Microscopic approach to the spectator matter fragmentation from 400 to 1000 AMeV
aa r X i v : . [ nu c l - t h ] D ec Microscopic approach to the spectator matterfragmentation from 400 to 1000 AMeV
Yogesh K. Vermani and Rajeev K. Puri ∗ Department of Physics, Panjab University,Chandigarh-160014, India.July 25, 2018
Abstract
A study of multifragmentation of gold nuclei is reported at incidentenergies of 400, 600 and 1000 MeV/nucleon using microscopic theory.The present calculations are done within the framework of quantummolecular dynamics (QMD) model. The clusterization is performedwith advanced sophisticated algorithm namely simulated annealingclusterization algorithm (SACA) along with conventional spatial cor-relation method. A quantitative comparison of mean multiplicity ofintermediate mass fragments with experimental findings of ALADiNgroup gives excellent agreement showing the ability of SACA methodto reproduce the fragment yields. It also emphasizes the importanceof clustering criterion in describing the fragmentation process withinsemi-classical model. ∗ [email protected] Introduction
A highly excited system formed in a nucleus-nucleus collision, as a rule, isexpected to break into several pieces consisting of free nucleons, light chargedparticles (LCP’s), intermediate mass fragments (IMF’s) as well as heavierresidue. This phenomenon of breaking of colliding nuclei into several piecesis known as multifragmentation [1, 2, 3, 4, 5, 6, 7]. Due to its complexdynamics, mechanism behind this picture of ’explosive’ break up (into severalentities) is not yet known completely.At low incident energies, excitation energy available to the system is verysmall. Therefore, larger impact of collisions is needed to break the systeminto pieces of different sizes. In other words, fruitful destruction is possibleonly for the central collisions. On the other hand, mutual correlations amongnucleons are preserved in peripheral collisions, therefore, not much deviationfrom the initial picture will be seen. In contrast, excitation energy depositedin the system is very large at higher incident energies. Therefore, centralcollisions break the matter into very smaller pieces and rarely one sees inter-mediate mass fragments or heavy mass fragments in these events. Maximumnumber of IMF’s can only be seen at semi-central impact parameters. Largenumber of experiments have witnessed this trend of fragmentation at variousincident energies and impact parameters. This change in the behavior offragment pattern is also termed as a rise and fall in the multifragmentation[6, 8, 9, 10].As we go further towards higher incident energies, maximal of IMF multi-plicity starts shifting towards peripheral geometries. Such trends have beenfound and reported in several recent experiments of ALADiN - collabora-tion [2, 6]. In addition, manyfold aspects of spectator matter fragmentationhave also been studied for the collision of Au + Au on ALADiN set upat incident energies varying between 150 and 600 MeV/nucleon. Recently,2NDRA experiments extended the energy domain covering the incident ener-gies between 40 and 150 MeV/nucleon [11]. The sole motivation for all theseexperiments was the fantastic physics that may emerge from the disintegra-tion of excited systems leading to the expansion of matter to low densities.This onset of multifragmentation and afterward transition to vaporizationphase has also been linked to the concept of liquid-gas phase transition ofnuclear matter [7, 12, 13]. Such critical behavior is, however, reported to beinfluenced by the finite size effects [3, 14].All these experimental studies characterize the evolution of heavy-ion re-action from dominant multifragment-decay channel to complete disassemblyinto light charge particles (LCP’s) and free nucleons sometimes also termedas ’vaporization’. The very recent study by Puri and Kumar [15] analyzed the Ca + Ca reaction for incident energies between 20 and 1000 MeV/nucleonand over entire impact parameter range. They predicted a clear rise and fallof multiplicity in incident energy and impact parameter planes.On the theoretical front, not much success has been reported to repro-duce the ALADiN experimental data [6, 16, 17]. Theoretical approacheswhich follow the evolution of target and projectile to complete disassemblyof nuclear matter needs secondary algorithm to clusterize the phase space.Even afterburners have also been employed to extract fragments. The presentstudy aims to check whether microscopic reaction models can explain the uni-versality reported by ALADiN group [8] in spectator fragmentation or not.Molecular dynamical models QMD [4] and QPD [18] were found to explainsome of the features of this experimental data [6]. This questions the valid-ity of molecular dynamics models (MDM). The fallacy was largely attributedto the lack of advanced secondary clustering models [19, 20, 21]. The clus-tering criterion is one of the basic ingredients that may control the reactionmechanism in semi-classical models like quantum molecular dynamics model.3ecently, a novel clusterization algorithm based on the energy minimiza-tion criteria namely simulated annealing clusterization algorithm (SACA)was proposed [20]. As a first attempt, results with this algorithm were quitepromising one [20]. In ref. [20], Au + Au reaction was studied at inci-dent energy of 600 MeV/nucleon. Based on the ALADiN results, there oneassumed that fragment pattern does not change above 400 MeV/nucleon.Therefore, it remains to be seen whether QMD model can reproduce thisuniversality feature or not. We plan to address this situation in this letter.We apply this algorithm to ALADiN data at incident energies of 400, 600and 1000 MeV/nucleon in order to see whether our approach can explainthe rise and fall phenomenon and universal behavior in spectator fragmen-tation at such higher incident energies. It is worth mentioning that SACAalgorithm has been robust against experimental data at lower tail of incidentenergies. In our earlier studies [22], SACA method was reported to reproducethe charge yields at incident energies between 25 and 200 AMeV. In this anal-ysis, O+Ag/Br reactions were taken [22]. In another study, SACA methodwas tested against INDRA experimental data at 50 AMeV [23]. In this study,Xe+Sn reaction was subjected to multifragmentation and various variablessuch as charge, proton like and IMFs yields, angular distribution, averagekinetic energies etc. were analyzed. SACA method explained all these ob-servables quite nicley, whereas conventional method failed badly [23]. Due tothe fact that interaction energy among fragments is ignored, this approachof SACA can not be applied to incident energies below above mentioned one.To study fragmentation in Au+Au reaction, we followed nuclear collisionswithin QMD model [4]. The phase space thus generated is clusterized usingadvanced SACA method. 4 SACA formalism
To generate the phase space of nucleons, we use quantum molecular dynamics (QMD) model. For the details of the QMD model, reader is referred torefs. [4, 20]. The next essential step is to clusterize the phase space stored atvarious time steps in each event. The extensively used approach assumes thecorrelating nucleons to belong to same fragment if their centers are closer than4 fm i.e. | r α − r β |≤ f m . It may often lead to wrong results if applied athigher densities and hence can’t address the time scale of multifragmentation.This approach is labeled as minimum spanning tree (MST) algorithm.In our latest approach, fragments are constructed based on the energycorrelations. The pre-clusters obtained with the MST method are subjectedto a binding energy condition [20, 24]: ζ i = 1 N f N f X α =1 "r(cid:16) p α − P cmN f (cid:17) + m α − m α + 12 N f X β = α V αβ ( r α , r β ) < E Bind , (1)with E bind = -4.0 MeV if N f ≥ E bind = 0 otherwise. In eq. (1), N f is the number of nucleons in a fragment and P cmN f is the center-of-massmomentum of the fragment. The requirement of a minimum binding energyexcludes the loosely bound fragments which will decay at later stage.To look for the most bound configuration (MBC), we start from a randomconfiguration which is chosen by dividing whole system into few fragments.The energy of each cluster is calculated by summing over all the nucleonspresent in that cluster using eq. (1). Note that we neglect the interactionbetween the fragments. The total energy calculated in this way will differfrom the total energy of the system [24].5et the total energy of a configuration k be E k (= P i N f ζ i ), where N f is the number of nucleons in a fragment and ζ i is the energy per nucleonof that fragment. Suppose a new configuration k ′ (which is obtained by(a)transferring a nucleon from randomly chosen fragment to another frag-ment or by (b) setting a nucleon free, or by (c) absorbing a free nucleoninto a fragment) has a total energy E k ′ . If the difference between the oldand new configuration ∆ E (= E k ′ − E k ) is negative, the new configurationis always accepted. If not, the new configuration k ′ may nevertheless beaccepted with a probability of exp ( − ∆ E/υ ), where υ is called the controlparameter. This procedure is known as Metropolis algorithm. The controlparameter is decreased in small steps. This algorithm will yield eventuallythe most bound configuration (MBC). Since this combination of a Metropolisalgorithm with slowly decreasing control parameter υ is known as simulatedannealing , so our approach is dubbed as simulated annealing clusterizationalgorithm (SACA). For more details, we refer the reader to ref. [24]. For the present study, we use a soft equation of state (EoS) along withstandard energy-dependent n-n cross section [25]. The soft EoS has beenadvocated by many studies [5, 6, 16, 17, 20, 26]. The phase space is generatedand stored at many time steps and is then subjected to the above mentionedclusterization procedures. To address the time scale of multifragmentation ofspectator matter, we employed SACA method as well as spatial correlationmethod ( i.e.
MST).The density of environment is often correlated with the prediction ofbreaking of nuclear matter into pieces. One can also look density distribu-tion in coordinate space to investigate the formation of fragments. We here6ompute average density of system as : h ρ i = * A T + A P A T + A P X i =1 A T + A P X j>i πL ) / × e − ( r i ( t ) − r j ( t )) / L E , (2)with r i and r j being the position coordinates of i th and j th nucleons.The Gaussian width L is fixed with standard value of 1.08 f m . Figure 1 (toppanel) shows the time evolution of average nuclear density h ρ/ρ o i for Au+Ausystem at incident energies of 400, 600 and 1000 MeV/nucleon and at an im-pact parameter of 6 fm. The average nuclear density reaches its maximalaround 25 fm/c. This time domain also witnesses the maximum collisionrate and nuclear interactions which are going on between among target andprojectile nucleons. This maximal density shifts towards later times as wego down the incident energies. The fine point is that there is an insignificantchange in the density profile while enhancing the incident energy by the fac-tor of 2.5 times i.e. going from 400 to 1000 MeV/nucleon. At the final stageof the reaction, we don’t see any significant change with the incident energy.The bottom panel of fig. 1 shows the time evolution of the heaviest fragment h A max i using MST and SACA techniques. The MST method gives one bigcluster at the time of maximum density, whereas one sees striking ability ofSACA method in identifying the heaviest fragment quite early when violentphase of the reaction still continues. This suggests that evolution of multi-fragmentation is an intricate process. In other words, fragmentation startsat quite early stage when nucleons are still interacting among themselvesvigorously. The early recognition of heaviest fragment h A max i rules out itsformation out of the neck region. i.e. geometrical overlap between projectileand target. This suggests the emission of h A max i from the spectator region.Similar trends of transition from the participant to spectator fragmentationhas also been observed and reported by ALADiN-collaboration [8]. This find-7ng also confronts the common standpoint of thermal origin of fragments i.e. fragments are created after the thermalization sets in. Further after violentphase of reaction is over ( i.e. after 60 fm/c), binding energy of all clustersin SACA method is greater than E Bind , the minimum binding energy neededto bind the group of nucleons into cluster. Fragments after time 60 fm/cleave the reaction zone without nucleon-nucleon correlations being destroyedfurther. Hence fragment configuration obtained at the earlier time can becompared with experimental data. Strikingly, earlier detection of fragments(not shown here) at all incident energies upto 1000 MeV/nucleon gives uspossibility to look into the n-n interactions when nuclear matter is still hotand dense. Further, one is also free from the problem of stability of frag-ments. The failure of MST method to detect the fragments also questionsits validity at incident energies as high as 1000 MeV/nucleon. Simple cor-relations method fails to detect the fragments even at these high excitationenergies. The further rise in h A max i after 60 fm/c using SACA technique isdue to the reabsorption of surrounding light fragments by the heavier frag-ments. We see that heavier h A max i survive at smaller incident energies thanat higher incident energies. The capability of QMD model clubbed withSACA method is illustrated in fig. 2 where we display the mean multiplicityof intermediate mass fragments h N IMF i as a function of impact parameterof the reaction. Also shown are the results obtained with MST method. Ourmodel calculations with SACA method are in close agreement with ALADiNdata [8] for Au + Au reaction at all incident energies 400, 600 and 1000MeV/nucleon. As seen in the fig. 2, we also achieved a reasonable reproduc-tion of the shape of impact parameter dependence of h N IMF i . Due to shallowminima sometimes, we also see second minima before 60 fm/c in peripheralcollisions. We show also the calculations at these minima marked as (*). Wesee that fragment structure at these minima is further closer to the data.8urther the peak value of h N IMF i and the corresponding impact parameterb is also well estimated with QMD + SACA method. The prominent featureof the spectator decay is the invariant nature of the IMF distribution with re-spect to the bombarding energy. The SACA method successfully reproducedthe universal nature of spectator fragmentation at all the three bombardingenergies. It is worth interesting to note that these universal features observedin multifragmentation of gold nuclei persist upto much higher bombardingenergies than explored in this work [27]. In contrary, the normal spatial cor-relation method fails badly to explain the production of intermediate massfragments at all incident energies. This questions the validity of MST methodin explaining the fragmentation pattern in heavy-ion collisions. We have studied multifragment-emission in Au + Au reaction at inci-dent energies of 400, 600 and 1000 MeV/nucleon where ALADiN experimentsshowed the universality in the production of intermediate mass fragments.For this study, we employed QMD model clubbed with energy minimiza-tion algorithm (SACA) along with conventional spatial correlation method.Our findings reveal that SACA is able to reproduce the universal nature ofmultifragmentation of excited spectator over entire impact parameter-energyplane whereas spatial correlation method failed to reproduce the IMF mul-tiplicity. This is for the first time that QMD + SACA approach is able toreproduce the entire energy domain. It also shows that mass and multiplic-ity of spectator fragments remain invariant to range of bombarding energies.This also resolved the earlier discrepancy where QMD model underestimatedthe fragment yield [6, 16] at large impact parameters even after 200 fm/c. Inour case, SACA method is successful in breaking the spectator matter into9ntermediate mass fragments. Our results show that the QMD model con-tains necessary ingredients to describe the physics of spectator decay. Theclustering algorithm one uses, however, holds the key tenet to explain thereaction mechanism. This work was supported by a research grant from Department of Scienceand Technology, Government of India.
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Figure Captions
FIG. 1.
Top panel: Time evolution of the average nucleon density h ρ/ρ o i reached in Au + Au collision. Bottom panel: The heaviest fragment h A max i obtained with SACA and MST analysis as a function of time in Au + Au collision. FIG. 2.
The mean multiplicity of intermediate mass fragments h N IMF i asa function of impact parameter b for the reaction of Au + Au . The modelcalculations with SACA (solid squares) and MST (open triangles) methodsare compared with experimental data (open circles) reported by ALADiNgroup [8]. 12
50 100 150 200050100150200250300350400 (cid:1)(cid:2)(cid:3)(cid:0)(cid:4)(cid:5)
MST
E= 400 MeV/ nucl. E= 600 MeV/ nucl. E= 1000 MeV/ nucl. E= 400 MeV/ nucl. E= 600 MeV/ nucl. E= 1000 MeV/ nucl. time (fm/c) (cid:6) A m ax (cid:7) SACA
Au + Au E= 400 MeV/ nucl. E= 600 MeV/ nucl. E= 1000 MeV/ nucl. b=6 fm (cid:8) r / r o (cid:9) E=1000 MeV/nucl. b (fm)
ALADiN SACA (60 fm/c) SACA (t min ) MST
E=600 MeV/nucl.
E=400 MeV/nucl.
Au + Au (cid:10) N I M F (cid:11)(cid:11)