Sequential fissions of heavy nuclear systems

In Xe+Sn central collisions from 12 to 20 MeV/A measured with the INDRA 4$\pi$ multidetector, the three-fragment exit channel occurs with a significant cross section. In this contribution, we show that these fragments arise from two successive binary splittings of a heavy composite system. Strong Coulomb proximity effects are observed in the three-fragment final state. By comparison with Coulomb trajectory calculations, we show that the time scale between the consecutive break-ups decreases with increasing bombarding energy, becoming compatible with quasi-simultaneous multifragmentation above 18 MeV/A.


Introduction
In central heavy-ion collisions at beam energies between 25 and 100 MeV/A, production of many nuclear fragments is observed. The fragment production is compatible with the simultaneous break-up of finite pieces of excited nuclear matter [1]. This so-called "nuclear multifragmentation" is a fascinating process which has been widely studied by the INDRA collaboration, notably in 129 Xe+ nat Sn central collisions [2][3][4][5][6][7][8][9][10][11]. But the energy required for the onset of multifragmentation is still an open question.
With the recent data on 129 Xe+ nat Sn reaction at energies between 8 and 20 MeV/A, Chbihi et al. [12] have shown that at the lowest beam energy (8 MeV/A) central collisions lead mainly to two fragments in the exit channel. From 12 MeV/A, the three-fragment exit channel becomes significant. However these fragments might be produced by sequential fission [13], ternary fission [14] or multifragmentation [1]. In this contribution, we determine the order and time scale of three fragment emission and show the evolution of the deexcitation process from hot sequential fission to multifragmentation.
arranged in 17 rings centered on the beam axis, covers 90% of the solid angle, and can identify in charge fragments from hydrogen to uranium with low thresholds.
In this analysis, we considered only fusion-like events with three heavy fragments (Z > 10) in the exit channel. The measured fragments in each event are sorted according to their atomic number such that Z 1 Z 2 Z 3 . This classification is introduced only to facilitate the presentation of the data.
3 From sequential to simultaneous break-up

Qualitative evolution
First we will show qualitatively the evolution of the decay process from two splittings well separated in time towards simultaneous fragmentation. If two successive independent splittings occur, three possible sequences of splittings have to be considered. For instance, in one sequence, the first splitting leads to a fragment of charge Z 1 and another fragment which, later, undergoes fission leading to Z 2 and Z 3 . Let us call this sequence 1. The sequences 2 and 3 are deduced by circular permutation.
Bizard et al. [13] proposed a method to show qualitatively the nature of the process. To test the compatibility of an event with the sequence of splittings i, we compare the experimental relative velocities with those expected for two successive fissions. For each event we build the following quantities: where i = 1, 2, 3; v exp αβ is the experimental relative velocity between fragments α and β; and v viola αβ is the expected relative velocity for fission, taken from the Viola systematic [16,17]. The first (second) term in Eq.(1) refers to the first (second) splitting. The lower the value of P i , the larger the probability of the considered event to have been generated by the sequence of splittings i. The three values of P i are calculated for each event and represented in Dalitz plots (Fig.1). In this diagram, the distance of each point from the three sides of the triangle reflects the relative values of P 1 , P 2 , and P 3 . At 12 MeV/A bombarding energy ( Fig.1(a)), events populate mainly three branches parallel to the edges of the Dalitz plot, which correspond to the three sequences of sequential break-up (P i P j , P k ). Simultaneous break-up events would be located close to the centre of this plot (P i ∼ P j ∼ P k ), where few events are observed. The strong accumulations of events on the corners correspond to particular kinematic configurations where we are not able to disentangle two sequences (P i ∼ P j P k ). Consequently, for this energy, fragments arise mainly from two sequential splittings. With increasing beam energy ( Fig.1(b-d)), the three branches are still present but become closer and closer to the centre of the Dalitz plot. This means that fragment production becomes more and more simultaneous. In other words, when increasing the beam energy the deexcitation process evolves continuously from two sequential splittings towards simultaneous fragmentation. In the following we will quantify this effect by measuring the time δt between the two splittings. First we must determine, event by event, in which order fragments have been produced.

Sequence of splittings
To identify the sequence of splittings, we only consider the second separation step. For each event, we compare the relative velocity between each pair of fragments with that expected for fission taken from the Viola systematics. The pair with the most Viola-like relative velocity is considered to have been produced during the second splitting. We can therefore trivially deduce that the remaining fragment was emitted first. This procedure amounts to computing, for each event, the three following values: which corresponds to the second term of Eq.(1). The smallest value of χ i determines the sequence i of splittings. Once the sequence of splittings is known event by event, fragments can be sorted according to their order of production and the intermediate system can be reconstructed. Let us now call Z f 1 and Z f 2 , the two nuclei coming from the first splitting. The heaviest fragment Z f 2 breaks later in Z s 1 and Z s 2 . Mean charges of all fragments are presented in Tab.1. For each reconstructed splitting, we also compute the charge asymmetry δZ i = (Z i 2 − Z i 1 )/(Z i 2 + Z i 1 ) . Mean charges and asymmetries are comparable for all beam energies (Tab.1). In addition, the mean asymmetry of the first splitting δZ f is significantly larger than the quasi-symmetric entrance channel. It is a strong indication that the first  Figure 3. Correlation between the inter-splitting angle and the relative velocity of the second splitting for: (triangles down) 12MeV/A, (circles) 15MeV/A, (triangles up) 18MeV/A, (squares) 20MeV/A. stage of the reactions is an incomplete fusion of projectile and target nuclei, leading to the formation of heavy composite systems with atomic numbers at least as large as the values of Z src (no attempt was made to correct fragment charges for pre-or post-scission evaporation of charged particles).

Decrease of the inter-splitting time
To measure the inter-splitting time, we used the correlation between the inter-splitting angle θ and the relative velocity of the second splitting: v s 12 = v s 1 − v s 2 (Fig.2). For long inter-splitting times the second splitting occurs far from the first emitted fragment. The relative velocity v s 12 is then only determined by the mutual repulsion between Z s 1 and Z s 2 and should not depend on the relative orientation of the two splittings. In other words, for long inter-splitting times v s 12 should be independent of θ. However, for short inter-splitting time the second splitting occurs close to the first emitted fragment. The relative velocity v s 12 is modified by the Coulomb field of Z f 1 and depends on the relative orientation of the two splittings. In this case, v s 12 should present a maximum for θ = 90˚. We used this Coulomb proximity effect as a chronometer to measure the inter-splitting time δt.
The experimental correlation between v s 12 and θ is presented in Fig.3, for all beam energies. These correlations present a maximum at θ ∼ 90˚, which increases with increasing beam energy. We quantify this effect by the Coulomb distortion parameter δv = v s 12 (90 • ) − v s 12 (0 • ), which increases with the beam energy (Fig.4). It indicates that the second splitting occured closer and closer to the first emitted fragment.
To translate δv in terms of inter-splitting time δt, we performed simple Coulomb trajectory calculations for three fragments using mean charges given in Tab.1. We simulated sequential break-ups and we computed δv by varying δt to get a calibration function. Finally, we obtained the evolution of the inter-splitting time as a function of the beam energy (Fig.5). At 12 MeV/A, δt is of the order of 2 × 10 −21 s and decreases by a factor eight over the studied bombarding energy range. Our trajectory calculations show that below δt ∼ 0.5 × 10 −21 s it is no longer meaningful to speak of sequential fission. Indeed, the two nuclei resulting from the first splitting do not have sufficient time to move apart beyond the range of the nuclear forces before the second splitting occurs. This inter-splitting time is reached around 18 MeV/A. Therefore, the decrease of δt with increasing beam energy shows the continuous evolution of the decay mechanism, from hot sequential fission toward multifragmentation.

Conclusion
In this contribution, we have investigated the three-fragment exit channel in 129 Xe+ nat Sn central collisions from 12 to 20 MeV/A. These fragments arise mainly from two successive splittings which are compatible with sequential fissions of heavy composite systems. We estimated the time between the two successive fissions by Coulomb chronometry. Starting from δt ∼ 2 × 10 −21 s at 12 MeV/A, the inter-splitting time decreases by a factor eight over the studied bombarding energy range, becoming compatible with simultaneous multifragmentation above 18 MeV/A.