Fission in Inverse Kinematics A path to new experimental observables

Historically, experimental fission studies were based on reactions in direct kinematics with fixed target-like fissioning systems. Besides its advantages, this technique su↵ers from some drawbacks such as the difficulty of producing exotic fissioning systems and the seldom measurement of the fragment atomic number. Inverse kinematic provides an alternative to ease these issues and o↵ers a new set of experimental observables that improves our level of information about the fission process, including an unprecedented access to the scission point. In this document, we review some of the observables obtained from the experimental campaign based on inverse kinematics, performed at VAMOS/GANIL.


Introduction
Nuclear fission is a complex phenomenon in which the nucleus is deformed and heated, collective energy is dissipated in the form of intrinsic degrees of freedom, while the pairing and shell structure guide the process until two excited fragments are produced and accelerated due to their Coulomb repulsion.
Since its discovery, almost 80 years ago [1,2], our current knowledge of fission was built on the interpretation of observables almost exclusively gathered in direct kinematics experiments: a heavy-nuclide target is bombarded by lighter projectiles that produce a target-like fissioning system.The properties of the fission products are measured to form the ensemble of experimental observables.Among them, direct kinematics allows the study of the fission probability, the fragments total kinetic energy, T KE, and mass distributions, their angular distribution, or the neutron evaporation.Together with the characterization of the fissioning system in terms of isotopic identification, excitation energy, etc., they shape the current image of fission: a process that runs along a potential energy formed by macroscopic and microscopic features of the nucleus, which depends sensitively on the number of protons and/or neutrons involved, and opens isolated deformation paths leading to di↵erent configurations at the scission point, called modes or channels [3,4].
In spite of a wide opportunity of the fission setup in the direct kinematics method, it su↵ers from the following drawbacks: the small velocities of the emitted fragments disturb a good measurement of their the atomic number Z and impose restrictions to the amount of material in the ?e-mail: manuel.fresco@usc.es?? Present address: IPN Orsay, IN2P3/CNRS-UPS, F-91406 Orsay Cedex, France target, while the use of direct kinematics limits systematic measurements to fissioning systems close in proton and neutron numbers to those of the target species.These difficulties can be overcome with the use of inverse kinematics: exotic nuclei can be produced with in-flight techniques and later guided to a secondary reaction target where fission can be induced; the measurement of the fragments Z distribution is also facilitated by their higher velocity when emitted from the moving fissioning systems.
The pioneering fission experiments with inverse kinematics at GSI [5] o↵ered the measurement of Z and T KE distributions for a large number of neutron-deficient actinides [6], along the systematic measurement of the evenodd staggering [7], which is another scarce observable.Together, these observables revealed a fresh look on the role of protons and dissipation in the fission process.This is particularly interesting considering that shells are revealed in the number of protons and neutrons, and their e↵ect is less evident in the mass distributions.
The last upgrade in the use of inverse kinematics was to include also the mass of the fragments, A, in the set of observables.The use of magnetic spectrometers allowed such improvement within two of the current experimental campaigns: SOFIA at GSI, which measures electromagnetic-induced fission of neutron-deficient systems [8,9] with the help of the large dipole magnet AL-ADIN; and the fission campaign at GANIL, where fissioning systems around 238 U are studied through transferand fusion-induced fission using the VAMOS spectrometer [10,11].In the following, we discuss the collection of observables measured in the VAMOS/GANIL experiments.

Transfer-and Fusion-Induced Fission in Inverse Kinematics at GANIL
As mentioned in the previous section, the fission campaign carried out at GANIL focused on the study of fission induced by direct reactions measured in the VAMOS variable mode spectrometer.A 238 U beam was shoot towards a 12 C target with 6.1 AMeV, an energy slightly above the Coulomb barrier, assuring the cross sections for transfer and fusion channels to reach tenths and hundreds of mb, respectively.Transfer reactions produced beam-like fissioning systems from U to Cm within a range of excitation energy, E ⇤ FS , below some 30 MeV, while fusion reactions resulted in 250 Cf with a fixed excitation energy of around 45 MeV.
The experimental setup can be laid out in two stages: the first one includes SPIDER, a set of two annular, double-sided, stripped silicon detectors devoted to the reconstruction of the direct reaction through the identification of the target-like recoil and the measurement of its angle and energy.A small part of the EXOGAM gamma detector is also placed to measure prompt gamma emission from the fragments.The second stage is based on the VAMOS spectrometer and its focal detectors, which detect one of the fission fragments for the isotope identification as well as the flight angle and velocity measurements.Figure 1 shows the VAMOS setup during the first run of the campaign.Further details can be found in Refs.[10][11][12] The VAMOS setup o↵ers a collection of observables carrying information from both the fissioning system and the produced fragments.For each fission event, the fissioning system is identified, and its E ⇤ FS , angle, and velocity are measured with the reconstruction of the direct reaction.Concerning the fission fragments, the use of inverse kinematics allowed, for the first time, the isotopic identification of the full fission fragment distribution [10].Taking advantage of the features of transfer reactions, the evolution of the fission-fragment distributions on the Z-A plane can be studied as a function of the fissioning nuclides and their excitation energies [13].In addition to Z and A, the fragments velocities and angles with respect to the beam direction are also measured, which, together with the identification and kinematics of the fissioning system, can be translated to the fissioning system reference frame.The simultaneous measurement of these characteristics allows to build new observables, o↵ering new insights into the fission process.In the following sections we review and discuss some of these new observables.

Neutron distribution and average neutron excess
The simultaneous measurement of the fragments Z and A distributions makes it possible to correlate both observables and build the neutron-number distribution for each event j, as N j =A j -Z j .Such distribution corresponds to the fragments after post-scission neutron evaporation and thus it reflects the e↵ect of neutron shells modified by the release of evaporated neutrons.Figure 2 shows the Z and post-evaporation N distributions for a set of systems with di↵erent average E ⇤ FS .The Z distributions of these systems show very similar features on the heavy fragment, with a clear preference for producing Z ⇠52, 54 fragments.However, the N distributions are more difficult to interpret.The pronounced peak around N=82 might be a combination of the influence of the neutron shell and the accumulation of the end point of the neutron evaporation chains of its neighbors, which is coherent with the known minimum of evaporation for A ⇠130 (see [14,15], for instance).
A degree of the magicity of nuclei a↵ecting the fragments distribution can be represented by the neutron excess, defined as: where Y(A, Z) is the normalized yield of the A, Z fragment.Figure 3 shows the hNi/Z distribution of the systems measured in the second run of the campaign.Despite the e↵ect of neutron evaporation, a clear polarization is observed.Its shape evolve with E ⇤ FS until disappearing for 250 Cf, produced with E ⇤ FS =46 MeV, suggesting that these features might be the result of the shell structure of the fragments.To explore this possibility we plot, in the same figure, the lines corresponding to proton and neutron numbers of relevant spherical and deformed shells [16].
We can see that the low-E ⇤ FS systems tend to approach the crossing of Z=50, N=82 spherical shells in the heavy fragments, and the crossing of Z=44, N=64 deformed shells in the light fragments.It is also worthy of note an accumulation around this point for all the systems, regardless of their E ⇤ FS .The region around Z=44, N=64 is known to yield a high neutron multiplicity in the post-scission evaporation [14] but, at the same time, it was also observed to remain almost una↵ected by the E ⇤ FS of the fissioning system [17].

Observables at the scission point
As mentioned in Sec. 2, the velocity of the fragments in the reference frame of the fissioning system can be determined with the reconstruction of the direct reaction producing the fissioning system and the measurement of the velocities of the fragments in the laboratory frame.Assuming that post-scission evaporation does not modify the velocity of the fragments, and applying the momentum conservation, the masses of the fragments at scission can be deduced [18].In the case of the VAMOS/GANIL campaign, one fragment per fission event is measured and thus the observables calculated at scission are expressed as average quantities as a function of the fragment Z.Consequently, and unless stated otherwise, all quantities referred in this work to fragment properties are functions of the fragment Z.A full discussion of the method applied to these data can be found in [19].
Figure 4 shows the di↵erence between the postevaporation and pre-evaporation hNi/Z for 240 Pu with E ⇤ FS =9 MeV and 250 Cf with E ⇤ FS =42 MeV fission, both measured in the first run of the campaign.As we discussed in the previous section, the main features of preevaporation hNi/Z can be observed for low-E ⇤ FS systems.High-E ⇤ FS systems present a complete di↵erent picture before and after evaporation: the almost flat hNi/Z distribution of the post-evaporation 250 Cf data (E ⇤ FS =42 MeV) is an outcome of the subtle counterbalance between the preevaporation hNi/Z value and the neutron emission, since both of them increase with Z.We can observe a clear change of slope around Z=54 that might be associated with the di↵erent evolution of deformed single-particle states in the regions between 44<Z<54 and 55<Z.
The access to Z, A, and velocity of the fragments at scission, combined with the identification of the fissioning system and its E ⇤ FS , also permits to perform the energy balance between the input and output of the fission reaction.The input energy corresponds to the sum of the mass of the fissioning system, M FS , and its E ⇤ FS .Considering no emission from saddle to scission, this is transformed into the ground-state masses of the fragments, M 1 and M 2 , their total excitation energy, T XE, defined as the sum of the intrinsic and deformation energy of the fragments, and the total kinetic energy T KE, which is the sum of the Coulomb repulsion energy and the pre-scission kinetic energy.The energy balance is then It is important to note that while M FS +E ⇤ FS is obtained from the reconstruction of the direct reaction, this quantity is, in (solid symbols); the black line is the T KE measured from 239 Pu(n th ,f) [15].Bottom panel: T XE g.s. ; the black line is the T XE g.s.estimated from 239 Pu(n th ,f) [15].

general, accurate for E ⇤
FS below the sum of the neutron separation energy and the fission barrier.For higher energies, an estimation of the pre-saddle evaporation is needed.As explained, M 1 and M 2 are obtained from the momentum conservation in the fissioning system reference frame, while T KE is calculated with the velocities in the same reference frame.From these quantities is possible to calculate the distribution of both the average T XE g.s. and the Q g.s.value at the ground state of the reaction, defined as Figure 5 shows the resulting T KE, Q g.s., and T XE g.s. at scission for 240 Pu with E ⇤ FS =9 MeV and 250 Cf with E ⇤ FS =42 MeV, both from the first run of the campaign.A first glance at the top panel shows a very similar behavior for both T KE and Q g.s. in the case of the low-energy fission of 240 Pu.This similarity was already hinted in previous studies of cold fission in light actinides, where the contribution of T XE is at minimum [20].In the case of high-energy fission of 250 Cf, the similarity between T KE and Q g.s.disappears around the symmetry, where a di↵erence builds up.Concerning T XE g.s., the bottom panel of Fig. 5 shows an almost flat distribution for low energy fission, with an increase of almost 10 MeV around the symmetry.However, the case of high-energy fission shows the T XE g.s.growing from ⇠ 25 MeV at asymmetric splits to ⇠ 45 MeV around the symmetry.The green symbols correspond to results of the energy density functional model used in [28].Bottom panel: Quadrupole deformation β (black dots) displayed with its moving average (blue dashed line).The hatched areas correspond to neutron (red) and proton (blue) spherical and deformed shells.The short-dashed black line is the average deformation at the ground state.
The results of T KE and T XE g.s. in 240 Pu are also compared in Fig. 5 with previous studies of 239 Pu(n th ,f) [15].The overall values of T KE and T XE g.s. are in agreement with the present data with the exception of the symmetric split, where a larger T KE and a smaller T XE g.s. are obtained in the present data.This di↵erence is likely to be a combination of the larger excitation energy of the fissioning system and the limited resolution of the present experiment for symmetric splits due to the lower statistics.

Energetics and configuration at scission
The fission observables discussed so far permit, combined with reasonable assumptions, to extract more information on the scission configuration of the fragments, such as their excitation energy, deformation, and distance.The details of such calculation can be found in [23] The measured T XE is the sum of the excitation energy E ⇤ i of each fragment i. 1 These can be separated into each fragment deformation energy E ⇤,def i and intrinsic energy, which is the sum of the initial excitation energy above the Experimentally, T XE is fully released in the form of neutron and γ evaporation: where ⌫ i is the neutron multiplicity, " i is the neutron kinetic energy, E γ i is the energy released by γ emission, and Q n i is the di↵erence between pre-and post-evaporation fragment masses.Previous data show that the average " is approximately symmetric with respect to the fragments split [14], and that E γ i follows the behavior of ⌫ i with a total value similar to the neutron separation energy S n [24].These properties allow to express Eq. 5 as ⌫ i and Q n i are measured observables, while S n is calculated as the average neutron separation energy for each split.
The combination of Eqs. 4 and 6 permits to deduce i is shared between the fragments in a regime of statistical equilibrium, following the prescription of Refs.[25,26].Concerning the dissipation energy, in this work we followed the prescription of Ref. [27], where the dissipation is a fixed fraction of the available T XE. 2 These approximations and prescriptions are valid for low-energy fission of actinides and, in our case, we applied them to the 240 Pu data at E ⇤ FS =9 MeV.The top panel in Fig. 6 shows the distribution of the excitation energy E ⇤ i and the fraction dedicated to deformation, E ⇤,def i .The E ⇤ i data is compared with a recent calculation based on a real-time, microscopic framework of energy density functional [28], also applied to lowenergy fission of 240 Pu.We can see what the authors interpret as a quasi-spherical slightly-excited heavy fragment 2 A complete discussion of the convenience of these prescriptions can be found in [23].around Z=52 and a highly-deformed highly-excited light one around Z=42.There is a fair discrepancy with our results: at Z ⇠52 we find deformed fragments excited up to 20 MeV, while at Z ⇠42 we have similarly deformed fragments with an excitation energy of ⇠10 MeV.
It is also worthy to mention the relatively high multiplicity around Z=50: the literature shows multiplicities of the order of ⌫ ⇠0.5 around the corresponding masses (see [15,17] for example).In the present case, the excess of ⌫ can be explained, at least in its most part, with the E ⇤ FS induced by the transfer reaction.Other factors, such as an increase of the deformation or a lower fission barrier due to the angular momentum, might also play a role.
The bottom panel of Fig. 6 shows the translation of E ⇤,def into quadrupole deformation β by computing the increase in energy of the Weizsäcker liquid-drop massformula for variations in the surface and Coulomb terms due to small quadrupole deformations, following the prescription of Swiatecki [29].We can see that β retains the well-known sawtooth shape already observed in ⌫ and E ⇤ , while it runs through Z=44, N=64, 88 deformed shells.It is also interesting that, although β is "pulled" towards lower values around Z=50, N=82 spherical shells, the deformation is relatively high.The competition between single particle structure and the macroscopic potential, which is particularly strong close to symmetry, seems to hinder the e↵ect of these spherical shells.
Once the deformation of the fragments are known, the tip distance between the fragments, d, can be extracted with the measured T KE, which is the sum of the contribution from the Coulomb repulsion between the fragments, E k,C , and the pre-scission kinetic energy, E k,pre : (7) In this case, we use E k,pre from a two-center shellmodel parameterization, calculated as a function of the fragment [30].The tip distance d is then deduced from E k,C following the Cohen-Swiatecki formula [31].The resulting distance d is displayed in Fig. 7.We observe values between 4 and 5 fm, further than the "standard" 2-3 fm of low-energy fission of actinides [3,16,[32][33][34][35] but similar to the values used in recent scission-point models [36].We can also see that configurations around Z 1 =42, Z 2 =50 are more compact than those at symmetric and very asymmetric splits.This minimum coincides with a maximum in T KE, suggesting a correlated behavior between both the T KE and d, and also with shell e↵ects.
The T XE discussed so far corresponds to the total excitation energy gained by the fragments at scission with respect to their ground states.We can recover the potential energy surface by expressing the total excitation energy with respect to the deformed ground states of the fragments (see for example [21,22]).Together with the calculated E ⇤,def , β i , and d, they can be combined to outline the shape of the potential energy PE landscape that the system experiences at scission, defined as with the distance between the centers of the fragments d cm : to obtain the PE-d cm -Z correlation at scission.The result is displayed in Fig. 8.The potential energies at scission appear with a prominent minimum of some 10 MeV smaller than the asymmetric splits, which coincides with an elongated shape.At the same time, the distance between fragments is still smallest around Z 1 =42, Z 2 =50, meaning the system breaks sooner when a↵ected by microscopic e↵ects.The fact that the macroscopic potential at symmetry is observed in the depth of the landscape with no apparent structure e↵ects suggests a strong connection between these e↵ects and the elongation that the system can reach before splitting.

Conclusions
The use of inverse kinematics in new experimental campaigns allows to obtain simultaneously a number of observables that were historically difficult to access and always in separate experiments, such as the fragment A, Z, and velocity distributions.The correlation of these observables permitss to construct new quantities with improved sensitivity to fission properties.The neutron excess, and the T XE and T KE at scission are now available for different systems and as a function of the fission energy.In some cases, such as low-energy fission of actinides, the new observables can be extended to reveal the energy sharing and the configuration at scission, in terms of deformation and distance of the fragments.These results help to better understand the role of structure e↵ects in the process and contribute to the pool of experimental fission data to be compared with state-of-the-art models.

Figure 1 .
Figure 1.Schematic layout of the VAMOS setup during the first run of the fission campaign.After the transfer reaction, the target-like recoil (RN, black arrow) is measured in SPIDER, while one of the fission fragments (FF, red arrow) emitted after fission of the beam-like system is detected in the detector setup placed at the focal plane of the VAMOS spectrometer.

Figure 2 .
Figure 2. Fragment Z (top panel) and N (bottom panel) distributions for the fissioning systems measured in the second run of the VAMOS/GANIL campaign.The parentheses show the average excitation energy.

Figure 3 .
Figure 3. Fragment hNi/Z distribution for the fissioning systems measured in the second run of the VAMOS/GANIL campaign.The parentheses show the average excitation energy.The longdashed lines correspond to proton and neutron numbers related to spherical and deformed shells.

Figure 4 .
Figure 4. Pre-evaporation (open symbols) and post-evaporation (solid symbols) hNi/Z distributions for 240 Pu (blue dots) and 250 Cf (red squares) measured in the first run of the VA-MOS/GANIL campaign.The parentheses show the average excitation energy.

Figure 5 .
Figure 5.The figure shows the energy components for 240 Pu with E ⇤ FS =9 MeV (blue dots) and 250 Cf with E ⇤ FS =42 MeV (red squares).Top panel: T KE (open symbols) and Q g.s.(solid symbols); the black line is the T KE measured from 239 Pu(n th ,f)[15].Bottom panel: T XE g.s. ; the black line is the T XE g.s.estimated from 239 Pu(n th ,f)[15].

Figure 6 .
Figure 6.Top panel: E ⇤ (green long-dashed line) and E ⇤,def (black dots) of fragments produced in low-energy fission of 240 Pu.The blue dashed line is a moving average of the E ⇤,def data.The green symbols correspond to results of the energy density functional model used in[28].Bottom panel: Quadrupole deformation β (black dots) displayed with its moving average (blue dashed line).The hatched areas correspond to neutron (red) and proton (blue) spherical and deformed shells.The short-dashed black line is the average deformation at the ground state.

Figure 7 .
Figure 7. Tip distance between the fragments (symbols).The red dashed line is a moving average of the data.

Figure 8 .
Figure 8. Potential landscape at scission.The white dashed line corresponds to a moving average of the scission point as a function of PE, d cm , and Z.The white solid line shows the projection onto the d cm -Z plane.