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Evaporation channel as a tool to study fission dynamics
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ScienceDirect Nuclear Physics A 971 (2018) 21–34 www.elsevier.com/locate/nuclphysa
Evaporation channel as a tool to study fission dynamics A. Di Nitto a,b,∗ , E. Vardaci a,c , G. La Rana a,c , P.N. Nadtochy a,d , G. Prete e a INFN, Sezione di Napoli, 80126 Napoli, Italy b Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany c Dipartimento di Fisica, Università degli Studi di Napoli “Federico II”, 80126 Napoli, Italy d Omsk State Technical University, Mira prospekt 11, 644050 Omsk, Russia e INFN, Laboratori Nazionali di Legnaro, 35020 Legnaro (Padova), Italy
Received 30 November 2017; received in revised form 15 January 2018; accepted 15 January 2018
Abstract The dynamics of the fission process is expected to affect the evaporation residue cross section because of the fission hindrance due to the nuclear viscosity. Systems of intermediate fissility constitute a suitable environment for testing such hypothesis since they are characterized by evaporation residue cross sections comparable or larger than the fission ones. Observables related to emitted charged particles, due to their relatively high emission probability, can be used to put stringent constraints on models describing the excited nucleus decay and to recognize the effects of fission dynamics. In this work model simulations are compared with the experimental data collected via the 32 S + 100 Mo reaction at Elab = 200 MeV. Consequently we pointed out, exploring an extended set of evaporation channel observables, the limits of the statistical model and the large improvement obtained with a dynamical model. Moreover we stress the importance of using an apparatus covering a large fraction of 4π to extract observables. Finally, we discuss the opportunity to measure more sensitive observables by a new detection device in operation at LNL. © 2018 Elsevier B.V. All rights reserved. Keywords: 132 Ce; Fusion-evaporation; Fusion-fission; Statistical model; 3-D Langevin equations
* Corresponding author at: Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany.
E-mail address: [email protected] (A. Di Nitto). https://doi.org/10.1016/j.nuclphysa.2018.01.008 0375-9474/© 2018 Elsevier B.V. All rights reserved.
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1. Introduction Since the discovery of nuclear fission in 1939 [1,2], a large effort was devoted to provide a realistic description of this complex phenomenon originated by the interplay of macroscopic and microscopic degrees of freedom in a nucleus. With the advent of heavy-ion accelerators the study of fission was extended to a new variety of nuclei produced in few-nucleon direct transfer reactions [3,4] or in complete fusion reactions (as in recent works [5,6]). And more recently new detailed information has been obtained in a study on fissioning systems at high excitation energies and low angular momenta produced in inverse kinematics by means of spallation reactions [7]. Toward heavier mass region the quasi-fission process suddenly appears and becomes the main antagonist to the fusion and consequently to the formation of Superheavy elements (see the reviews [8,9]). It is well established that fission is a slow process dominated by nuclear viscosity [10,11]. A very striking experimental evidence of this behavior is the excess of pre-scission light particles with respect to the predictions of the Statistical Model (SM), and its dependence on the excitation energy [12–14]. Phenomenological studies based on the SM predictions were carried out with the aim to estimate the fission delay time, and, in some cases, to extract the strength of nuclear viscosity. The estimates given by different authors predict a quite wide range of dissipation strengths and different dependencies on temperature and deformation (see reviews [15–17] and references therein). However, this kind of approach is founded on the reliability of the SM to reproduce the observables in the evaporation residue (ER) channel, and this has not yet been fully explored. The lack of experimental constraints to the SM appears to be, in several cases, one of the sources of controversial results. Indeed, the limits of this model have been evidenced by considering a large set of observables [18,19]. Dynamical models based on a stochastic approach combined with an evaporative model, for light particles and gamma quanta, seem to be a more suitable tool for the description of the collective evolution of excited nuclei [20,21]. Although much work has been devoted to fission dynamics, there are still many open questions: the timescale, the strength and nature of dissipation, the dependence on the temperature and shape of the fissioning system. The dynamics of the fission process is expected to affect the evaporation residue channel because of the fission hindrance due to nuclear viscosity. For this reason, the study of the evaporation residues channel can play a very important role. Systems of intermediate fissility constitute a suitable environment for measuring potentially informative observables related to the ER channel, being characterized by higher probability for charged particle emissions and integral ER cross section comparable with the fission one. In order to address fission dynamics such advantages were largely exploited by using as probes the light particles [22] and, only recently, by using the fission-fragment charge distribution [23]. However in order to fruitfully benchmark the existing models, the experimental uncertainties have to be minimized. Thereby larger angular coverage apparatuses are an essential add-on to step forward. Here we report on the analysis of the evaporation and fission decays of 132 Ce compound nuclei at Ex = 122 MeV, produced by the 200 MeV 32 S + 100 Mo reaction. For this system, ER and fusion-fission (FF) angular distributions and cross sections, light charged particle (LCP) multiplicities and spectra as well as ER-LCP angular correlations were measured in several experiments [18,24]. The measured quantities were compared with the SM calculations, carried out by changing many physical ingredients of the model, and with calculations of a Dynamical Model (DM) based on the 3-D Langevin equations. We found that the ER observables, especially
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Fig. 1. (Color online.) Schematic representation of a coincident event where the LCP is detected by a Ball telescope and ER by a PPAC. The emission angle of evaporation residue (θER ) is included between the beam direction and the line connecting the center of PPAC and the target. These two lines define the reaction plane π . The polar and azimuthal angles (θLCP ) and (φLCP ), respectively, define the direction of LCP impinging on a Ball telescope. The Ring at (θLCP ) is shown.
the LCP multiplicities and ER-LCP angular correlations, can be used not only to fix the SM parameters [18], but also to provide constraints on the ingredients describing the fission mechanism. The analysis based on the DM better reproduces the data [24]. However, there is still substantial room for improving the reliability of such conclusions. At this regards, in the present article we investigate which additional observable in the ER channel is affected by fission dynamics. Additional observables with such properties are for example the LCP partial multiplicity distributions and ER-LCP correlation angular distribution as function of evaporation residues angles. This article is organized as it follows. In Sec. 2 we provide key details of the experimental setup used to perform measurements on the system 32 S + 100 Mo at 200 MeV. In Sec. 3 the theoretical models used for simulations are briefly introduced. In Sec. 4 we show the experimental results and the comparison with the simulations. In particular we identify and propose further observables that guarantee additional constraints to the dynamics of models and that have not received much attention so far. In Sec. 5 we draw our conclusions. 2. Experimental setup The experiment was performed at the XTU-Tandem accelerator of Laboratori Nazionali di Legnaro. A pulsed beam of 32 S with an intensity of about 1–3 enA was used to bombard a self-supporting 100 Mo target 400 µg/cm2 thick. A beam burst with period of 800 ns and duration of about 3 ns was used. We used the Ball sector of 8π LP apparatus [25] to detect light charged particles and fission fragments, while the heavy residues were detected in a system of four parallel plate avalanche counters (PPAC) placed at forward angles. The Ball covers the polar angle from 34 to 165 degrees and consists of 7 rings of E-E telescopes placed co-axially around the beam direction. The schematic of a backward ring is shown in Fig. 1. Each ring contains 18 telescopes. Considering this geometry, the detectors in a ring have the same average polar angle with respect to the beam direction axis, and all together they cover the azimuthal angle from 0 to 360 degrees. As a whole the Ball covers a solid angle of about 80% of 4π by means of 125 telescopes. Each telescope is made by a first stage (E) of 300 µm thick Si detectors followed by a second stage (E) of 5 mm thick CsI(Tl) with photo-diode read out. The four PPAC modules were placed symmetrically with respect to the beam direction to measure evaporation residues. Each PPAC module was positioned at 4.5 degrees with respect to the beam direction and subtended a solid angle of 0.8 msr. A detailed description of the particle identification techniques adopted and performances obtained with this experimental setup are provided in previous papers [19,18].
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We collected the data by requiring several triggering conditions, as described in [18], to perform measurements of the single and coincidence yields in the same run. In this way we obtained an extended set of observables relative to the fusion-evaporation channel. It consists of proton and α particle multiplicities and energy spectra, as well as angular correlations among the LCP and ER. In a complementary experiment evaporation residues were detected with the setup PISOLO [26] which includes an electrostatic deflector and an Energy-Time of Flight system to separate evaporation residues from the beam particles. In this way the angular distribution of ERs was obtained, including the point at zero degree. A more detailed description of the full data set for the present comparative analysis can be found in ref. [18,24]. The experimental data are compared with simulations in section 4. 3. Models The extended set of observables, described above, was compared with the theoretical predictions provided by the SM code PACE2_N11 and a DM code. The computer code PACE2_N11 is an extensively modified version of the code PACE2 [27] that simulates the multistep de-excitation of the compound nucleus both through light particle evaporation and fission processes. Light particle evaporation is implemented according to the Hauser-Feshbach formulation and the fission probability is calculated by using the transitionstate model. Fission barriers are computed with the Finite Range Liquid Drop Model (FRLDM) [28]. The competition between the different decay modes is treated with a Monte Carlo approach. PACE2 was modified to implement several options for leading parameters, i.e. transmission coefficients, level density parameter and yrast line [18,19,29]. The description of the heavy-ion induced reactions by means of the Langevin approach was initiated first in the 1990s [30,16] to reproduce fission observables and then extended to heavier systems also to predict the production cross sections of new heavy and superheavy nuclei in recent years [31]. In the DM code, used in this work, the decays of the excited nuclei in the fission and evaporation channels are described with a stochastic approach code based on 3-D Langevin Equations [32]. Besides to simulate the angular and energy distribution of evaporated light particles, it was combined with the computer code LILITA_N11 [33]. The dynamical evolution of the nuclear shapes are described in terms of (c, h, α) parametrization [34]. Where the collective variables describing the fissioning/evaporating nuclei are: c elongation, h neck size at given c and α mass asymmetry. During the dynamical evolution, the excitation energy is dissipated, i.e. converted from the collective to intrinsic motion, and this process itself determines the fluctuations of the nuclear shapes. This process is relatively slow therefore it is simulated dividing its time-scale in steps of t = 10−25 s and calculating all the information concerning the shape, the excitation energy and the angular momentum of the nucleus. At each step if the scission configuration is reached the nucleus breaks in two fragments, otherwise light particle emission could occur. The light particle emission, taking into account the excitation energy and the angular momentum of the compound nucleus, is evaluated using the LILITA_N11 implemented as subroutine. The use of LILITA_N11 allows to adopt the same options for leading parameters (see Ref. [35,36]) implemented in PACE2_N11. Both the SM and DM codes were modified in order to produce an event-by-event output including energy and angular distributions in the laboratory system of emitted ER and light particles that, filtered according to the response function of 8 π LP, can be used for a direct comparison of predictions and experimental data.
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4. Results and discussion The main goal of the present analysis was to exploit the evaporation channel observables in order to put stringent constraints on the SM parameters included in the models used for fission dynamics studies. These parameters are crucial being also used to describe the evaporation of light pre-scission particles emitted by the compound nucleus along the path from its formation up to the scission in two fragments. So, in this work it is shown how the SM parameters influence the fission observables and how to avoid an incorrect definition of them. This latter is an essential step in order to solve the existing ambiguity on ingredients used for the description of fission dynamics, e.g. friction [21,30]. In a previous work the 32 S + 100 Mo reaction at 200 MeV extended data set was compared with the SM calculations [18], equivalent to the SM PACE2_N11. It was observed that if the analysis is limited to the pre-scission channel, as usually done in fission dynamics studies [37], LCP multiplicities and FF cross section can reasonably be well reproduced without any delay. From this result one could conclude that no dynamical effects take place in the decay of the CN. However, on one hand different combinations of the input parameters do not exclude the presence of a relatively small dynamical effect (i.e. a fission delay), as expected from the systematics [38]. On the other hand, the same model strongly overestimates the ER particle multiplicities and consequently cannot be considered as a reliable tool to estimate the fission time-scale through the pre-scission LCP multiplicities. Thus, we used the ER angular distributions and LCP evaporative spectra to limit the range of variability of the SM parameters included in the DM. Calculations considering different dissipation mechanisms were compared with this full set of experimental data and most of the observables were well reproduced [24]. Therefore, we determined the dissipation mechanisms and time-scales featuring the dynamical evolution towards the fission of excited 132 Ce nuclei. In the first part of this section we discuss the impact of different SM parameters on predictions of both the PACE2_N11 and the DM codes through the comparison with the experimental data. Therefore, we gradually introduce more and more exclusive observables to remark the differences between the predictions based on the statistical and dynamical approaches. Finally, the advantages offered by the new detection system for ERs in operation at LNL will be illustrated in the light of model predictions. The importance to explore very selective observables will be evidenced in the comparison with existing data and the benefits of new observables will be illustrated by comparing simulations filtered with the new detection system geometry. 4.1. Selection of SM and DM parameters In Fig. 2 we show the angular distribution of the evaporation residues, measured with the PISOLO electrostatic deflector, compared with the results of the statistical model calculations. The experimental statistical errors are of the order of the point size. By changing the leading parameters of the SM significant deviations are produced in the ER angular distributions as shown in Fig. 2. The leading parameters of the simulations were the LCP transmission coefficients, the level density parameters aν for particle evaporation, and the yrast line prescriptions. Transmission coefficients derived from the optical model (OM) [39–41] and from the fusion systematics (FS) [42] in combination with a level density parameters between A/6 and A/12 were used. The moment of inertia was calculated by adopting prescriptions for the yrast line with parameters
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Fig. 2. (Color online.) ER angular distribution of the reaction 32 S + 100 Mo. The experimental data (dots) [18] are compared with the SM calculations (top) performed with the four prescription of Table 1: a) and b) using the RLDM yrast line and OM transmission coefficients with aν = A/12 and aν = A/6, respectively; c) and d) using the RS yrast line and aν = A/6 with OM and FS transmission coefficients, respectively. Table 1 Prescriptions of SM parameter set adopted in the calculations for the 200 MeV 32 S + 100 Mo reaction. Prescriptions
aν
Yrast Line
Trans. Coef.
a) b) c) d)
A/12 A/6 A/6 A/6
RLDM RLDM RS RS
OM OM OM FS
from the Rotating Liquid Drop Model (RLDM) [28] or by assuming the compound nucleus as a the rigid sphere (RS) with r0 = 1.2 fm. The four different prescriptions used in the simulations shown in Fig. 2 are reported In Table 1. They are chosen among many combinations of the SM parameter values. The aim was to explore the full range of variability of the observables under examination. The ratio af/aν was kept equal to 1 due to relatively weak effects on the ER observables (see discussion in [18]). The comparison of ER angular distributions in Fig. 2a shows that the data are not well reproduced with prescriptions a), b) and c), whereas a reasonable good agreement can be obtained by adopting the prescription d). The fact that the RS approximation provides a better fit to the ER angular distribution may seem at variance with the expectation that the compound nuclei might be deformed. Indeed deformation effects might well be washed out on the ER angular distribution being them averaged over the long evaporation cascades and the broad window of angular momenta involved. Hence, we do not expect the ER angular distribution to be very sensitive to the deformation of the compound nucleus. It is therefore not unexpected that a RS approximation can reproduce the ER angular distribution alone. The RS approximation is considered in the following discussion because it provides the better reproduction of ER angular distribution. However, this observation cannot represent a strong indication for the nuclear shapes of evaporation residue nuclei. In fact, the effects of nuclear deformations on this observable are washed out being averaged on the large interval of angular momenta populating the fusion-evaporation channels. In Table 2 the LCP multiplicities and cross sections of ER and FF channels, measured using 8π LP and PISOLO and setups, respectively, are compared with a series of DM calculations. In these simulations the nuclear shape evolution as function of the time is described using the coupled Langevin equations as in Ref. [24]. Where the potential energy is calculated within the framework of a macroscopic model [28] and the inertia tensor by means of the Werner-Wheeler approximation for incompressible irrotational flow [10]. In the present analysis different prescriptions were used to calculate friction tensors. In particular we used one-body dissipation
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Table 2 ER and FF channel experimental LCP multiplicities and cross sections of the reaction 32 S + 100 Mo. The experimental observables are compared with the DM calculations performed using different set of parameters. Viscosity
aν
MpER
MαER
σ ER (mb)
MpF F
MαF F
σ F F (mb)
one-body ks = 0.1 ks = 0.25 ks = 0.5 ks = 1 ks = 1 ks = 1
A/6 A/6 A/6 A/6 A/7 A/8
1.26 1.24 1.22 1.20 1.20 1.20
0.55 0.54 0.54 0.56 0.61 0.70
706 762 770 793 797 802
0.021 0.026 0.042 0.052 0.059 0.075
0.013 0.016 0.017 0.030 0.041 0.061
230 174 166 143 139 134
two-body νo = 0.02 νo = 0.10 νo = 0.15 νo = 0.15
A/6 A/6 A/6 A/8
1.26 1.20 1.18 1.17
0.52 0.56 0.57 0.71
721 784 811 822
0.019 0.035 0.048 0.063
0.009 0.022 0.028 0.043
215 152 125 114
0.90 (0.14)
0.56 (0.09)
828 (50)
0.055 (0.007)
0.038 (0.005)
130 (13)
Exp.
[43] based on the “wall” and “wall-plus-window” formulas, with different reduction factors ks for the “wall” contribution, and the two-body dissipation [10], with different two-body friction constants νo . While, the parameters used in the subroutine LILITA_N11 to calculate energy and angular distributions of light particles and evaporation residues were kept constant according the prescription d), i.e. the one that better reproduces the ER angular distribution as shown in Fig. 2. However in order to evidence the influence of a typical SM parameter on the DM predictions, the level density parameter aν was varied from A/6 to A/8. If the analysis is limited only to the observables measured in the fission channel (pre-scission protons, MpF F , and α particles, MαF F , and fission cross section, σ F F ), the experimental data can be reasonably reproduced with high viscosity parameters or, by reducing the aν , with low viscosity parameters. In fact, the pre-scission particle multiplicities are largely increased changing from A/6 to A/8, e.g. the MαF F becomes up to about 2 times larger, and contemporary σ F F is reduced. The large variability observed as function of aν and also on the other SM parameters, see Ref. [18], lets essential the simultaneous investigation of the evaporation channel to avoid an incorrect estimation of the dissipation mechanism featuring the CN under investigation. Consequently, we investigated the ER channel observables to define the SM parameters. In fact, they are strongly influenced by SM parameters and are practical insensitive to the dissipation prescriptions and their intensities. For instance, they vary by a quantity that is below 20% by spanning the very large range of dissipation parameters shown in Table 2. So a value of aν within A/6–A/7 avoids the overestimation of α particles multiplicities in the evaporation channel produced by the value aν = A/8. Extending the comparison to the full set of observables available, the lower viscosity parameter values can be considered very unlikely because they always overestimate the σ F F and the MαER , and underestimate the σ ER . In conclusion, we obtain an overall reproduction of measured quantities using the one-body dissipation with ks = 1 and aν = A/6. We note a good reproduction of data also using two-body dissipation with νo ≈ 0.15 · 10−21 MeV s fm−3 . However, we excluded this value being unrealistically strong, i.e. very close to infinity viscosity. Indeed, also when other authors [30] used a similar value (νo = 0.15 · 10−21 MeV s fm−3 ) they well
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Fig. 3. (Color online.) ER angular distributions (left), proton (center) and α particle (right) energy spectra measured in coincidence with ERs. The experimental data (dots) are compared with the predictions of the statistical model prescription d) in Table 2 (solid line) and the dynamical model (dashed line).
reproduced σ F F , σ ER and the pre-scission neutron multiplicity, MnF F , but observed too large differences in some observables as for instance the kinetic energies of fission-fragments. In addition, this two-body prescription underestimates the MαF F . Unfortunately this discrepancy cannot be accommodated by reducing the level density parameter down to aν = A/8, because it produces an overestimation of MαER . Accordingly in the following sections we compared only the DM predictions performed using one-body dissipation with ks = 1 and aν = A/6 with the rest of measured observables. 4.2. Evaporation channel energy and angular distributions In Fig. 3 we compare the measured ER angular distributions, evaporative proton and α particle energy spectra with results of statistical and dynamical model simulations. Both models very well reproduce the ER angular distributions, the proton spectrum and the high energy side of α particle spectrum, whereas the low energy side of the latter is better reproduced with the SM. The same agreement holds also for the spectra measured at different angles. Therefore it is reasonable to conclude that an overall agreement of energy spectra and angular distributions of evaporation residues can be obtained by adopting both the SM and the DM. We would like to remark that in our calculations the deformation effects on LCP emission are considered only by changing the nuclear radius but keeping the nucleus as a rigid sphere. It is known that deformation effects of the emitting sources (ER’s and FF’s) can be very important during LCP emission both for lighter systems [44] and for heavier systems [45] in which LCP’s and neutrons were analyzed. In the present data the good reproduction of the many observables with RS approximation indicates that deformation effects can be of minor importance. However, we are working on this specific aspect by following two complementary paths: first, by searching for additional observables which might be potentially more sensitive to deformations and, second, by implementing in our codes deformations effects. The use of the other prescriptions in Table 1 produces large deviations not only in the evaporation residues distribution and fission cross sections, but also in the LCP multiplicities and energy spectra (see Table 2 and for further details Ref. [18]). The sensitivity reached in this comparison it is sufficient to strictly define the most appropriate SM parameter set to well reproduce the evaporation channel observables, but does not allow to say which of the two models better reproduces the experimental data. Indeed observables so far considered are only slightly influenced by the evaporation-fission competition.
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Fig. 4. (Color online.) Experimental and calculated protons (top) and α particles (bottom) differential multiplicity distributions as function of 8π LP LCP detector identification number. Black dots represent the experimental data. The blue (dashed) and orange (solid) lines are the SM and DM simulations, respectively. See the text for details.
Table 3 The experimental and calculated mean particle multiplicities in the ER channel together with the fission and evaporation cross sections.
DM SM Exp.
MER p
MER α
σ ER (mb)
σ FF (mb)
1.20 1.43 0.90(0.14)
0.56 0.72 0.56(0.09)
793 817 828(50)
143 139 130(13)
A more exclusive observables to constraint the reaction dynamics are the LCP differential multiplicities. These observables were obtained by normalizing the 8π LP ER-LCP coincidence yields to the number of ER single events divided by the individual LCP detector solid angle. The resulting differential multiplicities of LCP are shown in Fig. 4 as a function of the 8π LP LCP detector identification number. As mentioned before, for a fixed polar angle (θLCP ), the 18 telescopes of the ring span the azimuthal angle (φLCP ) from 0◦ to 360◦ . Hence, the oscillating behavior as function of the detector number is due to a combined effect of kinematics and of angular momentum of the composite system. In particular, the maxima correspond to events where ERs and LCPs are emitted in-plane and on opposite side with respect to the beam direction, whereas the minima occur when ERs and LCPs are emitted in-plane and on the same side with respect to the beam direction. In Fig. 4 are indicated the mean polar angles corresponding to the telescope detecting the LCPs, θLCP . By adopting different SM parameters the differential LCP multiplicities, and in particular those relative to the α particle distributions, will change not only in terms of the bulk shapes (see for instance the changing in the maxima to minima ratios in [18, 29]), but also in terms of absolute values. The results of the SM and DM calculations assuming the d) prescriptions, without any relative normalization, are compared with the experimental data. Using the SM we observe a consistent overestimation of both LCP multiplicities in the evaporation channel, while the evaporation and fission cross sections are well reproduced (see Table 3). The DM predictions well reproduce not only evaporation and fission cross sections, but also the evaporative α particle multiplicity; only the proton multiplicity is slightly overestimated. Therefore, by using the DM it is possible to obtain a better overall agreement in the evaporation channel that makes us confident on the use of such a model for studying the fission process.
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Fig. 5. (Color online.) Ratios between α particles and protons partial multiplicities obtained with 8π LP Ball filter as function of ER detection angles. The lines represent the SM calculations performed by using the four prescriptions of Table 1 and the DM calculation, the dot is the experimental value measured with 8π LP Ball and the PPAC at 4.5 degrees.
4.3. New observables High precision measurements of the evaporative LCP absolute multiplicities can be obtained only by measuring simultaneously ERs and LCPs emitted on the full solid angle. However, experimental apparatuses usually have limited angular coverage and only partial multiplicities are accessible. By this we refer to the case when differential multiplicities are directly measured only at one fixed trigger position (the angle of ER detector). The angle-integrated multiplicities, also named as “absolute multiplicities”, are commonly obtained by a model calculation that reproduces, at the best, the partial multiplicities. Another way to describe in short this procedure is to say that the partial multiplicities are integrated by the model. In this sense the absolute multiplicities are model dependent. This method also allows to remove the dependence on the single trigger detector position. We report in Table 3 the model dependent absolute multiplicities obtained following this procedure. We performed redundant measurements by using 4 symmetric PPACs around the beam direction. The redundancy is very useful because from one side assures the correct alignment of the beam impinging on the target, and from the other it makes it possible to apply a summing procedure to reduce statistical fluctuations. By using our setup, the partial multiplicities were obtained as the ratios among the yields of LCP in coincidence with ER’s and the yields of all ER’s detected in single by our PPAC’s. Afterward, the absolute multiplicities were extrapolated from the predictions that reproduce the distribution of partial multiplicities. Where we obtained the simulated partial multiplicities by filtering the event-by-event output of the simulation codes through the geometry of the 8π LP apparatus and the PPAC system mounted at 4.5 degrees. In order to further exploit the partial multiplicities as a function of the position of trigger detector, we computed the ratio between the proton and α particle partial multiplicities for different trigger positions. This new observable does not require the integration over the trigger angle and therefore can be directly compared to experimental data. The results of calculations for the different models used so far are shown in Fig. 5 together with the experimental point measured with our setup. This ratio is simultaneously model independent and very sensitive not only to the decay kinematics, but also to the competition between evaporative LCP and fission channels. The trends of the ratios show that partial multiplicities measured at various trigger angles allow to distinguish between different models and parameter values. In other words, the ratios of the partial multiplicities as function of the trigger position is an additional good probe for the fission
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Fig. 6. (Color online.) Right-hand panel: exploded view drawing (a) of the new PPAC system for evaporation residue detection. θ is the spanned angular range that depends on the mounting position. Narrower angular openings for the ions can be defined by combining this system with masks. In the 8π LP chamber, the minimum value of 3 degrees at around the forward direction is achieved by installing the PPAC at 60 cm distance from the target. Right-hand panel: photograph of the front side of the new PPAC system.
dynamics. The trend observed in the calculations is due to the interplay among many ingredients. Generally speaking, it is possible to observe a reduction of the proton partial multiplicities with the increase of θtrigger ; on the contrary the α particle multiplicities quickly increase. This behavior is explained by considering that ERs at larger θtrigger need more recoil, that is preferably provided by α particle evaporation. Therefore, larger θtrigger selects events with larger α particle multiplicities. Furthermore, α emission carries out more excitation energy with respect to lighter particles and the probability of a coincident proton emission will be reduced further by increasing the ratio at higher θtrigger . More complex is to explain the differences originated by each single SM parameter. For a fixed yrast line, increasing aν produces almost constant reduction in the ratios (see the prescriptions a) and b)). Going from the yrast line calculated using the RLDM to the yrast line assuming the nucleus as a rigid sphere with r0 = 1.2 fm (that means a decrease of the moment of inertia) we observe a decrease in the ratios, that becomes larger with θtrigger (see the prescriptions b) and c)). While by changing transmission coefficients, i.e. passing from OM to FS ones, there is a decrease of about a factor two in the ratio that is almost independent from the ER angles considered, (see prescription c) and d)). We observe that only combining all the effects producing a decrease in the α particle to proton multiplicity ratios is possible to improve the reproduction of experimental data. Thus, in agreement also with all the other observables, the prescription d) is the best set of parameters. By comparing the statistical model calculation with the dynamical one, only small differences are observed at θtrigger = 4.5 degrees. This experimental datum alone cannot provide a conclusive indication about which of the two models better describe the CN decay. However, the important conclusion is that measurements at different trigger angles can indeed constitute an additional probe to distinguish between SM and DM descriptions. Given this peculiar dependence of the partial multiplicities on the trigger angle, we have prepared a new system for ER detection, whose schematic is shown in the Fig. 6. It consists of two annular PPACs (front and rear) divided in 6 independent sectors with a wide area and an absorber foil mounted in between. The absorber is adapted to the experimental conditions with thickness sufficient to stop only ERs and to let the other lighter ions, like LCPs and elastic scattered beam particles, passing through it and reach the rear PPAC. The ERs are therefore selected using the rear PPAC signal as a veto. A mask can be mounted in front of the PPAC, as indicated by the three blue spots in the Fig. 6, in order to detect ER angular distribution between 3 and 8 degrees with respect to the beam. Furthermore, we would like to stress that thanks to the symmetric arrangement of the Ball telescopes and the PPAC sectors, also with this system
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it is possible to perform redundant measurements and get the consequent benefits. Therefore we planned to use this new PPAC system for our next experiments with 8π LP detection apparatus. 5. Summary and conclusions The larger charged particle multiplicities in pre-scission and evaporation channels occurring in the intermediate fissility composite systems make them well suited to get information on the fission process. In this work we studied the evaporative light charged particles emitted by the 132 Ce nuclei at E = 122 MeV by comparing the simulations with experimental data measured x with the 8π LP apparatus. We performed the simulations using the SM code PACE2_N11 and a dynamical model based on the 3-D Langevin equations. Evaporation residue angular distribution and LCP energy spectra were exploited to define the SM parameters for SM and DM simulations. The ER α particle and proton multiplicities are always overestimated by the PACE2_N11 code, irrespective of the SM parameters adopted. Such a failure would affect also the emission probability in the pre-scission channel, making the extraction of the fission time scale unreliable. A possible reason for the LCP multiplicities overestimation in the ER channel could be ascribed to a not properly accounted competition between the different decay channels. This problem would affect at the same time several ingredients of the SM such as level density and transmission coefficients. This hypothesis could be explored by enlarging the present data set with observables related to the neutron channel, such as neutron energy spectra in coincidence with ER and FF. Unfortunately, the neutron spectra in the ER channel are rarely collected. This remark indeed reinforce the fact that extended data set are essential to achieve reliable simulations to address the open questions on fission dynamics and strongly suggest the use of setups of high efficiency. Our analysis would greatly benefit of the measurement of neutrons in coincidence with ER and FF and will be pursued in the near future, possibly by using the neutron detector array NEDA [46], currently under construction. The possibility of using a detection setup which allows to measure partial multiplicities for different ER trigger positions gives access to a new observable which is sensitive to the different implementation of fission dynamics models. By means of an annular PPAC, the angular distribution of the ratios between the α particle and proton partial multiplicities can be measured over an extended angular coverage (see Fig. 5). From the experimental point of view this quantity is model independent and, according to the predictions, is extremely sensitive not only to the main (leading) evaporative parameters, but also to the fission process description included in the statistical and the dynamical model adopted in this work. In conclusion, the models strongly indicate that the exclusive observables of evaporation residue channel represent a powerful tool to progress in fission dynamics studies. Acknowledgements We would like to thank the anonymous referee for his helpful comments and constructive suggestions. References [1] L. Meitner, O.R. Frisch, Nature 143 (1939) 239, https://doi.org/10.1038/143471a0. [2] O. Hahn, F. Strassmann, Naturwissenschaften 27 (1939) 11, https://doi.org/10.1007/BF01488241.
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ScienceDirect Nuclear Physics A 971 (2018) 21–34 www.elsevier.com/locate/nuclphysa
Evaporation channel as a tool to study fission dynamics A. Di Nitto a,b,∗ , E. Vardaci a,c , G. La Rana a,c , P.N. Nadtochy a,d , G. Prete e a INFN, Sezione di Napoli, 80126 Napoli, Italy b Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany c Dipartimento di Fisica, Università degli Studi di Napoli “Federico II”, 80126 Napoli, Italy d Omsk State Technical University, Mira prospekt 11, 644050 Omsk, Russia e INFN, Laboratori Nazionali di Legnaro, 35020 Legnaro (Padova), Italy
Received 30 November 2017; received in revised form 15 January 2018; accepted 15 January 2018
Abstract The dynamics of the fission process is expected to affect the evaporation residue cross section because of the fission hindrance due to the nuclear viscosity. Systems of intermediate fissility constitute a suitable environment for testing such hypothesis since they are characterized by evaporation residue cross sections comparable or larger than the fission ones. Observables related to emitted charged particles, due to their relatively high emission probability, can be used to put stringent constraints on models describing the excited nucleus decay and to recognize the effects of fission dynamics. In this work model simulations are compared with the experimental data collected via the 32 S + 100 Mo reaction at Elab = 200 MeV. Consequently we pointed out, exploring an extended set of evaporation channel observables, the limits of the statistical model and the large improvement obtained with a dynamical model. Moreover we stress the importance of using an apparatus covering a large fraction of 4π to extract observables. Finally, we discuss the opportunity to measure more sensitive observables by a new detection device in operation at LNL. © 2018 Elsevier B.V. All rights reserved. Keywords: 132 Ce; Fusion-evaporation; Fusion-fission; Statistical model; 3-D Langevin equations
* Corresponding author at: Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany.
E-mail address: [email protected] (A. Di Nitto). https://doi.org/10.1016/j.nuclphysa.2018.01.008 0375-9474/© 2018 Elsevier B.V. All rights reserved.
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1. Introduction Since the discovery of nuclear fission in 1939 [1,2], a large effort was devoted to provide a realistic description of this complex phenomenon originated by the interplay of macroscopic and microscopic degrees of freedom in a nucleus. With the advent of heavy-ion accelerators the study of fission was extended to a new variety of nuclei produced in few-nucleon direct transfer reactions [3,4] or in complete fusion reactions (as in recent works [5,6]). And more recently new detailed information has been obtained in a study on fissioning systems at high excitation energies and low angular momenta produced in inverse kinematics by means of spallation reactions [7]. Toward heavier mass region the quasi-fission process suddenly appears and becomes the main antagonist to the fusion and consequently to the formation of Superheavy elements (see the reviews [8,9]). It is well established that fission is a slow process dominated by nuclear viscosity [10,11]. A very striking experimental evidence of this behavior is the excess of pre-scission light particles with respect to the predictions of the Statistical Model (SM), and its dependence on the excitation energy [12–14]. Phenomenological studies based on the SM predictions were carried out with the aim to estimate the fission delay time, and, in some cases, to extract the strength of nuclear viscosity. The estimates given by different authors predict a quite wide range of dissipation strengths and different dependencies on temperature and deformation (see reviews [15–17] and references therein). However, this kind of approach is founded on the reliability of the SM to reproduce the observables in the evaporation residue (ER) channel, and this has not yet been fully explored. The lack of experimental constraints to the SM appears to be, in several cases, one of the sources of controversial results. Indeed, the limits of this model have been evidenced by considering a large set of observables [18,19]. Dynamical models based on a stochastic approach combined with an evaporative model, for light particles and gamma quanta, seem to be a more suitable tool for the description of the collective evolution of excited nuclei [20,21]. Although much work has been devoted to fission dynamics, there are still many open questions: the timescale, the strength and nature of dissipation, the dependence on the temperature and shape of the fissioning system. The dynamics of the fission process is expected to affect the evaporation residue channel because of the fission hindrance due to nuclear viscosity. For this reason, the study of the evaporation residues channel can play a very important role. Systems of intermediate fissility constitute a suitable environment for measuring potentially informative observables related to the ER channel, being characterized by higher probability for charged particle emissions and integral ER cross section comparable with the fission one. In order to address fission dynamics such advantages were largely exploited by using as probes the light particles [22] and, only recently, by using the fission-fragment charge distribution [23]. However in order to fruitfully benchmark the existing models, the experimental uncertainties have to be minimized. Thereby larger angular coverage apparatuses are an essential add-on to step forward. Here we report on the analysis of the evaporation and fission decays of 132 Ce compound nuclei at Ex = 122 MeV, produced by the 200 MeV 32 S + 100 Mo reaction. For this system, ER and fusion-fission (FF) angular distributions and cross sections, light charged particle (LCP) multiplicities and spectra as well as ER-LCP angular correlations were measured in several experiments [18,24]. The measured quantities were compared with the SM calculations, carried out by changing many physical ingredients of the model, and with calculations of a Dynamical Model (DM) based on the 3-D Langevin equations. We found that the ER observables, especially
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Fig. 1. (Color online.) Schematic representation of a coincident event where the LCP is detected by a Ball telescope and ER by a PPAC. The emission angle of evaporation residue (θER ) is included between the beam direction and the line connecting the center of PPAC and the target. These two lines define the reaction plane π . The polar and azimuthal angles (θLCP ) and (φLCP ), respectively, define the direction of LCP impinging on a Ball telescope. The Ring at (θLCP ) is shown.
the LCP multiplicities and ER-LCP angular correlations, can be used not only to fix the SM parameters [18], but also to provide constraints on the ingredients describing the fission mechanism. The analysis based on the DM better reproduces the data [24]. However, there is still substantial room for improving the reliability of such conclusions. At this regards, in the present article we investigate which additional observable in the ER channel is affected by fission dynamics. Additional observables with such properties are for example the LCP partial multiplicity distributions and ER-LCP correlation angular distribution as function of evaporation residues angles. This article is organized as it follows. In Sec. 2 we provide key details of the experimental setup used to perform measurements on the system 32 S + 100 Mo at 200 MeV. In Sec. 3 the theoretical models used for simulations are briefly introduced. In Sec. 4 we show the experimental results and the comparison with the simulations. In particular we identify and propose further observables that guarantee additional constraints to the dynamics of models and that have not received much attention so far. In Sec. 5 we draw our conclusions. 2. Experimental setup The experiment was performed at the XTU-Tandem accelerator of Laboratori Nazionali di Legnaro. A pulsed beam of 32 S with an intensity of about 1–3 enA was used to bombard a self-supporting 100 Mo target 400 µg/cm2 thick. A beam burst with period of 800 ns and duration of about 3 ns was used. We used the Ball sector of 8π LP apparatus [25] to detect light charged particles and fission fragments, while the heavy residues were detected in a system of four parallel plate avalanche counters (PPAC) placed at forward angles. The Ball covers the polar angle from 34 to 165 degrees and consists of 7 rings of E-E telescopes placed co-axially around the beam direction. The schematic of a backward ring is shown in Fig. 1. Each ring contains 18 telescopes. Considering this geometry, the detectors in a ring have the same average polar angle with respect to the beam direction axis, and all together they cover the azimuthal angle from 0 to 360 degrees. As a whole the Ball covers a solid angle of about 80% of 4π by means of 125 telescopes. Each telescope is made by a first stage (E) of 300 µm thick Si detectors followed by a second stage (E) of 5 mm thick CsI(Tl) with photo-diode read out. The four PPAC modules were placed symmetrically with respect to the beam direction to measure evaporation residues. Each PPAC module was positioned at 4.5 degrees with respect to the beam direction and subtended a solid angle of 0.8 msr. A detailed description of the particle identification techniques adopted and performances obtained with this experimental setup are provided in previous papers [19,18].
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We collected the data by requiring several triggering conditions, as described in [18], to perform measurements of the single and coincidence yields in the same run. In this way we obtained an extended set of observables relative to the fusion-evaporation channel. It consists of proton and α particle multiplicities and energy spectra, as well as angular correlations among the LCP and ER. In a complementary experiment evaporation residues were detected with the setup PISOLO [26] which includes an electrostatic deflector and an Energy-Time of Flight system to separate evaporation residues from the beam particles. In this way the angular distribution of ERs was obtained, including the point at zero degree. A more detailed description of the full data set for the present comparative analysis can be found in ref. [18,24]. The experimental data are compared with simulations in section 4. 3. Models The extended set of observables, described above, was compared with the theoretical predictions provided by the SM code PACE2_N11 and a DM code. The computer code PACE2_N11 is an extensively modified version of the code PACE2 [27] that simulates the multistep de-excitation of the compound nucleus both through light particle evaporation and fission processes. Light particle evaporation is implemented according to the Hauser-Feshbach formulation and the fission probability is calculated by using the transitionstate model. Fission barriers are computed with the Finite Range Liquid Drop Model (FRLDM) [28]. The competition between the different decay modes is treated with a Monte Carlo approach. PACE2 was modified to implement several options for leading parameters, i.e. transmission coefficients, level density parameter and yrast line [18,19,29]. The description of the heavy-ion induced reactions by means of the Langevin approach was initiated first in the 1990s [30,16] to reproduce fission observables and then extended to heavier systems also to predict the production cross sections of new heavy and superheavy nuclei in recent years [31]. In the DM code, used in this work, the decays of the excited nuclei in the fission and evaporation channels are described with a stochastic approach code based on 3-D Langevin Equations [32]. Besides to simulate the angular and energy distribution of evaporated light particles, it was combined with the computer code LILITA_N11 [33]. The dynamical evolution of the nuclear shapes are described in terms of (c, h, α) parametrization [34]. Where the collective variables describing the fissioning/evaporating nuclei are: c elongation, h neck size at given c and α mass asymmetry. During the dynamical evolution, the excitation energy is dissipated, i.e. converted from the collective to intrinsic motion, and this process itself determines the fluctuations of the nuclear shapes. This process is relatively slow therefore it is simulated dividing its time-scale in steps of t = 10−25 s and calculating all the information concerning the shape, the excitation energy and the angular momentum of the nucleus. At each step if the scission configuration is reached the nucleus breaks in two fragments, otherwise light particle emission could occur. The light particle emission, taking into account the excitation energy and the angular momentum of the compound nucleus, is evaluated using the LILITA_N11 implemented as subroutine. The use of LILITA_N11 allows to adopt the same options for leading parameters (see Ref. [35,36]) implemented in PACE2_N11. Both the SM and DM codes were modified in order to produce an event-by-event output including energy and angular distributions in the laboratory system of emitted ER and light particles that, filtered according to the response function of 8 π LP, can be used for a direct comparison of predictions and experimental data.
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4. Results and discussion The main goal of the present analysis was to exploit the evaporation channel observables in order to put stringent constraints on the SM parameters included in the models used for fission dynamics studies. These parameters are crucial being also used to describe the evaporation of light pre-scission particles emitted by the compound nucleus along the path from its formation up to the scission in two fragments. So, in this work it is shown how the SM parameters influence the fission observables and how to avoid an incorrect definition of them. This latter is an essential step in order to solve the existing ambiguity on ingredients used for the description of fission dynamics, e.g. friction [21,30]. In a previous work the 32 S + 100 Mo reaction at 200 MeV extended data set was compared with the SM calculations [18], equivalent to the SM PACE2_N11. It was observed that if the analysis is limited to the pre-scission channel, as usually done in fission dynamics studies [37], LCP multiplicities and FF cross section can reasonably be well reproduced without any delay. From this result one could conclude that no dynamical effects take place in the decay of the CN. However, on one hand different combinations of the input parameters do not exclude the presence of a relatively small dynamical effect (i.e. a fission delay), as expected from the systematics [38]. On the other hand, the same model strongly overestimates the ER particle multiplicities and consequently cannot be considered as a reliable tool to estimate the fission time-scale through the pre-scission LCP multiplicities. Thus, we used the ER angular distributions and LCP evaporative spectra to limit the range of variability of the SM parameters included in the DM. Calculations considering different dissipation mechanisms were compared with this full set of experimental data and most of the observables were well reproduced [24]. Therefore, we determined the dissipation mechanisms and time-scales featuring the dynamical evolution towards the fission of excited 132 Ce nuclei. In the first part of this section we discuss the impact of different SM parameters on predictions of both the PACE2_N11 and the DM codes through the comparison with the experimental data. Therefore, we gradually introduce more and more exclusive observables to remark the differences between the predictions based on the statistical and dynamical approaches. Finally, the advantages offered by the new detection system for ERs in operation at LNL will be illustrated in the light of model predictions. The importance to explore very selective observables will be evidenced in the comparison with existing data and the benefits of new observables will be illustrated by comparing simulations filtered with the new detection system geometry. 4.1. Selection of SM and DM parameters In Fig. 2 we show the angular distribution of the evaporation residues, measured with the PISOLO electrostatic deflector, compared with the results of the statistical model calculations. The experimental statistical errors are of the order of the point size. By changing the leading parameters of the SM significant deviations are produced in the ER angular distributions as shown in Fig. 2. The leading parameters of the simulations were the LCP transmission coefficients, the level density parameters aν for particle evaporation, and the yrast line prescriptions. Transmission coefficients derived from the optical model (OM) [39–41] and from the fusion systematics (FS) [42] in combination with a level density parameters between A/6 and A/12 were used. The moment of inertia was calculated by adopting prescriptions for the yrast line with parameters
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Fig. 2. (Color online.) ER angular distribution of the reaction 32 S + 100 Mo. The experimental data (dots) [18] are compared with the SM calculations (top) performed with the four prescription of Table 1: a) and b) using the RLDM yrast line and OM transmission coefficients with aν = A/12 and aν = A/6, respectively; c) and d) using the RS yrast line and aν = A/6 with OM and FS transmission coefficients, respectively. Table 1 Prescriptions of SM parameter set adopted in the calculations for the 200 MeV 32 S + 100 Mo reaction. Prescriptions
aν
Yrast Line
Trans. Coef.
a) b) c) d)
A/12 A/6 A/6 A/6
RLDM RLDM RS RS
OM OM OM FS
from the Rotating Liquid Drop Model (RLDM) [28] or by assuming the compound nucleus as a the rigid sphere (RS) with r0 = 1.2 fm. The four different prescriptions used in the simulations shown in Fig. 2 are reported In Table 1. They are chosen among many combinations of the SM parameter values. The aim was to explore the full range of variability of the observables under examination. The ratio af/aν was kept equal to 1 due to relatively weak effects on the ER observables (see discussion in [18]). The comparison of ER angular distributions in Fig. 2a shows that the data are not well reproduced with prescriptions a), b) and c), whereas a reasonable good agreement can be obtained by adopting the prescription d). The fact that the RS approximation provides a better fit to the ER angular distribution may seem at variance with the expectation that the compound nuclei might be deformed. Indeed deformation effects might well be washed out on the ER angular distribution being them averaged over the long evaporation cascades and the broad window of angular momenta involved. Hence, we do not expect the ER angular distribution to be very sensitive to the deformation of the compound nucleus. It is therefore not unexpected that a RS approximation can reproduce the ER angular distribution alone. The RS approximation is considered in the following discussion because it provides the better reproduction of ER angular distribution. However, this observation cannot represent a strong indication for the nuclear shapes of evaporation residue nuclei. In fact, the effects of nuclear deformations on this observable are washed out being averaged on the large interval of angular momenta populating the fusion-evaporation channels. In Table 2 the LCP multiplicities and cross sections of ER and FF channels, measured using 8π LP and PISOLO and setups, respectively, are compared with a series of DM calculations. In these simulations the nuclear shape evolution as function of the time is described using the coupled Langevin equations as in Ref. [24]. Where the potential energy is calculated within the framework of a macroscopic model [28] and the inertia tensor by means of the Werner-Wheeler approximation for incompressible irrotational flow [10]. In the present analysis different prescriptions were used to calculate friction tensors. In particular we used one-body dissipation
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Table 2 ER and FF channel experimental LCP multiplicities and cross sections of the reaction 32 S + 100 Mo. The experimental observables are compared with the DM calculations performed using different set of parameters. Viscosity
aν
MpER
MαER
σ ER (mb)
MpF F
MαF F
σ F F (mb)
one-body ks = 0.1 ks = 0.25 ks = 0.5 ks = 1 ks = 1 ks = 1
A/6 A/6 A/6 A/6 A/7 A/8
1.26 1.24 1.22 1.20 1.20 1.20
0.55 0.54 0.54 0.56 0.61 0.70
706 762 770 793 797 802
0.021 0.026 0.042 0.052 0.059 0.075
0.013 0.016 0.017 0.030 0.041 0.061
230 174 166 143 139 134
two-body νo = 0.02 νo = 0.10 νo = 0.15 νo = 0.15
A/6 A/6 A/6 A/8
1.26 1.20 1.18 1.17
0.52 0.56 0.57 0.71
721 784 811 822
0.019 0.035 0.048 0.063
0.009 0.022 0.028 0.043
215 152 125 114
0.90 (0.14)
0.56 (0.09)
828 (50)
0.055 (0.007)
0.038 (0.005)
130 (13)
Exp.
[43] based on the “wall” and “wall-plus-window” formulas, with different reduction factors ks for the “wall” contribution, and the two-body dissipation [10], with different two-body friction constants νo . While, the parameters used in the subroutine LILITA_N11 to calculate energy and angular distributions of light particles and evaporation residues were kept constant according the prescription d), i.e. the one that better reproduces the ER angular distribution as shown in Fig. 2. However in order to evidence the influence of a typical SM parameter on the DM predictions, the level density parameter aν was varied from A/6 to A/8. If the analysis is limited only to the observables measured in the fission channel (pre-scission protons, MpF F , and α particles, MαF F , and fission cross section, σ F F ), the experimental data can be reasonably reproduced with high viscosity parameters or, by reducing the aν , with low viscosity parameters. In fact, the pre-scission particle multiplicities are largely increased changing from A/6 to A/8, e.g. the MαF F becomes up to about 2 times larger, and contemporary σ F F is reduced. The large variability observed as function of aν and also on the other SM parameters, see Ref. [18], lets essential the simultaneous investigation of the evaporation channel to avoid an incorrect estimation of the dissipation mechanism featuring the CN under investigation. Consequently, we investigated the ER channel observables to define the SM parameters. In fact, they are strongly influenced by SM parameters and are practical insensitive to the dissipation prescriptions and their intensities. For instance, they vary by a quantity that is below 20% by spanning the very large range of dissipation parameters shown in Table 2. So a value of aν within A/6–A/7 avoids the overestimation of α particles multiplicities in the evaporation channel produced by the value aν = A/8. Extending the comparison to the full set of observables available, the lower viscosity parameter values can be considered very unlikely because they always overestimate the σ F F and the MαER , and underestimate the σ ER . In conclusion, we obtain an overall reproduction of measured quantities using the one-body dissipation with ks = 1 and aν = A/6. We note a good reproduction of data also using two-body dissipation with νo ≈ 0.15 · 10−21 MeV s fm−3 . However, we excluded this value being unrealistically strong, i.e. very close to infinity viscosity. Indeed, also when other authors [30] used a similar value (νo = 0.15 · 10−21 MeV s fm−3 ) they well
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Fig. 3. (Color online.) ER angular distributions (left), proton (center) and α particle (right) energy spectra measured in coincidence with ERs. The experimental data (dots) are compared with the predictions of the statistical model prescription d) in Table 2 (solid line) and the dynamical model (dashed line).
reproduced σ F F , σ ER and the pre-scission neutron multiplicity, MnF F , but observed too large differences in some observables as for instance the kinetic energies of fission-fragments. In addition, this two-body prescription underestimates the MαF F . Unfortunately this discrepancy cannot be accommodated by reducing the level density parameter down to aν = A/8, because it produces an overestimation of MαER . Accordingly in the following sections we compared only the DM predictions performed using one-body dissipation with ks = 1 and aν = A/6 with the rest of measured observables. 4.2. Evaporation channel energy and angular distributions In Fig. 3 we compare the measured ER angular distributions, evaporative proton and α particle energy spectra with results of statistical and dynamical model simulations. Both models very well reproduce the ER angular distributions, the proton spectrum and the high energy side of α particle spectrum, whereas the low energy side of the latter is better reproduced with the SM. The same agreement holds also for the spectra measured at different angles. Therefore it is reasonable to conclude that an overall agreement of energy spectra and angular distributions of evaporation residues can be obtained by adopting both the SM and the DM. We would like to remark that in our calculations the deformation effects on LCP emission are considered only by changing the nuclear radius but keeping the nucleus as a rigid sphere. It is known that deformation effects of the emitting sources (ER’s and FF’s) can be very important during LCP emission both for lighter systems [44] and for heavier systems [45] in which LCP’s and neutrons were analyzed. In the present data the good reproduction of the many observables with RS approximation indicates that deformation effects can be of minor importance. However, we are working on this specific aspect by following two complementary paths: first, by searching for additional observables which might be potentially more sensitive to deformations and, second, by implementing in our codes deformations effects. The use of the other prescriptions in Table 1 produces large deviations not only in the evaporation residues distribution and fission cross sections, but also in the LCP multiplicities and energy spectra (see Table 2 and for further details Ref. [18]). The sensitivity reached in this comparison it is sufficient to strictly define the most appropriate SM parameter set to well reproduce the evaporation channel observables, but does not allow to say which of the two models better reproduces the experimental data. Indeed observables so far considered are only slightly influenced by the evaporation-fission competition.
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Fig. 4. (Color online.) Experimental and calculated protons (top) and α particles (bottom) differential multiplicity distributions as function of 8π LP LCP detector identification number. Black dots represent the experimental data. The blue (dashed) and orange (solid) lines are the SM and DM simulations, respectively. See the text for details.
Table 3 The experimental and calculated mean particle multiplicities in the ER channel together with the fission and evaporation cross sections.
DM SM Exp.
MER p
MER α
σ ER (mb)
σ FF (mb)
1.20 1.43 0.90(0.14)
0.56 0.72 0.56(0.09)
793 817 828(50)
143 139 130(13)
A more exclusive observables to constraint the reaction dynamics are the LCP differential multiplicities. These observables were obtained by normalizing the 8π LP ER-LCP coincidence yields to the number of ER single events divided by the individual LCP detector solid angle. The resulting differential multiplicities of LCP are shown in Fig. 4 as a function of the 8π LP LCP detector identification number. As mentioned before, for a fixed polar angle (θLCP ), the 18 telescopes of the ring span the azimuthal angle (φLCP ) from 0◦ to 360◦ . Hence, the oscillating behavior as function of the detector number is due to a combined effect of kinematics and of angular momentum of the composite system. In particular, the maxima correspond to events where ERs and LCPs are emitted in-plane and on opposite side with respect to the beam direction, whereas the minima occur when ERs and LCPs are emitted in-plane and on the same side with respect to the beam direction. In Fig. 4 are indicated the mean polar angles corresponding to the telescope detecting the LCPs, θLCP . By adopting different SM parameters the differential LCP multiplicities, and in particular those relative to the α particle distributions, will change not only in terms of the bulk shapes (see for instance the changing in the maxima to minima ratios in [18, 29]), but also in terms of absolute values. The results of the SM and DM calculations assuming the d) prescriptions, without any relative normalization, are compared with the experimental data. Using the SM we observe a consistent overestimation of both LCP multiplicities in the evaporation channel, while the evaporation and fission cross sections are well reproduced (see Table 3). The DM predictions well reproduce not only evaporation and fission cross sections, but also the evaporative α particle multiplicity; only the proton multiplicity is slightly overestimated. Therefore, by using the DM it is possible to obtain a better overall agreement in the evaporation channel that makes us confident on the use of such a model for studying the fission process.
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Fig. 5. (Color online.) Ratios between α particles and protons partial multiplicities obtained with 8π LP Ball filter as function of ER detection angles. The lines represent the SM calculations performed by using the four prescriptions of Table 1 and the DM calculation, the dot is the experimental value measured with 8π LP Ball and the PPAC at 4.5 degrees.
4.3. New observables High precision measurements of the evaporative LCP absolute multiplicities can be obtained only by measuring simultaneously ERs and LCPs emitted on the full solid angle. However, experimental apparatuses usually have limited angular coverage and only partial multiplicities are accessible. By this we refer to the case when differential multiplicities are directly measured only at one fixed trigger position (the angle of ER detector). The angle-integrated multiplicities, also named as “absolute multiplicities”, are commonly obtained by a model calculation that reproduces, at the best, the partial multiplicities. Another way to describe in short this procedure is to say that the partial multiplicities are integrated by the model. In this sense the absolute multiplicities are model dependent. This method also allows to remove the dependence on the single trigger detector position. We report in Table 3 the model dependent absolute multiplicities obtained following this procedure. We performed redundant measurements by using 4 symmetric PPACs around the beam direction. The redundancy is very useful because from one side assures the correct alignment of the beam impinging on the target, and from the other it makes it possible to apply a summing procedure to reduce statistical fluctuations. By using our setup, the partial multiplicities were obtained as the ratios among the yields of LCP in coincidence with ER’s and the yields of all ER’s detected in single by our PPAC’s. Afterward, the absolute multiplicities were extrapolated from the predictions that reproduce the distribution of partial multiplicities. Where we obtained the simulated partial multiplicities by filtering the event-by-event output of the simulation codes through the geometry of the 8π LP apparatus and the PPAC system mounted at 4.5 degrees. In order to further exploit the partial multiplicities as a function of the position of trigger detector, we computed the ratio between the proton and α particle partial multiplicities for different trigger positions. This new observable does not require the integration over the trigger angle and therefore can be directly compared to experimental data. The results of calculations for the different models used so far are shown in Fig. 5 together with the experimental point measured with our setup. This ratio is simultaneously model independent and very sensitive not only to the decay kinematics, but also to the competition between evaporative LCP and fission channels. The trends of the ratios show that partial multiplicities measured at various trigger angles allow to distinguish between different models and parameter values. In other words, the ratios of the partial multiplicities as function of the trigger position is an additional good probe for the fission
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Fig. 6. (Color online.) Right-hand panel: exploded view drawing (a) of the new PPAC system for evaporation residue detection. θ is the spanned angular range that depends on the mounting position. Narrower angular openings for the ions can be defined by combining this system with masks. In the 8π LP chamber, the minimum value of 3 degrees at around the forward direction is achieved by installing the PPAC at 60 cm distance from the target. Right-hand panel: photograph of the front side of the new PPAC system.
dynamics. The trend observed in the calculations is due to the interplay among many ingredients. Generally speaking, it is possible to observe a reduction of the proton partial multiplicities with the increase of θtrigger ; on the contrary the α particle multiplicities quickly increase. This behavior is explained by considering that ERs at larger θtrigger need more recoil, that is preferably provided by α particle evaporation. Therefore, larger θtrigger selects events with larger α particle multiplicities. Furthermore, α emission carries out more excitation energy with respect to lighter particles and the probability of a coincident proton emission will be reduced further by increasing the ratio at higher θtrigger . More complex is to explain the differences originated by each single SM parameter. For a fixed yrast line, increasing aν produces almost constant reduction in the ratios (see the prescriptions a) and b)). Going from the yrast line calculated using the RLDM to the yrast line assuming the nucleus as a rigid sphere with r0 = 1.2 fm (that means a decrease of the moment of inertia) we observe a decrease in the ratios, that becomes larger with θtrigger (see the prescriptions b) and c)). While by changing transmission coefficients, i.e. passing from OM to FS ones, there is a decrease of about a factor two in the ratio that is almost independent from the ER angles considered, (see prescription c) and d)). We observe that only combining all the effects producing a decrease in the α particle to proton multiplicity ratios is possible to improve the reproduction of experimental data. Thus, in agreement also with all the other observables, the prescription d) is the best set of parameters. By comparing the statistical model calculation with the dynamical one, only small differences are observed at θtrigger = 4.5 degrees. This experimental datum alone cannot provide a conclusive indication about which of the two models better describe the CN decay. However, the important conclusion is that measurements at different trigger angles can indeed constitute an additional probe to distinguish between SM and DM descriptions. Given this peculiar dependence of the partial multiplicities on the trigger angle, we have prepared a new system for ER detection, whose schematic is shown in the Fig. 6. It consists of two annular PPACs (front and rear) divided in 6 independent sectors with a wide area and an absorber foil mounted in between. The absorber is adapted to the experimental conditions with thickness sufficient to stop only ERs and to let the other lighter ions, like LCPs and elastic scattered beam particles, passing through it and reach the rear PPAC. The ERs are therefore selected using the rear PPAC signal as a veto. A mask can be mounted in front of the PPAC, as indicated by the three blue spots in the Fig. 6, in order to detect ER angular distribution between 3 and 8 degrees with respect to the beam. Furthermore, we would like to stress that thanks to the symmetric arrangement of the Ball telescopes and the PPAC sectors, also with this system
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it is possible to perform redundant measurements and get the consequent benefits. Therefore we planned to use this new PPAC system for our next experiments with 8π LP detection apparatus. 5. Summary and conclusions The larger charged particle multiplicities in pre-scission and evaporation channels occurring in the intermediate fissility composite systems make them well suited to get information on the fission process. In this work we studied the evaporative light charged particles emitted by the 132 Ce nuclei at E = 122 MeV by comparing the simulations with experimental data measured x with the 8π LP apparatus. We performed the simulations using the SM code PACE2_N11 and a dynamical model based on the 3-D Langevin equations. Evaporation residue angular distribution and LCP energy spectra were exploited to define the SM parameters for SM and DM simulations. The ER α particle and proton multiplicities are always overestimated by the PACE2_N11 code, irrespective of the SM parameters adopted. Such a failure would affect also the emission probability in the pre-scission channel, making the extraction of the fission time scale unreliable. A possible reason for the LCP multiplicities overestimation in the ER channel could be ascribed to a not properly accounted competition between the different decay channels. This problem would affect at the same time several ingredients of the SM such as level density and transmission coefficients. This hypothesis could be explored by enlarging the present data set with observables related to the neutron channel, such as neutron energy spectra in coincidence with ER and FF. Unfortunately, the neutron spectra in the ER channel are rarely collected. This remark indeed reinforce the fact that extended data set are essential to achieve reliable simulations to address the open questions on fission dynamics and strongly suggest the use of setups of high efficiency. Our analysis would greatly benefit of the measurement of neutrons in coincidence with ER and FF and will be pursued in the near future, possibly by using the neutron detector array NEDA [46], currently under construction. The possibility of using a detection setup which allows to measure partial multiplicities for different ER trigger positions gives access to a new observable which is sensitive to the different implementation of fission dynamics models. By means of an annular PPAC, the angular distribution of the ratios between the α particle and proton partial multiplicities can be measured over an extended angular coverage (see Fig. 5). From the experimental point of view this quantity is model independent and, according to the predictions, is extremely sensitive not only to the main (leading) evaporative parameters, but also to the fission process description included in the statistical and the dynamical model adopted in this work. In conclusion, the models strongly indicate that the exclusive observables of evaporation residue channel represent a powerful tool to progress in fission dynamics studies. Acknowledgements We would like to thank the anonymous referee for his helpful comments and constructive suggestions. References [1] L. Meitner, O.R. Frisch, Nature 143 (1939) 239, https://doi.org/10.1038/143471a0. [2] O. Hahn, F. Strassmann, Naturwissenschaften 27 (1939) 11, https://doi.org/10.1007/BF01488241.
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