Angular momentum of doubly magic 132 Sn ﬁssion product : experimental and theoretical aspects

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Introduction
Nuclear fission is such a complex phenomenon that despite more than 80 years of intensive research, we are still unable to predict the main observables of fission from a consistent and robust fission model. For example, the question of how fission fragments acquire their spin (total angular momentum) is still unclear, as demonstrated by the recent workshop on this topic [1]. This question is however important because the spin of the fission fragments and the available excitation energy at scission affect the de-excitation process and has therefore an impact on the characteristics of the emitted prompt neutrons and gamma rays.
In this paper, we focus on the spin of the doubly magic 132 Sn nucleus produced after neutron evaporation in the case of the thermal-neutron-induced fission of 235 U and 241 Pu. In the first section, we will briefly recall how the spin of 132 Sn was determined as a function of its kinetic energy thanks to the isomeric ratio measurements. In section 2, we discuss the role of the intrinsic excitation energy available at scission on the mechanism of spin generation. * e-mail: olivier.serot@cea.fr From these considerations, we investigate the average spin of 132 Sn as a function of the incident neutron energy during a low-energy fission reaction (before the second-chance fission).

13Sn total angular momentum as a function of its kinetic energy
Isomeric Ratio measurements One way to assess the spin of fission products is to measure their isomeric ratio, as proposed a long time ago by Huizenga and Vandenbosch [2,3]. So, in the frame of the collaboration between the 'Laboratoire de Physique Subatomique et Corpusculaire' (LPSC, France), the 'Institut Laue Langevin' (ILL, France) and the CEA-Cadarache (France), experimental campaigns were performed on the LOHENGRIN recoil mass spectrometer of the Institut Laue-Langevin with the aim of measuring, after prompt neutron emission, the dependence of the isomeric ratio with the kinetic energy (KE). The isomeric ratio is defined by: where R 132m S n and R 132gs S n are respectively the production rate of the isomeric state and ground state. In the present study, the isomeric state is characterized by its energy E 1 =4.847 MeV, its spin parity J π =8 + and its half life τ=2 µs, while the ground state is characterized by its energy E=0 MeV, its spin parity J π =0 + and its half life τ=39.7 s. The experiment on the 235 U(n th ,f) reaction was performed in 2015 and is reported in Ref. [4], while the experiment on the 241 Pu(n th ,f) reaction was carried out in 2019 (see Refs. [5,6]). Results on both reactions are plotted in Fig. 1. In this figure, the 132 Sn kinetic energy (KE) distributions (after prompt neutron emission) measured thanks to the mass spectrometer LOHEGRIN are also shown. By weighing IR(KE) with this KE-distribution, we obtained the average isomeric ratio IR which is reported in Tab. 1.

Spin assessment
In order to assess the spin of the 132 Sn fission product from our measured isomeric ratio, we use the Monte Carlo FIFRELIN code [7,8]. This code is capable to simulate the de-excitation of a nucleus by means of the Hauser-Feshbach statistical theory [9]. Hence, for an initial state defined by its excitation energy and spin-parity . Schematic description of the γ-decays simulation performed with the FIFRELIN code. The calculation starts from a given state (below S n and above E 1 ) characterized by an excitation energy E * and a spin-parity J π .
(E*, J π ) as shown in Fig. 2, the probability to feed the isomeric and the ground states can be calculated and the IR can be deduced. Since we want to reproduce the measured IR after prompt neutron emission, FIFRELIN calculations were limited to the following energy range (by step of 0.5 MeV): E 1 < E * < S n where (E 1 is the energy of the isomeric state and S n =7.3 MeV the neutron separation energy), and for J π varying from 0 ± to 30 ± . Assuming that the spin distribution of the 132 Sn fission product follows a Rayleigh-type distribution (see Eq. 2) and using a Bayesian statistical analysis, we were able to extract, for each kinetic energy, the best spin cutoff parameter σ 2 , i.e. the one which best reproduces the experimental isomeric ratio.
The detailed procedure used to extract the spin of the fission product can be found in Ref. [4]. The dependence of the 132 Sn average spin with its kinetic energy for both 235 U(n th ,f) and 241 Pu(n th ,f) reactions is shown in Fig. 3. For both reactions, a similar trend can be observed, i.e a rather flat behavior at low kinetic energy and then a decrease of the spin with increasing KE. The average spins J obtained by weighting results shown in Fig. 3 with the 132 Sn kinetic energy distribution (blue curve of the Fig. 1) Table 1. Average isomeric ratio and spin of the 132 Sn produced after prompt neutron emission from 235 U(n th ,f) and 241 Pu(n th ,f) reactions are given in Tab. 1. Two additional comments can be mentioned: • Since the probability to emit the prompt neutrons is very low for the doubly magic 132 Sn, the spin distributions before and after prompt neutron emission are close to each other; • Our results (see Table 1), are consistent with the recent experimental data obtained by Wilson [10], where an average spin between 3.5 and 4 was found around the mass 130 for the three investigated fissioning systems. They are also consistent with the calculation of Marevic [11] that predicts an average spin (before prompt neutron) around 2.5 in the mass region 132. The same order of magnitude was found from the theoretical work proposed by Randrup [12].

Dependence of J with the Kinetic Energy
In order to interpret, at least qualitatively, the dependence of J with the kinetic energy, Fig. 4 can be helpful. On this figure, proposed by Thomas [13] and continued by Gönnenwein [14], the energy components (deformation, coulomb and intrinsic) are plotted for a given fission fragment pair, as a function of the elongation of the system at scission. On the x-axis, the deformation starts from two spherical compact fission fragments up to two deformed fission fragments configuration. These two extreme cases are called respectively "cold compact" and "cold deformed" because in both cases no intrinsic excitation energy is available. An "intermediate" case is also considered, where the light fragment is strongly deformed, while the heavy one ( 132 Sn) is spherical. Note that the sum of the three energy components is constant and must be equal to the Q of the reaction (horizontal line). Since the stiffness parameter which describes the resistance of a nucleus against deformation is very high for the doubly magic 132 Sn fission product, the phase space between "intermediate" and "cold deformed fission" configurations (black points in Fig. 4) will almost never be reached during the fission process. From the figure, we see that when the KE (V coul ) increases (from the right to the left), then the intrinsic excitation energy decreases, as also observed is not accessible when the heavy nucleus is a doubly magic nucleus ( 132 Sn). Bottom: Evolution of the 132 Sn average spin with its kinetic energy, as measured on LOHENGRIN. This evolution seems to be strongly correlated with the available intrinsic excitation energy at scission.
for the average spin (Fig. 3). In other word, the average spin seems to be generated mainly by the available intrinsic excitation energy at scission.

Average spin as a function of E n
In this sub-section, we try to calculate the dependence of the 132 Sn isomeric ratio with the incident neutron energy, E n , for the 235 U(n,f) reaction. For that, the four free parameters available in FIFRELIN are tuned in order to reproduce the total prompt neutron multiplicity ν T OT given in the JEFF-3.3 library (see upper part of Fig. 5). FIFRELIN calculations are performed for three incident neutron energies (before the second-chance fission): thermal, 2 MeV and 5 MeV. It is generaly admitted that the additional energy brought by the incident neutron appears mainly under the form of intrinsic energy (not deformation energy) and goes essentially into the heavy fragment (see for example Ref. [15]). As a consequence, the average prompt neutron multiplicity of the light fission fragment group stays rather constant, while the average prompt neutron multiplicity of the heavy fission fragment group is increasing with E n . This phenomenom, observed experimentally (see Ref. [16] for example) is well reproduced by FIFRELIN as shown by the red and green lines of the upper part of Fig. 5. Similar calculations were already performed with FIFRELIN on the 237 Np(n,f) reaction [17]. Since, as seen in the previous sub-section, the spin of the doubly magic 132 Sn nucleus is mainly generated by excitation of single-particle states at scission, we expect therefore a strong increase of J with the incident neutron energy. It is what we observe from the FIFRELIN calculations (see the lower part of the Fig. 5), where the isomeric ratios and the average spins are plotted as a function of E n .