Study of fragmentation using clusterization algorithm with realistic binding energies
Yogesh K. Vermani, Jatinder K. Dhawan, Supriya Goyal, Rajeev K. Puri, J. Aichelin
aa r X i v : . [ nu c l - t h ] D ec Study of fragmentation using clusterizationalgorithm with realistic binding energies
Yogesh K. Vermani, Jatinder K. Dhawan, Supriya Goyal and Rajeev K. Puri ∗ Department of Physics, Panjab University,Chandigarh-160014, India.J. AichelinSUBATECH - IN2P3/CNRS - Ecole des Mines de Nantes4, rue Alfred Kastler, F-44072 Nantes, Cedex 03, FranceAugust 10, 2018
Abstract
We here study fragmentation using simulated annealing clusterizationalgorithm (SACA) with binding energy at a microscopic level. In anearlier version, a constant binding energy (4 MeV/nucleon) was used. Weimprove this binding energy criterion by calculating the binding energy ofdifferent clusters using modified Bethe-Weizs¨acker mass (BWM) formula.We also compare our calculations with experimental data of ALADiNgroup. Nearly no effect is visible of this modification.
In the recent years, several theoretical attempts [1, 2, 3, 4, 5] have been reportedon spectator matter fragmentation observed in relativistic heavy-ion (HI) re-actions using ALADiN set up [6, 7, 8, 9]. The multifragmentation has beenthought to be one of the important phenomena for the understanding of phasetransition in nuclei and nuclear equation of state. The multiplicity of interme-diate mass fragments (IMFs) in central collisions is reported to first increase ∗ [email protected] E ≈
100 MeV/nucleon [7, 10] and thendecline afterwards indicating a complete disassembly of nuclear matter. At rela-tivistic energies, IMF emission becomes preferential only at peripheral collisions[6, 7, 8, 9, 11] where system has relatively low excitation energy. The low energyheavy-ion collisions are dominated by the phenomena such as the deep-inelasticscattering and fusion-fission. The fireball-spectator picture, however, emergesand dominates the physics at relativistic energies where the formation of heav-ier clusters is a rather unusual phenomenon. The most complete experimentsof ALADiN collaboration have shown that fragment emission pattern remainsalmost unchanged above the incident energy of 400 MeV/nucleon for a givenprojectile-target combination [9]. This observation is also very often termed asuniversality of the fragmentation emission and has been discussed in the liter-ature extensively [6, 8, 9, 12]. In these experiments, the correlation betweenIMF multiplicity and impact parameter ‘b’ suggests a picture of transition fromevaporation to complete disassembly with increasing violence of the collision[6, 8, 11, 13]. At higher incident energies, one also expects complete disassem-bly or vaporization of the colliding matter [7, 10, 14].It has been reported earlier that fragment multiplicities predicted by quan-tum molecular dynamics (QMD) model [15, 16, 17] coupled with conventionalclustering technique such as minimum spanning tree (MST) algorithm are sig-nificantly underestimated for larger values of impact parameters [3, 8, 7]. Thechoice of different nuclear incompressibilities ( i.e equations of state) was foundto have only marginal influence on the predicted IMF multiplicities and lightcharged particles yield [8]. About decade ago, Dorso et al. [18] advanced a newalgorithm in which fragments if already formed can be identified earlier. Thescope of this approach was limited to light systems like Ca+Ca where only a fewfragments are produced. The results indicated a quite early formation. How-ever, for the understanding of multifragmentation, the multifragment eventsobserved in the collision of heavy systems have to be analyzed. Unfortunately,the computing time for the algorithm employed [18] increase by roughly N!,where N is the number of nucleons in the system. Hence a completely newnumerical procedure was invented to extend the approach to larger and morerelevant systems [19]. Due to small surface, heavier nuclei are close to nuclearmatter and hence are ideal to study the physics.The basic principle behind this algorithm [19] is that fragment structure was2chieved via energy minimization using simulated annealing technique whichyields maximum binding energy of the system consisting of fragments of allsizes produced in a reaction. In this algorithm, each cluster is subjected to abinding energy check. As a first attempt, a constant average binding energycheck of -4 MeV/nucleon was employed for the all clusters. This algorithm (la-beled as simulated annealing clusterization algorithm i.e. SACA) yielded quiteencouraging results. For instance, one could explain the fragment distributionfor the reactions of O+ Ag/Br at incident energies 25-200 AMeV [20, 21]. Forthe first time, this microscopic approach [3] could also reproduce the fragmentmultiplicities in Au + Au reaction at E=600 AMeV measured by ALADiNcollaboration. It is worth mentioning that the MST approach failed badly toreproduce this experimental trend [3, 8, 7]. A comparison of SACA (withoutbinding energy cut of -4 MeV/nucleon) and one developed by Dorso et al . [18]yielded the same results for lighter colliding nuclei.As discussed above, each fragment in SACA method was subjected to aconstant binding energy of -4 MeV/nucleon. We know that the binding energydepends on the mass of the fragment/nucleus. One is always wondering whetherthis criterion of average binding energy is justified or not. In this paper, we wishto address the above question by subjecting each fragment to its true bindingenergy that has now been measured to a very precise level with reference tounstable and stable isobars, proton-rich and neutron-rich nuclei. We shall showthat this improvement does not yield different results justifying the validity ofthe algorithm.We employ quantum molecular dynamics (QMD) model as primary modelto follow the time evolution of nucleons. Section 2 describes the primary QMDmodel along with details of simulated annealing clusterization algorithm (SACA)and its extension. Section 3 deals with the calculations and illustrative results,which are summarized in section 4. The quantum molecular dynamics is an n- body theory that simulates the heavy-ion reactions between 30 AMeV and 1AGeV on event by event basis. This is3ased on a molecular dynamics picture where nucleons interact via two andthree-body interactions. The explicit two and three-body interactions preservethe fluctuations and correlations which are important for n -body phenomenonsuch as multifragmentation [15, 16, 17]. Nucleons follow the classical trajectoriesobtained by Hamilton’s equations of motion:˙ r α = ∇ p α hHi , α = 1 , ..., N ;˙ p α = −∇ r α hHi , α = 1 , ..., N. (1)Here, nucleons interact via n-n interactions and stochastic elastic and inelasticcollisions. For further details of the model, the reader is referred to Ref. [15]. As discussed in the previous section, earlier versions of clustering algorithmsuch as minimum spanning tree (MST) rely on the spatial correlation principleto identify the fragment configuration [15]. In this algorithm, two nucleons areconsidered to be a part of the same fragment if their inter-nucleon distance issmaller than r C (in fm) . One generally takes 2 ≤ r C ≤
4. Naturally, it cannotaddress the time scale of fragmentation. This failure led to the developmentof more sophisticated algorithm based on the simulated annealing technique.This approach, known as simulated annealing clusterization algorithm (SACA),is based on the principle of energy minimization which requires that a group ofnucleons can form a bound fragment if their total fragment energy per nucleon ζ i is below certain binding energy E bind i.e. ζ i = 1 N f N f X α =1 q(cid:0) p α − P N f (cid:1) + m α − m α + 12 N f X β = α V αβ ( r α , r β ) < − E Bind . (2)In the original SACA version [19], we take E bind = 4.0 AMeV if N f ≥ E bind = 0 otherwise. In this equation, N f is the number of nucleons in afragment, P N f is the average momentum of the nucleons bound in the fragment.To find the most bound configuration, we start with a random configuration andthe energy of each cluster is calculated using Eq. (2). Let the total energy of aconfiguration k be E k (= P i N f ζ i ), with ζ i is the energy per nucleon associatedwith that fragment.Now to generate new configuration k ′ , we assume that this can be achievedby (a) transferring a nucleon from some randomly chosen fragment to another4ragment, by (b) setting a nucleon free or, by (c) absorbing a free nucleon into afragment] has total energy E ′ k . If the difference between energies of the old andthe new configurations, ∆ E (= E ′ k − E k ) is negative, the new configuration isalways accepted. If not, the new configuration k ′ may nevertheless be acceptedwith a probability of exp ( − ∆ E/c ), where ‘c’ is called control parameter. Thisprocedure is known as Metropolis algorithm. The control parameter is decreasedin small steps. This algorithm will yield eventually the most bound configura-tion (MBC). Since this combination of Metropolis algorithm with decreasingcontrol parameter is known as simulated annealing, this approach was dubbedas simulated annealing clusterization algorithm (SACA) [19]. The present algo-rithm with a constant average binding energy check is labeled as SACA (1.1).For further details, we refer the reader to Refs. [19, 20, 21, 22]. In Fig. 1,we show the calculated multiplicities of different fragments as well as the meansize of the largest fragment h A max i as a function of different binding energycuts using original SACA. The multiplicities of light charged particles LCPs[2 ≤ A ≤
4] and size of the largest fragment h A max i remains almost unaffectedby changing the binding energy cut. However, the multiplicities of mediummass fragments MMFs [5 ≤ A ≤
20] and intermediate mass fragments IMFs[5 ≤ A ≤
65] show strong sensitivity towards the imposed binding energy. Fromthis analysis, it would be interesting to study the effect of binding energy onaverage fragment production. The choice of proper binding energy can be basedon either experimental information or on theoretical information. Since experi-mental information is range bound, we shall use theoretical formulation.One of the earlier attempts to reproduce the gross features of nuclear bindingenergies was made by Weizs¨acker et al. [23]. The Bethe-Weizs¨acker (BW) massformula for the binding energy of a nucleus reads as [24]: E bind = a v N f − a s N / f − a c N zf ( N zf − N / f − a sym ( N f − N zf ) N f + δ. (3)Here, N zf stands for the proton number of a fragment. The various termsinvolved in this mass formula are the volume, surface, Coulomb, asymmetryand pairing terms. The strength of different parameters is: a v =15.777 MeV, a s =18.34 MeV, a c =0.71 MeV and a sym =23.21 MeV respectively [24]. The pair-ing term δ is given by: δ = + a p N − / f f or even N zf and even N nf , (4) δ = − a p N − / f f or odd N zf and odd N nf , (5)5 | Binding Energy | (MeV/nucleon) E=600 MeV/nucleonb=12 fm
Au + Au L C P s MM F s I M F s (cid:1) A m ax (cid:2) Figure 1: The average mass of the heaviest fragment h A max i , mean multiplicitiesof light charged particles LCPs, medium mass fragments MMFs, and interme-diate mass fragments IMFs as a function of binding energy check imposed forthe reaction of Au + Au at 600 MeV/nucleon and at an impact parameterof 12 fm. δ = 0 f or odd N f nuclei, (6)with a p = 12 MeV and N nf being the neutron number of a fragment. This for-mula reproduces the binding energy of stable nuclei but faces serious problemfor light nuclei along the drip line and with nuclei having rich neutron or protoncontent. The inadequacy of BW mass formula for lighter nuclei was removed by6amanta et al. [24] by modifying its asymmetry and pairing terms. This mod-ified formula was dubbed as modified Bethe-Weizs¨acker mass (BWM) formula[24]. The beauty of BWM formula lies in its ability to reproduce the bindingenergies for light nuclei near the drip line [24]. For a large number of unstableisobars, isotones and halo nuclei, it was shown in Ref. [24] that this modifiedformula reproduces the experimental binding energies quite precisely. In theBWM formula, the binding energy of a fragment is defined as [24]: E bind = a v N f − a s N / f − a c N zf ( N zf − N / f − a sym ( N f − N zf ) N f (1+ e − Nf / ) + δ new . (7)The strength of various parameters now reads: a v =15.777 MeV, a s =18.34 MeV, a c =0.71 MeV and a sym =23.21 MeV, respectively. The pairing term δ new is givenby: δ new = + a p N − / f (1 − e − N f / ) f or even N zf and even N nf , (8) δ new = − a p N − / f (1 − e − N f / ) f or odd N zf and odd N nf , (9) δ new = 0 f or odd N f nuclei, (10)with a p = 12 MeV.We extend the SACA method by incorporating this binding energy formuladuring the formation of the clusters. Each fragment at the end of the pro-cedure is subjected to this new binding energy (Eq.(7)) instead of a constant-4 MeV/nucleon binding energy. Any fragment that fails to fulfil the abovebinding energy criterion is treated as a group of free nucleons. At the end, allfragments are properly bound. This version is labeled as SACA (2.1). We havealso tested the spectrum for actual experimental binding energies [25]. Onlysmall difference is seen for lighter fragments only. We simulated the collisions of Au + Au at incident energy of 600 AMeVusing a soft equation of state along with standard energy dependent n-n crosssection. We display in Fig. 2, the average mass of the largest fragment h A max i ,mean multiplicities of free nucleons, light charged particles LCPs [2 ≤ A ≤ ≤ A ≤ ≤ A ≤
65] and intermediate mass fragments IMFs [5 ≤ A ≤
65] as afunction of time for the reaction of Au + Au at 600 AMeV and impact7arameter 12 fm. As expected, h A max i is nearly independent of the bindingenergy criterion, whereas insignificant influence can be seen on the multiplicitiesof free nucleons, LCPs, MMFs and IMFs. Similar trends were also observed forthe central reaction of Au + Au at 600 AMeV.To further explore the characteristics of fragment structure obtained withmodified SACA (2.1), we show in Fig. 3, the impact parameter dependence ofmean multiplicities of various fragments. This will also help to understand theproper energy deposition in the spectator matter. The result obtained withSACA (1.1) and SACA (2.1) are displayed for the reaction of Au + Au at 600 AMeV as a function of impact parameter. The time for realization ofdifferent fragments was chosen to be 60 fm/c. This is the time when h A max i hasminimum size and configuration realized at this stage is most bound [19]. Incentral collisions, SACA (2.1) predicts smaller h A max i , whereas trend reversesin the peripheral collisions. As a result, free nucleons also behave accordingly.The yields of IMFs and MMFs do not reduce appreciably for central as wellas peripheral geometries using extended version of SACA. This is due to thefact that fragments recognized by SACA method are properly bound, therefore,simple cut also yields same results.We also attempted to confront our present calculations using extended clus-terization approach SACA (2.1) (at t=60 fm/c) with experimental data of AL-ADiN group [9] for the reaction of Au (600 AMeV) + Au. In Fig. 4, we showthe mean IMF multiplicity h N IMF i (in upper panel) and average charge of thelargest fragment h Z max i (in lower panel) as a function of impact parameterat 600 AMeV. The calculations with the original SACA (1.1) version are alsoshown for comparison. All calculations were subjected to experimental cuts offorward hemisphere. The h N IMF i and h Z max i obtained with different versionsof SACA are quite close to each other and to the experimental data. It justifiesthe use of average binding energy within above algorithm. We have also cal-culated the yields at incident energies of 400 and 1000 MeV/nucleon. Similarresults are also obtained at these incident energies. Summarizing the work, we have proposed an extension to SACA method byincorporating the binding energy of individual fragments calculated from the8
Au + Au F ree N u c l . E=600 MeV/nucleonb=12 fm L C P s MM F s (cid:0) A m ax (cid:3) H M F s t (fm/c) I M F s SACA (1.1) SACA (2.1)
Figure 2: The average mass of heaviest fragment h A max i and the mean multi-plicities of various kinds of fragments as a function of time for the reaction of Au + Au at 600 MeV/nucleon and at an impact parameter of 12 fm. Thesolid and dashed lines depict the results due to original SACA and its extension.9 E=600 MeV/nucleon
Au + Au (cid:4) A m ax (cid:5) F ree N u c l . H M F s L C P s MM F s b (fm) I M F s SACA (1.1) SACA (2.1)
Figure 3: The impact parameter dependence of average size of the heaviestfragment h A max i and mean multiplicities of various kinds of fragments for thereaction of Au + Au at incident energy 600 MeV/nucelon. The solid anddashed curves depict results of SACA (1.1) and SACA (2.1), respectively.10 〈 N I M F 〉〈 Z m ax 〉 Data SACA (1.1) (60 fm/c) SACA (2.1) (60 fm/c) b (fm)
E= 600 MeV/nucleon
Au + Au Figure 4: The mean IMF multiplicity (top panel) and average charge of theheaviest fragment (bottom panel) as a function of impact parameter. Opencircles depict the experimental data points [9].modified Bethe-Weizs¨acker mass (BWM) formula. Based on our calculations,we noticed that this extension has little effect on the fragment multiplicities andmean size of the largest fragment at 60 fm/c as well as at asymptotic times. Inperipheral collisions, new extension reduces the IMF yield, thereby increasingthe size of h A max i marginally. Both versions of SACA are clearly close to eachother and to ALADiN data. 11ne of the authors (Y. K. V) would like to thank Drs. W. Trautmann and C.Samanta for interesting and constructive discussions. This work is supported bya research grant from Council of Scientific and Industrial Research, Governmentof India, vide grant no. 7167/NS-EMR-II/2006. References [1] Barz H, Bauer W, Bondorf J P, Botvina A S, Donangelo R, Schulz H, andSneppen K 1993
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