## 2 Materials and Methods
### 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 \({}^{40}\)Ar + \({}^{208}\)Pb at the E\({}_{\rm lab}\) = 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 \({}^{246}\)Fm [66] and in \(\alpha\) accompanied cold ternary fission in \({}^{257}\)Fm [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 \(A_{1}\geq A_{2}\geq A_{3}\). 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 \(A_{1}\geq A_{2}\geq A_{3}\) 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 \({}^{48}\)Ca. Among the combinations including \({}^{48}\)Ca, the \({}^{132}\)Sn + \({}^{68}\)Zn + \({}^{48}\)Ca tripartition has been considered in more detail in Section 2.2.3 to present how the DTF and STF products can be experimentally separated.