Decay of excited nuclei produced in 78,82 Kr + 40 Ca reactions at 5.5 MeV/nucleon
G. Ademard, J.P. Wieleczko, J. Gomez Del Campo, M. La Commara, E. Bonnet, M. Vigilante, A. Chbihi, J.D. Frankland, E. Rosato, G. Spadaccini, Sh.A. Kalandarov, C. Beck, S. Barlini, B. Borderie, R. Bougault, R. Dayras, G. De Angelis, J. De Sanctis, V.L. Kravchuk, P. Lautesse, N. Le Neindre, J. Moisan, A. D'onofrio, M. Parlog, D. Pierroutsakou, M.F. Rivet, M. Romoli, R. Roy, G.G. Adamian, N.V. Antonenko
aa r X i v : . [ nu c l - e x ] A p r Decay of excited nuclei produced in , Kr + Ca reactions at 5.5 MeV/nucleon
G. Ademard, J.P. Wieleczko, ∗ J. Gomez del Campo, M. La Commara,
3, 4
E. Bonnet, M. Vigilante,
3, 4
A. Chbihi, J.D. Frankland, E. Rosato,
3, 4
G. Spadaccini,
3, 4
Sh.A. Kalandarov,
C. Beck, S. Barlini, B. Borderie, R. Bougault, R. Dayras, G. De Angelis, J. De Sanctis, V.L. Kravchuk, P. Lautesse, N. Le Neindre, J. Moisan,
1, 15
A. D’Onofrio, M. Parlog, D. Pierroutsakou, M.F. Rivet, M. Romoli, R. Roy, G.G. Adamian,
5, 6 and N.V. Antonenko Grand Acc´el´erateur National d’Ions Lourds (GANIL),CEA/DSM-CNRS/IN2P3, Boulevard H. Becquerel, F-14076, Caen, France Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Dipartimento di Scienze Fisiche, Universit`a di Napoli ”Federico II”, I-80126, Napoli, Italy INFN, Sezione di Napoli, I-80126, Napoli, Italy Joint Institute for Nuclear Research, 141980 Dubna, Russia Institute of Nuclear Physics, 702132 Tashkent, Uzbekistan IPHC, IN2P3-CNRS, F-67037, Strasbourg Cedex2, France INFN, Sezione di Firenze, I-50125 Firenze, Italy IPNO, IN2P3-CNRS and Universit´e Paris-Sud 11, F-91406, Orsay Cedex, France LPC, IN2P3-CNRS, ENSICAEN and Universit´e, F-14050, Caen Cedex, France CEA, IRFU, SPhN, CEA/Saclay, F-91191, Gif-sur-Yvette Cedex, France INFN, LNL, I-35020 Legnaro (Padova) Italy INFN, Sezione di Bologna, I-40127 Bologna, Italy IPNL, IN2P3-CNRS et Universit´e, F-69622, Villeurbanne Cedex, France Laboratoire de Physique Nucl´eaire, Universit´e de Laval, Qu´ebec, Canada Dipartimento di Scienze Ambientali, Seconda Universit`a di Napoli, I-81100, Caserta, Italy (Dated: September 12, 2018)Decay modes of excited nuclei are investigated in , Kr + Ca reactions at 5.5 MeV/nucleon.Charged products were measured by means of the 4 π INDRA array. Kinetic-energy spectra and an-gular distributions of fragments with atomic number 3 ≤ Z ≤
28 indicate a high degree of relaxationand are compatible with a fission-like phenomenon. Persistence of structure effects is evidenced fromelemental cross-sections ( σ Z ) as well as a strong odd-even-staggering (o-e-s) of the light-fragmentyields. The magnitude of the staggering does not significantly depend on the neutron content of theemitting system. Fragment-particle coincidences suggest that the light partners in very asymmetricfission are emitted either cold or at excitation energies below the particle emission thresholds. Theevaporation residue cross-section of the Kr + Ca reaction is slightly higher than the one mea-sured in Kr + Ca reaction. The fission-like component is larger by ∼
25% for the reaction havingthe lowest neutron-to-proton ratio. These experimental features are confronted to the predictionsof theoretical models. The Hauser-Feshbach approach including the emission of fragments up to Z = 14 in their ground states as well as excited states does not account for the main features of σ Z .For both reactions, the transition-state formalism reasonably reproduces the Z -distribution of thefragments with charge 12 ≤ Z ≤
28. However, this model strongly overestimates the light-fragmentcross-sections and does not explain the o-e-s of the yields for 6 ≤ Z ≤
10. The shape of the whole Z -distribution and the o-e-s of the light-fragment yields are satisfactorily reproduced within thedinuclear system framework which treats the competition between evaporation, fusion-fission andquasifission processes. The model suggests that heavy fragments come mainly from quasifission whilelight fragments are predominantly populated by fusion. An underestimation of the cross sectionsfor 16 ≤ Z ≤
22 could signal a mechanism in addition to the capture process.
PACS numbers: 24.60.Dr, 24.10.Pa, 25.70.Gh
I. INTRODUCTION
Heavy-ion induced reactions are appropriate to explorethe response of nuclei under stress of different nature andto delineate the degrees of freedom at work in the var-ious bombarding energy domains. The regime of warmmedium-mass ( A ∼ − ∗ Electronic address: [email protected] formed in fusion reactions at incident energies below 10MeV/nucleon is characterized by the predominant role ofthe angular momentum of the emitting nuclei and of themass (charge) asymmetry degree of freedom. An abun-dant literature has reported that the CN decay modespopulate the whole mass (charge) range from evaporatedlight particles (like n, p , α ) up to the symmetric fission,and the intermediate-mass fragments (IMF) in betweenthe two extremes [1–5]. From the accumulated data onecould identify two basic features of the final products: thecharge distribution evolves from a U-shape at low angu-lar momentum (with a minimum at symmetry) towardsa bell shape at high angular momentum (with a maxi-mum around symmetric fission) [2]; a staggering of thefragment cross-sections σ Z is superimposed on this globalfeature, with a magnitude which depends on the size ofthe emitting nuclei and which increases as the neutron-to-proton N/Z ratio of the emitter decreases [3, 7]. It hasbeen suggested that the staggering effects reflect someproperties of nuclei involved at the end of the disinte-gration cascade [8]. Indeed, a plausible explanation ofthe staggering of σ Z would be that structure effects per-sist in the production mechanism and that fragments areemitted relatively cold, otherwise the subsequent decaywould have blurred the fluctuations of the yields. More-over, the neutron content of the emitter manifests itselfin the magnitude of the IMF cross-sections as shown inRefs. [3, 7, 9]. This raises the question of the N/Z depen-dence of the decay channels which is a relatively unknownand very attractive topic in the context of radioactivebeam facilities.On the theoretical side, sophisticated approaches havebeen developed to explain the complex facets of the dis-integration modes. Many features of the light-particleemission are satisfactorily understood within the Hauser-Feshbach formalism [10] emphasizing the role of the avail-able phase space at each step of the whole cascade [11].On the other hand, the mechanism at the origin of thefragment emission from CN has stimulated numerous ap-proaches as for example: the multi-step Hauser-Feshbachmodel including emission channels up to Ca [12]; thetransition -state model describing IMF emission as asym-metric fission [4, 13]; the dynamical cluster-decay modelassuming pre-formed cluster [14, 15]; the dinuclear sys-tem model aiming to treat the competition between theevaporation channel and the binary-decay channels as-sociated to fusion and quasifission processes [16]. Thoseapproaches are based on distinct hypotheses as well asfundamental nuclear ingredients such as the level den-sity or the fission barriers to describe the thermal andcollective properties that rule the competition betweenCN decay modes. It is worth noticing that the
N/Z degree of freedom is expected to play a crucial role onthese quantities. For example, the level-density param-eter is related to the effective mass, a property of theeffective nucleon-nucleon interaction that is sensitive tothe neutron-proton composition of the nuclei; the fissionbarriers depend strongly on the symmetry energy thatis weakly constrained by experimental data [17]. There-fore, new experimental data on decay channels of nucleiat high angular momenta and different
N/Z ratio aresorely needed.Besides the decay stage, the phase of CN formation hasits own crucial interest. Indeed, since more than threedecades, a rich wealth of data has revealed the complex-ity of the fusion process and of the collisional stage pre-ceding the CN formation. For example, extensive exper-imental and theoretical investigations have shown thatfusion mechanism at the vicinity of the barrier [18] is drastically influenced by the internal structure and
N/Z ratio of the participating nuclei. Moreover, a large bodyof data for a wide variety of systems has demonstratedthe role of dynamical effects on the fusion process andthe strong inhibition of the CN formation by quasifis-sion (QF). This phenomenon corresponds to the separa-tion of the partners after a significant rearrangement ofthe mass and charge degree of freedom [19–24]. Inter-estingly, in medium-mass systems, it has been recentlyshown [16], that the competition between fusion-fissionand quasifission mechanisms strongly depends on the an-gular momentum. This calls for new data to extent ourknowledge on the influence of the dynamics on fusionprocess in this mass region.Finally, we would like to stress that an accurate pre-diction of the IMF cross-sections has important conse-quences. Indeed, one could perform spectroscopic stud-ies of the residual nuclei left in excited states after thefragment emission. This kind of experiment has shownthe strong selectivity of the C emission with respectto the 3 α channel [25]. An evident area for such stud-ies is around the doubly magic Sn since these nucleiare extremely difficult to reach by means of the stan-dard fusion-evaporation method. However, a recent at-tempt [26] has suggested that the C emission from a
Ba CN formed in the Ni + Ni fusion reaction at ∼ , Kr + Ca reactions at5.5 MeV/nucleon incident energy. This energy regimeis well adapted to form nuclei in a controlled way interms of excitation energy since the incomplete-fusionprocess or pre-equilibrium emission are expected to benegligible. Exclusive measurements on an event-by-eventbasis are required to provide a characterization of themechanism. Therefore a 4 π detection apparatus withlow energy thresholds and charge identification of theproducts is needed. The combination of both INDRAarray [27] and the technique of the reverse kinematicspermit us to collect high quality data on evaporation-residues and elemental cross-sections of fragments. Ourdata set, obtained with a projectile pair differing by fourneutrons, gives new insights on the influence of the neu-tron content on decay mechanisms that allows us to eval-uate the respective merits of very popular theoretical ap-proaches. Some preliminary results have been recentlypresented [28]. Here we concentrate on main features ofthe heavy products, and the study of the light-particleemission will be presented in a forthcoming paper.In Table I are grouped some quantities characterizingthe , Kr + Ca reactions at 5.5 MeV/nucleon incidentenergy. CN excitation energies E ⋆ have been calculatedusing mass tables [29]. l graz ~ is the grazing angular mo-mentum given by semi-classical formula. l pocket ~ is theangular momentum at which the pocket in the interac-tion potential vanishes. The potential is calculated asin Ref. [30]. J cr ~ is the maximum angular momentumfor capture process as deduced from the dinuclear sys-tem (DNS) calculations (see Sect. V for details). N/Z isthe neutron-to-proton ratio of the reaction and V B is thefusion barrier [30]. Others interaction potential choices,like those compared in [31], give similar l pocket and V B values. As reported in Table I, the total available kineticenergy in the center-of-mass (c.m.) E c.m. is well abovethe fusion barrier and the grazing angular momentumis large with respect to l pocket ~ . Thus, in the reactionsunder study, we expect that the fusion process will bemainly governed by the inner pocket of the potential andto a lesser extent by the external fusion barrier. TABLE I: Quantities characterizing the studied reactions. Kr + Ca Kr + Ca E ⋆ (MeV) 99 107 E c.m. / V B V B (MeV) 91.2 90.3 N/Z l graz
96 100 l pocket
70 73 J cr
73 75
The organization of the paper is as follows: the experi-mental procedures are described in Sec. II. Experimentalresults are shown in Sec. III for the inclusive data andin Sec. IV for the fragment-light particle coincidences.Sec. V deals with comparisons to statistical and dynam-ical calculations. The conclusions of this work are givenin Sec. VI.
II. EXPERIMENTAL PROCEDURES
The experiment was performed at the GANIL facilityin Caen. Beams of , Kr projectiles with energies of5.5 MeV/nucleon impinged on self-supporting 1 mg/cm thick Ca targets. The targets were prepared from highpurity foils by rolling. The contaminants, mostly oxy-gen and tantalum, were negligible as thoroughly checkedduring the data analysis.The kinetic energy and atomic number of the ejectileswere measured by means of the 4 π INDRA array. Thereverse kinematics confers to the reaction products a fo-cussing at forward angles and a momentum boost in thelaboratory frame. For the experimental data reportedhere, a significant part of the reaction products is emit-ted from 3 ◦ to 45 ◦ . In this range, the INDRA arrayis made of 8 rings comprising detection modules with FIG. 1: (Color online) Two-dimensional plot combining theenergy deposited in the ionization chamber (vertical axis) andin the silicon detector (horizontal axis) for fragments emittedat 10 ◦ ≤ θ lab ≤ ◦ measured in the Kr + Ca reactionat 5.5 MeV/nucleon. three layers: an ionization chamber (IC) operated with50 mbar (30 mbar) of C F gas for 3 ◦ ≤ θ lab ≤ ◦ (27 ◦ ≤ θ lab ≤ ◦ ), respectively; a 300 µ m thick sili-con detector (Si); a 14 or 10 cm length CsI(Tl) scintilla-tor. The medium and backward angles from 45 ◦ to 176 ◦ are covered by 8 rings equipped with IC/CsI(Tl) detec-tors, the ICs being operated with 30 mbar of C F gas.For the calibration of the CsI at backward angles, onemodule per ring is equipped with a Si(80 µ m)/SiLi(2000 µ m) telescope inserted between IC and CsI. The energycalibration of the various layers was obtained by meansof alpha particles emitted from a Cf source and fromthe elastic scattering of projectiles having various ener-gies ( , , Kr , As , Cr , Mo ) selectedthanks to the CIME cyclotron. Energy calibration of thedetectors ensured on accuracy of within 5%.The intensity of the beams was adjusted in order tomaintain a low probability for pile-up of the events andthe data acquisition dead time below 25%. The reac-tion products were measured event-by-event by using tworecording modes, a minimum-bias trigger based on thenumber M of fired telescopes. The first mode ( M ≥ M ≥
2) per-mits to accumulate high statistics for the reactions ofinterest.The kinetic energy and the atomic number of thedetected products were deduced from the energy de-posited in the IC and Si detectors, corrected for the en-ergy losses in the target as well as in the dead zonesof the various detection layers [32]. A typical exampleof a two-dimensional calculated spectrum used for the Z -identification is shown in Fig. 1 where the horizontal(vertical) axis represents the energy deposited in the Si(IC) detector, respectively. These data were obtained at10 ◦ ≤ θ lab ≤ ◦ . Although only the fragments emit-ted in the forward hemisphere in the c.m. are collected,one recognises the typical pattern of reaction productsin reverse kinematics. The ridges associated to differentatomic number are seen from Z = 6 up to Z = 37. Theproducts with charge 3 ≤ Z ≤ Z = 16 with a stronger magnitude forfragments with charge Z ≤
10. Moreover, we clearly see aquasi-elastic component around Z = 36 which manifestswith a higher statistics.Event-by-event Z -identification of each detected prod-uct was achieved by projecting data such as that of Fig. 1onto lines which were drawn so as to follow the ridge foreach Z . Charge resolution of one unit was obtained upto Z = 37 for high-energy fragments. Identification forlow-energy fragments was assured by calculations basedon energy-loss tables, with a resolution of few chargeunits [33]. Then we build two calculated spectra repre-senting the total kinetic energy in the laboratory frame E tot (the total charge Z tot ) obtained by summing up thekinetic energy (the atomic number) of each particle iden-tified in the event, respectively. In the following steps ofthe analysis, we kept only the events satisfying Z tot ≤
60 and E tot ≤ E lab , where E lab is the bombarding en-ergy. The limit on Z tot slightly exceeds the total availablecharge ( Z tot = 60) to take into account the uncertaintyon the charge identification. Applying such criteria en-ables us to control the event pile-up and double countingof the elastic scattering has been evaluated to be less than4 × − . Consequently, the number of events comprisingparticles coming from two distinct reactions is negligible. III. EXPERIMENTAL RESULTSA. Kinematical features
Another piece of information on the reaction mech-anism can be obtained from the kinetic-energy spectraof the ejectiles. The transformation into the center-of-mass frame was obtained by means of an event-by-eventanalysis. Fig. 2 shows some representative examples ofthe c.m. kinetic-energy spectra of fragments with theindicated atomic number from Z = 6 to Z = 24 scat-tered at 7 ◦ ≤ θ lab ≤ ◦ in the Kr + Ca reactionat 5.5 MeV/nucleon. A Gaussian-like distribution (lines
20 40 60 80 -2 -1 Z=6Z=7Z=8
20 40 60 80 -2 -1 Z=12Z=14Z=16
20 40 60 80 -2-1 Z=9Z=10Z=11
20 40 60 80 -2-1 Z=18Z=20Z=24 (MeV) c.m. ε ( a r b . un i t s ) c . m . ε d Ω / d σ d FIG. 2: (Color online) Center-of-mass kinetic-energy spec-tra of fragments with indicated atomic number from Z = 6to Z = 24 produced in the Kr + Ca reaction at 5.5MeV/nucleon and detected at 7 ◦ ≤ θ lab ≤ ◦ . Lines repre-sent the results of a fit with a Gaussian function. Statisticalerrors are shown. in Fig. 2) reproduces rather well the experimental dataover a large energy range. Such a feature could be re-lated to secondary emission of light particles or/and toshape fluctuations with the associated variations of theCoulomb barrier.For each fragment, the c.m. average velocity < V c.m. > was deduced from the average kinetic energy assuming amass number given by an empirical formula [34]. Theresults are reported in Fig. 3 for various laboratory an-gles corresponding to the average values of the detectionrings. For a given Z , < V c.m. > is roughly the same re-gardless of the emission angle except for Z ≤
12 at themost forward angles. We thus conclude that a high de-gree of relaxation of the relative kinetic energy has beenreached prior to the breakup of the excited nuclear sys-tem. < V c.m. > follows a quasi-linear decreases withincreasing atomic charge Z . This feature is well docu-mented ([3, 4, 13]), and is interpreted as a signature ofa binary process dominated by the Coulomb interactionbetween the considered fragment and its complementarypartner. The total average kinetic energy for symmetricdivision ( < T KE sym > = 81 ± Z = 28) isconsistent ( E K = 83.4 MeV for the Ba nucleus) witha recent compilation on the total kinetic energy releasein the fission phenomenon [35]. < V c . m . > ( c m n s - ) Z < θ lab > = 5.7 o < θ lab > = 8.5 o < θ lab > = 11.8 o FIG. 3: (Color online) Experimental c.m. average velocity < V c.m. > of fragments with atomic number 6 ≤ Z ≤ Kr + Ca reaction at 5.5MeV/nucleon. -2 -1
0 20 40 60 80 100 d σ / d Ω c . m . ( a r b . un i t s ) < θ c.m. > (deg)Z = 32 Z = 33 exp(- a θ c.m. ) FIG. 4: (Color online) Angular distributions of fragmentswith atomic number Z = 32 and 33 produced in the Kr + Ca reaction at 5.5 MeV/nucleon. The lines areexponential functions to guide the eye.
B. Angular distributions
Valuable information on the production mechanismcould be extracted from the angular distributions of thefragments. These distributions are obtained by integrat-ing the kinetic-energy spectra. Some typical examplesare given in Figs. 4 and 5 for various fragments.The angular distributions of the fragments with atomicnumber close to the projectile one ( Z = 36) are stronglypeaked at forward angles as shown in Fig. 4. These prod-ucts arise from direct two-body reactions or deep inelas-tic collisions in which nucleons are transferred into oremitted from the projectile. Indeed, in peripheral colli-sions the target-like products are expected to be ejectedin the backward hemisphere of the c.m., while those com-ing from the projectile would be strongly focused in theforward hemisphere. Fig. 4 illustrates such a behaviourfor Z = 32 and Z = 33 for which the angular dis- d σ / d Ω c . m . ( a r b . un i t s ) < θ c.m. > (deg) Z = 10Z = 12Z = 14 0 20 40 60 80Z = 16Z = 20Z = 24 θ c.m. FIG. 5: (Color online) Angular distributions of fragmentswith charge Z = 10 , , , , ,
24 produced in the Kr + Ca reaction at 5.5 MeV/nucleon. Dashed linesare 1 / sin θ c . m . functions that have been normalized to theexperimental data at < θ lab > = 5.7 ◦ , corresponding to < θ c.m. > = 12 ◦ –17 ◦ . Error bars are inside the symbols. tributions dσ/d Ω c . m . exhibit a strong decrease. More-over, one observes two components corresponding pre-sumably to quasi-elastic reactions at the most forwardangles and deep-inelastic collisions which dominate for θ c.m. & ◦ . The continuous line in Fig. 4 represents anexponential function that follows the experimental datafor θ c.m. & ◦ .In Fig. 5 we present the angular distributions dσ /dΩ c . m . for fragments with atomic number Z =10 , , , , ,
24 produced in the Kr + Ca reac-tion. In spite of a measurement over a limited angularrange in the laboratory frame, the reverse kinematics al-lows to define unambiguously the shape of the angulardistributions in the c.m. frame. In contrast with thepreviously observed feature for fragments with Z ∼ / sin θ c . m . dependence(shown as dashed lines in Fig. 5). This signs a high de-gree of equilibration. Indeed, in heavy-ion reactions, CNwhich undergo fission have generally high angular mo-mentum and the angular distributions of the fission frag-ments would show a 1 / sin θ c . m . shape. However, thiskind of behaviour is not a sufficient condition to ensurea CN formation. In fact, in quasifission (QF) process,the reactants retain some memory of the entrance chan-nel which manifests in a strong anisotropy of the an-gular distribution [23]. Our apparatus does not allowan accurate measurement of the angular distributions ofthe fragments scattered at angles close to the beam di-rection. This prevents a dedicated investigation of theanisotropy. Thus at this stage of the analysis of the an-gular distributions presented in Fig. 5, one concludes thatthe predominant mode of the fragment production is thedisintegration either of a long-lived system or CN. Z Z Ca Kr+ FIG. 6: (Color online) Experimental correlation betweenthe two biggest fragments Z and Z with Z ≥ Z and48 ≤ Z tot ≤ C. Fragment-fragment coincidences
The correlations between the charge of the fragmentsare instructive since they permit to check the binary na-ture of the mechanism. In the present work, an even-by-event analysis was performed in order to extract the twobiggest fragments, i . e . those having the highest atomicnumbers Z and Z (with Z ≥ Z ) in each event.Fig. 6 shows the Z vs Z correlation measured in the Kr + Ca reaction in the case of events satisfying thecriterion 48 ≤ Z tot ≤
60. The lower limit is applied toexclude the events in which one of the two fragmentshas not been detected. The upper limit take into ac-count the uncertainty on the Z -identification (see Sec. II).The highest yields are localised in two regions: Z ∼ Z ∼ Z , Z ∼ Z = 56), reflecting the light-particle emissionfrom the fragments, or/and from the composite systembefore splitting. The linear correlation between Z and Z illustrates the binary nature of the mechanism. Here,the term binary means that the major part of the nu-cleons available in the reaction is distributed in the twobiggest measured fragments.As far as kinetic-energy spectra, angular distributionsof the fragments and fragment-fragment coincidences areconcerned, the same conclusions hold for Kr + Careaction.
D. Cross sections
The absolute differential cross-sections dσ/d Ω c . m . wereobtained from the normalization with respect to the elas-tic scattering. To select the appropriate angle for normal-ization purposes, both grazing angle and angular distri-bution of the elastic scattering were deduced from opticalmodel calculations [38]. To do so, a set of optical parame-ters was extracted from the study of the Ar + Se reactionat 5 MeV/nucleon [39] which is similar to those studied inthe present work. From the analysis, we deduced that thegrazing angle is about θ lab = 20 ◦ (around θ c.m. = 55 ◦ ).Moreover, σ/σ Ruth ( θ lab ) = 1 for θ lab ≤ ◦ . Thus theRutherford differential cross-section of the elastic scat-tering was integrated over the range 7 ◦ ≤ θ lab ≤ ◦ to get the normalization factor. Then the absolute to-tal cross-sections of the fragments with atomic number3 ≤ Z ≤
28 were obtained by angular integration assum-ing a 1 / sin θ c . m . shape as indicated in Sec. III.B. Thisprocedure could not be suited to the non-measured partof the angular distribution close to the beam direction,but the weight of this angular domain is negligible.In the following, we concentrate on the decay be-haviour of a long-lived system, and consequently the crosssections of the quasi-elastic component are not discussedhere due to the exponential shape of the angular distri-butions, akin to a fast process.The inclusive cross-sections σ Z of fragments withatomic number 3 ≤ Z ≤
28 are shown in Fig. 7 forthe Kr + Ca (solid squares) and Kr + Ca (opensquares) reactions. Note that the Be cross-sections aredepleted due to the contribution of the non-identified Befragment. The σ Z distributions for both systems exhibita maximum around Z = 26, a value close to half of theavailable charge. Such a feature indicates that these el-ements come either from the symmetric fission of CN orfrom a class of collisions in which a strong relaxation ofthe entrance channel mass-asymmetry has been reached. σ Z ( m b ) Z Kr+ Ca Kr+ Ca FIG. 7: (Color online) Experimental cross-sections for frag-ments with atomic number 3 ≤ Z ≤
28 emitted in the Kr + Ca (full squares) and Kr + Ca (open squares)reactions at 5.5 MeV/nucleon.
Moreover, except for 3 ≤ Z ≤ σ Z measured in the Kr + Ca system is systematically lower and the yieldsaround the symmetric splitting are about 25% smaller forthe system having the highest neutron-to-proton ratio.Such a lowering of the cross section for the symmetricsplitting as the neutron content of the emitter increasesis also observed in , , Kr + C reactions [3]. This
N/Z dependence would be consistent with the expecta-tions of the liquid-drop model in which the fission bar-rier of a neutron-poor CN is expected to be smaller thanfor the neutron-rich one, providing that these fission-likefragments originate from CN decay.A strong odd-even-staggering (o-e-s) of the σ Z for frag-ments with Z ≤
10 is visible, and this effect is still presentfor higher Z with a smaller amplitude. Fluctuations infragment yields have already been observed in a widerange of reactions, from CN regime to spallation reac-tions [3, 7, 8, 36, 37]. It is worth noticing that the stag-gering in the yields of light clusters shown in Fig. 7 isvery similar to the one observed for systems in the samerange of mass, excitation energy and angular momen-tum [2, 5]. This would indicate that the staggering isnot preferentially driven by microscopic properties of thecomplementary partners since they are different for eachstudied reaction.As shown in Fig. 7, the o-e-s for light fragments isroughly the same for both reactions and is about a fac-tor 3. Such a result is at variance with , , Kr + C data [3] for which the o-e-s decreases for neutron-rich CN. As far as the entrance channels are concerned,the main difference between those data and the presentones comes from the magnitude of the spin that couldbe transferred into the composite system. Thus, the o-e-s of the light-fragment yields could be influenced bythe spin which would induce different compactness of thescission-configurations and thus a sensitivity to structureproperties in the deformation space.As suggested by the shape of the Z -distribution, thehigh partial waves in the entrance channel should havefed the fragment emission mechanism. However, thecross sections of the light clusters (Li, B) are astonish-ingly low. Indeed, in Nb + Be, C reactions [4] inwhich low angular momentum were involved, the crosssections of the light clusters are of the same order ofmagnitude or even higher than in our measurements. Apossible explanation would be that at high angular mo-mentum a large part of the flux has been deviated froma CN formation. Such a possibility will be discussed inSec. V.The cross sections of the fission-like products, σ expfiss ,were obtained by summing up the yields of the frag-ments in a range of atomic number 3 ≤ Z ≤
26. Theupper limit corresponds to the atomic number of thefragments produced with the highest cross-section andtakes into account qualitatively the secondary decay oflight charged particles (see Fig. 6). Thus, consideringthe range 3 ≤ Z ≤
26 we obtain σ expfiss = 447 ±
46 mb( σ expfiss = 332 ±
35 mb) for the Kr + Ca ( Kr + Ca) reactions, respectively. We remind here that we havetermed as fission-like products those with an angular dis-tribution akin to that of a long-lived system, and σ expfiss could contain both CN and QF contributions.The ER component is identified thanks to a ∆ E − E two-dimensional plot using the energy deposited in theIC and Si detectors. Absolute differential cross-sections dσ ER /d Ω lab are deduced from the normalization withrespect to the elastic scattering. Since dσ ER /d Ω lab ≈ exp[ − k sin θ lab ] [40], the experimental distribution is ex-trapolated towards the beam direction, and σ expER could beextracted. Extensive simulations using statistical codePACE4 [41] were performed to check this procedure. Weobtain σ expER = 539 ±
100 mb ( σ expER = 492 ±
90 mb) forthe Kr + Ca ( Kr + Ca) reactions, respectively.These cross sections together with σ expfiss are gathered inTable II.The sum of the fission-like and ER cross-sections de-fines the experimental capture cross-sections σ expcapt = σ expER + σ expfiss and we measured σ expcapt = 986 ±
110 mb( σ expcapt = 824 ±
97 mb) for the Kr + Ca ( Kr + Ca)reaction, respectively. By using the sharp cut-off approx-imation formula σ expcapt ( E c . m . ) = π ~ µE c . m . J max X J =0 (2 J + 1)= π ~ µE c . m . ( J max + 1) , (1)we obtained J expmax = 75 ± ±
4) for the Kr + Ca( Kr + Ca) reaction, respectively.
TABLE II: Measured and calculated evaporation residues andfission-like cross-sections. See Sec. V. for details of the calcu-lations performed with GEMINI and DNS codes.(mb) Kr + Ca Kr + Ca σ expfiss ±
46 332 ± σ expE.R. ±
100 492 ± σ geminifiss
600 547 σ geminiE.R.
237 285 σ DNSfiss
349 208 σ DNSE.R.
601 638
From the ER cross-sections we have calculated the re-duced quantity Λ ER = 2 µE c . m . σ/ ( π ~ ), in which the de-pendence on the entrance channel is removed. In theliterature we have extracted the same quantity for re-actions similar to those studied here. The Λ ER val-ues for , Kr + Ca reactions are compatible withthe data for quasi-symmetric entrance channel such as,for example, Ni + Ni [42] or Cr + Fe [44] andmass-asymmetric as S + Ge [43] reaction. Howeverthe Λ ER values for , Kr + Ca reactions are smallerthan the one extracted for other mass-asymmetric sys-tems such as O + Mo [44] and S +
Mo [45]. Thiswould indicate a different boundary between evaporationand fission-like channels in the J -space as a function ofthe mass-asymmetry of the entrance channel, as for ex-ample when fusion and quasifission processes competewith each other.The capture cross-section in Kr + Ca reaction ishigher than the one measured in Kr + Ca reaction.This behaviour is at variance with observations in thevicinity of the Coulomb barrier for systems with similarmasses ([46–48]). Considering these measurements at thehighest bombarding energy ( ∼
10% above the Coulombbarrier), σ expcapt of a neutron-rich system ( S + Zr) ex-ceeds by ∼
25% the capture cross-section of a neutron-poor system ( S + Zr) and the same trend is observedfor the S + , Zr reactions. However, in these casesthe cross sections of fission-like products were negligiblewhile this decay mode accounts for almost 50% of σ expcapt inthe , Kr + Ca reactions at 5.5 MeV/nucleon. In thereactions studied here, the difference in σ expcapt is mainlydue to the fission-like component, leading to a smallercapture cross-section for the Kr + Ca system. Theconfrontation with the predictions of theoretical modelswill bring more information to discuss this aspect.
IV. FRAGMENT-PARTICLE COINCIDENCEMEASUREMENTS
To better understand the fragment emission mecha-nism and to get more insights on the o-e-s of the light-fragment yields, we have performed an event-by-eventanalysis of the light charged particles (LCPs) in coinci-dence with fragments. In the first step, we calculated foreach fragment the relative velocity between that fragmentand each detected LCP of the event. Then we consider anew frame with one axis corresponding to the directionof the fragment velocity in the c.m. frame and the planeperpendicular to this axis. Finally, we projected the rel-ative velocities previously calculated onto this new frameand deduced the component parallel ( V k ) and perpendic-ular ( V ⊥ ) with respect to the direction of the fragmentvelocity in the c.m. frame. In doing so, for fragments ofa given Z , having different emission angles in the c.m.,the procedure enables to construct a common referenceframe for the LCPs in coincidence with these fragments.We have seen the binary nature of the fragment produc-tion with a small amount of particles emitted meanwhile.Thus, the kick induced by the emitted particles shouldbe small and one could assume that fragments are flyingback-to-back in the center-of-mass. Then, the emissiondirection of one fragment defines the recoil direction ofits complementary partner. With such a method appliedto an ensemble of reactions, the particles emitted by one fragment with a constant velocity value will draw onecircle centered at the origin of the reference frame in aV k -V ⊥ plot.Fig. 8 presents typical examples of V k − V ⊥ diagramsfor α -C (first row), α -Si (second row) and α -Fe (thirdrow) coincidences measured in the Kr + Ca reac-tion . The black circles represent the average velocitiestaken from systematics compiled by Parker et al. [49].For α -C coincidences, the relative velocities draw a cir-cular region (akin of a Coulomb ring) which is centeredat the origin when they are projected into the frame(termed as Compl-frame) of the complementary part-ner of the C nuclei (top right panel) whereas no sucha circular region centered at the origin can be seen whenthe relative velocities are plotted in the frame (termedas Z-frame) of the light partner (top left panel). For Z = 14 and 26, both fragments emit light-particles asillustrated by the two circles centered at both referenceframes. Thus, we observe the change of behaviour of thelight-particle emission from very asymmetric ( Z = 6) toasymmetric ( Z = 14) and almost symmetric ( Z = 26)fragmentation. The same conclusions hold for fragment-proton coincidences. Thus, in Kr + Ca reactions at -4 -2 0 2 4 ) - ( c m n s V -4-2024 -4 -2 0 2 4 ) - ( c m n s V -4-2024 ) -1 ( cm ns V -4 -2 0 2 4 ) - ( c m n s V -4-2024 Z= 26, Z-frame -4 -2 0 2 4 -4-2024 -4 -2 0 2 4-4-2024 ) -1 ( cm ns V -4 -2 0 2 4 -4-2024 FIG. 8: (Color online) V k -V ⊥ diagrams of alpha particlesdetected in coincidence with C (first row), Si (second row) andFe (third row) fragments produced in Kr + Ca reactionat 5.5 MeV/nucleon (see text). The velocities are calculatedin the reference frames of the light fragment (left panels) andof the complementary fragment (right panels) ◦ ≤ θ lab ≤ ◦ .Such a limited angular range prevents to extract quan-titative information on emission characteristics such asmultiplicity of light-charged particles associated to eachfragment pair. This kind of analysis will be presented ina forthcoming paper.The broken dashed line in Fig. 9 shows the proton sep-aration energy S p calculated for the most abundant ele-ment given by the mass tables. A strong o-e-s is observedfor S p with roughly the same magnitude over the range6 ≤ Z ≤
28. It is worth noticing that the o-e-s of S p and σ Z are in phase each other. For light fragments both σ Z and S p are larger for even- Z . One can make an estima-tion of the excitation energy E ∗ Z stored in the fragments.The total kinetic energy released in the binary fragmen-tation could be deduced from the kinetic energy of thelight partner for which the mass number is calculated as-suming that its N/Z ratio is the same as the compositesystem. By assuming a rigid rotation and a thermal equi-librium between both partners one can deduce E ∗ Z fromthe energy balance. The results of such calculations areshown in Fig. 9 for an initial angular momentum of 40(thin line) and 60 (thick line). E ∗ Z increases almost con-stantly from about 8 MeV for Z = 8 to about 30 MeV for Z = 28. The staggering of E ∗ Z is due to the fact that iso-topic distribution for a given Z is not taken into account.The values of E ∗ Z for Z ≤
12 are below 15 MeV, i . e . donot exceed the separation energy. One should note thatthe particle-fragment Coulomb barrier is not included, asit would have been done to estimate the emission energythresholds. However, taking into account the Coulombbarrier would not change drastically the pattern since theCoulomb energy grows smoothly with the atomic numberof the fragment.The attenuation of the staggering of σ Z for fragmentshaving large Z would be related to a blurring due tolight-particle emission as suggested by the coincidencedata and by the estimation of E ∗ for symmetric frag-mentation. Same conclusions hold when considering theseparation energy of alpha particles. Thus, the σ Z forlight fragments reflect the persistence of structure effectsin asymmetric fragmentation. This could be associatedto a microscopic contribution to the potential energy sur-face which is a key ingredient in determining the fragmentyields and/or to specific properties of the level density atenergy below the particle emission thresholds. Such in-fluences need further investigations. σ Z ( m b ) E n e r g y ( M e V ) Z Kr+ CaS p E z* ( l = 40 )E z* ( l = 60 )E z* ( DNS ) FIG. 9: (Color online) Experimental cross-sections for frag-ments emitted in Kr + Ca (solid squares) reactions at 5.5MeV/nucleon. The broken dashed line represents the protonseparation energy. Thin (thick) lines refer to the excitationenergy stored in the fragment assuming an initial spin of 40(60) respectively. Dotted line shows the DNS calculations.
V. COMPARISON WITH MODELS
In this section we compared data and the predictionsof three theoretical approaches: two of them describe thedecay modes of CN while the third one treats the dynam-ical evolution of the interacting partners and the physicsgoverning the CN formation. Comparison of preliminarydata and the dynamical cluster-decay model assumingpre-formed clusters [14] has been presented in Ref [15].
A. Comparison with BUSCO
The Hauser-Feschbach approach is very successful incomputing the light-particle emission from CN. In theBUSCO code [12], this formalism has been extended tothe IMF emission in their ground states as well as excitedstates. In the version of the code we used in the presentwork, the emission of fragments up to Z = 14 has beenincorporated. It should be noticed that fission channelis not taken into account. However, the model containsinteresting features which justify the comparison to thepresent data, providing that the CN spin-distribution isgiven by the sharp cut-off approximation with J max keptas a free parameter.The decay width of a channel α from a CN formed ata spin J is given by [11, 12] P Jα = X l α Z T lα ( ǫ α ) ρ ( E ∗ CN − ǫ α , J ) dǫ α . (2)In Eq. 2, T l α are the optical-model transmission coeffi-cients evaluated at the relative kinetic energy ǫ α in theemitter frame and ρ is the Fermi-gas model level densityof the daughter nuclei computed with the prescription of0 σ Z ( m b ) Z Kr+ CaJ max = 60J max = 37
FIG. 10: (Color online) Experimental cross-sections forfragments emitted in the Kr + Ca reaction at 5.5MeV/nucleon (squares), compared to BUSCO calculationsassuming a J -distribution given by the sharp cut-off approx-imation with J max = 60 (thick line) and J max = 37 (dashedline). Calculations have been performed with a level-densityparameter a = A/ . − . Ref. [53]. The transmission coefficients have been param-eterized by a Fermi function T lα ( ǫ α ) = (1 + exp[ − ( B lα − ǫ α ) / ∆ α B lα ]) − , where B lα = B + ~ l α ( l α + ) / µ R α . The parameters B , R α and ∆ α are obtained from thebest fits of optical-model transmission coefficients. Thepredictions of the model have been successfully comparedto data in medium-mass CN region [9, 12, 25].The present calculations were performed using a level-density parameter a = A/ . − and a sharp cut-offapproximation with J max = 60 as a starting guess. Theresults of the BUSCO calculations for the Kr + Careaction are symbolized by a thick line in Fig. 10.The model fails to reproduce the features of the Z -distribution, although an odd-even staggering as in thedata is seen for Z ≤
8. For Z ≤
14 one observes a globaldecreasing of the calculated σ Z at variance with data.More specifically, the cross section of C is overestimatedby a factor 30, while σ Z for 8 ≤ Z ≤
12 are overestimatedwithin a factor of 2 to 6. A calculation assuming J max =37 (dashed line in Fig. 10) in order to reproduce σ Z for Clargely misses the yields of the other species. Taking a J -distribution with a diffuseness around J max instead of asharp cut-off approximation, or making different choicesof the level-density parameter do not improve the predic-tions of the model.Since the interaction barriers play a crucial role inthe competition between the decay channels, we com-pared the calculated kinetic-energy spectra of the frag-ments to the experimental data. In the BUSCO code,the kinetic-energy spectra result from the folding of the optical-model transmission coefficients and the level den-sity. Thus the shape of the spectra is a good test ofthe calculation. The comparison of theoretical and mea-sured spectra is presented in Fig. 11 for Z = 6 , , Z , the calculation was normalized to the inte-gral of the kinetic energy distribution. The agreementis very good for the mean kinetic energy. However, thecalculated width of the distribution is smaller. The sameconclusion holds for other fragments. Improvement of thecalculated kinetic-energy spectra could be obtained by afine-tuning of the parameters, but the isotopic distribu-tion is unknown and such a fitting procedure would notbe under control. We thus conclude that the basic ingre-dients to estimate the kinematics seem to be reasonablyimplemented.A possible explanation of the disagreement with theexperiment would be the too small number of excitedstates n ex incorporated into the calculation. Indeed, for C nucleus, n ex = 5 are included up to 16.7 MeV; for O, n ex = 7 up to 19.2 MeV and n ex = 7 up to 18 MeVfor Si. Such a reduced number of excited states maystrongly affect the fragment cross-sections, more specifi-cally the yields of light clusters with respect to the heavyones, and the production of odd- and even- Z and/or N nuclei. For example there are 60 states below 8.32 MeVin F, 103 states below 13.97 MeV in Ne, 160 statesbelow 8.19 MeV in Al and 62 states below 11.59 MeVin Si [50]. Considering a small number of excited states n ex , the code BUSCO would amplify the effect of the Q -values and barriers which could explain the abrupt de-crease of the cross sections of the light fragment. Addi-tion of further excited states could be envisaged but theupper limit of the fragments to be considered in the cal-culation and the treatment of the fission channel are stillimportant open questions yet to be resolved.
20 40 60 80 -1 Z=6
20 40 60 80 -1 Z=8 dataBUSCO
20 40 60 80 -1 Z=10 (MeV) c.m. ε ( a r b . un i t s ) c . m . ε d Ω / d σ d FIG. 11: (Color online) Kinetic energy spectra in c.m. framefor Z = 6 , ,
10 emitted in the Kr + Ca reaction. His-tograms are data and dashed lines are the results from theBUSCO calculations using J max = 60 and a level-density pa-rameter a = A/ . − . Calculations were normalized todata assuming the same integral for each Z . B. Comparison with GEMINI
In their work, N. Bohr and A.J. Wheeler [51] recog-nized that the fission probability of a nucleus is gov-erned by the number of states above the fission barrierand the saddle-configuration plays the role of a transi-tion state between the CN and the scission-configuration.Moretto [52] extended this concept to the asymmetric-fission mechanism. The GEMINI code [34] combinesHauser-Feschbach and transition-state formalisms to de-scribe the disintegration of a hot CN by emission of prod-ucts spanning the whole mass (charge) range from neu-tron to the fragment corresponding to the symmetric fis-sion. The evaporation channels include n, p, d, t , He and α particles. The emission of fragments with Z ≥ Z, A )from a CN at excitation energy E ∗ CN and spin J is writtenas: Γ Z,A ( E ∗ CN , J )= 12 πρ Z E ∗ CN − E sad ( J )0 ρ sad ( U sad , J ) dǫ, where U sad = E ∗ CN − E sad ( J ) − ǫ and ρ sad are the thermalenergy and the level density calculated at the conditionalsaddle-point configuration, respectively. ǫ is the kineticenergy and E sad ( J ) is the energy of the saddle-point con-figuration calculated in the finite-range liquid-drop modelof Sierk. Nuclear level densities are given by the Fermi-gas formula for a fixed angular momentum J as follows ρ sad ( U sad , J ) ∝ (2 J + 1) U sad exp[2 p ( aU sad )] . In the model, the angular momentum J lim ~ at whichthe fission barrier disappears is 69 ~ for the Ba nu-cleus and 74 ~ for the Ba nucleus. In the case of the
Ba nucleus, J lim is higher than J expmax deduced fromdata, while J lim < J expmax for the Ba nucleus. Conse-quently, the calculations have been performed assuminga sharp cut-off for the angular momentum distributionwith J max = J lim = 69 for the Kr + Ca reactionand J max = J expmax = 70 for the Kr + Ca reaction.Results of the calculations are reported in Fig. 12a forthe Kr + Ca system and in Fig. 12b for the Kr + Ca reaction. As a first attempt we adopt a level-densityparameter a = A/ − . The thick line in Fig. 12apresents the predictions for the disintegration of BaCN assuming J max = 69. The shape of the Z -distributionfor 12 ≤ Z ≤
28 is reasonably reproduced, although themodel systematically underestimates the fragment yieldsin the range 18 ≤ Z ≤
26 by roughly 20%. A bet-ter agreement could be obtained by scaling the fission barriers but the examination of the whole Z -distributionis more instructive. Indeed, the model overestimatesby about a factor 10 the sum of the cross-sections for3 ≤ Z ≤
11. The difference comes mainly from the veryhigh Li cross-section, while C and O calculated yields arelarger by about a factor 3. To give a flavour of the J max -dependence of the Z -distribution, results for J max = 55and J max = 45 are shown as dashed and dotted lines,respectively. C (Ne) yields are in satisfactory agreementfor J max = 45 (55) but in both cases the whole shapeis not correctly reproduced. This conclusion does notdepend on the sharp cut-off approximation. Indeed, asmooth transition around J max would degrade the globalagreement since such spin-distribution tends to depopu-late the region around the symmetry and conversely toincrease the yield for Z around 16–20. In this way thenet effect would be an increase of the width of the Z -distribution and thus the agreement would become worse.Moreover, no major influence is observed by varying thelevel-density parameter from A/ A/
10 MeV − . Re-garding the staggering of the yields, one could observe arelatively good agreement above Z =10, but the odd-eveneffect is not at all reproduced for the light fragments. Thesame conclusions could be written from the predictions ofthe disintegration of a Ba CN (thick line in Fig. 12b).In the range 12 ≤ Z ≤
28, the model reproduces the ex-perimental data both in shape of the Z -distribution andmagnitude of the cross sections. As for the Kr + Careaction, the model fails to reproduce the Z -distributionfor 3 ≤ Z ≤ σ Z ( m b ) Z a) J max = 69J max = 55J max = 45 Kr+ Ca σ Z ( m b ) Z a) J max = 69J max = 55J max = 45
2 6 10 14 18 22 26b) J max = 70 Kr+ Ca
2 6 10 14 18 22 26b) J max = 70 FIG. 12: (Color online) a) Experimental cross-sections forfragments emitted in the Kr + Ca reaction (full squares),compared to the predictions of the GEMINI code assumingdifferent maximum angular momenta : J max = 69 (thickline), J max = 55 (dashed line) and J max = 45 (dotted line);b) Experimental cross-sections for fragments emitted in the Kr + Ca reaction (open squares), compared to the predic-tions of the GEMINI code assuming J max = 70 (thick line).Calculations were performed taking a = A/ − for thelevel-density parameter. Z -distributions for light fragmentstogether with an overestimation of their yields mightbe due to a low barrier for mass-asymmetric fission.For medium-mass nuclei there is a quasi–degeneracy ofsaddle- and scission-configurations, thus the total kineticenergy of the fragments is tightly related to the barrier.Considering the energy balance, a lower potential en-ergy would correspond to higher excitation energy in theprimary fragments. From the calculations, we deducedthe primary Z -distribution before secondary decays andthe multiplicity of the particles emitted from each frag-ments. A careful analysis of the results indicates that,for 3 ≤ Z ≤
11, the initial smooth behaviour of the Z -distribution is modified by an emission of protons and α particles which finally induces the fluctuations of the cal-culated yields shown in Fig. 12a,b. Thus, in the model,the fluctuations of the yields for light fragments are re-lated to secondary emission of light particles, in contra-diction with our data.Last, the calculated ER cross-sections σ GEMINIER forboth systems (reported in Table II) are in the 250-300 mbrange depending on the assumptions on level-density pa-rameter. These values are lower by about a factor 2 withrespect to the experimental data. The low σ GEMINIER val-ues could be related to the mass-asymmetric barrier thatleads to enhance the light-fragment emission with respectto the evaporation of light particles.Consequently, since the Z -distribution mainly reflectsthe evolution of the barrier profile as a function of themass-asymmetry and angular momentum, the compari-son with data would indicate a failure of the model todescribe the boundary between asymmetric and sym-metric fission at high angular momentum and that thelandscape of the potential energy surface around sym-metry would be steeper than the one implemented inthe GEMINI code. These conclusions hold if the de-cay products are unambiguously associated to CN dis-integration. In this case, other potential-energy surfacessuch as the one recently developed [54] might have a bet-ter behaviour around symmetry as indicated in a recentinvestigation [55]. C. Comparison with the dinuclear system (DNS)model
Both approaches presented in previous subsectionstreat the decay of an initial CN and disregard the col-lisional stage leading to its formation. However, a largebody of data has reported on the competition betweenthe fusion and the quasifission phenomena, the latter cor-responding to the capture of interacting partners with asignificant flow of matter and kinetic energy followed bya reseparation without being trapped in the CN config-uration. For the interpretation of these two kinds of re-actions, the new concept of the DNS has been developedand successfully compared to collisions involving massivenuclei [56]. This model has been recently applied [16] to the decay of medium-mass excited nuclei formed at rel-atively low angular momentum. Here we compared thepredictions of the DNS model to our data which indicatea strong relaxation at relatively high angular momentumand moderate excitation energy. A detailed descriptionof the model can be found in [16, 56]; only the mostsalient features are outlined.The DNS model describes the evolution of the inter-acting nuclei along two degrees of freedom; the relativedistance R between the center of the nuclei; the chargeand mass-asymmetry degrees of freedom, which are de-fined here by the charge Z and mass A of the light part-ner of the DNS. After the dissipation of kinetic energyand angular momentum of the relative motion, the DNSis trapped in the pocket of the interaction potential be-tween partners. Then, a statistical equilibrium is reachedin the mass-asymmetry coordinate so that the formationprobability P Z,A of each DNS or CN configuration de-pends on the potential energy U ( R m , Z, A, J ), calculatedwith respect to the potential energy of the rotational CNwhere R m is the location of the minimum in the interac-tion potential. After the capture stage, there are nucleondrift and nucleon diffusion between the nuclei which con-stitute the DNS. Then, the excited DNS can decay witha probability P RZ,A in the R -coordinate if the local exci-tation energy of the DNS is high enough to overcome thebarrier in the nucleus-nucleus potential. Ultimately, thesystem evolves either towards a CN configuration thatsubsequently decays, or to a DNS configuration. Thelatter process, in which a two-body configuration is keptall along the trajectory, is the quasifission phenomenon.The emission probability W Z,A ( E ∗ CN , J ) of a fragment( Z , A ) is calculated as the product of the DNS formationprobability and the DNS decay probability: W Z,A ( E ∗ CN , J ) = P Z,A P RZ,A P Z ′ ,A ′ P Z ′ ,A ′ P RZ ′ ,A ′ , where the indexes Z ′ and A ′ go over all possible channelsfrom the neutron evaporation to the symmetric splitting.The probability P Z,A is the equilibrium limit of themaster equation (see [16, 56] for details) given by P Z,A ( E ∗ CN , J )= exp[ − U ( R m , Z, A, J ) /T CN ( J )]1 + P Z ′ =2 ,A ′ exp[ − U ( R m , Z ′ , A ′ , J ) /T CN ( J )] . The quasifission barrier B qfR , calculated as the differencebetween the bottom of the inner pocket and the top ofthe external barrier, prevents the decay of the DNS alongthe R -degree of freedom with the weight P RZ,A given as P RZ,A ∼ exp[ − B qfR ( Z, A, J ) /T Z,A ( J )] .T CN ( J ) and T Z,A ( J ) are the temperatures of the CN andthe DNS, respectively. The Fermi-gas model is employedto compute the temperature, with a level-density param-eter a taken as the high excitation limit of Ref. [57] that3means a = 0 . A +0 . A / . With this prescription weobtained a = 17 .
34 MeV − for the Ba nuclei, equiva-lent to a = A/ . − , a value close to those we usedin BUSCO and GEMINI calculations.In the DNS model, all the trajectories leading to CNand QF processes represent the capture phenomenon.The pocket in the nucleus-nucleus potential disappears atsome critical value J = J cr and the DNS formation is nolonger possible at J > J cr . The critical value J cr deter-mines the capture cross-section. The dominant reactionmechanism (CN or QF) strongly depends on the angularmomentum. For the reactions studied here, the drivingpotential at low angular momentum shows that CN con-figuration is energetically more favorable than any DNSconfiguration. At higher angular momentum, the drivingpotential has a minimum at the symmetric DNS and thecharge (mass)-drift pushes the system towards symmetricconfiguration. Consequently CN configuration becomesenergetically less favorable and the high partial waveslead to QF. However, both mechanisms coexist in a widerange of angular momenta. For example, in the case ofthe Kr + Ca reaction at 5.5 MeV/nucleon, the evap-oration residue component accounts for about 10% of thepartial cross-section at J = 65.There are two important facets of the model. First,no a priori assumption is made on the relaxation of the N/Z degree of freedom. Indeed the
N/Z -equilibrationis reached when the DNS is trapped. Secondly, theconnection between binary decay and evaporation chan-nel is provided in a straightforward way by the mass-asymmetry coordinate. So, in the DNS model, the com-petition between the decay channels is treated in a com-mon framework.Fig. 13a (Fig. 13b) compared DNS predictions anddata for the Kr + Ca ( Kr + Ca) reaction, re-spectively. For both reactions, the largest value of theangular momentum J max is taken as the critical value J cr according to the model. For the Kr + Ca sys-tem, J max = 70 is the value deduced from the mea-sured total cross-section. Predictions with J max = 65 for Kr + Ca reaction are shown for the sake of compar-ison. Last, the Be cross-section has been removed fromthe results of the calculations to permit the comparisonwith data.We observe a spectacular improvement with respectto the predictions of the BUSCO and GEMINI codes.Indeed, the DNS model satisfactorily reproduces themain features of the Z -distributions. For both reactions,the shape of the Z -distributions, the strong odd-even-staggering for 5 ≤ Z ≤
10, the small cross-sections oflight fragments as well as σ Z around Z = 28 are wellreproduced. However, for 16 ≤ Z ≤
22 the DNS modelunderestimates the fragment cross-sections by about afactor 2 to 3. Since the whole capture cross-section isconsidered, no improvement could be obtained within thepresent version of the model. Nevertheless, as reported inTable I, J cr values of the DNS model are coherent with l pocket calculated using the proximity potential. More- over, the ER cross-sections predicted by the DNS model σ DNSER (see Table II) are compatible with data, althoughthe dependence of the ER cross-section on the neutron-to-proton ratio does not follow the same trend as the oneseen in the experiment. Thus, the depletion observedin the calculated yields for 16 ≤ Z ≤
22 might sig-nal, in addition to the capture process, the presence of aclass of deep-inelastic collisions associated to an incom-plete relaxation of the entrance channel mass (charge)-asymmetry, and presumably localized in a J -window justabove J cr . In this case the yields of the products near theentrance channel ( Z = 20) can exceed the predictions ofthe DNS model.The staggering of the yields decreases as the atomicnumber increases in agreement with the experimentalfindings. Since the pairing energy of the DNS light nu-cleus decreases with increasing mass number A , the odd-even effect becomes weaker for larger Z -values. More-over, the magnitude of the staggering is also influencedby the excitation energy stored in the primary fragments(see dotted line in Fig. 9). For nuclei with Z .
10 thecalculated average excitation energy is below the particleemission threshold and these nuclei do not decay furtherexcept by γ -emission which is not taken into account inthe present version of the model. For heavy fragments,the average excitation energy and spin are high enoughto open-up the decay by light particles which strongly at-tenuates the odd-even structures of the Z -distributions.Such results agree with our conclusions from the analysisof the fragment-particle coincidences.In agreement with data, σ Z for fragments with Z < Kr + Ca reaction. This can be ex-plained by their smaller mass-asymmetric decay barriers σ Z ( m b ) Z a) Kr+ CaJ max = 73J max = 65
2 6 10 14 18 22 26b) Kr+ CaJ max = 75J max = 70
FIG. 13: (Color online) Comparison between measured andcalculated cross-sections. The calculated results with J max =65 ( J max = 73) for the Kr + Ca reaction and J max =70 ( J max = 75) for the Kr + Ca reaction are shown bydashed (solid) lines in panel a (b), respectively. Full (open)squares are data from the Kr + Ca ( Kr + Ca) reac-tion, respectively. CONe
MgSSi
ArCa Ti σ Z ( E c . m . , J ) ( m b ) JJ σ Z ( E c . m . , J ) ( m b ) CrNiFe
FIG. 14: Partial cross-sections of the indicated fragmentsas a function of the angular momentum for the Kr + Careaction at 5.5 MeV/nucleon. for the reaction induced with Kr projectile.The calculated yields for 3 ≤ Z ≤
10 show a largeodd-even-staggering of about a factor 10. Such o-e-s ismuch bigger than the experimental results and is mainlydue to a strong underestimation of the odd- Z yields ofB, N and F while the C and O yields are well repro-duced. The low predicted yields of the light fragmentswith odd- Z could be related to the prescription for thestatic deformation for odd-nuclei which enters into thenucleus-nucleus potential. Reasonable changes of staticdeformation would have minor effects on the yields. An-other possibility would be the interplay between somemicroscopic properties (such as pairing interaction forexample) and deformation experienced by the dinuclearsystem en route to separation. Data would indicate anattenuation of these properties with deformation. Fi-nally, the nuclear level densities below separation energycould play a role in the competition between channelssince they could still retain some structure behaviourswhich are not included in the Fermi-gas approach [50].Comparing the calculated cross-sections for J max =65 and 73 ( J max = 70 and 75) for the Kr + Ca( Kr + Ca) reactions (see Figs. 13a, b), one can de-duce that the contribution from high-partial waves to theyields for Z ≤
10 is negligible. The calculated partial pro-duction cross-sections σ Z ( E c . m . , J ) for some fragments from C to Ar are shown in Fig. 14 for the Kr + Careaction at 5.5 MeV. We observed that most of the lightfragments, as for example C, O or Ne, comes from an-gular momenta around J ~ ≈ ~ . On the contrary,most of the heavy fragments as for example Cr, Fe or Niis associated to partial waves around J max . It is worthnoting that σ Z ( E c . m . , J ) develops two components forfragments with large Z showing a population throughboth CN and quasifission mechanisms. Examination ofthe results leads to the conclusion that QF is the domi-nant decay channel for heavy fragments while light frag-ments are predominantly populated by CN. Thus, theangular momentum strongly influences the competitionbetween the binary decay channels and, correspondingly,the probability of light-fragment emission. One shouldalso remind that the careful identification of the originof the binary decay products is a prerequisite before ex-tracting information such as viscosity or fission barriersfrom fitting data. Thus, it would be very instructiveto probe the competition between CN and QF compo-nents in the same mass region by studying small mass-asymmetric reactions where the flux going to CN is ex-pected to dominate over a large range of incident partialwaves. Experiments using a spin-spectrometer with highcapabilities [58] could be appropriate for such investiga-tions.The DNS model provides a good framework to describeboth qualitatively and quantitatively fusion- evaporationcross-sections as well as the main features of the yieldsof the light fragments such as C or O. The calculationsconfirm what we have deduced from the analysis of thefragment-light particle coincidences. The excitation ener-gies and spins left in the heavy partners (Sn, Cd) after Cor O emission are very high and since these heavy nucleiare neutron-deficient, the secondary emission of light par-ticles leads to the formation of residues of masses A ∼ VI. SUMMARY AND CONCLUSIONS
We have presented the results of a study on decaymodes of excited nuclei formed in , Kr + Ca reac-tions at 5.5 MeV/nucleon. The 4 π INDRA array whichis very well suited to study the fate of violent colli-sions [60], has been exploited here for the first time inlow bombarding energy regime. The kinetic-energy spec-tra, the angular distributions and the Z -distribution forfragments with 3 ≤ Z ≤
28 show the characteristicsof fission-like phenomenon. Analysis of the fragment-particle coincidences indicates that light partners in veryasymmetric fission are produced either cold or at excita-tion energies below the particle emission thresholds. Weobserve a persistence of structure effects from elemen-tal cross-sections with a strong odd-even-staggering for5the lightest fragments. The magnitude of the staggeringdoes not significantly depend on the neutron-to-protonratio of the emitting system. The ER cross-section ofthe Kr + Ca reaction is slightly higher than the onemeasured in the Kr + Ca reaction. The fission-likecomponent is larger by ∼
25% for the reaction having thelowest neutron-to-proton ratio. Last, the cross sectionsof the light clusters (Li, Be, B) are astonishingly low.These experimental features were compared to the pre-dictions of various theoretical approaches assuming ei-ther the formation of CN (BUSCO, GEMINI) or describ-ing both the collisional stage preceding the CN forma-tion and the competition with quasifission process (DNSmodel). The better global agreement is obtained withinthe DNS framework. For the , Kr + Ca reactionsat 5.5 MeV/nucleon, the DNS model describes quantita-tively the ER cross-sections, the odd-even-staggering ofthe light fragments and their low cross sections as wellas a large portion of σ Z for 12 ≤ Z ≤
28. Finally, thefeatures of the charge distribution for 3 ≤ Z ≤
28 are con-sistent with a strong competition between fusion-fissionand quasifission processes. Examination of the resultssuggest that the quasifission mechanism is the dominantproduction mode for heavy fragments while light clustersare predominantly populated by decay of CN.The confrontation with data confirms the crucial roleof the mass (charge)-asymmetry degree of freedom on thedisintegration of excited nuclei. Moreover the potentialenergy surface that governs the evolution of the system must contain the contribution of microscopic propertiesof nuclei such as pairing interaction, shell effects or staticdeformations. The interplay between the mass (charge)-asymmetry and
N/Z -degrees of freedom and their mu-tual influence on the competition between fusion evapo-ration reactions and binary decays is yet to be explored.The advent of powerful ISOL facilities will undoubtedlyprovide very well adapted opportunities to bring new in-sights on the respective role of the mass-asymmetry and
N/Z -degree of freedom during strongly dissipative colli-sions such as fusion and quasifission processes.
VII. AKNOWLEDGMENTS
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