fission of fermium isotopes has been studied in prompt neutron emission in ²⁴⁶Fm [66] and in $\alpha$ accompanied cold ternary fission in ²⁵⁷Fm [67].

First and foremost, we explored the possibility of TTF according to the Three Cluster Model (TCM) that pictures ternary fission of heavy nuclei as a clustering effect [41,42]. The Ternary Fission Potential (TFP) is computed from the values of the binding energies of the fragments, Coulomb interaction, and Yukawa plus exponential nuclear attractive potential among the three fragments. Out of a large number of possible three body tripartitions, the possible fragment combinations are reduced to a subset of 299,774 combinations by applying the condition $A_1\ge A_2\ge A_3$. In the model, the geometrical arrangement of the three fragments at the contact stage is limited to be collinear or equatorial [42,68]. For both scission configurations, the TFPs satisfying the condition $A_1\ge A_2\ge A_3$ have been calculated as a function of the three fragments atomic numbers ($Z_1$, $Z_2$, and $Z_3$) and are presented in Figure 2 as a Dalitz plot.

In collinear configurations, the region of absolute minima corresponds to the emission of a heavy fragment with $Z_1\approx 76$, $Z_2\approx 22$, and $Z_3\approx 2$. In the equatorial configurations, the region of minima, in addition to the very asymmetric masses common to the collinear configurations, includes also more symmetric configurations up to ($Z_1=Z_2=50$). However, the TFP is shown in both configurations as well as local minima for $Z_1$, $Z_2$, and $Z_3$ values of 50, 30, and 20, respectively, as indicated by arrows in Figure 2. These atomic numbers correspond to Sn, Zn, and Ca nuclei, which are the magic nuclei candidates for TTF in our investigation. In particular we focused on 8 TTF combinations. These combinations are characterized by having two out of three fragments as double magic nuclei to follow the indication of the calculations presented in [29,32,69]. The TFP of these combinations for collinear and equatorial configurations are presented in Figure 3. The deepest minimum corresponds to combinations with ⁴⁸Ca. Among the combinations including ⁴⁸Ca, the ¹³²Sn + ⁶⁸Zn + ⁴⁸Ca tripartition has been considered in more detail in Section 2.2.3 to present how the DTF and STF products can be experimentally separated.

2.2.1. Energy Distributions

The ternary fragment kinetic energies and emission angles in DTF and STF mechanisms are reported in Figure 4 and Figure 5. They are calculated according to the equations described in Section 2.1. As input parameters, we considered the following: (i) All mass and atomic numbers of the above-discussed three-fragment combinations to calculate nuclear masses and reaction Q-values; (ii) Kinetic energies ranging from 1 to 350 MeV with 1 MeV step and emission angles ranging from 0 to $-180°$ with $5°$ step for the heavy fragment $A_1$; (iii) Emission angles ranging from 0 to $180°$ with $5°$ step for the medium fragment $A_2$. Only configurations with fragments $A_1$ and $A_2$ emitted on opposite sides with respect to the beam direction have been selected. This constraint is chosen to simulate the detector geometry of a typical setup (see later in Section 2.4).

As shown in Figure 4, kinetic energies larger than 270 MeV appear only in DTF mechanisms. Since the kinetic energy is mainly of Coulomb origin, this effect is related to the strength of the repulsion at the specific stage at which the repulsion occurs and on the atomic number of the fragments. Clearly, for DTF the highest energy of the third fragment is consistent with the repulsion of the medium and heavy fragments with lowest kinetic energies.

For the STF, Coulomb repulsion acts during two steps. In the first step, Coulomb repulsion is responsible for the velocity of the fragments $A_1$ and $A_{23}$. The second step produces a repulsion of nearly equal strength. The configuration that guarantees very low energy of $A_2$ and $A_3$ is therefore not possible. The overlapped kinematics imposes further constraints. Consequently, the heavier fragment can reach energies up to 350 MeV, and the configurations with low $E_1$ and $E_2$ are not possible.

2.2.2. Angular Distributions

The decay mechanisms largely affect also the angular correlations of the reaction products. By considering only the ternary decay occurring in-plane the resulting emission angles in the laboratory system are presented in Figure 5. The fragment angle (${\theta }_3$), as function of the heavier fragment angles, ${\theta }_1$ and ${\theta }_2$, is computed when the fragments are emitted on opposite sides with respect to the beam direction. In DTF calculations, once the light fragment direction (${\theta }_3$) is fixed (i.e., a detector is mounted at some fixed angle), there exists an interval of ${\theta }_1$ and ${\theta }_2$ corresponding to it. By analyzing the DTF distribution in detail, we observe that the fragment $A_3$ is emitted at the backward angles (${\theta }_3>90°$) when the two heavier fragments move at small angles in the forward (${\theta }_1=-50°$–$0°$ and ${\theta }_2=0°$–$50°$). This behavior depends not only on the decay mechanism but also on the fact that the beam energy is not very high compared to the Coulomb repulsion energies involved in the process.

Completely different is the behavior of TTF events produced by the STF mechanism. Indeed, for a fixed ${\theta }_1$ value, the ${\theta }_2$ and ${\theta }_3$ values are very similar, and their ranges of variability are narrower than those observed in DTF calculations. Furthermore, the two heaviest fragments are never emitted at the same angle, i.e., if ${\theta }_1=-40°$, ${\theta }_2$ and ${\theta }_3$ would be more than $100°$ and vice versa. Obviously, this is a consequence of the two-step process and its kinematics, where the $A_1$ and $A_{23}$ fragments move apart in opposite directions in the center of the mass frame. The only exception occurs for the cases of ${\theta }_1=-{\theta }_2$ at around $70°$. Therefore, to collect STF events very stringent constraints should be considered for positioning the detectors.

2.2.3. Disentangling of DTF and STF in the ¹³²Sn + ⁶⁸Zn + ⁴⁸Ca Tripartition

To gain insight on how to optimize the experimental conditions to collect fragments from TTF, the energy and angular distributions of the tripartition ¹³²Sn + ⁶⁸Zn + ⁴⁸Ca has been taken as a reference.

The computed energy distributions of these three fragments are shown in the left and right panels of Figure 6 for DTF and STF, respectively. We observe that the kinetic energies of the middle (⁶⁸Zn) and heavy (¹³²Sn) fragments in most DTF cases can reach much lower kinetic energies (down to 1–50 MeV) with respect to those in STF, where the fragments are emitted with the maximum amplitudes at 120 MeV and 140 MeV, respectively. We underline that the peaks in the distributions are due to the facts the same energy can be obtained from different combinations of energies and angles of the other two fragments. Thus, being the total available energy the same for both processes, the energies of light fragments (⁴⁸Ca) result much larger in DTF. The kinetic energy of ⁴⁸Ca fragments in DTF can go up to 350 MeV, and the maximum probability is at around 180 MeV, whereas in STF, the ⁴⁸Ca kinetic energy does not exceed 260 MeV, and the maximum probability is reached at below 100 MeV.

Although these distributions show marked differences, they imply the simultaneous measurement of energy, mass, and atomic numbers over a large angular range if the aim is the disentanglement of the origin of the production mechanism. Thus, for planning an experimental investigation the limited angular coverage of the used detectors has to be taken into account.

In Figure 7, the angular correlations between ⁴⁸Ca and ⁶⁸Zn fragments are shown. The angular correlations are presented for DTF and STF for fixed values of the ¹³²Sn emission angle. We note that for ¹³²Sn emitted at the forward direction (${\theta }_1\ge -90°$), the same tern of angles does not correspond to both the DTF and STF events, i.e., they are separable simply by collecting coincidence events from detector placed at different angles. However, to define the mechanism responsible for the TF it is convenient to perform measurements in which we can collect products of both mechanisms simultaneously and exploit other observables to disentangle each contribution.

The results shown in Figure 7, for a specific tripartition, reflect what has been previously shown in Figure 5, i.e., wider distributions correspond to DTF whereas, in STF, only very narrow regions are found for the ⁶⁸Zn and ⁴⁸Ca angles.

This is a general trend except for the configurations in which ${\theta }_1=-{\theta }_2$. Also, in this case, the ${\theta }_3$ distribution of DTF is narrower, as shown in Figure 8. Thus, with detectors covering only a small section of the solid angle, it would be possible to measure the ${\theta }_3$ distribution. Particularly interesting are the events collected in the two-coincidence mode from detectors in ${\theta }_1=-{\theta }_2=-70°$ configuration because they include middle and heavy fragments produced by both the DTF and STF mechanisms.

The dynamics of the DTF and STF processes also affect the energy of the fragments. At the separation stages, indeed, the different Coulomb repulsion drives the trajectories and determines the observed kinetic energy distributions. In Figure 9, we present the energies ⁶⁸Zn and ⁴⁸Ca for fixed ¹³²Sn angles (same values considered in Figure 7).

Being the angular distribution of DTF fragments broader a similar trend is expected for the kinetic energy intervals. Thus, we observe ⁶⁸Zn and ⁴⁸Ca kinetic energies distributed over a wide interval ranging from few MeV up to about 300 MeV and form 50 MeV up to 350 MeV (blue lines), respectively, whereas the narrower angular distributions of STF produce strong correlations between the Zn and Ca kinetic energies as indicated by the red lines. In Figure 9, we observe that the phase space of ⁶⁸Zn and ⁴⁸Ca energies for the STF and DTF are not overlapping except for a few points corresponding to the ¹³²Sn emitted at ${\theta }_1=-90°$, where an overlap in the angular distributions is also observed. Consequently, the kinetic energies of fragments are observables sensitive to the tripartition mechanism. The indications of the competition among the two ternary mechanisms, obtained by performing measurements of angular distributions, can be confirmed by measuring the energy spectra of a single fragment in coincidence with ¹³²Sn measured at a fixed angle.

The correlations presented so far allow us to identify guidelines for selecting observables and detector locations if exclusive measurements have to be planned to disentangle fragments from DTF or STF mechanisms.

In Figure 10 the ⁴⁸Ca angular and energy distributions, under the condition of angular symmetric emission of the medium and heavy fragment, i.e., ${\theta }_1=-{\theta }_2=-70°$, are presented. The ⁴⁸Ca fragments are emitted at $60°\pm 10°$ and $90°\pm 15°$ for DTF and STF, respectively. Furthermore, these events correspond also to well separated kinetic energy ranges 40–190 MeV and 240–310 MeV for STF and DTF, respectively, as shown in Figure 10b. Therefore, by including two detectors covering these angles a solid event-by-event identification of DTF and SFT products is possible.

2.3. Collection of Binary Fission Events

TTF is considered as a rare process compared to BF. Although the total energy released in ternary events is larger than that of BF, ternary probability is hindered by larger barriers. Sophisticated three-center shell model calculations [70] indicate a significant reduction of the ternary fission barrier with the increase in the nuclear mass. Furthermore, it is expected that this barrier becomes rather low (or vanishes completely) in the case of SHN. Therefore, by considering that the ternary fission decay can be rather probable in very heavy nuclei [29], it is interesting not only to determine what is the possible decay mechanisms taking place but also its occurrence with respect to binary fission. The simultaneous measurement of ternary and binary events would allow us to reach this goal.

In this contest, the ⁴⁰Ar + ²⁰⁸Pb reaction at 230 MeV is convenient because the folding angles goes from $140°$, for the symmetric fission, to $130°$, for the asymmetric fission involving the ⁷⁸Ni fragment, as shown in Figure 11.

The events of other possible asymmetric channels have not been plotted because they will overlap with BF, elastic, and QE events, whose fragment masses are very close to ¹³²Sn and ²⁰⁸Pb doubly magic nuclei.

BF events can be collected with two detectors symmetrical placed at $70°$ on the opposite side of the beam direction. By covering the angle at about $60°$ in coincidence with the detector at $-70°$ also the very asymmetric fission events would be collected.

2.4. Detection Apparatus

We present here an example of an arrangement of existing detectors, inspired by the correlations discussed above, to perform measurements aimed at highlighting the occurrence of DTF and STF.

The apparatus is shown in Figure 12. It consists of 4 detection elements mounted in the horizontal plane (our reaction plane) and centered around the target. The polar angles with respect to the beam direction (the z-axis) are positive anticlockwise. The four elements are named as follows:

(i)

A TOF arm at $-70°\pm 5°$;

(ii)

A TOF arm at $70°\pm 5°$;

(iii)

A string of 7 telescope detectors covering $50°$ and centered at $90°$;

(iv)

A system of a TOF arm combined with a squared array of telescope detectors at $60°\pm 5°$.

Each TOF arm consists of a start and a stop detector that is position sensitive and allows the ion to pass through with a relatively small energy loss. The telescope detector consists of two stages and is made by a thin (20 $\mathsf{\mu }$m) and a thick ($300\mathsf{\mu }$m) Si detectors. A string of 7 telescopes, mounted in a single raw, would be used for the construction of the detection element (iii). The same telescopes are mounted in a squared array downstream the stop detector of arm (iv) which will be able to collect therefore TOF and residual energy of the particles passing through.

Each telescope can be used to identify the charge of the lighter ion by exploiting the $\Delta E-E$ technique. The use of several telescopes in a raw (element (iii)) allows accessing the angular distributions over a wide angular range centered at $90°$. Example of expected angular and energy distributions for the light fragment in STF events are shown in Figure 10. At the forward angles, the narrower angular distribution calculated for the DTF events, instead, can be measured by exploiting the high spatial resolution of the TOF detector mounted upstream in the element (iv).

The TOF detectors, featuring the (i), (ii), and (iv) detection elements, can be realized by using two TOSCA units [71] coaxially mounted downstream and acting as start and stop detectors, respectively. The TOSCA units represent the core detection elements. These detectors, developed for a wide variety of experiments as those described in [72,73], have been successfully adopted for the study of binary fission and quasifission in 2023 and of multinucleon transfer reactions in 2024 at JYFL and GSI, respectively. Each unit measures the time and the (x, y) position of charged particles passing through a thin layer mounted orthogonal to the reaction plane. The TOSCA units, represented with triangular prisms in Figure 12, offer state-of-the-art spatial and time resolutions. By performing coincidence measurements between the two TOSCA units of each TOF detector, the velocity vectors of particles can be determined. By crossing thin layers (few $\mathsf{\mu }$m) of plastic material, the variations of trajectories and energies of particles are minimal, so further independent measurements can be performed. Thus, by mounting the telescope array downstream the stop detector in element (iv) we can also measure the mass, charge, and energy of light fragments with a single detection element.

To collect valuable data, an effective trigger scheme able to collect events produced by different decay processes should be arranged. A single event occurs when each sensitive part of a detection element produces a valid signal, i.e., the detectors of each element should be arranged in AND mode. Then, by considering the different number of products and distributions featuring them, the OR mode between the following event conditions can be adopted:

(a)

Coincidences between (i) and (ii) elements;

(b)

Coincidences between (i) and (iii) elements;

(d)

Coincidences between (i), (ii), and (iii) elements;

(e)

Single event form (iv) and (iii) elements.

The trigger conditions a and b will provide the data for separating BF and TF decays discussed in Section 2.3, while the trigger conditions c and d will be used to investigate the observables featuring the DTF and STF mechanisms, respectively, as the energy and angular correlations of the fission fragments. The condition e will be used to characterize fragments from the different processes and will be useful for the experimental measurements of the detection efficiency.

2.5. Rate Estimates

In order to evidence the differences among competing processes following the collisions, sufficient statistics have to be collected. The TTF involving lighter fragment mass $A>23$, for the ⁴⁰Ar + ²⁰⁸Pb reaction at 230 MeV, has been evaluated to be around 5 mb. This value is determined by considering a TTF to BF cross section ratio of about 0.7%, according to the ⁴⁰Ar-induced reactions systematics with a beam energy of 230 MeV on ²⁰⁸Pb target [49,51], and a fusion–fission cross section of 700 mb, calculated with the Bass model. Furthermore, we considered the sharing among the possible tripartitions energetically allowed. Although this is a very large number, by considering the increase in the probability of emitting two doubly-magic nuclei per fission decay and $A_3>23$, we can expect a branching of 1% for the ¹³²Sn + ⁶⁸Zn + ⁴⁸Ca tripartition. This tripartition, indeed, coincides with one of the deepest minima of TFP built by considering both DTF and STF mechanisms, as discussed in Section 2.2. So we can estimate a cross section of 50 $\mathsf{\mu }$b for this specific tripartition. This gross estimate is corroborated by the 9.4 $\mathsf{\mu }$b of ²⁸Mg in coincidence with two fission fragments measured in the ⁴He + ²³⁸U reaction at $E^{*}$ = 118 MeV [74]. This value, according to the systematics [23], is, indeed, expected to be lower than ours because it involves tripartitions with only one doubly-magic nucleus, and it is performed at lower beam energy and in a composite system with lower fissility parameter (${\mathrm{Z}}^2/\mathrm{A}$), 36.6 instead of 40.3 for the ⁴⁰Ar + ²⁰⁸Pb reaction.

Along with the production yield, the detection efficiency also has to be considered. The proposed setup is intended for the collection of both double and triple coincidence events. The double events are necessary for the BF yield and can be used to provide a description of the TTF by estimating the properties of missing fragments, as done in several previous experiments. However, a definitive identification of TTF requires the direct and simultaneous measurement of three fragments. By considering the experimental efficiency of 4% for double coincidence events, typical of the two-arm TOF spectrometer with similar size and intrinsic efficiency of those considered here, we can assume 0.8% as minimum detection efficiency for triple coincidence events. The deduced efficiency is in line with the value obtained in a very recent experiment performed with the TOF-spectrometer CORSET that was arranged for measuring ternary reaction products [54].

In conclusion, by considering the estimated cross section and detection efficiency, and assuming the use of a beam intensity of 20 pnA ($1.2\times {10}^{11}$ pps) on a ²⁰⁸Pb target $0.2\mathsf{\mu }\mathrm{g}/{\mathrm{cm}}^2$ thick, about 200 events of ¹³²Sn + ⁶⁸Zn + ⁴⁸Ca tripartition can be collected per day. Therefore, in the plots (Figure 13, Figure 14 and Figure 15), extracted by considering double and triple coincidence modes, 2000 events can be collected in 10 days of beam time provided by existing accelerator facilities such as LNL (Italy) or JYFL (Finland). The cross sections for BF, which are an order of magnitude larger, as well as the larger detection efficiency, makes the collection of thousands of BF events achievable in a few hours with the experimental conditions described above.

3. Geant4—Simulation Results

The geometry of the detection apparatus includes both active and passive elements. Both of these, along with the detector geometry, have been included in a simulation based on the Geant4 [75,76]. Geant4 is a versatile toolkit used to simulate the passage of particles through matter. It was developed for high energy physics to construct virtual experimental setups with detectors, and now it is used by a wider community. In this work, Geant4 was used to model a setup for the detection of ternary decays of composite systems. The simulation tracked the ions’ time of interaction, position, and energy loss in the detector, providing essential insights into their interaction with the detector materials. The G4EmLivermorePhysics class was employed, which is ideal for modeling electromagnetic interactions of ions with energies in the tens of MeV range. This class includes precise energy loss data for the ions and detection materials. The decay processes were managed using G4DecayPhysics class, which handled the potential radioactive decay of unstable ions. These combined models ensured accurate simulation of the ions’ interaction with the detector, including the tracking of their decay when applicable.

The aim is to filter the simulated events with a realistic example of a detection setup and evaluate the feasibility of an experiment. The complete simulated setup includes the four detection elements described in Section 2.4.

By including the resolution and efficiency of existing devices we determined the capability to separate events produced by the different processes taking place as well as the geometrical constraints in shaping such a detection apparatus.

3.1. Correlated Velocities in forward Detectors

According to the above discussion two TOF detectors placed at $-70°\pm 5°$ and $70°\pm 5°$ will measure the time and position of the particle passing through a thin ($190\mathsf{\mu }{\mathrm{g}/\mathrm{cm}}^2$) Mylar layer.

The time and flight path experimentally measured in similar devices with lower performances show Gaussian distributions featured by 150 ps and 2 mm full width at half-maximum (FWHM), see [53]. Therefore, the time and flight path values resulting from the simulations were smeared with Gaussian distributions whose widths correspond to the aforementioned FWHM values. The resulting velocity distributions ($v_1$ and $v_2$) with the typical Gaussian distributions are shown in Figure 13a,b. In the velocity spectra the peaks corresponding to the products of different mechanisms result as well separated. The separation and tagging of events results in being much more effective by simultaneously considering the two-velocity correlation plot, as indicated by the well separated loci shown in Figure 13c.

3.2. Identification of Light Ions at Backward Direction for STF Characterization

A string of two-stage telescopes centered at around $90°$ and covering a wide polar angle represents the element (iii). This detection element is mainly included for the characterization of STF events by measuring the angular and energy distributions of light fragments shown in Figure 10. The granularity of the array will provide the angular distribution, while the use of two-stage telescopes is motivated by the interest in both measuring the energies and identifying the Z of lighter fragments. Then, we have considered for the first stage a 20 $\mathsf{\mu }$m thick Si detector, which produces the ($\Delta E$) signal and only partially slows down the light ions, and for the second stage, a 300 $\mathsf{\mu }$m thick Si detector that is sufficiently thick for collecting the residual energies ($E_{res}$) of the ions expected. The dependence of the energy resolution of the Si detector on atomic mass (A) and energy (E) of heavy ions can be taken into account by considering the empirical formula proposed in [77]:

$$\begin{array}{ccc}& & {\displaystyle \frac{\Delta E}{E}}=5\times {10}^{-3}\left({\displaystyle \frac{1+0.01A}{\sqrt{\frac{E}{A}}}}\right).\hfill \end{array}$$

Accordingly, we considered an upper limit for our measurements of 900 keV, corresponding to ions with mass A = 50 at E = 300 MeV. The energy resolutions of detectors have been introduced in our simulations with the same approach described for the time and position resolutions of the TOF detector. The energy spectra at $90°\pm 15°$ for ions with masses ranging from A = 48 to A = 176 are shown in Figure 14a,b for the thin ($\Delta E$) and thick ($E_{res}$) detectors, respectively. The two peaks in the $E_{res}$ spectrum correspond to the light fragments of ⁴⁸Ca with energies larger than 100 MeV as those produced by DTF and STF decay. Indeed, only low Z ions can pass through the 20 $\mathsf{\mu }$m Si layer of $\Delta E$ detector. The heavier fragments from SymBF and AsymBF reactions, due the larger energy loss, will be stopped and their kinetic energies will be collected in the $\Delta E$ detector only.

The energy peaks of the fragments from different processes are well separated in the ($\Delta E$) spectra, see Figure 14a. Although the use of a single thin detector would be sufficient to disentangle them, the use of a second detector stage is important for two reasons: (i) By measuring the total kinetic energies of all products, it is possible to achieve a comprehensive description of the reaction mechanism without making any further assumption on the reaction channel and kinematics; (ii) By using the $\Delta E-E$ technique for the identification of the Z of the lightest fragments, it is possible to separate fragments having the same energy and mass but different Z. For instance, in Figure 16, we present a $\Delta E-E$ plot for several fragments of mass number A = 48 and energy ranging from 125 to 350 MeV.

3.3. Identification of Light Ions in Forward Direction for DTF Characterization

The detection element (iv) placed in the most forward direction consists of a TOF arm combined with a square array of two-stage telescopes. This detection element is required to collect the light fragments of DTF events when the heavier and middle mass are emitted at $-70°$ and $70°$, respectively. Therefore, to cover the maximum emission angle of the lightest fragments, as shown in Figure 10a, detection element (iv) is centered at $60°\pm 5°$. With this element, we intend to measure simultaneously and independently the velocity vectors and energies of the ions. The method adopted for simulating these quantities is the same used in the other detection elements, see Section 3.1 and Section 3.2. As before, we also consider the same values of time, position, and energy resolution for the active parts. By performing a measurement with this detection element, a clear identification of all ions is achievable. Also, in this case, the energy measurement obtained with the thin detectors would be sufficient to disentangle the products of different processes. This confirms that the trajectories after passing through the thin layers of TOF detectors are practically undisturbed. However, in order to obtain complete information on the angular distribution of these light ions emitted with a narrower distribution, the use of a TOF detector sensitive to the position would provide very valuable data. Furthermore, an even better clear separation can be easily reached by combining the information collected by using all sensitive elements. An idea of the separation reachable can be provided by looking at the $\Delta E-v$ matrix shown in Figure 15.

By considering the capability to perform a direct mass identification given by this element, i.e., obtained by the independent measurement of the velocity and energy of the ions, further indication can be achieved. For instance it will be possible to determine the masses most abundantly produced by all involved processes.

4. Discussion and Conclusions

The existing information on ternary fission has been mainly collected by studying the spontaneous and neutron-induced fission when the influence of shell effects are not damped by excitation energies. By studying nuclei in the region of masses around A = 250 large changes in mass and TKE distributions of binary decay products have been evidenced. These changes are related to the dynamical evolution of the processes. Also, the dynamical evolution of the ternary fission is still debated. The peculiarities of this region of nuclei, accessible in heavy-ion-induced reactions, due to the reduction of fission barriers could be favorable also for investigating the ternary fission.

In this work, we have studied the kinematics of the possible decay processes of ²⁴⁸Fm* compound nuclei produced in the ⁴⁰Ar + ²⁰⁸Pb reaction at 230 MeV with the aim to define the observables to distinguish DTF, STF, and BF events.

The fragments’ energy and angular distributions have been calculated for the most probable ternary fragmentations chosen by the corresponding minima in TFP surfaces. It is found that by fixing the emission angles of the two heavier fragments the resulting energy and angular correlations are very different in DTF and STF processes. To define specific experimental conditions, the ternary fragmentation ¹³²Sn + ⁶⁸Zn + ⁴⁸Ca, corresponding to a minimum value in the TFP surface due to the two doubly magic products, has been considered. The comparative analysis of the products’ angular and energy distributions has been considered for distinguishing the events originating from the different TF processes and symmetric and asymmetric binary fission events expected to be the main decay channels of the reaction. Thus, a multi-coincidence apparatus involving state-of-art detection elements, such as the TOSCA units and two-stage Si detector telescope array, has been proposed for collecting valuable data.

The calculations filtered by implementing a realistic apparatus response function within the Geant4 simulation toolkit framework have been used to simulate the experimental observables. The simulation results indicate that binary and ternary fission events can be simultaneously measured and clearly separated by exploiting the differences in the energy, velocity, and angular distributions of the fragments.

Regarding the estimated yields of TF, the efficiency of the proposed setup and the available beam intensity indicate the possibility of collecting valuable data by considering the systematics on ⁴⁰Ar-induced fission reactions. Then, in conclusion, the proposed experimental investigation seems to be well suited to evidence the TF in heavy-ion-induced reactions, determine the ratio between TF and BF occurrences, and progress in the knowledge of TF fission dynamics.

We do not exclude the possibility that during ternary fission, neutrons can be produced. It would be useful to measure their multiplicities because the ternary neutrons can feed the cycles of reactions used for nuclear energy production and neutron transport as summarized in [78,79].