Size Effects in Heavy Ions Fragmentation A. Barran˜´on ∗; J. A. L´opez †; C. Dorso. ‡ 3 October, 20th, 2002 0 0 2 n Abstract a J Rise-PlateauCaloriccurvesfordifferentHeavyIoncollisionshavebeen 2 obtained, in the range of experimental observations. Limit temperature decreases when the residual size is increased, in agreement with recent 2 theoretical analysis of experimental results reported by other Collabo- v rations. Besides, promptly emitted particles influence on temperature 1 plateauisshown. LATINObinaryinteractionsemiclassical modelisused 0 to reproduce the inter-nucleonic forces via Pandharipande Potential and 0 fragments are detected with an Early Cluster Recognition Algorithm. 1 0 3 1 Introduction 0 / h M. Rousseau et. al. have studied light charged particle emission from a t compound nucleus of 40Ca obtained from 28Si+12C reaction, providing - l evidenceofbinaryemission of8Be,inadatafittingscenarioformed bya c u sequential evaporation process, a subsequent fragment emission from the n 40Ca compound nucleusand finally the residual nucleusfreeze-out [1]. : In the case of inverse kinetic reactions, fragmentation occurs in two v stages, namely a first pre-equilibrium stage where participants interact i X anddepositexcitationenergyinthespectators,andasecondstagewhere the excited spectator residual decays [2]. r a Consideringsmallexcitations,thereisadelayinfissionfragmentemis- sionintherangeof1000fm/candforlargerexcitationsthechargedparti- clesareemittedatthesametimeasheavyfragments(Z =8-16),with imf anintegralrupturetimeintherangeof200fm/cafterthermalization [3]. Kolomietz et. al. have obtained “rise-plateau-rise isobaric caloric curves intheThomas-FermiapproximationwithaneffectiveforceSkM,showing that in the plateau thereis liquid-gas coexistence [4]. Gourio et. al. have obtained light particle emission times around 200 fm/c, usingtwo-particlecorrelation functionsinordertoanalyzeINDRA and GANIL data [5]. D’Enterr´ıa et. al. have obtained similar caloric curves studying Brehmstahlung photon emission in Heavy Ion Collisions ∗Universidad Auto´noma Metropolitana. Unidad Azcapotzalco. Av. San Pablo 124, Col. Reynosa-Tamaulipas,MexicoCity. email: [email protected] †Dept. ofPhysics,TheUniversityofTexas atElPaso. ElPaso,TX,79968 ‡Dep. deF´ısica,UniversidaddeBuenosAires. Buenos Aires,Argentina 1 Size effects in Heavy Ions Fragmentation 2 using a thermal model [6]. D’Agostino et. al. have shown the useful- ness of an abnormal change in kinetic energy variance as a first order phase transition signature [7]. Nuclear collisions for energies in tenths of meVs/nucleon produce excited systems that break-upinto several inter- mediatesizefragments[8]. Thesemultifragmentationphenomena,similar to those occurring in inertial confinement reactions [9] and in the nanos- tructuressurfacesynthesis[10],couldbetheadaptation ofamacroscopic phase transition toa finite and transient system. In the realm of nuclear physics, the possibility of reaching a critical behavior in heavy ions collisions has motivated several studieson critical exponents. Thefirstonefocusedonproton-Xeandproton-Arcollisionsby Purdue group [11, 12] and others afterwards [13]. More recently,modern detectiontechnologyallowed caloriccurveexperimentalcomputation[14, 15] , i.e. the relation between system temperature and excitation energy at fragmentation time. Nevertheless, the influence of residual finite size, a transient evolution and experimental limitations due to observing only final fragments, turn out to be problematic. As explained by Pochodzalla y Trautmann [16] , there are difficulties in the reconstruction of laid down energy via exit channels and therefore in caloric curvecomputation. Otherproblemsarise from the fluctuations in system size [17], “side feeding” effects on final mass distribution, and in theuse of final spectra modified by collision evolution [18]. Asa matterof fact, experimentsusing different“thermometers” have reached contradictory caloric curves such as the typical rise-plateau-rise plot [19], a peculiar “rise-plateau” caloric curve [20], and even a “rise- rise”caloriccurvewithouttheplateaucharacteristicofafirst-orderphase transition [21]. Thesuppressionofthefinaltemperaturegrowthcouldberelatedeither toalowdensityrupture[22],ortoahighfragmentationenergydestroying hot intermediate fragments [16]. With so manyvariables affecting thenuclear caloric curvedetermina- tion,thequestiontoansweriswhetheritispossibletoextractthiscaloric curveinnuclearcollisions. Morerecently,thiswasdealtwithaMolecular Dynamics study in nuclear systems inside of a container [23] and for a freely expandingclassical system [24]. Evenwhenthosecomputationscannotbecompared,neitherbetween them nor with experiments, those exercises were able to obtain a caloric curve. Sugawa and Horiuchi [23], used Antisymmetrized Molecular Dy- namics,inordertostudyauniformlyexcited,containedsystemwithcon- stant pressure, and obtained a “rise-plateau-rise” caloric curve. On the other side, Strachan and Dorso [24] used a uniformly excited Lennard- Jones system with free expansion and obtained a caloric curve with a “rise-plateau” shape. The difference between these results arises from the collective expansion manifest in finite systems (but absent in infinite andcontainedsystems),actingasanenergyreservoirandavoidingafinal temperature rise [24]. And Gobet et. al. have obtained event by event “rise-plateau-rise” caloriccurvesforcollisions ofhydrogenclustersionswithHeliumtargets, consideringthiscaloriccurveasafirst-orderphasetransitionsignaturein a finite system. The plateau is interpreted as latent heat associated with Size effects in Heavy Ions Fragmentation 3 this phase transition [25]. Notwithstanding their differences, these exercises suggest that frag- menting small systems may provide information about the caloric curve. Nevertheless, in regards of the fundamental role played by the geometry of the fragmenting system, i.e. contained systems versus free-expansions systems,asubtlequestionmightbethatoftheroleplayedbythecollision incaloriccurvecomputations. Namely,theinfluenceofinducedlargecor- relations due to non linear dynamics on the obstacles found to compute the caloric curve. Inthisstudyacomputationisperformedthatreproducesexperimental collisionsinordertoobtainthecaloriccurveandexaminethedependence of thelimit temperature on theresidual size. We use the weaponry developed in [24] to focus on colliding excited systems in order to obtain the caloric curve and its dependence on the residual size. The manuscript is organized in the following way: after describing the model used, the fragment recognition algorithm and the persistence in section II, Section III establishes an effective way to com- putethecollision dynamicstages and section IV studiesthetemperature in the fragmenting system participant region. Caloric curve dependence onresidualsizeisshowninsectionV,andsomefinalconclusionsclosethe article in Section VI. 2 Latino Model. LATINO Model [26] uses the semi-classical approximation to simulate Heavy Ions collisions via a binary interaction Pandharipande potential. This potential is formed by a linear combination of Yukawa potentials, with thecoefficients fixed in order to reproduce the properties of nuclear matter. Clusters are detected with an Early Cluster Recognition Algo- rithm that optimizes the configurations in energy space. Ground states areproducedgeneratingarandomconfigurationinphasespace,gradually reducing the velocities of the particles confined in a parabolic potential, until thetheoretical bindingenergy is reached. Duetoobscuritiesinthebreakupprocess,acomplexmodelisneeded with the auxiliary tools required. Collision evolution is modeled with a MolecularDynamicscode,thoughsinceMolecularDynamicsoperatesata nucleoniclevel,itisnecessarytotransformparticleinformationintermsof fragment informationviaafragmentrecognition algorithm. Usingcaloric curveasaphasetransitionsignature,requirestheknowledgeofthebreak up time, using a property known as partition “persistence”. Molecular Dynamicsadvantages to studynuclear collisions havebeen established previously. In this work we study the time evolution of cen- tral collisions numerical simulation. Target consists of a 3D droplet in its “ground state”. The projectile is a randomly oriented droplet in its “ground state”, boosted on thetarget with different kineticenergies. Numericalintegrationoftheequationsofmotionisperformedthrough a Verlet algorithm at time intervals that ensure energy conservation at least in a 0.05 %. Fig. 1 shows theevolution of a typicalexperiment. The range for the projectile kinetic energy starts from E = 800 beam Size effects in Heavy Ions Fragmentation 4 MeV inthecenterofmassreferencesystemwithabouttwohundredcol- lisionsforeachprojectileenergy. Coveringawideenergyrange,weobtain different dynamical evolutions: starting from events where the projectile isabsorbedbythedropletsurfaceuptoeventswithanexponentialdecay mass spectra. As shown in Fig. 2, the shapes of the spectra converge to a power law in this range of intermediate energies. The temperature of theparticipantregion iscomputed usingKineticGas Theory andsystem excitation is calculated with the energy deposited in the residual. When theprojectile energy isincreased, themultiplicityis also increased as the system enters intothe phasecoexistence region (Fig. 3b) [27]. With the fragments detected in phase-space, Mean Velocity Transfer is computed in the following way : MVTi = |vk,i(t+dt)−vk,i(t)| (1) k X 2.1 Fragment Recognition In order to obtain information about the fragments, Molecular Dynam- ics microscopic data must be analyzed through a Fragment Recognition Algorithm. Although there are many, we will now describe the one used here. In the “Minimum Spanning Tree in Energy” algorithm (MSTE), a given set of particles i,j,...,k belongs to the same cluster Ci if: ∀i ∈ Ci, exists j ∈ Ci/eij ≤ 0 where eij = V(rij) +(pi − pj)2/4µ, and µ is the reduced mass of the couple i,j. MSTE searches configuration corre- lations between particles, considering their relative moment. Due to its sensitivity torecognize early particles, it is extremely useful tostudy the pre-equilibrium energy distribution of theparticipant particles. 2.2 Partition Persistence In order to use the caloric curve as a phase transition signature, both systemexcitationandtemperatureatphasetransitionareneeded. Hence, break-uptimeisrequired,andiscalculatedintermsofthefragmentation time,τ ,namelyatimewhenfragmentsevaporateonlysomemonomers. ff τ canbedetectedcomparingpartitionsatdifferenttimes,whichcanbe ff done using the“Microscopic Persistence Coefficient”, P [28]: P[X,Y]= 1 niai (2) clni cl bi X whereX ≡{Ci}andY ≡{Ci′}arePdifferentpartitions,biisthenumberof particlepairsintheclusterCi ofpartitionX,ai isthenumberofparticle pairs belonging to cluster Ci that remain paired in a given cluster Cj′ of partition Y, ni is thenumberof particles in cluster Ci. P[X,Y] isequalto 1if themicroscopic composition of partition X is equal to that of partition Y, and tends to 0 when none of the particles belonging to a given cluster X remain together in any cluster of Y. It is convenient to studythetime evolution for thequantities: Pˆ+[X(t)]≡hP[X(t),X(t→∞)]i (3) events Size effects in Heavy Ions Fragmentation 5 Pˆ−[X(t)]≡hP[X(t→∞),X(t)]i (4) events Pˆdt[X(t)]≡hP[X(t),X(t+dt)]i (5) events where X(t) represents a partition computed at time t, X(t → ∞) is an asymptotic partition, and hi represents an average on the total set events of collisions. In simple terms, Pˆ+[X(t)] determines whether all particles bound at time t remain boundasymptotically. In a similar fashion Pˆ−[X(t)] measures the reciprocal value , i.e., the degree in which an asymptotical partition is contained in a partition occurring at time t. In conjunction, Pˆ+ y Pˆ− can be used to analyze theevolutionofthemicroscopiccompositionofthemostboundpartition, until it reaches its asymptotic form. FinallyPˆdt[X(t)]providesinformationonthegivenpartitionactivity, andcanbethususedtodefinethefragmentationtime,t ,onceitreaches ff some degree of stabilization. It is useful to work with some normalized quantities: P+[X(t)]=Pˆ+[X(t)]/ P X(t→∞),X′(t→∞) (6) events D h iE P−[X(t)]=Pˆ−[X(t)]/ P X(t),X′(t) (7) events D h iE Pdt[X(t)]=Pˆdt[X(t)]/ P X(t+dt),X′(t+dt) (8) events whereX′ isthepartitionformedDbyhtheclustersbelongingitEopartitionX, andwhereeachfragmenthasevaporatedaparticle. Normalizedquantities compare real values of Pˆ′s with an evaporative level of reference. P+ , P− and Pdt, refer to distinct persistence coefficients, namely forwards, backwards and differential, respectively. 3 Dynamic Evolution Armed with the tools introduced before (MD, MSTE and P+ , P− and P ) we proceed to study the dynamical evolution of the collisions de- dt scribed insection II.AnalyzingtheMeanVelocity Transfer, wecan char- acterize the collision stages and identify early emitted particles. This allows tostudytheexcitation energyandthedetection ofthefragmenta- tion time. A Collision Stages Two colliding stages are observed, with an initial highly colliding stage produced when the projectile hits the droplet surface and the energy is distributed chaotically. Collisions at this initial stage form a shock wave responsible for the prompt emission of light energetic particles from the surface. As the shock wave travels into the droplet, it produces den- sity fluctuations and internal fractures as a consequence of the momen- tumtransferredandinitiatesdisorderedcollisionsleadingtoanexcitation thermalization. Size effects in Heavy Ions Fragmentation 6 In order to reach a deeperunderstanding of this process, we compute the “Mean Velocity Transfer”, defined by: N Mj(t)= |(~vi(t+dt)−~vi(t))·eˆj| (9) *Xi=1 +events wherejdenotesincidentand normaldirectionsand eˆj isaunitaryvector in these directions. Fig. 5 shows time evolution of Mx for distinct values of E for Ni+Agcentral collision. beam Isotropiccollisions(disorderedcolliding mode)areresponsibleofthe momentum redistribution among the particles remaining in the system. Thisenergyheatsthesystemandconstitutesanexpandingcollectivemo- tion. Thedisorderedcolliding modeistheonlyonepresent foruniformly excitedsystemswhereit isresponsibleoffragmentation anditsoutwards flow disperses thefragments [24, 29]. MSTE algorithm can be used to study the size of the MSTE biggest fragment,totalmultiplicityandpersistencecoefficients. Figs. 3a,bandc show time evolution of these three quantities. The description suggested is that, due to an initial violent collision, some particles acquire enough energy to be released of thevicinity or interior of a cluster. This reduces themassofthebiggestMSTEclusterinthisearlystageandincreasesthe totalmultiplicity. Thistendencyismaintaineduntilthemeanmomentum for each particle permits the binding of particles configurationally close. Aftertheinitialreductionoftheinitialbiggestfragmentsize,acoalescent behavior promotes the multiplicity decrease and the biggest cluster size grows until it reaches a maximum. This time signs the end of the initial energy distribution and can beused todefinethe pre-equilibrium time. Persistence coefficients can be used to understand how the partition reaches its asymptotic microscopic composition. Fig. 3c shows the time evolution of P+[X(t)] and P−[X(t)] computed using the MSTE parti- tions for a E . beam Backwards Persistence Coefficient shows initially a decrease due to the fragment formation stage, followed by a subsequent increase due to the dynamic re-absorption lasting until tpre. High values of P− indicate that more particles belonging to a given cluster remain paired at time t. Consistently, P+ coefficient shows a plateau during this re-absorption stage that is extended until time tpre, followed by a monotonic increase duetoanevaporativedynamicsoftheMSTEclusters. Duringthisstage, MSTEalgorithmdetectsalargebiggestcluster,revealingthatthesystem is still dense. In summary,an initial stage characterized bytheexistence ofashock wave releasing particles from the surface, the reduction of the biggest fragment size and an increase of themultiplicity (i.e. decreases P−). Af- terwards, the shock wave crosses the droplet distributing uniformly the energy (Mx ∼ My) and a coalescent behavior is installed signed by the increaseofP− anddeletingthememoryoftheentrancechannel,reducing the multiplicity and augmenting the size of the biggest fragment until it reachesa∼80%−90%ofthetotalmass. Thisisfollowed byanevapora- tivedynamicsoftheclustersasindicatedbyamonotonicincreaseofP+. Fig. 4showscollidingstagesforNi+Agwithanenergyequalto1600MeV. Size effects in Heavy Ions Fragmentation 7 A first stage ends with a peak in the kinetic energy transported by light particles, signed by a peak in the black curve. A second stage ends with the attenuation of the intermediate fragments production (pink curve). Between these peaks, a peak in Mean Velocity Transfer (orange curve) is observed. Hence, Mean Velocity Transfer (MVT) produces instabilities leading to light particles emission and subsequent intermediate fragment emission. Once the MVT is stabilized, a final stage follows characterized by theemission of light fragments and system freeze-out. 4 Temperature The next step to obtain a caloric curveis the computation of thesystem temperature at fragmentation time, using the kinetic energy of nucleons in the participant region, relative to the center of mass. This seems to be justified as long as the excited droplets reach thermal equilibrium at break-up[24]. Nevertheless,thereexistobjectionswithrespecttotheuse of thermodynamicconcepts in small finitetransient systems [30, 31]. Now we definethis temperature and study its time evolution. A Participant Region Temperature. Incentralnuclearcollisions,shockwavesorthogonaltothebeamdirection arise,separatingcompressednuclearmatterfromcoldnuclearmatter. As the projectile penetrates into the target, a region with both high density andexcitation isformed. Thecomplexcollectively expandsinadirection normaltothebeamduetothefact thatpressureisnullintheexteriorof the system and therefore expansion in beam direction is decreased. Adiscontinuityappearsinthecontactpointbetweennuclei,withboth nuclei traveling in opposite directions with respect to the center of mass, that evolves into two shock waves traveling in opposite directions across the projectile and the target [32]. A study based on the dynamic model of several fluids, showed that most of the energy associated with the stopping derived from transverse collective motion, is used for mid-rapidity fragment production [33]. In the contact region between both nuclei, frictional forces promote energy dissipation and particle deflection, since projectile participant nucleons interact several times with target participant nucleons. [34]. All along this discontinuity, transport effects are expected leading to a soft change in theproperties instead of a discontinuous transition [35]. In this participant region a temperature can be obtained, considering the nucleons in this region and computing the temperature provided by the kinetic energy of these nucleons measured with respect to the center of mass. Promptly emitted, not reabsorbed nucleons are excluded, using the MSTE Early Cluster Recognition Algorithm for thissake. B PEPS and Energy Excitation. Asithappensintheexperimentalcase,theobservedpresenceofpromptly emitted particles(PEP) makes possible the a priori computation of the Size effects in Heavy Ions Fragmentation 8 fraction of beam energytransformed intoexcitation energy. Thiscompli- cation is absent in computations for both contained or infinite systems, and requires a sensible definition of the excitation energy that should depend on thePEPs. PEPs can be defined as unbound light clusters (with mass ≤ 4) de- tected by the algorithm at time tpre and that remain unbound at any posterior time (i.e. no re-absorption). Once PEPs are defined, their ki- neticenergycan beusedtoestimatetheenergyremaining inthesystem. Figs. 6a and 6b show that the transported energy by PEPs and the numberofPEPsasafunctionofthebeamenergy,respectively. Withthe numberofPEPsandtheirenergyquantified,theenergyremaininginthe system after time tpre can bealso obtained. Fig. 6c shows that energy is deposited in the target as a function of the beam energy. These figures show that the fraction of available energy leaving the system, as a consequence of this prompt emission, is considerable. Even more, the energy remaining in the system shows a saturation behavior indicating that there exists a limit for the quantity of energy that can be transferred in a collision, which does not occur in uniformly excited systems. C Fragmentation Time Afterthepre-equilibriumtime,tpre(cf. sectionIIIA),theenergyremain- ing in the system is completely distributed, and the density correlations r−pinducedbytheinitialshockwavebegintoconstituteacollectiveex- pansion. Eventually,thisexpansionwill transformtheinitialfluctuations into fragments well definedin space r, recognizable for MSTE algorithm. Caloric curves should reflect thesystem state along thephase transition, andweconsiderthistimeasthefragmentation time,t ,associated with ff the stabilization of MSTE density fluctuations 5 Caloric Curves and Residual Size Oncethecollisiondynamicsimulationisperformed,promptfragmentsare detected,fragmentation timesareidentifiedandexcitation energyaswell as temperature are computed,a caloric curvecan beobtained. Formally, this quantity investigated experimentally [14, 15, 19, 20, 21, 36] as well as computationally [23, 24, 37], is the functional relation between the temperature of the system and its excitation energy. In this study we extend this analysis toPandharipande droplets excited by collisions. Figs. 7-9 show the caloric curves computed for a wide energy range of excited collision systemsbuiltfrom participant region temperaturesat timet . Ascanbeseen,caloriccurvesaresimilartothoseobtainedfrom ff uniformly excited systems [37]. Once again, the relevant characteristic is the almost constant temperature behavior in fragmentation region. In otherwords,collisiondataprovidea“rise-plateau”caloriccurve. Besides, these caloric curves portray a limit temperature that diminishes as the residual size increases. Size effects in Heavy Ions Fragmentation 9 6 Conclusions. Event byevent “Riseplateau” Caloric curveshavebeen obtained for dif- ferentresidualsizebinsforAg+AgCentralHeavyIonCollision. Temper- ature is computed using Kinetic Gas Theory in the Participant Region, in the moment when the persistence reaches a value close to one. When the residual size increases, a clear decrease of the limit temperature is observed. This is in agreement with recent studies of experimental data performed byNatowitz et.al. [38]. Thereforewehaveconfirmedusingadynamicalmodel,recentanalysis ofexperimentaldata,whereadecreaseofthelimittemperatureisobserved as theresidual size is increased. 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