## 2 Materials and Methods
### DTF and STF Differences in Energy and Angular Distributions
#### 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 (\(\theta_{3}>90^{\circ}\)) when the two heavier fragments move at small angles in the forward (\(\theta_{1}=-50^{\circ}\)-\(0^{\circ}\) and \(\theta_{2}=0^{\circ}\)-\(50^{\circ}\)). 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 \(\theta_{1}=-40^{\circ}\), \(\theta_{2}\) and \(\theta_{3}\) would be more than \(100^{\circ}\) 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 \(70^{\circ}\). Therefore, to collect STF events very stringent constraints should be considered for positioning the detectors. Figure 5: Emission angles of light fragment \(\theta_{3}\) as a function of the heavy (\(\theta_{1}\)) and medium (\(\theta_{2}\)) mass angles in the laboratory system calculated considering the DTF (**a**) and STF (**b**) mechanisms. In the calculations, \(A_{1}\geq A_{2}\geq A_{3}\) and two of the three fragments are doubly magic nuclei. See the text for more details. .2.3 Disentangling of DTF and STF in the \({}^{132}\)Sn + \({}^{68}\)Zn + \({}^{48}\)Ca Tripartition  
To gain insight on how to optimize the experimental conditions to collect fragments from TTF, the energy and angular distributions of the tripartition \({}^{132}\)Sn + \({}^{68}\)Zn + \({}^{48}\)Ca 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 (\({}^{68}\)Zn) and heavy (\({}^{132}\)Sn) 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 (\({}^{48}\)Ca) result much larger in DTF. The kinetic energy of \({}^{48}\)Ca fragments in DTF can go up to 350 MeV, and the maximum probability is at around 180 MeV, whereas in STF, the \({}^{48}\)Ca 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.