A new "3D Calorimetry" of hot nuclei
E. Vient, L. Manduci, E. Legouée, L. Augey, E. Bonnet, B. Borderie, R. Bougault, A. Chbihi, D. Dell'Aquila, Q. Fable, L. Francalanza, J.D. Frankland, E. Galichet, D. Gruyer, D. Guinet, M. Henri, M. La Commara, G. Lehaut, N. Le Neindre, I. Lombardo, O. Lopez, P. Marini, M. Parlog, M. F. Rivet, E. Rosato, R. Roy, P. St-Onge, G. Spadaccini, G. Verde, M. Vigilante
AA new “3D Calorimetry” of hot nuclei
E. Vient, ∗ L. Manduci,
2, 1
E. Legou´ee, L. Augey, E. Bonnet, B. Borderie, R. Bougault, A. Chbihi, D.Dell’Aquila,
4, 6
Q. Fable, L. Francalanza, J.D. Frankland, E. Galichet,
4, 7
D. Gruyer,
1, 8
D. Guinet, M. Henri, M. La Commara, G. Lehaut, N. Le Neindre, I. Lombardo,
6, 10
O. Lopez, P. Marini, M. Pˆarlog,
1, 12
M. F. Rivet, † E. Rosato, † R. Roy, P. St-Onge,
13, 5
G. Spadaccini, G. Verde,
4, 10 and M. Vigilante Normandie Univ, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, F-14000 Caen, France ´Ecole des Applications Militaires de l’ ´Energie Atomique, B.P. 19, F-50115 Cherbourg, France SUBATECH UMR 6457, IMT Atlantique, Universit´e de Nantes, CNRS-IN2P3, 44300 Nantes, France Institut de Physique Nucl´eaire, CNRS/IN2P3, Univ. Paris-Sud,Universit´e Paris-Saclay, F-91406 Orsay cedex, France Grand Acc´el´erateur National d’Ions Lourds (GANIL),CEA/DRF-CNRS/IN2P3, Bvd. Henri Becquerel, 14076 Caen, France Dipartimento di Fisica ’E. Pancini’ and Sezione INFN,Universit´a di Napoli ’Federico II’, I-80126 Napoli, Italy Conservatoire National des Arts et M´etiers, F-75141 Paris Cedex 03, France Sezione INFN di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy IPNL/IN2P3 et Universit´e de Lyon/Universit´e Claude Bernard Lyon1,43 Bd du 11 novembre 1918 F69622 Villeurbanne Cedex, France INFN - Sezione Catania, via Santa Sofia 64, 95123 Catania, Italy CEA, DAM, DIF, F-91297 Arpajon, France Hulubei National Institute for R & D in Physics and Nuclear Engineering (IFIN-HH),P.O.BOX MG-6, RO-76900 Bucharest-M`agurele, Romania Laboratoire de Physique Nucl´eaire, Universit´e Laval, Qu´ebec, Canada G1K 7P4 (Dated: October 14, 2018)In the domain of Fermi energy, it is extremely complex to isolate experimentally fragments andparticles issued from the cooling of a hot nucleus produced during a heavy ion collision. This paperpresents a new method to characterize more precisely hot Quasi-Projectiles. It tries to take intoaccount as accurately as possible the distortions generated by all the other potential participants inthe nuclear reaction. It is quantitatively shown that this method is a major improvement respectto classic calorimetries used with a 4 π detector array. By detailing and deconvolving the differentsteps of the reconstitution of the hot nucleus, this study shows also the respective role played bythe experimental device and the event selection criteria on the quality of the determination of QPcharacteristics. PACS numbers: 24.10.-i ; 24.10.Pa ; 25.70.-z ; 25.70.Lm ; 25.70.Mn
I. INTRODUCTION
The only way to study experimental nuclear thermody-namics is to produce hot nuclei during nuclear collisions.The hot nuclei are obtained thus in extremely violentand complex conditions. In the domain of Fermi energy,we observe clearly for the energy dissipation a compe-tition between nuclear mean field and nucleon-nucleoninteraction [1, 2]. The collisions present mostly a strongbinary character preserving a very strong memory of theentrance channel [3]. The process of deeply inelastic dif-fusion becomes the dominant phenomenon [4–8]. It isaccompanied by an important emission of light particles,thermally unbalanced [8–10]. But we also observe animportant production of I ntermediate M ass F ragments( IMFs ) at the interface of the two colliding nuclei. Thislatter is usually called
Neck Emission [8, 11–15]. Fusion ∗ [email protected]; http://caeinfo.in2p3.fr/ † deceased is also observed but its cross section becomes low for sym-metric collisions above 30 A.MeV [3, 16–19]. Obviously,the respective cross sections of these various processeschange according to the system, the incident energy andthe impact parameter. The formed hot nuclei de-excitein many particles and fragments. Only a 4 π array canallow to study such physical processes because of its de-tection capabilities. We know that it is fundamental inthermodynamics or statistical mechanics that the stud-ied system has to be perfectly defined and characterized.This is the major experimental challenge encountered bynuclear physicists working in the domain of Fermi en-ergy. For this reason, the main purpose of this paper is topresent, understand and validate a new method of char-acterization of an excited Q uasi- P rojectile ( QP ) with a4 π experimental set-up in this energy range. By char-acterization, we mean: to determine its charge, mass,velocity and excitation energy. We want also to improve,possibly, these “measurements” compared to the existingmethods [20]. This study is made with the 4 π array IN-DRA [21] for the system Xe + Sn at 50 A.MeV. In thefirst section, we will discuss how to select the products of a r X i v : . [ phy s i c s . d a t a - a n ] S e p the Quasi-Projectile decay among all the particles pro-duced during the reaction. In the following section, wewill present the principles of our calorimetry. In the lastsection, we will study this new experimental calorimetryby comparing to a standard calorimetry. II. IMPROVED SELECTION OF THEEVAPORATED PARTICLESA. Light particle characterization : influences ofthe reference frame and of the experimental set-up
Here we discuss firstly how the reference frame usedin the analysis and the experimental set-up may influ-ence the spatial and energetic characteristics of the L ight C harged P articles ( LCPs ) and consequently the deter-mination of the excitation energy of the hot nuclei andthe energetic spectrum slopes of these particles. For
BeamDirectionReaction Plane V QP φ V α θ spin Figure 1: Definitions of the polar angle θ spin and of the az-imuthal angle φ of light emitted particle (the azimuthal angle φ presented in the figure is negative). this study, the SIMON event generator, developed byD.Durand [22], was used. It is set to supply only purebinary collisions Xe + Sn at 50 A.MeV without pre-equilibrium particles. We will study different angulardistributions of emitted particles. Figure 1 shows theframe used to study the LCP angular distributions andallows to define the different used angles. The polar angle θ spin is defined as the angle between the vector normalto the reaction plane and the velocity vector of a par-ticle emitted by the Quasi-Projectile in the QP frame.The azimuthal angle φ is defined as the angle betweenthe QP velocity vector in the frame of the center of mass(c.m.) and the normal projection of the velocity vectorof the emitted particle on the reaction plane. The az-imuthal angle is defined positive when the projection ofthe velocity vector of the particle on the reaction plane islocated on the “left” with respect to the direction of theQP velocity vector. A particle emitted in the reactionplane and in the direction of the QP velocity vector willhave φ = 0 ◦ and θ spin = 90 ◦ . Figure 2 shows two graphsbuilt in the real initial frame of the emitting source, i.e. the QP (which we will call “True Frame”). The events arenot filtered at this stage. Therefore, we assume the use ofa perfect detector. The particles were solely evaporated Figure 2: a) Slopes of energy spectra of alphas evaporated bythe QP, obtained by fit for various domains of φ , accordingto the associated mean φ . This is done for various selectionsaccording to E ∗ /A of the QP. The red line corresponds tothe true initial temperature of the QP, associated with eachrange of excitation energy per nucleon. b) Mean multiplic-ities of the light particles emitted by the QP, generated bySIMON, according to φ , without experimental filter (the ve-locity vector being calculated in the true initial frame of theQP). by the QP according to SIMON. The graphs of the figure2-a show the slopes obtained by a maxwellian fit of theenergy spectra of the alphas, for various angular domainsof φ and for ten bins of 1 MeV of the QP excitation en-ergy per nucleon. The collisions are increasingly violentby going from left to right and downwards in the figure.The graph of the figure 2-b presents the mean multiplicityof the light charged particles according to the azimuthalangle φ for various ranges of the excitation energy pernucleon. We find in both cases the expected result forthe emission by a thermalized nucleus, i.e. flat angulardistributions for all the light particles and an uniformmeasured slope whatever φ . This guarantees the validityand the coherence of SIMON with respect to the treat-ment of the isotropic QP decay. Figure 3: a) φ distributions of the light particles emitted bythe QP supplied by SIMON after the INDRA filter (the veloc-ity vector of the particle being calculated here in the rebuiltframe for various selections according to E tr ). b) φ distri-butions of the light particles emitted by the QP, generatedby SIMON, after passage in the INDRA filter, for various se-lections according to E tr . The considered events are called“complete events” (the velocity vector of each particle beingcalculated in the true initial frame of the QP). To see and understand the impact of the used exper- imental set-up, INDRA, we coupled to the generatedevents by SIMON a software which integrates and simu-lates all the phases of the ion detection by INDRA (de-tection efficiency, kinetic energy resolution, angular reso-lution, isotopic identification). This fundamental stage issummarized in terms of the “INDRA experimental filter”.A correct calorimetry requires the complete detection ofthe nuclear reaction products. To try to achieve this goal,we required a constraint on the total detected charge andmomentum (80 % of the initial value) as done in [23, 24].We also aim at understanding the influence of this eventselection concerning the calorimetry. These events arecalled hereafter “complete events”. Indeed, because wedo not have an experimental direct access to the exci-tation energy of the nucleus, we have chosen the totaltransverse kinetic energy of light charged particles E tr as experimental selector of the violence of the collision[10, 25]. In the figure 3-b, for the complete events, wecan observe precisely the sole influence of the INDRAexperimental filter on the spatial distribution of particlesevaporated by the QP (the frame used here is the trueframe of the QP). This selection of complete events in-volves an important deformation of the φ distribution ofevaporated particles, even if the velocity vector of evapo-rated particles is defined in the true initial frame. Thereare apparently more light particles emitted on the rightthan on the left with respect to the QP velocity vector,for the peripheral collisions. In fact, the criterion of com-pleteness implies that we only keep events for which theemission of the light particles allowed the QP residue toavoid the forward of the detector, made to let pass thebeam. This effect is all the more strong as the QP isforward focused.Now, as shown in the figure 3-a, if the velocity vectoris defined in the rebuilt frame (built considering interme-diate mass fragments and heavy fragments located in theforward hemisphere respect to the c.m.), we can observethat the effect is even more amplified. What is observedin the figure 3 is the“right-left effect”described in [23, 24].This important deformation involves a breaking of therevolution symmetry for the evaporation around the axispassing by the center of the QP and perpendicular to thereaction plan, in the velocity space.We can wonder whether this apparent effect is just anartifact related to SIMON. This is why, we present in thefigure 4 the mean multiplicities of the light charged parti-cles, located in the front of the center of mass, accordingto φ , obtained with the experimental data recorded bythe INDRA array. We observe indeed the same trend,even enhanced because of pre-equilibrium emission (nottaken into account in SIMON) which is preferentially lo-cated between the two partners of the collision, thereforeat angles φ near 180 ◦ . The θ spin distributions are alsomodified by the selection criteria and the INDRA exper-imental filter. In figure 5, we present a comparison ofthe multiplicity distributions of particles located in thefront of the center of mass according to the cos ( θ spin )obtained with data (figure 5-a) and pure binary SIMON Figure 4: Experimental azimuthal distributions of the lightcharged particles located in the forward hemisphere of thecenter of mass (their velocity vector being calculated in therebuilt frame) for various E tr bins, obtained experimentallywith INDRA. The considered events are events said “com-plete”. simulations (figure 5-b) for complete events, we selectedsemi-peripheral collisions by using the LCP total trans-verse kinetic energy. We find in both cases similar re-sults : there is no apparent symmetry in φ . Only theangular domains 0 ◦ - 30 ◦ and 30 ◦ - 60 ◦ are similar. Infact, these trends were already seen in a study made bySteckmeyer et al. in [23], where the authors showed howthe “right-left effect” and the experimental method of re-construction (only from the intermediate mass fragmentsand heavy fragments) deform the symmetric distributionexpected for LCP emissions by an hot rotating source.The required completeness favors the events presentinga pronounced “right-left effect”, for the peripheral col-lisions. It was also shown in this study [23] that thearea of space, located at the right front in the frameof the reconstructed QP, is strongly polluted by a pre-equilibrium emission (for semi-peripheral and central col-lisions). There is also a slight pollution of the front ofcenter of mass by the Quasi-Target emission, as alreadynoted in the reference [24].Now, for complete events, we will make an equivalentstudy concerning the energy characteristics of the LCPsin the frame of the emitting nucleus. We study firstthe effect of the filter (in the figure 6-b) then the cu-mulated effect of the filter and of the reconstruction ofthe frame velocity vector (in the figure 6-a). The filterand the associated criteria of completeness imply an ap- Figure 5: a) Multiplicity distributions of the LCPs, located inthe front of the center of mass, according to the cos ( θ spin ) forvarious φ bins. They are obtained for data selection accord-ing to the transverse energy corresponding to semi-peripheralcollisions and also for complete events. b) The same dis-tributions for data supplied by SIMON with the same eventselection (the frame of the source is reconstructed by the ex-perimental method). parent modulation of the mean kinetic energy accordingto φ , mainly for the peripheral collisions. In the figure6-a, the addition of the reconstruction effect involves aright-left skewness of the modulation, still related to the“right-left effect”. This phenomenon is more importantfor alpha particles. But it is also observed for the otherlight particles (p, d, t, He). It still exists when thestudied physical quantity is not the mean kinetic energy
Figure 6: a) Mean energy distributions of the alphas evapo-rated by the QP according to φ generated by SIMON afterINDRA filter (their velocity vector being calculated in therebuilt frame) for various E tr bins. b) Mean energy dis-tributions of the alphas evaporated by the QP according to φ , generated by SIMON after INDRA filter (their velocityvector being calculated in the true initial frame) for various E tr bins. For a) and b) , the considered events are knownas “complete”. The red line corresponds to a value twice theexpected temperature of the hot nucleus (with a density pa-rameter equal to 10). but the temperature determined by maxwellian fits of thekinetic energy spectra. Figure 7 shows the experimen-tal data to compare with the simulation results of thefigure 6-a. There is an important difference due to thepre-equilibrium emission around -180 ◦ / 180 ◦ . Only thearea 0 ◦ - 60 ◦ in φ , symbolized by a transparent light blueband, seems partly consistent with the mean kinetic en- Figure 7: Mean energy distributions of the alphas evaporatedby the QP according to φ and the various selections accordingto E tr for the experimental data obtained with INDRA. Theconsidered events are events known as “complete”. The redline corresponds to 2 times the measured temperature. ergy expected because of evaporation as displayed by thehorizontal red line. This trend is also found for the otherlight particles not shown here for the sake of simplicity.These observations are confirmed by figure 8 where weshow the experimental energetic spectra of tritons, de-fined in the reconstructed frame of the QP for various φ bins and a selection in transverse energy correspond-ing to semi-peripheral collisions. We see again that theparticles emitted only in an azimuthal domain between-30 ◦ and 60 ◦ give spectra compatible with a pure thermalemission. This is also seen for the other light particles.Other studies have been made with the event generatorHIPSE [26], which treats more carefully the LCP produc-tions at mid-rapidity than SIMON. They also show thatonly a very limited angular domain in the front of theQP is almost not polluted [27]. This type of results hadbeen also pointed out using Landau-Vlasov calculationsbut for smaller systems in the reference [28]. B. The importance of selecting the reactionmechanism
As already discussed in the introduction, in heavy ionreactions at intermediate energy it is possible to observedifferent reaction mechanisms. We defined selection cri-teria in order to disentangle between pure binary colli-sions followed by a standard statistical decay and colli-sions with neck formation. We will call the former
Sta-
Figure 8: Kinetic energy distributions of the tritons in thereconstructed frame of the QP, for various angular selectionsaccording to φ obtained by INDRA collaboration for the sys-tem Xe + Sn at 50 A.MeV. The events are complete andcorrespond to semi-peripheral collisions. tistical collision and the latter Neck Emission . It wasshown in [27, 29, 30] that a valid criterion, used to iden-tify an event as
Statistical collision, is the presence ofthe second heaviest fragment in the front of the centerof mass, in the forward hemisphere of the QP frame (seefigure 9).
Neck Emission events are those with mid-rapidity emission (nuclear matter between the two part-ners of the binary collision). This discrimination allowsto isolate binary collisions followed by statistical emis-sion. This latter seems to be a more adapted scenariowith respect to the bases of our reconstruction methodof the hot nucleus. It was already shown and discussed inreferences [27, 29, 30] the fact that these two mechanismscorrespond to well different dynamics. The process of en-ergy dissipation may have different origins as reported inthe reference [29]. In figure 10, we selected the violence ofthe collision using the variable E tr normalized to theavailable energy in the center of mass in the directionperpendicular to the beam. We use also the variable η which characterizes the charge asymmetry between thetwo heaviest fragments in the front of the center of mass. η = ( Z max − Z max ) / ( Z max + Z max ) (1)This variable is used to obtain a signal of bimodalityto characterize an eventual liquid-gas phase transition innuclei [31–33]. Figure 10 shows, for various violences ofcollision (different E tr domains), the distribution of thecosine θ rel , the angle formed between the relative velocityvector of the two heaviest fragments in the front of the Figure 9: Diagram plotting the selection criteria of the twostudied reaction mechanisms. center of mass (see definition in equation 2) and the veloc-ity vector of the reconstructed source. We limit ourselvesto present a typical case here: collisions for the systemXe + Sn at 50 A.MeV, having a medium asymmetry be-tween the two biggest fragments ( i.e. , 0 . < η (cid:54) . −−→ V rel = −→ V Z max − −→ V Z max (2)We can note that the events, labeled as Statistical , Xe + Sn 50 A MeV: DATA c oun t s c oun t s c oun t s c oun t s c oun t s c oun t s c oun t s c oun t s cos( θ Rel ) c oun t s cos( θ Rel ) c oun t s Neck + Backward StatForward StatNeckBackward Stat . Etr12 Bin No 3Etr12 Bin No 1 Etr12 Bin No 2Etr12 Bin No 5 Etr12 Bin No 4Etr12 Bin No 6 Etr12 Bin No 7 Etr12 Bin No 8Etr12 Bin No 9 Etr12 Bin No 10
Figure 10: Distributions of the cosine of the angle θ rel , pre-sented for events having a medium mass asymmetry betweenthe two biggest fragments of the event and for various se-lections: the curve (blue full line) corresponds to the NeckEmission , the curve (red dashed line) at left corresponds tothe
Statistical emission in the front of the QP frame and thisone (small black dashed line) at the right corresponds to thebackward QP emission, the curve (pink line and point) corre-sponds to the
Pure Neck Emission for which the statisticalcontribution was subtracted. The violence of the collision in-creases when one goes from left to right and downwards. present a flat distribution, if we neglect certain problemsof angular acceptance, which are more visible for the pe-ripheral collisions. The events of type
Neck (blue fulllines) present rather a focusing of the relative velocityvector in the direction of the velocity vector of the recon-structed source; this focusing effect is increasingly largeras the collision is less violent. To improve the selection ofthis contribution, we may subtract the statistical back-ward distribution, for which the second heaviest fragmentwould be emitted backward with respect to the QP. Thisback statistical distribution should be the symmetricaldistribution of the events known as
Statistical , such aswe select them, with respect to the zero abscissa of thedistribution (black dashed line). It must give positiverelative angle values taking into account the definition ofthe relative velocity vector. We obtain thus after sub-traction, a proper cosine θ rel distribution of the eventscorresponding to the genuine Neck Emission , i.e. PureNeck Emission (pink line and point in figure 10). Thedistribution shape does not change, but the proportionof events can be estimated more carefully; this contribu-tion tends to disappear when the violence of the collisiongrows. In fact, we choose to keep both mechanisms to seeif differences appear when we apply our new calorimetry.
III. THE NEW “3D CALORIMETRY”A. Determination of the spatial domain of emissionby the QP
The study reported in the reference [24] and the con-clusions drawn in the sub-section II A convinced us thatto correctly characterize the QP de-excitation, we mustuse a quite restricted spatial domain. From the angu-lar definitions of φ and of θ spin given in the sub-sectionII A, we consider as particles emitted actually andsolely by the QP: the particles located in the an-gular azimuthal domain included between 0 ◦ and60 ◦ in the reconstructed QP frame . It thus corre-sponds to one-sixth of the total solid angle as we can see itin figure 11. This spatial domain is, by definition, linkedto the QP vector velocity reconstructed in the frame ofthe center of mass, and it will change according to thevarious selections on the violence of the collision. TheQuasi-Projectile is then reconstructed, event by event,by assigning to each particle a given probability to beemitted, which was determined using all the informationobtained in the spatial domain defined above. This kindof calorimetry was partially used in a simplistic way in[34]. B. Calculation of the emission probabilities by theQP
To apply this 3D calorimetry and define the emissionprobability by the QP, we assume that the process of de-
Figure 11: Diagram allowing to visualize the considered spa-tial domain to define the probability of emission by the QPfor all types of particles. excitation of the QP presents a symmetry of revolutionaround the perpendicular axis to the reaction plane de-scribed by the reconstructed velocity vector of the QPand the velocity vector of the initial projectile. We de-fined, for φ varying from -180 ◦ to 180 ◦ , six areas 60 ◦ wide.At first, for each one, we built all the polar angular dis-tributions as well as the energy distributions for all thetypes of detected particles, defined in the frame of the re-constructed QP. This is made by using all the light parti-cles and the intermediate mass fragments with the excep-tion of the two heaviest fragments emitted in the front ofthe center of mass (the probability is one for both nucleicoming from the QP). From these distributions, for anangular domain between φ and φ , we determine an ex-perimental emission probability by the QP for a particleof kinetic energy E k at an angle θ spin and in an azimuthalangular domain ∆ φ : P rob ( E k , θ spin , ∆ φ = φ − φ ).We assume at first that, for any particle, the proba-bility to be emitted in a polar angle θ spin is independentfrom the kinetic energy and vice versa. This means thatwe neglect the influence of the angular momentum onthe distributions of kinetic energy. Considering this hy-pothesis, we can thus calculate the probability from thefollowing relation: P rob ( E k , θ spin , ∆ φ ) = P rob ( E k , ∆ φ ) × P rob ( θ spin , ∆ φ )(3)From this last relation and from our choice of selectionof particles emitted by the QP, we deduce that the ex-perimental probability is given by the following relation: P rob ( E k , θ spin , φ − φ ) = dN ( E k , ◦ − ◦ ) dE k dN ( E k ,φ − φ ) dE k × dN ( θ spin , ◦ − ◦ ) d cos θ spin dN ( θ,φ − φ ) d cos θ spin (4)This calculation is done also according to the violence ofthe collision, the mechanism of reaction and the asymme-try between the two heaviest fragments in the front. Fur-thermore, the chosen completeness criteria are here lessdrastic than previously and focused on the reconstructionof the QP. The studied events are here complete events Figure 12: Proportions of the various types of reaction mech-anisms according to the “experimental impact parameter” (vi-olence of the collision) for collisions Xe + Sn at 50 A.MeV. to the front of the center of mass. For these events onecan observe in figure 12, the proportion of the differentselections of mechanism and decay taken into account inthis study on the basis of the violence of the collision.We observe a forward statistical contribution which isrelatively constant around 30-35 % (we remind that thisnumber must be doubled to get the right proportion ofevents corresponding to the statistical emission).We thus obtain experimental probability functions de-pending on the following variables:
P rob ( E k , θ spin , φ − φ , E tr N orm, η, M echanism, Z, A )for the particles of charge Z (cid:54)
3, identified also by mass.
P rob ( E k , θ spin , φ − φ , E tr N orm, η, M echanism, Z )for the particles of charge
Z > E c , θ and φ . C. QP reconstruction
For each particle ( Z n , A n , −→ P n ) detected in an event,we determine its kinetic energy, the polar and azimuthalangle of its velocity vector, defined in the reconstructedframe. We deduce thus its probability P rob n to be emit-ted by the QP from the experimental functions of prob-ability (see Eq 4). We then associate this probability tothis particle to reconstruct the characteristics of the QP. The QP charge can be estimated as follows: Z QP = multot (cid:88) n =1 P rob n × Z n (5)For the mass, we need to make some hypotheses. Weassume that the QP keeps the isotopic ratio of the initialprojectile and that nuclei follow the valley of stability.The mass conservation allows us to deduce the number ofneutrons produced by the QP as indicated by the relationbelow. A QP = Z QP × /
54 = multot (cid:88) n =1 P rob n × A n + N neutron (6)We can then deduce the reaction Q-Value. Q = E b ( A QP , Z QP ) − multot (cid:88) n =1 P rob n × E b ( A n , Z n ) − N neutron × E b (1 ,
0) (7) E b ( A n , Z n ) is the binding energy of the nucleus A n Z n X .We determine the QP velocity vector in the frame of thecenter of mass of the reaction only from charged particlesby using the following expression: −→ V QP = multot (cid:80) n =1 P rob n × −→ P n ( A QP − N neutron ) (8)with −→ P n the linear momentum of the n th particle in theframe of the c.m..We can then calculate the QP excitation energy: E ∗ QP = multot (cid:88) n =1 P rob n × E kn + N neutron ×(cid:104) E k (cid:105) p + α − Q − E kQP (9)being E kn the kinetic energy of the n th particle in theframe of the center of mass, (cid:104) E k (cid:105) p + α the mean kineticenergy of the neutrons deduced from those of the protonsand alphas (Coulomb energy is subtracted) and finallywith E kQP the QP kinetic energy in the center of mass. IV. COMPARISON WITH A CLASSICCALORIMETRY
As a rough guide, we will compare this “3D calorime-try” with the “Standard Calorimetry” based on a tech-nique doubling the particle contribution in the front ofthe QP, described in [20] [23] [24] [35].Figure 13 shows the average evolution of the QP chargeand excitation energy per nucleon for the various selec-tions of interest. Figure 13-a shows the excitation energyper nucleon for the two different calorimetries and thedissipated energy per nucleon during the collision. This
Xe + Sn 50 A MeV
3D CalorimetryStandard CalorimetryDissipation E * / A ( M e V ) E * / A ( M e V ) E * / A ( M e V ) StatHigh Asym NeckHigh AsymStatMedium Asym NeckMedium AsymStatSmall Asym NeckSmall Asym a
3D CalorimetryStandard Calorimetry Q P C ha r ge Q P C ha r ge Etrans12 Norm Q P C ha r ge Etrans12 Norm
StatHigh Asym NeckHigh AsymStatMedium Asym NeckMedium AsymStatSmall Asym NeckSmall Asym b . Figure 13: a) Average correlation between the measured QPexcitation energy per nucleon and the normalized transverseenergy of LCPs for various selections of mechanism and asym-metry for collisions Xe + Sn at 50 A.MeV. b) Average correla-tion between the QP reconstructed charge and the normalizedtransverse energy of LCPs for various selections of mechanismand asymmetry for collisions Xe + Sn at 50 A.MeV (the blackline indicates the projectile charge). latter is in fact determined from the measured QP veloc-ity by our method. As indicated in the reference [3], wecan obtain the quantity of the initial incident energy inMeV, which is dissipated per nucleon during the collisionby the following expression: E Dissipated A = E CM ( A P roj + A T ar ) − × V rel c × amu c (10)with E CM the available energy in the center of mass of the reaction and V rel the relative velocity between part-ners of the collision.Here, as the system is symmetric, we assume that V rel (cid:39) × V QP , with V QP velocity of the QP in theframe of the center of mass. The equation 10 is valid onlyfor symmetric systems and assumes that pre-equilibriumemission is symmetric in the frame of the center of mass.This energy thus represents the maximum energy whichcan be stored by the QP. Figure 13-a gives, for eachkind of mechanism and each asymmetry, the curves rel-ative to the excitation energy per nucleon obtained bythe 3D calorimetry (red circles), the standard one (greensquares) and the dissipated energy per nucleon (blue tri-angles). In this figure, one can notice immediately thatthere is a qualitative visible improvement of the measure-ments. With the new calorimetry, for the most periph-eral collisions, we obtain lower limit values which seemreasonable: they are close to E ∗ QP = 0 A.MeV for the ex-citation energy per nucleon and Z QP = 54 for the chargeof the QP (figure 13-b), which is not always the casefor the standard calorimetry. Moreover, contrary to thestandard method the measured excitation energy neverexceeds the estimated dissipated energy. There is thus areasonable coherence between the measured QP velocityand the measured excitation energy per nucleon.Regarding the measurement of the QP charge in Figure13-b, the very different measurements, obtained with thestandard calorimetry between the events called Statis-tical and the events with a
Neck Emission , disappearcompletely with the new 3D calorimetry. The indicatedstandard deviations are also clearly smaller.This simple comparison is not naturally enough to val-idate this new calorimetry but gives some evidences thatthis 3D calorimetry improves a lot the measurement ofQP ( E ∗ QP and Z QP ). It seems essential to study it bymeans of the most realistic possible simulation and thiswill be the subject of a next paper. V. CONCLUSIONS
By provoking extremely violent collisions of heavy ions,nuclear physicists try to modify the internal energy ofnucleus. In the domain of Fermi energy how this energy isdeposited and stored is still a subject of many discussions.This is mainly due to the fact that it is very difficult toprove experimentally in an unquestionable way that ahot nucleus, thermodynamically well equilibrated, wasformed. In this context, we described the foundations ofa method of QP reconstruction and reminded difficultieswhich can intervene.We chose to follow an experimental approach trying tosolve us gradually the difficulties. For that purpose, weused the event generator SIMON and a simulation pro-gram, reproducing as accurately as possible the behaviorof our experimental device.First, being placed in the framework of binary col-lisions generated by SIMON without pre-equilibrium,0we study the de-excitation of an isolated hot Quasi-Projectile. We observed important spatial and energeticdistortions concerning the evaporated particles by theQP. We showed the fundamental roles played by the ex-perimental device and the recoil effects.To make a correct calorimetry, we need to detect allthe evaporated fragments and particles by the QP. There-fore, we must use events said “complete events”. For thevery peripheral collisions, the detection of the QP residueremains difficult because of the INDRA forward accep-tance. For collisions when few particles are evaporated,they must have a large transverse linear momentum todeviate correctly the QP residue allowing its detection.“Complete events” correspond mainly to these type ofevent for peripheral collisions. It is difficult to com-pensate for the recoil effect due to the emission of thefirst evaporated particle. Consequently, we do not findisotropic spatial and energetic distributions but a recoileffect called “right-left”, which is dominant for peripheralcollisions. It involves indirectly also difficulties in the de-termination of the source velocity, when only IMFs andheavy fragments are used to reconstruct the frame of thehot nucleus. There is clearly a difference between thecenter of mass of the IMFs and that of the LCPs, whichinvolves an apparent energetic contribution of LCPs toolarge in the reconstructed frame of the QP. All these factsare confirmed with the real experimental data and evenamplified by pre-equilibrium contribution.Second, we presented a new calorimetric protocol tocharacterize the Quasi-Projectile. This calorimetry isbased on the experimental determination of an emissionprobability starting from the physical characteristics ofparticles in a restricted domain of the velocity space. Wewanted that this one can take into account the influenceof pre-equilibrium particles and of a possible contributionof the mid-rapidity such the Neck Emission.In this study, we added supplementary criteria of eventselections to observe the robustness of this calorimetry.We differentiated the collisions with Neck Emission fromthe others. We also took into account the asymmetrybetween the two biggest fragments in the front of thecenter of mass.To demonstrate completely the interest of the 3Dcalorimetry, we compared it with a standard method con-sisting doubling of the light charged particles located inthe front of the reconstructed frame. This last one seemsclearly inaccurate. They tend to give excitation ener-gies per nucleon too large in comparison to the apparentdissipation, as it can be observed clearly in the figure13. They also provide too wide distributions of the ex-citation energy per nucleon or the charge for the recon-structed QP [24]. This brings moreover into question thequality of selections done with an excitation energy ob-tained with standard calorimetric methods like this. Thenew calorimetry gives better estimates of the excitationenergy per nucleon than previous methods. But it can al-low, by its intrinsic hypotheses (two particle sources), tomake solely a correct physical characterization of binary collisions, i.e. peripheral or semi-peripheral collisions.To complete our conclusion on this study, we can noticethat a real improvement of the quantitative characteriza-tion of the QP can be obtained only, either by an effectivecorrection of the complex distortions generated by the ex-perimental device and the criteria of event selection, orby the construction of a 4 π detector, which would havea higher granularity, a forward efficient detection and anexcellent isotopic resolution, improving the identificationand the kinematic characterization of particles. We fi-nally have to keep in mind that all these conclusions canbe drawn in principle only for the studied system, Xe +Sn, and are to be confirmed for the others. This workconfirms on the other hand in a clear way that a compar-ison between a theoretical model and experimental datamakes sense in this domain of physics only if the theoret-ical model is passed through a software filter simulatingthe totality of the detector response.1 [1] G. Lehaut et al., Phys. Rev. Lett. , 232701 (2010).[2] O. Lopez et al., Phys. Rev. C , 064602 (2014).[3] V. M´etivier et al., Nuclear Physics A , 357 (2000).[4] J. C. Steckmeyer et al., Phys. Rev. Lett. , 4895 (1996).[5] G. Casini et al., Phys. Rev. Lett. , 3364 (1991).[6] R. J. Charity et al., Zeitschrift f¨ur Physik A Hadrons andNuclei , 53 (1991).[7] D. Jouan et al., Zeitschrift f¨ur Physik A Hadrons andNuclei , 63 (1991).[8] L. Gingras et al., Phys. Rev. C , 061604 (2002).[9] D. Dor´e et al., Phys. Rev. C , 034612 (2001).[10] J. P´eter et al., Physics Letters B , 187 (1990).[11] M. Di Toro, A. Olmi, and R. Roy, The European PhysicalJournal A - Hadrons and Nuclei , 65 (2006).[12] L. Stuttg´e et al., Nuclear Physics A , 511 (1992).[13] G. Casini et al., Phys. Rev. Lett. , 2567 (1993).[14] C. P. Montoya et al., Phys. Rev. Lett. , 3070 (1994).[15] J. F. Lecolley et al., Physics Letters B , 202 (1995).[16] P. Eudes, Z. Basrak, F. S´ebille, V. de la Mota, andG. Royer, Phys. Rev. C , 034609 (2014).[17] S. C. Jeong et al., Nuclear Physics A , 208 (1996).[18] N. Marie et al., Physics Letters B , 15 (1997).[19] L. Beaulieu et al., Phys. Rev. Lett. , 462 (1996).[20] V. Viola and R. Bougault, The European Physical Jour-nal A - Hadrons and Nuclei , 215 (2006). [21] J. Pouthas et al., Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers,Detectors and Associated Equipment , 418 (1995).[22] D. Durand, Nuclear Physics A , 266 (1992).[23] J. C. Steckmeyer et al., Nuclear Physics A , 537(2001).[24] E. Vient et al., Nuclear Physics A , 555 (2002).[25] J. (cid:32)Lukasik et al., Phys. Rev. C , 1906 (1997).[26] D. Lacroix, A. Van Lauwe, and D. Durand, Phys. Rev.C , 054604 (2004).[27] E. Vient, M´emoire d’habilitation `a diriger des recherches,Universit´e de Caen (2006).[28] P. Eudes, Z. Basrak, and F. S´ebille, Phys. Rev. C ,2003 (1997).[29] J. Colin et al., Phys. Rev. C , 064603 (2003).[30] J. Normand, Ph.D. thesis, Universit´e de Caen (2001).[31] M. Pichon et al., Nuclear Physics A , 267 (2006).[32] E. Bonnet et al., Phys. Rev. Lett. , 072701 (2009).[33] M. Bruno, F. Gulminelli, F. Cannata, M. D’Agostino,F. Gramegna, and G. Vannini, Nuclear Physics A ,48 (2008).[34] E. Vient et al., Nuclear Physics A , 588 (1994).[35] E. Bonnet et al., Phys. Rev. Lett.105