New signatures of phase transition from Statistical Models of Nuclear multifragmentation
NNew signatures of phase transition fromStatistical Models of Nuclear multifragmentation
G. Chaudhuri, S. Mallik, P. Das, S. Das Gupta
Abstract : The study of liquid-gas phase transition in heavy ion collisions has gener-ated a lot of interest amongst the nuclear physicists in the recent years. In heavy ioncollisions, there is no direct way of measuring the state variables like entropy, pres-sure, energy and hence unambiguous characterization of phase transition becomesdi๏ฌcult. This work proposes new signatures of phase transition that can be extractedfrom the observables which are easily accessible in experiments. It is observed thatthe temperature dependence of the ๏ฌrst order derivative of the order parameters innuclear liquid gas phase transition exhibit similar behavior as that of the variationof speci๏ฌc heat at constant volume ๐ถ ๐ฃ which is an established signature of ๏ฌrst or-der phase transition. This motivates us to propose these derivatives as con๏ฌrmatorysignals of liquid-gas phase transition. The measurement of these signals in easilyfeasible in most experiments as compared to the other signatures like speci๏ฌc heat,caloric curve or bimodality. Total multiplicity, size of largest cluster are some of theorder parameters which have been studied. Statistical Models based on canonicalensemble and lattice gas model has been used for the study. This temperature wherethe peak appears is designated to be the transition temperature and the e๏ฌect of cer-tain parameters on this has also been examined. The multiplicity derivative signatureproposed in this work has been further con๏ฌrmed by other theoretical models as wellas in experimental study. G. ChaudhuriVariable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, Indiae-mail: [email protected]
S.MallikVariable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, IndiaP.DasVariable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, IndiaS. Das GuptaPhysics Department, McGill University, Montrรฉal, Canada H3A 2T8 1 a r X i v : . [ nu c l - t h ] J a n G. Chaudhuri, S. Mallik, P. Das, S. Das Gupta
The phenomenon of liquid-gas phase transition occurring in heavy ion collisions atintermediate energies is a subject of contemporary interest [1, 2, 3, 4, 5, 6, 7, 8].The nature of nucleon-nucleon strong interaction potential, which is an attractiveone with a repulsive core is very similar to the Van der Waals potential[4] except forthe magnitude. This type of interaction explains the phenomenon of phase transitionin ordinary liquid and hence similar observation is very much expected in nuclearsystems also. In ordinary liquids, when it is heated , the temperature rises till theboiling point is reached after which it remains constant till the whole amount ofliquid is converted to gas. Similarly in order to observe phase transition in nuclearsystem, one has to pump energy to the system and only possible way is by means ofnucleus-nucleus collision. In very high energy heavy ion collision at high density andtemperature, hadronic matter transforms to the Quark Gluon Plasma (QGP) phase.At the intermediate energy regime, nuclear multifragmentation is the dominantmechanism which can be related to a liquid-gas kind of transition at sub-saturationnuclear density . Theoretical models of multifragmentation predict the existenceof phase transition in in๏ฌnite nuclear matter. Experimental signatures also indicatechange of state and this can be interpreted as ๏ฌnite size counterpart of the 1st orderphase transition in nuclear matter. Di๏ฌerent signatures of this transition have beenstudied extensively both theoretically as well as experimentally [4, 5, 7, 8, 9]. Thevariation of excitation energy and speci๏ฌc heat with temperature are two well studiedsignatures theoretically in order to detect the ๏ฌrst order phase transition[10, 11, 12]. T A m a x ( a ) ( b )( c ) ( d ) T E * / A T M S / A T Fig. 1
Variation of (a)Total Multiplicity M (b)excitation energy ๐ธ โ / ๐ด (MeV/A), (c) entropy pernucleon S/A (d) average size of the largest cluster ๐ด ๐๐๐ฅ with temperature T (MeV) for the fragmentsproduced in the fragmentation of an ideal one-component system of size A=500.itle Suppressed Due to Excessive Length 3 Phase transition is usually characterized by the speci๏ฌc behaviour of state vari-ables like pressure, density, energy, entropy etc [13, 14]. The order of phase transition,according to Ehrenfest, is determined by the lowest order derivative of free energythat shows a discontinuity. In heavy ion collisions there is no direct way of accessingthese state variables and hence unambiguous detection of phase transition becomesdi๏ฌcult. The present work is motivated by this limitation and aims at looking forsignatures of phase transition that can be extracted from the observables which areeasily accessible in experiments. Ideally phase transition exists in the thermody-namic limit and for a ๏ฌrst order one, entropy should have a ๏ฌnite discontinuity andspeci๏ฌc heat a divergence at the phase transition temperature. In ๏ฌnite nuclei thediscontinuity or divergence is replaced by sudden jump or maxima. The variationof total multiplicity or size of the largest cluster ( ๐ ๐๐๐ฅ ) with temperature is verymuch similar to that of entropy or excitation energy(caloric curve) with temperatureand this can be seen from Fig. 1 Hence the ๏ฌrst order derivative of these observ-ables with temperature is expected to behave in a similar way as those of entropy orenergy. This observation led to the investigation of the nature of the derivatives ofthese multifragmentation observables which can be easily measured in experiments.Encouraging results have been obtained from this study and it has been observedthat ๏ฌrst order derivative of the order parameters related to the total multiplicity,largest cluster size (produced in heavy ion collisions) exhibit similar behavior as thatof the variation of speci๏ฌc heat at constant volume ๐ถ ๐ฃ which is an established sig-nature of ๏ฌrst order phase transition. This motivates us to propose these derivativesof total multiplicity, largest cluster size[15, 16, 17, 18] as con๏ฌrmatory signals ofliquid-gas phase transition. Another observable we have proposed here is related tothe di๏ฌerence (normalized) between the sizes of the ๏ฌrst and second largest clusterswhich also serve as order parameter for phase transition in nuclear fragmentationand has been studied experimentally [19, ? ] too. The derivatives of all these peakat the same temperature as speci๏ฌc heat and hence can con๏ฌrm the phase transitionin the fragmentation process. The measurement of these signals in easily feasiblein most experiments as compared to the other signatures like speci๏ฌc heat, caloriccurve or bimodality. This temperature where the peak appears is designated to bethe transition temperature and the e๏ฌect of certain parameters on this has also beenexamined.We have used mainly used statistical model based on canonical ensemble whichis better known by the Canonical Thermodynamical Model (CTM)[21] in order tostudy the fragmentation of nuclei. In such models of nuclear disassembly it is as-sumed that because of multiple nucleon-nucleon collisions a statistical equilibriumis reached and disintegration pattern is solely decided by the statistical weights in theavailable phase space. The temperature rises and the system expands from normaldensity and composites are formed on the way to disassembly as a result of density๏ฌuctuation. As the system reaches between three to six times the normal volume,the interactions between composites become unimportant (except for the long rangeCoulomb interaction) and one can do a statistical equilibrium calculation to obtainthe yields of composites at a volume called the freeze-out volume. This model can beimplemented in di๏ฌerent statistical ensembles(microcanonical, canonical and grand G. Chaudhuri, S. Mallik, P. Das, S. Das Gupta canonical [1, 3, 21]. In our calculation, the partitioning into available channels issolved in the canonical ensemble where the number of particles in the nuclear sys-tem is ๏ฌnite (as it would be in experiments).The study is done for di๏ฌerent nuclearsizes, freeze out volumes and temperatures. Since Coloumb interaction is long rangeand suppresses the signatures of phase transition, hence we have switched o๏ฌ theColoumb force in some part of our study in order to have better idea of the signatures.In such cases we have considered symmetric nuclear matter and no distinction ismade between neutron and proton. In addition to CTM we have also used the LatticeGas Model [22] recently developed in our group in order to study the multiplicityderivative signal. This model uses geometry similar to the percolation model butis much more elaborate with insertion of an Hamiltonian. Both the thermodynamicand the lattice gas model con๏ฌrmed the multiplicity derivative as signature of ๏ฌrstorder phase transition in nuclear multifragmentation.We have given brief description of the models used in our calculation in the nextsection. After that the results displaying the new signatures are proposed in details.The last section gives the summary of our work.
In this section we describe brie๏ฌy the canonical thermodynamical model which isbrie๏ฌy designated as CTM. We assume that a system with ๐ด nucleons and ๐ protons at temperature ๐ , has expanded to a higher than normal volume and thepartitioning into di๏ฌerent composites can be calculated according to the rules ofequilibrium statistical mechanics. In a canonical model, the partitioning is donesuch that all partitions have the correct ๐ด , ๐ (equivalently ๐ , ๐ ). Details of theimplementation of the canonical model can be found elsewhere [21]; here we givethe essentials necessary to follow the present work.The canonical partition function is given by ๐ ๐ ,๐ = โ๏ธ (cid:214) ๐ ๐ ๐ผ,๐ฝ
๐ผ ,๐ฝ ๐ ๐ผ ,๐ฝ ! (1)Here the sum is over all possible channels of break-up (the number of such channelsis enormous) which satisfy ๐ = (cid:205) ๐ผ ร ๐ ๐ผ ,๐ฝ and ๐ = (cid:205) ๐ฝ ร ๐ ๐ผ ,๐ฝ ; ๐ ๐ผ ,๐ฝ is the partitionfunction of one composite with neutron number ๐ผ and proton number ๐ฝ respectivelyand ๐ ๐ผ ,๐ฝ is the number of this composite in the given channel. The one-body partitionfunction ๐ ๐ผ ,๐ฝ is a product of two parts: one arising from the translational motion ofthe composite and another from the intrinsic partition function of the composite: ๐ ๐ผ ,๐ฝ = ๐ ๐ โ ( ๐๐๐ ) / ๐ด / ร ๐ง ๐ผ ,๐ฝ ( ๐๐๐ก ) (2) itle Suppressed Due to Excessive Length 5 Here ๐ด = ๐ผ + ๐ฝ is the mass number of the composite and ๐ ๐ is the volume availablefor translational motion; ๐ ๐ will be less than ๐ , the volume to which the systemhas expanded at break up. We use ๐ ๐ = ๐ โ ๐ , where ๐ is the normal volume ofnucleus with ๐ protons and ๐ neutrons. In this calculation we have used a fairlytypical value ๐ = ๐ .The probability of a given channel ๐ ( (cid:174) ๐ ๐ผ ,๐ฝ ) โก ๐ ( ๐ , , ๐ , , ๐ , ......๐ ๐ผ ,๐ฝ ....... ) isgiven by ๐ ( (cid:174) ๐ ๐ผ ,๐ฝ ) = ๐ ๐ ,๐ (cid:214) ๐ ๐ ๐ผ,๐ฝ
๐ผ ,๐ฝ ๐ ๐ผ ,๐ฝ ! (3)The average number of composites with ๐ผ neutrons and ๐ฝ protons is seen easily fromthe above equation to be (cid:104) ๐ ๐ผ ,๐ฝ (cid:105) = ๐ ๐ผ ,๐ฝ ๐ ๐ โ ๐ผ ,๐ โ ๐ฝ ๐ ๐ ,๐ (4)The constraints ๐ = (cid:205) ๐ผ ร ๐ ๐ผ ,๐ฝ and ๐ = (cid:205) ๐ฝ ร ๐ ๐ผ ,๐ฝ can be used to obtain di๏ฌerentlooking but equivalent recursion relations for partition functions ๐ ๐ ,๐ = ๐ โ๏ธ ๐ผ ,๐ฝ ๐ผ๐ ๐ผ ,๐ฝ ๐ ๐ โ ๐ผ ,๐ โ ๐ฝ (5)These recursion relations allow one to calculate ๐ ๐ ,๐ We list now the properties of the composites used in this work. The proton andthe neutron are fundamental building blocks thus ๐ง , ( ๐๐๐ก ) = ๐ง , ( ๐๐๐ก ) = He and He we use ๐ง ๐ผ ,๐ฝ ( ๐๐๐ก ) = ( ๐ ๐ผ ,๐ฝ + ) exp (โ ๐ฝ๐ธ ๐ผ ,๐ฝ ( ๐๐ )) where ๐ฝ = / ๐, ๐ธ
๐ผ ,๐ฝ ( ๐๐ ) is the groundstate energy of the composite and ( ๐ ๐ผ ,๐ฝ + ) is the experimental spin degeneracyof the ground state. Excited states for these very low mass nuclei are not included.For mass number ๐ด = ๐ด = ๐ผ + ๐ฝ ) ๐ง ๐ผ ,๐ฝ ( ๐๐๐ก ) = exp 1 ๐ [ ๐ ๐ด โ ๐ ( ๐ ) ๐ด / โ ๐ ๐ฝ ๐ด / โ ๐ถ ๐ ( ๐ผ โ ๐ฝ ) ๐ด + ๐ ๐ด๐ ] (6)The derivation of this equation is given in several places [3, 21] so we will not repeatthe arguments here. The expression includes the volume energy, the temperaturedependent surface energy, the Coulomb energy and the symmetry energy. The term ๐ ๐ด๐ represents contribution from excited states since the composites are at a non-zerotemperature.We also have to state which nuclei are included in computing ๐ ๐ ,๐ (eq.(17)).For ๐ผ, ๐ฝ , (the neutron and the proton number) we include a ridge along the line ofstability. The liquid-drop formula above also gives neutron and proton drip lines andthe results shown here include all nuclei within the boundaries.
G. Chaudhuri, S. Mallik, P. Das, S. Das Gupta
The long range Coulomb interaction between di๏ฌerent composites can be includedin an approximation called the Wigner-Seitz approximation. We incorporate thisfollowing the scheme set up in [3].
The statistical multifragmentation model described above calculates the properties ofthe collision averaged system that can be approximated by an equilibrium ensemble.Ideally, one would like to measure the properties of excited primary fragments afteremission in order to extract information about the collisions and compare directlywith the equilibrium predictions of the model. However, the time scale of a nuclearreaction(10 โ ๐ ) is much shorter than the time scale for particle detection (10 โ ๐ ).Before reaching the detectors, most fragments decay to stable isotopes in theirground states. Thus before any model simulations can be compared to experimentaldata, it is indispensable to have a model that simulates sequential decays. A MonteCarlo technique is employed to follow all decay chains until the resulting products areunable to undergo further decay. For the purposes of the sequential decay calculationsthe excited primary fragments generated by the statistical model calculations aretaken as the compound nucleus input to the evaporation code. Hence, every primaryfragment is decayed as a separate event.We consider the deexcitation of a primary fragment of mass ๐ด , charge ๐ andtemperature ๐ . The successive particle emission from the hot primary fragmentsis assumed to be the basic deexcitation mechanism. For each event of the primarybreakup simulation, the entire chain of evaporation and secondary breakup eventsis Monte Carlo simulated. The standard Weisskopf evaporation scheme is used totake into account evaporation of nucleons, ๐ , ๐ก , ๐ป๐ and ๐ผ . The decays of particlestable excited states via gamma rays were also taken into account for the sequentialdecay process and for the calculation of the ๏ฌnal ground state yields. We have alsoconsidered ๏ฌssion as a deexcitation channel though for the nuclei of mass < ฮ ๐ at which a particle of type ๐ is emitted. The di๏ฌerent equations for calculation of particle, gamma and ๏ฌssionwidths is given in details in [23] and we will skip them here. Once the emissionwidths are known, it is required to establish the emission algorithm which decideswhether a particle is being emitted from the compound nucleus. This is done [24]by ๏ฌrst calculating the ratio ๐ฅ = ๐ / ๐ ๐ก๐๐ก where ๐ ๐ก๐๐ก = โ / ฮ ๐ก๐๐ก , ฮ ๐ก๐๐ก = (cid:205) ๐ ฮ ๐ and ๐ = ๐, ๐, ๐, ๐ก, ๐ป๐ , ๐ผ, ๐พ or ๏ฌssion and then performing Monte-Carlo sampling froma uniformly distributed set of random numbers. In the case that a particle is emitted,the type of the emitted particle is next decided by a Monte Carlo selection with theweights ฮ ๐ / ฮ ๐ก๐๐ก (partial widths). The energy of the emitted particle is then obtainedby another Monte Carlo sampling of its energy spectrum. The energy, mass andcharge of the nucleus is adjusted after each emission. This procedure is followed foreach of the primary fragment produced at a ๏ฌxed temperature and then repeated over itle Suppressed Due to Excessive Length 7 a large ensemble and the observables are calculated from the ensemble averages.The number and type of particles emitted and the ๏ฌnal decay product in each eventis registered and are taken into account properly keeping in mind the overall chargeand baryon number conservation. The Lattice Gas Model is considerably more complicated than the percolationmodel[22] but expositions of the model exist [8, 25, 26, 8] and we refer to [26]for details. Let ๐ด = ๐ + ๐ be the number of nucleons in the system that dissociates.We consider ๐ท cubic boxes where each cubic box has volume ( . / . ) ๐ ๐ . ๐ท is larger than ๐ด (they have the same value in bond percolation model). Here ๐ท / ๐ด = ๐ ๐ / ๐ where ๐ is the normal volume of a nucleus with ๐ด nucleons and ๐ ๐ is the freeze-out volume where partitioning of nucleons into clusters is computed.For nuclear forces one adopts nearest neighbor interactions. Following normal prac-tice, we use neutron-proton interactions ๐ฃ ๐๐ =-5.33 MeV and set ๐ฃ ๐๐ = ๐ฃ ๐ ๐ =0.0.Coulomb interaction between protons is included. Each cube can contain 1 or 0nucleon. There is a very large number of con๏ฌgurations that are possible (a con๏ฌgu-ration designates which cubes are occupied by neutrons, which by protons and whichare empty; we sometimes call a con๏ฌguration an event). Each con๏ฌguration has anenergy. If a temperature is speci๏ฌed, the occupation probability of each con๏ฌgura-tion is proportional to its energy: P โ exp(-E/T). This is achieved by Monte-Carlosampling using Metropolis algorithm.Calculation of clusters need further work. Once an event is chosen we ascribe toeach nucleon a momentum. Momentum of each nucleon is picked by Monte-Carlosampling of a Maxwell-Boltzmann distribution for the prescribed temperature T.Two neighboring nucleons are part of the same cluster if (cid:174) ๐ ๐ / ๐ + ๐ < ๐ is ๐ฃ ๐๐ or ๐ฃ ๐๐ or ๐ฃ ๐ ๐ . Here (cid:174) ๐ ๐ is the relative momentum of the two nucleons and ๐ isthe reduced mass. If nucleon ๐ is bound with nucleon ๐ and ๐ with ๐ then ๐, ๐, ๐ arepart of the same cluster. At each temperature we calculate 50,000 events to obtainaverage energy < ๐ธ > and average multiplicity ๐ ๐ (where ๐ is the mass numberof the cluster) of all clusters. A cluster with 1 nucleon is a monomer, one with 2nucleons is a dimer and so on. The total multiplicity is ๐ = (cid:205) ๐ ๐ and (cid:205) ๐๐ ๐ = ๐ด where ๐ด = ๐ + ๐ is the mass number of the dissociating system. The variation of total multiplicity with temperature for fragmenting systems ascalculated by CTM is very similar to that of entropy. This motivated us to look forthe derivative of multiplicity which is expected to behave similarly as the derivativeof entropy w.r.t temperature which is nothing but the speci๏ฌc heat at constant volume
G. Chaudhuri, S. Mallik, P. Das, S. Das Gupta E * / A ( M e V ) * / A ( M e V ) T ( M e V ) ( a )Z = 8 2 N = 1 2 6
Z = 8 2 N = 1 2 6 ( b )
T ( M e V )
T ( M e V ) E * / A ( M e V ) ( c )A = 2 0 8 * / A ( M e V ) T ( M e V ) dM/dT (MeV-1)M dM/dT (MeV-1) ( d )A = 2 0 8 M Fig. 2
Variation of multiplicity M ((a) and (c)) and dM/dT ((b) and (d)) with temperature (bottomx axes) and excitation per nucleon (top x axes) from the CTM calculation for fragmenting systemshaving Z =82 and N =126 ((a) and (b)). (c) and (d) represent the same but for a hypotheticalsystem of one kind of particle with no Coulomb interaction but the same mass number (A =208). ๐ธ โ = ๐ธ โ ๐ธ , where ๐ธ is the ground-state energy of the dissociating system in the liquid dropmodel whose parameters are given in Ref. [21] ๐ถ ๐ . Unlike the entropy, one can measure the total multiplicity ๐ = (cid:205) ๐ ๐ ( ๐ beingthe mass number of the composites) with 4 ๐ detectors in the laboratory.In CTMthe derivative of ๐ with ๐ as a function of ๐ is seen to have a maximum. Fig. 2(left panel) shows the total multiplicity for fragmenting system having proton number( ๐ )=82 and neutron number ( ๐ )=126 and and its derivative ๐๐ / ๐๐ (the right panel).Results for both real nuclei and the one for one kind of particles have been displayedin order to emphasize the e๏ฌects of Coulomb interaction. The rise and the peak aremuch sharper in absence of Coulomb interaction clearly indicating the role of thelong range interaction in suppressing the signatures of phase transition. The featuresbecome less sharp as in ๐ =28 and ๐ =30, as the system size decreases (Fig. 3),In the next two ๏ฌgures Fig. 4 and Fig. 5 we compare dM/dT and ๐ถ ๐ for the twosame systems as used in 2 and ?? . We also consider the situation where the Coulombis switched o๏ฌ. The peak in ๐๐ / ๐๐ coincides with the maximum of speci๏ฌc heat atconstant volume ๐ถ ๐ฃ as a function of temperature for all the cases. Its an establishedfact that speci๏ฌc heat at constant volume peaks at the transition temperature and thisis a signature of 1st order phase transition. Hence based on our results as presentedin 4 and 5, we conclude that dM/dT can be a signature of phase transition and theadvantage is it gives an exact value of the transition temperature where the maximumof dM/dT occurs. Next we calculate the entropy since its well known that it showsa sharp rise near the transition temperature ., We have compared the temperature itle Suppressed Due to Excessive Length 9 E * / A ( M e V ) T ( M e V )T ( M e V ) T ( M e V ) E * / A ( M e V ) E * / A ( M e V )E * / A ( M e V ) T ( M e V ) ( a )
Z = 2 8 N = 3 0 dM/dT (MeV-1)
Z = 2 8 N = 3 0 ( b ) ( c )
A = 5 8 dM/dT (MeV-1)MM ( d )
A = 5 8
Fig. 3
Same as Fig. 2 but the fragmenting systems are Z =28 and N =30 ((a) and (b)) and A =58((c) and (d)).
Fig. 4
Variation of dM/dT (red solid lines) and ๐ถ ๐ฃ (green dashed lines) with temperature fromCTM for fragmenting systems having Z =82 and N =126 (a) and for hypothetical systems of onekind of particle with no Coulomb interaction of mass number A =208 (b). To draw dM/dT and ๐ถ ๐ฃ in the same scale, Cv is normalized by a factor of 1/50. [Reprinted with permission from S Mallik,G. Chaudhuri, P. Das, S. Das Gupta, Phys. Rev. C , 061601 , 2017 (R)] Copyright (2020) fromthe American Physical Society variation of dM/dT and the entropy for the fragmenting system Z=82, N=126, andalso for an ideal condition, neglecting the Coulomb interaction. This is displayed inin the next plot 6. It is evident, that for both the cases, the entropy changes rapidly inthat regime of temperature scale where dM/dT exhibits a maximum.. The presenceof Coulomb interaction in a real system, only, smears the rise of entropy and the peak in dM/dT. This behaviour, further, establishes that dM/dT is indeed a signatureof phase transition. Z = 2 8 N = 3 0 ( a )
T ( M e V )( b )
A = 5 8
T ( M e V )
Fig. 5
Same as in Fig. 4, but the fragmenting systems are Z =28 and N =30 (a) and A =58 (b).[Reprinted with permission from S Mallik, G. Chaudhuri, P. Das, S. Das Gupta, Phys. Rev. C ,061601 , 2017 (R)] Copyright (2020) from the American Physical Society The multiplicity of the intermediate mass fragments ( ๐ ๐ผ ๐ ๐น ) in heavy ion col-lisions strongly con๏ฌrms the process of multifragmentation [3]. It is an importantobservable of multifragmentation, which is measured in the experiment, sometimes,instead of the total multiplicity M. Therefore, we wanted to perform a similar teston the derivative of ๐ ๐ผ ๐ ๐น . We have plotted the variation of ๐ ๐ผ ๐ ๐น and its temper-ature derivative with temperature for the system Z=82, N=126 in 7, and compared ๐๐ ๐ผ ๐ ๐น /dT with ๐ถ ๐ . ๐ ๐ผ ๐ ๐น and ๐๐ ๐ผ ๐ ๐น /dT display a similar behaviour as that ofthe total multiplicity and its derivative except for the fact that the peak position of itsderivative do not coincide with that of ๐ถ ๐ . This is expected because the calculationof ๐ถ ๐ involves all the fragments irrespective of their mass or charge, but in ๐ ๐ผ ๐ ๐น ,only selected fragments are included.Last but not the least we would like to study the e๏ฌect of secondary decay on theexcited fragments formed after multifragmentation. In a heavy ion collision, when anucleus breaks up through the process of nuclear multifragmentation, the resultingcomposites are called primary fragments. The primary fragments are excited ingeneral, and lose excitation through sequential two-body decay, and thus change thetotal multiplicity. The ๏ฌnal cold fragments, called secondary fragments, are detectedin the laboratory.The fragments that we are dealing with in our study (using CTM), are primaryfragments. The secondary decay may a๏ฌect the total multiplicity in such a way thatmight change the behaviour of multiplicity discussed above. As we are interested inthe experimental signature, we investigate the e๏ฌect of the secondary decay in ourcalculation , and do the same study with the multiplicity of the secondary fragments.We have plotted the multiplicities of the primary and the secondary fragments andtheir derivatives in 8. It is apparent that the e๏ฌect of secondary decay does not alterour previous observation. Moreover, it enhances the signals, the total multiplicitychanges more rapidly, and the peak in ๐๐ / ๐๐ is sharper in case of the secondary itle Suppressed Due to Excessive Length 11 Z = 8 2 N = 1 2 6 ( a ) 2 4 6 8 1 002 04 06 0 ( b )
T ( M e V )
A = 2 0 8
Fig. 6
Variation of entropy (blue dashed lines) and dM/dT (red solid lines) with temperaturefrom CTM for fragmenting systems having Z =82and N =126 (a) and for hypothetical system ofone kind of particle with no Coulomb interaction of mass number A =208 (b). To draw S anddM/dT in the same scale, S is normalized by a factor of 1/20 for Z =82 and N =126 system and1/50 for hypothetical system of one kind of particle. [Reprinted with permission from S Mallik, G.Chaudhuri, P. Das, S. Das Gupta, Phys. Rev. C , 061601 , 2017 (R)] Copyright (2020) from theAmerican Physical Society Z = 8 2 N = 1 2 6 ( a )
Z = 8 2 N = 1 2 8 ( b )
T ( M e V )
Fig. 7
Variation of intermediate-mass fragment(IMF) multiplicity ๐ ๐ผ ๐๐น (a) and ๏ฌrst-orderderivative of IMF multiplicity ๐๐ ๐ผ ๐๐น / ๐๐ (b) with temperature from CTM calculation for frag-menting systems having Z =82 and N =126. Variation of ๐ถ ๐ฃ with temperature (T) is shown bygreen dashed line in (b). To draw d ๐ ๐ผ ๐๐น / ๐๐ and ๐ถ ๐ฃ in the same scale, ๐ถ ๐ฃ is normalized by afactor of 1/100. [Reprinted with permission from S Mallik, G. Chaudhuri, P. Das, S. Das Gupta,Phys. Rev. C , 061601 , 2017 (R)] Copyright (2020) from the American Physical Society fragments. Thus the maxima of multiplicity derivative can be extracted successfullythrough experiments with an unaltered transition temperature.In order to further test multiplicity derivative as a possible signature for 1storder phase transition, we have carried out the investigation using the Lattice-gasModel[22]. This is shown in 9 We have plotted ๐ and its derivative against thetemperature ๐ in the left panel. ๐ shows a rise and the derivative shows a peakas expected. Plots of ๐๐ / ๐๐ and ๐ < ๐ธ > / ๐๐ are shown in 9(right panel). ๐ถ ๐ฃ goes through a maximum at some temperature which is a hallmark of ๏ฌrst orderphase transition and this occurs at the same temperature where ๐๐ / ๐๐ maximises.This is remarkably similar to results from CTM corroborating the evidence that theappearance of a maximum in ๐๐ / ๐๐ is indicative of a ๏ฌrst order phase transition.Our dM/dT (MeV-1)M Z = 2 8 N = 3 0 ( a ) ( b )Z = 2 8 N = 3 0
T ( M e V )
Fig. 8
E๏ฌect of secondary decay on M (a) and dM/dT (b) for fragmenting systems having Z=28 and N =30. Red solid lines show the results after the multifragmentation stage (calculatedfrom CTM), whereas blue dashed lines represent the results after secondary decay of the excitedfragments. [Reprinted with permission from S Mallik, G. Chaudhuri, P. Das, S. Das Gupta, Phys.Rev. C , 061601 , 2017 (R)] Copyright (2020) from the American Physical Society Fig. 9
Variation of M (a)and dM/dT (b) (red solid lines) and ๐ถ ๐ฃ with temperature from lattice gasmodel at ๐ท = ๐ =82 and ๐ =126. (c) d ๐ /d ๐ (red solidlines) and ๐ถ ๐ฃ (green dashed lines) with T ; to draw them in the same scale, ๐ถ ๐ฃ is normalised by afactor of 1 /
10; d ๐ /d ๐ is unit of MeV โ . [Reprinted with permission from S. Das Gupta,S. Mallikand G. Chaudhuri Phys. Rev. C , 044605, 2018] Copyright (2020) from the American PhysicalSociety proposed signal of multiplicity derivative ๐๐ / ๐๐ was tested and veri๏ฌed in di๏ฌerentstatistical and dynamical models like the statistical multifragmentation model (SMM)[28, 29], Quantum Molecular Dynamics (QMD) model [30] and Nuclear statisticalEquilibrium (NSE) model[31]. Our theoretical proposition of this signal got furthersupport when it was experimentally veri๏ฌed recently and tested using three di๏ฌerentreactions ๐ด๐ + ๐๐ , ๐ด๐ + ๐ด๐ and ๐ด๐ + ๐๐ at 47 Mev/n [32]. itle Suppressed Due to Excessive Length 13 ( b ) a2 M S T ( M e V ) ( g ) dM/dT ( h ) CV T ( M e V ) ( a ) ( e ) amax ( f ) ( d ) ( c ) damax/dT a2 Fig. 10
Variation of (a) a ๐๐๐ฅ , (b) a , (c) M, (d) S, (e) - ๐ a ๐๐๐ฅ / ๐๐ , (f) - ๐ a / ๐๐ , (g) dM/dT and(h) ๐ถ ๐ฃ with temperature for fragmenting system of mass A=200. V f = 6 V ( a ) ( b ) T ( M e V ) A = 2 0 0 T ( M e V )
Fig. 11
Variation of ๐ a ๐๐๐ฅ / ๐๐ with temperature (a) at constant freeze-out volume ๐ ๐ = ๐ but for three fragmenting system of mass 50 (blue dotted line), 100 (red dashed line) and 200(black solid line) and (b) for same fragmenting system of mass 200 but at three constant freeze-outvolumes ๐ ๐ = ๐ (magenta dotted line), ๐ ๐ = ๐ (black solid line) anf ๐ ๐ = ๐ (green dashedline). The average size of the largest cluster (cid:104) ๐ด ๐๐๐ฅ (cid:105) formed in the fragmentation ofthe excited nuclei acts as an order parameter for 1st order phase transition. Thevariable ๐ which is a measure of the di๏ฌerence between the average size of the1st ( (cid:104) ๐ด ๐๐๐ฅ (cid:105) ) and the 2nd ( (cid:104) ๐ด ๐๐๐ฅ โ (cid:105) ) largest cluster size divided by the sum of TP (MeV) ( a )V f = 6 V A = 2 0 0( b ) TP (MeV) V f / V A Fig. 12
Dependence of the peak position of - ๐ a ๐๐๐ฅ / ๐๐ , - ๐ a / ๐๐ , dM/dT and ๐ถ ๐ฃ on fragmentingsystem size (a) and freeze-out volume (b). these two ( ๐ = (cid:104) ๐ด ๐๐๐ฅ (cid:105) โ (cid:104) ๐ด ๐๐๐ฅ โ (cid:105)(cid:104) ๐ด ๐๐๐ฅ (cid:105) + (cid:104) ๐ด ๐๐๐ฅ โ (cid:105) ) also has similar behaviour as that of (cid:104) ๐ด ๐๐๐ฅ (cid:105) . Sothis observable which is measured in some experiments can also act as an orderparameter. The analytical expressions leading to the calculation of the average sizeof 1st and 2nd largest cluster can be found in [33]. Now, we will concentrate on theseobservables in order to study their variation with temperature. We consider an idealsystem of A=200 identical nucleons with no Coulomb force acting between themin order have better idea of these proposed signatures. Left panels of 10((a) to (d))display the variations of the four variables, the normalised size of the average largestcluster a ๐๐๐ฅ ( a ๐๐๐ฅ = (cid:104) ๐ด ๐๐๐ฅ (cid:105) ๐ด ) , a , total multiplicity M and entropy per particle(S/A) with temperature. a ๐๐๐ฅ and a are almost constant and assume a value โ a ๐๐๐ฅ and a displaysimilar behaviour as that of the multiplicity and the entropy; the sudden jump (orfall) of these four variables occur almost at the same temperature around 6 MeV. Thissimilarity motivates us to investigate the behaviour of the derivatives of a ๐๐๐ฅ and a .In the right panel of f ?? , temperature derivatives of all the four quantities are plottedas function of temperature. In the right bottom panel 10(h), we have plotted ๐ถ ๐ ,which is related to the temperature derivative of the entropy (S). The derivatives of a ๐๐๐ฅ and a exhibit maxima just like total multiplicity and speci๏ฌc heat, and almostat the same temperature, which we call the transition temperature. This establishesthese two variables as signatures of phase transition. This signature is much easierto access both theoretically and experimentally as compared to the bimodality in theprobability distribution of the largest cluster. The later has been used so far in orderto detect the existence of phase transition in nuclear multifragmentation but to detecttwo peaks(bimodality) of equal height in a distribution at a particular temperature itle Suppressed Due to Excessive Length 15 (or excitation energy) is far more a di๏ฌcult job than to simply calculate the derivativein its size with temperature or excitation energy. We strongly believe that this newproposed signature related to the largest cluster size will de๏ฌnitely provide a greatimpetus to the study of liquid gas phase transition in heavy ion collisions.Next, we have examined how the transition temperature varies with the sourcesize and the freeze-out volume. We have plotted the variation of d a ๐๐๐ฅ /dT with Tfor three di๏ฌerent fragmenting systems of size A=50, 100, 200 at a ๏ฌxed freeze-outvolume ๐ ๐ = ๐ in 11(a), and the same for three freeze-out volume ๐ ๐ =3 ๐ , 4 ๐ ,8 ๐ with ๏ฌxed source A=200 in 11(b). We see that the peaks are sharper for themore massive source and the higher freeze-out volume. The position of the peak isobserved to shift to the higher temperature region for the bigger source size, andthe lower temperature side for the greater freeze-out volume. This implies that thesmaller system fragments more easily at a lower transition temperature as comparedto its bigger counterparts. The peak also becomes sharper for bigger sources whichonce again proves that phase transition signals are enhanced in larger systems. Forfreeze-out volume, the result that we have obtained is expected, since higher freeze-out volume (lower density) will favour the disintegration of the nucleus, resulting inlower transition temperature.At the end, we have plotted the transition temperatures as a function of system sizeat ๏ฌxed freeze-out volume (left panel (a)), and as a function of freeze-out volume fora ๏ฌxed system (right panel (b)) in f ?? . In each panel, four di๏ฌerent sets of transitiontemperatures are plotted. Those sets are obtained from the position of the maximain ๐ a ๐๐๐ฅ / ๐๐ , ๐ a / ๐๐ , dM/dT and ๐ถ ๐ . The transition temperatures obtained fromall the four observables give consistent results. Small di๏ฌerences between them canbe attributed to the ๏ฌniteness of the fragmenting system. This work introduces some new signatures of nuclear liquid gas phase transitionwhich can be measured easily and more accurately in experiments. The observableschosen were the total multiplicity, average largest cluster size ๐ ๐๐๐ฅ and a normalisedvariable ๐ which assume distinctly di๏ฌerent values in liquid and the gas phases thusserving as order parameters of the transition. The variation of these observaableswith temperature is very much similar to that of entropy or excitation energy andhence the temperature derivatives behave as speci๏ฌc heat at constant volume ( ๐ถ ๐ฃ )which is an established signature of phase transition. Transition temperature canbe identi๏ฌed from the position of the maxima of these derivatives analogous tothat of ๐ถ ๐ฃ . Multiplicity derivative ( ๐๐ / ๐๐ ) serves as a robust signal with very goodperformance even in presence of Coloumb interaction which is long range and therebysuppresses signatures of phase transition. This signature not only persists but getsenhanced after secondary decay of the primary hot fragments and thus can be easilydetected in experiments which measures total multiplicity. This signal was proposedusing the canonical thermodynamical model and was later con๏ฌrmed by us using the lattice -gas model. This was very recently veri๏ฌed in other theoretical modelsas well as in experiment. The other two observables concerning the derivatives ofthe largest and second largest clusters also peak at the same temperature as that of ๐ถ ๐ฃ and can be considered as signals of 1st order phase transition in one componentsystem switching o๏ฌ the Coloumb interaction. It is sometimes easier to measure thesize of largest cluster than to count the total multiplicity covering all the fragmentsproduced in the experiment. The e๏ฌect of source size and freeze-out volume on thetransition temperature is also studied using the one-component model. The extensionof this study for real nuclei will be a part of our future work. References
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