Isospin dependent hybrid model for studying isoscaling in heavy ion collisions around the Fermi energy domain
aa r X i v : . [ nu c l - t h ] S e p Isospin dependent hybrid model for studying isoscaling in heavy ion collisions aroundthe Fermi energy domain
S. Mallik ∗ and G. Chaudhuri
1, 2 Physics Group, Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India
Investigation of observables from nuclear multifragmentation reactions depending on isospin led tothe development of a hybrid model. The mass and charge distribution as well as isotopic distributionwas studied using this model for
Sn+
Sn reaction as well as
Sn+
Sn reactions at differentenergies. The agreement of the results obtained from the model with those from experimental dataconfirms the accuracy of the model. Isoscaling coefficients were extracted from these observableswhich can throw light on the symmetry energy coefficient. Another important facet of this modelis that temperature of the studied reaction can be directly extracted using this model.
PACS numbers:
I. INTRODUCTION
The study of isospin dependent observables in nuclearmultifragmentation reaction around the Fermi energy do-main is a subject of contemporary interest [1, 2]. Differ-ent statistical models like Statistical MultifragmentationModel (SMM) [3], Canonical Thermodynamical Model[2, 4], have been explored to investigate and verify thephenomenon of isoscaling [5–9] as observed in experi-ments [10–15]. The main motivation behind inclusionof isospin in the transport model based on Boltzmann-Uehling-Uhlenbeck (BUU) equation [2, 16] was to studythe isospin dependent observables in this framework. Ahybrid model was developed in our group few years backin order to study the central collision of Xe on Sn[17]at beam energies around the Fermi energy domain andthe results were compared successfully with experimen-tal data [18]. This work is an extension of the earlierone where isospin degree of freedom is incorporated inthe transport model based on the Boltzmann-Uehling-Uhlenbeck (BUU) approach. This distinction betweenneutron and proton in the entrance channel enabled usto investigate isospin dependent observables in the exitchannels. Good agreement with experimental data ver-ified our approach (isospin dependent model) within areasonable accuracy.The models for explaining nuclear multifragmentationreactions can be broadly categorised under two heads. (i)Statistical Models [3, 4, 19] and (ii) Dynamical Models[16, 20, 21]. The statistical models which are based onphase space considerations has nice clusterization tech-niques included in them; the disadvantage of these mod-els being that they assume some initial conditions liketemperature, source size, freeze out volume etc. Theseconditions are either obtained from experimental observ-ables or parameterized. The dynamical models are basedon more microscopic calculations which deal with time ∗ Electronic address: [email protected] evolution of the nucleons in phase space. These modelswhen coupled to the initial stage of the statistical mod-els can fix these parameters from realistic considerations.Our present work is based on such hybrid model wherethe initial part of the nucleus nucleus collision is analysedusing the BUU equation while the clusterization part istaken care of by the Canonical Thermodynamical model.The significance of this model is that one can extract thetemperature of the studied reaction directly bypassing allambiguities.The excitation of the colliding system is calculated byusing dynamical Boltzmann-Uehling-Uhlenbeck (BUU)approach [16, 22] with appropriate consideration of pre-equilibrium emission. Then the disassembly of this ex-cited system is analysed by Canonical Thermodynami-cal model (CTM) [4]. The decay of excited fragments,which are produced in multifragmentation stage is calcu-lated by the Weisskopf evaporation model [23]. Charge,mass and isotopic distributions are the different observ-ables which have been examined using this model for
Sn+
Sn reaction as well as
Sn+
Sn reaction at50MeV/nucleon [24] subsequently compared with exper-imental data. We study central collisions around fermienergy domain which are extensively used for produc-ing neutron rich isotopes and for studying nuclear liq-uid gas phase transition. But in this particular workour main focus will be centered around the isospin de-pendent observables . More specifically we will exam-ine the isotopic distributions of different elements fromthe two reactions and finally investigate if the well stud-ied phenomenon of isoscaling emerges from our isospindependent hybrid model. Isoscaling[6–15, 25–28] is animportant technique observed in certain reactions whichdepends crucially on the isospin of the system and thuscan throw light on the symmetry energy[1]. This aspecthas motivated both the the theoreticians and the exper-imentalists to study isoscaling and its relation with thesymmetry energy coefficient. The present work is anothereffort in this direction. The fact that temperature can beestimated using this hybrid model facilitated the verifi-cation of similar temperature assumption made in theisoscaling equation[6–8, 10]. In the next section we willbriefly describe our model and then present the resultsin the subsequent section.
II. BASICS OF THE MODEL
The theoretical calculation consists of three dif-ferent stages: (i) Initial condition determination byisospin dependent Boltzmann-Uehling-Uhlenbeck model(BUU@VECC-McGill), (ii) fragmentation by canonicalthermodynamical model and (iii) decay of excited frag-ments by evaporation model.The BUU@VECC-McGill transport model calculation[29–31] for heavy ion collisions starts with two nucleiin their respective ground states approaching each otherwith specified velocities. For calculating ground state en-ergies and densities an isospin dependent Thomas-Fermimodel is developed separately which is briefly discussedin the Appendix section. The Thomas-Fermi phase spacedistribution is then sampled using Monte-Carlo techniqueby choosing test particles (we use N test = 100 for eachneutron or proton) with appropriate positions and mo-menta.In the center of mass frame, the test particles of theprojectile and the target nuclei are boosted towards eachother. Simulations are done in a 200 × × f m box.At t=0 fm/c the projectile and target nuclei are centeredat (100 fm,100 fm,90 fm) and (100 fm,100 fm,110 fm).The test particles move in a mean-field U ( ρ p ( ~r ) , ρ n ( ~r ))and occasionally suffer two-body collisions, with proba-bility determined by the nucleon-nucleon scattering crosssection. For each collision, Pauli blocking is checked andif the final states are allowed the momenta of the collid-ing particles are changed. The mean field potential usedfor this work is given by U ( ρ p ( ~r ) , ρ n ( ~r )) n/p = aρ ( ~r ) + bρ σ ( ~r )+ C sym ( ρ n ( ~r ) − ρ p ( ~r )) τ z + c ∇ r ρ ( ~r ) + 12 (1 − τ z ) U c (1)where ρ ( ~r ) = ρ p ( ~r ) + ρ n ( ~r ); ρ n ( ~r ) and ρ p ( ~r ) are neutronand proton densities at ~r and τ z is the zth componentof the isospin degree of freedom, which is 1 or − U c is the standard Coulombinteraction potential and the derivative term does notaffect nuclear matter properties but in a finite system itproduces quite realistic diffuse surfaces and liquid dropbinding energies. This can be achieved for A =-2230.0MeV f m , B =2577.85 MeV f m / , σ =7/6, ρ = 0 . c =-137.5 MeV f m [32]. Co-efficient for isospin termis C sym =200 MeV- f m . The mean-field propagation isdone by using the lattice Hamiltonian Vlasov methodwhich conserves energy and momentum very accurately[32, 33]. Two body collisions are calculated as in Ap-pendix B of ref. [16], except that the pion channels are closed, as there will not be any pion production at 50MeV/nucleon.One can calculate the excitation energy from projectilebeam energy by direct kinematics by assuming that theprojectile and the target fuse together. In that case theexcitation energy is too high as a measure of the exci-tation energy of the system which multifragments. Pre-equilibrium particles which are not part of the multifrag-menting system carry off a significant part of the energy.To get a better measure of excitation of the fragmentingsystem the pre-equilibrium particles can be identified af-ter BUU simulation at the freeze-out stage and can betaken out. In different multifragmentation experiments,it is observed that after pre-equilibrium emission around75% to 80% of the total mass creates the fragmenting sys-tem [34–36]. To make it consistent with the other exitingworks on the Sn+
Sn and
Sn+
Sn reactions at50 MeV/nucleon, we choose the test particles which cre-ate 75% of the total mass from the most central denseregion.At the end of the transport calculation in the freeze-out stage, the positions and momenta of the test particlesforming central dense region are known; hence from thesepositions and momenta, one can calculate the potentialand kinetic energies respectively. By adding kinetic andpotential energy the excited state energy of the clustercan be obtained. It is observed that excited state energybecome almost same for t ≥
100 fm/c [17]. Thereforeone can stop transport simulation at any time t ≥ t = 200 fm/c However, to knowthe excitation one needs to calculate the ground statestate energy also. This is done by applying the ThomasFermi method for a spherical (ground state) nucleus hav-ing mass equal to the cluster mass. Then subtracting theground state energy, the excitation is obtained. Know-ing mass and excitation of the fragmentating system, thefreeze-out temperature is calculated by using the canon-ical thermodynamic model CTM [4, 37, 38].Indeed CTM can be used to calculate the averageexcitation per nucleon for a given temperature and massnumber, and the relation is inversed to get the tempera-ture from the output of the dynamical stage[17, 22, 39].In CTM, it is assumed that a system with Z protonsand N neutrons at temperature T , has expanded to ahigher than normal volume where the partitioning intodifferent composites can be calculated according to therules of equilibrium statistical mechanics. According tothis model, the average number of composites with N neutrons and Z protons can be calculated from, h n N,Z i = ω N,Z Q N − N,Z − Z Q N ,Z (2)where, ω N,Z is the partition function of one compositewith N neutrons and Z protons and Q N ,Z is the to-tal partition function which can be calculated from therecursion relation, Q N ,Z = 1 N X N,Z
N ω
N,Z Q N − N,Z − Z (3)The description of ω N,Z and details of CTM can befound in Ref. [4].The excited fragments produced after multifragmen-tation decay to their stable ground states. They can γ -decay to shed energy but may also decay by lightparticle emission to lower mass nuclei. We include emis-sions of n, p, d, t, He and He. Particle-decay widths areobtained using Weisskopf’s evaporation theory. Fissionis also included as a de-excitation channel though for thenuclei of A<
100 its role will be quite insignificant. Thedetails of the evaporation stage are described in Ref. [23].
III. RESULTS
We have done calculations for the
Sn+
Sn and
Sn+
Sn reaction for projectile beam energy 50MeV/nucleon. In Fig. 1 we have plotted the charge (left)and mass (right) distributions of the fragments. In thegiven excitation energy regime, the distribution decreaseswith charge or mass number. Results have been com-pared with experimental data and good agreement hasbeen obtained as is visible from the figure. This estab-lishes the success of our model where the initial part hasbeen described by the isospin dependent BUU equationfollowed by the canonical thermodynamical model(CTM)for the deexcitation part. This motivated us to further -4 -2 Sn+ Sn d M / d ( s r - ) -4 -2 Sn+ Sn -4 -2 Sn+ Sn Z 0 5 10 15 2010 -4 -2 Sn+ Sn A FIG. 1: Theoretical (red dashed lines) charge (left panels)and mass distribution (right panels) of
Sn on
Sn (upperpanels) and
Sn on
Sn reaction (lower panels) reaction at50 MeV/nucleon.The experimental data are shown by blacksquares. -4 -3 -2 -1 d M / d ( s r - ) Z=6
Sn+ Sn -4 -3 -2 -1 Z=8
Sn+ Sn
10 12 1410 -4 -3 -2 Z=6
Sn+ Sn
14 16 18 20 2210 -4 -3 -2 -1 Z=8
Sn+ Sn A FIG. 2: Theoretical (red dashed lines) isotopic distributions(red dashed lines) at Z =6 (left panels) and 8 (right panels)for Sn on
Sn (upper panels) and
Sn on
Sn reac-tion (lower panels) reaction reaction at 50 MeV/nucleon.Theexperimental data are shown by black squares. probe into the details and thereby study the isotopicdistribution of some elements. In Fig. 2 have plot-ted the isotopic distribution of carbon and oxygen from
Sn+
Sn (upper panels) and
Sn+
Sn (lower pan-els) reactions. The production cross-section varies widelyover orders of magnitude and our model could success-fully predict this large change. The behaviour as seenfor
Sn+
Sn reaction is pretty similar as that of
Sn+
Sn , the difference being that the cross sectionof neutron rich isotopes are more for the neutron richreaction for obvious reasons. Here too the model calcu-lation could do good justice to the experimental data.In Fig.3 we have displayed the isoscaling which is theratio of the yields of the same fragment from the neu-tron rich to that of the neutron less isotope reaction. Inthe left panel, the odd Z ones are plotted while the evenones are plotted in the right panel just for the sake ofclarity. The plots are really nice with the lines approxi-mately parallel to each other as it should be if the law ofisoscaling is obeyed. The slope of these parallel lines issomewhat related to the symmetry energy coefficients α and β as given by the following isoscaling equation. R ( N, Z ) = h n N,Z i / h n N,Z i = C exp( µ n − µ n T N + µ z − µ z T Z )= C exp( αN + βZ ) (4)where R is the ratio of the yields from two reactionsand C is some arbitrary constant, µ n ’s and µ z ’s areneutron and proton chemical potential of the twofragmenting sources at freeze-out condition. One vital Z=8Z=6Z=4Z=2 N Z=7Z=5Z=3 R ( N , Z ) Z=1
FIG. 3: Isotopic ratios( R ) of multiplicities of fragments( N, Z ) where reaction 1 and 2 are
Sn on
Sn and
Snon
Sn respectively. For both reaction the projectile beamenergy is 50 MeV/nucleon. The left panel shows the ratios asfunction of neutron number N for fixed Z values, while theright panel displays the ratios as function of proton number Z for fixed neutron numbers ( N ). The red dashed lines aredrawn through the best fits of the theoretically calculated ra-tios (red circles). The experimental data are shown by blacksquares. assumption made while deriving the above equation isthat freeze-out temperature ( T ) of both the reactionsare same. The concept of temperature is quite familiarin heavy ion physics and it is usually calculated [40–44]from double isotope ratio method [45] or kinetic en-ergy spectra of emitted particles. But in both cases,sequential decay from higher energy states [46], Fermimotion [47], pre-equilibrium emission etc complicate thescenario of temperature measurement and the responseof different thermometers is sometimes contradictory[48, 49]. The advantage of using this hybrid modelcalculation is that one can estimate the temperatureof the intermediate energy heavy ion reactions directlyfrom here which bypasses all such problems. It isdirectly obtained from this isospin dependent hybridmodel calculation that, T for the Sn+
Sn reactionis 5.04 MeV while that for the
Sn+
Sn reactionis 5.08 MeV. The values are extremely close and thisconfirms strongly the assumption made for applyingisoscaling equation. The fact that temperature can be
Isoscaling Theoretical ExperimentalParameter α β -0.49 -0.42TABLE I: Best fit values of the isoscaling parameters α and β for the two reactions Sn on
Sn and
Sn on
Sn. Thevalues obtained from the slope of the primary and secondaryfragments as well as the experimental values are tabulated.The theoretical values of α and β are extracted for fragmentshaving Z N directly calculated from this model enabled the testingas well as verification of this assumption.The linear fits to the data points as obtained fromour model as well as experimental data can be used toextract the values of α and β which have been tabulated(Table 1). The closeness of the values extracted fromour model with those from the experiment establishesthe validity of our model. IV. DISCUSSIONS
The study of isospin dependent observables, morespecifically isoscaling is done by hybrid model. Thedynamical approach takes care of the initial stages ofthe reaction and the fragmentation of the excited systemis described by the statistical model. This hybrid modelis much economical but at the same time based onappropriate physical considerations at different stagesof the reaction. The introduction of isospin in themodel led us to study the observables dependent onisospin in order to check the accuracy of the model.Nice fits to the experimental data of isospin distributionfrom the two reactions confirms the validity of themodel. Isoscaling is also nicely displayed by the ratiosof yields from the two reactions and the straight linefits to the same have been used to extract the isoscalingcoefficients. One significance of using this hybrid modelis direct estimation of temperature of the reactionsbeing studied. This feature enabled us to confirm theimportant assumption of isoscaling equation, that is,temperature of both the reactions are very close. Thesatisfactory performance of the model motivates us touse it in order to probe experimental data of otherisospin dependent observables in future. This modelcan also be extended in future in order to study projec-tile fragmentation reactions in the higher energy domain.
V. APPENDIX: ISOSPIN DEPENDENTTHOMAS-FERMI MODEL
Consider a nucleus of Z protons and N neutrons. Thetotal energy (non-relativistic) of the system for mean fieldgiven in eq. 1 can be expressed as E = 3 h m (cid:20) π (cid:21) / ( Z ρ p ( r ) / d r + Z ρ n ( r ) / d r ) + a Z ρ ( r ) d r + bσ + 1 Z ρ σ +1 ( r ) d r + c Z ρ ( ~r ) ∇ r ρ ( r ) d r + C sym Z ( ρ n ( r ) − ρ p ( r )) d r + 14 πǫ Z Z ρ p ( ~r ) ρ p ( ~r ′ ) | ~r − ~r ′ | d rd r ′ (5)And the particle number conservation gives, Z ρ p ( r ) d r = Z Z ρ n ( r ) d r = N (6)By applying the variational method of energy minimiza-tion under the constraint of total proton and neutronconservation and assuming spherical symmetry, one canget the two Thomas-Fermi equations h m (cid:20) π (cid:21) ρ p ( r ) + n aρ ( r ) + bρ σ ( r ) + c ∇ r ρ ( r ) o − C sym n ρ p ( r ) − ρ n ( r ) o + 14 πǫ Z ρ p ( ~r ′ ) | ~r − ~r ′ | d r ′ − λ p = 0(7) h m (cid:20) π (cid:21) ρ n ( r ) + n aρ ( r ) + bρ σ ( r ) + c ∇ r ρ ( r ) o + C sym n ρ p ( r ) − ρ n ( r ) o − λ n = 0(8) where λ p and λ n are Lagrange undetermined multiplierfor proton and neutron conservation respectively. Tosolve these two coupled equations simultaneously, onecan put y p ( r ) = rρ p ( r ) and y n ( r ) = rρ n ( r ), so y p ( r ) and y n ( r ) vanishes both at r = 0 and r = ∞ i.e. above twocoupled equations become boundary value problem.Numerically one have to start from a guess proton andneutron density profile (for example, we have startedwith Myers density profile[50]) and guess value of λ p and λ n and by applying multidimensional Newton’s method(at different r values) for coupled equations 7 and 8 theground state neutron and proton density profile can beobtained. Then by using Monte-Carlo technique theinitial position and momenta of the proton and neu-tron test particles as well as ground state energy can beobtained from the calculated ground state density profile. [1] Bao-An Li and Wolf-Udo Schroder, Isospin Physics inHeavy-Ion Collisions at Intermediate Energies , Nova Sci-ence Pub. Inc. (2001).[2] S. Das Gupta, S. Mallik and G. Chaudhuri,
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