Recent progress of nuclear liquid gas phase transition

Vol 15 No. 1NUCLEAR SCIENCE AND TECHNIQUESFebruary 2004Recent progress of nuclear liquid gas phase transitionMA Yu-Gang, SHEN Wen-Qing(Shanghui institute of Applied Physics, the Chinese Academy of Sciences. Shanghai 201800Abstract Recent progress on nuclear liquid gas phase transition( LGPT) has been reviewed, especially for thesignals of LGPT in heavy ion collisions. These signals include the power-law charge distribution, cluster emissionrate, nuclear Zipf law. bimodality, the largest fluctuation of the fragments, A-scaling, caloric curve, phase coexis-tence diagram, critical temperature, critical exponent analysis, negative specific heat capacity and spinodal instabilityetc. The systematic works of the authors on experimental and theoretical LGPT are also introducedKeywords Nuclear liquid gas phase transition, Fragment yield distribution, Fisher droplet model, Zipf-type lawFluctuation observables, Caloric curve, Phase coexistence diagram, Critical exponent, Spinodal instabilityCLC numbers 0414. 13, 0571.61 Introductioncritical exponents in the charge or mass distribution ofe multifragmentation system' can be explained asPhase transition and critical phenomenon is an an evidence of phase transition. More recently, theextensively debatable subject in the natural sciences. negative microcanonical heat capacity was experiRecently, the same concept was introduced intomentally observed in nuclear fragmentation"whichastronomical objectsand the microscopic systems, relates to the liquid-gas phase transition, and insuch as in atomic cluster 2) and nuclei, of which the atomic cluster!2 which relates to solid to liquid phasenuclei, as a microscopic finite-size system,areover, some evidencetracting more and more nuclear experimentalists to of spinodal decomposition in nuclear multifragmenta-search for the liquid-gas phase transition ( LGPT)and tion was recently obtained experimentally, 4 whichinvestigate its behavior. To date, various experimental shows the presence of liquid-gas phase coexistenceevidences have cumulated which seem to be related to region and gives a strong argument in favor ofthe nuclear phase transition. For instance, violent existence of first-order liquid-gas phase transition inheavy-ion collisions break the nuclei into several in- finite nuclear systems. A-scaling of the largest fragmass fragments(IMF), which can be ment was also investigated recently and it shows to beviewed as a critical phenomenon as observed in fluid, a good probe to detect the phase change. The nuatomic and other systems. It prompts a possible sig- clear Zipf's law and multiplicity information entropynature on the liquid-gas phase transition in the nuclear ()were defined and proposed to diagnose the onsesystem. The sudden opening of the nuclear multifrag- of liquid-gas phase transition. 6. 7Phase coexistencementation and vaporization I channels can be inter- diagram was also constructed based on EoS data. 8. 191preted as the signature of the boundaries of phaseMeanwhile, several theoretical models have beenmixture. In addition, the plateau of the nuclear caloric developed to treat such a phase transition in the nucurve'in a certain excitation energy range gives a clear disassembly, e.g. percolation model, lattice gaspossible indication of a first-order phase transition 6.7 model, statistical multifragmentation model and mo-as predicted in the framework of statistical equilib- lecular dynamics model etc (e.g. see the recent reviewrium models. 81 On the other hand. the extraction of articles 20-22 and references therein ).In this article,weSupported by the National Natural Science Foundation of China for theTV凵中国煤化工9310NmmdNatural Science Foundation of China(No. 19705012), the Sciencedation ofangha (No97QA14038)and the Major State Basic Research Development ProgrCNMHReceived date: 2003-12-24MA Yu-Gang et al Recent progress of nuclear liquid gas phase transitionwill introduce the recent progress on nuclear LGPT in06experimental observation and theoretical treatment.The paper is organized as follows. Section 2 in-oduces the selected models for investigating the nuclear liquid gas phase transition, which include themean field theory, percolation model, statistical multifragmentation model, lattice gas model (LGM)andclassical molecular dynamics( CMD)model etc. Sec02tion 3 presents the extensive signatures for LGPT. Fi-nally a simple summary is given in Section 42 Selected models for nuclear LgPt2.1 Nuclear mean field theoryNuclear matter is an idealized system of equalL⊥⊥L_LLnumber of neutrons N and protons Z. The system is00020040060080.100.12014very large and the Coulomb interaction between pro-tons is switched off. due to the existence of coulombinteraction, stable systems are scarce after mass num-ig.I Nuclear matter equation of state with Skyrme interation with compressibility 201 Me V. In ascending order the iso-ber A= N+Z> 260, so no known nuclei approach the thermalstherm) and 17 Mev. The coexistence curve obtained from allimit of nuclear matter. However, the extrapolation Maxwell construction is shown. The vertical line is drawn atfrom known nuclei causes one to deduce that nuclear assumed freeze-out density 0.04 fm". The dot-dash line is ob-matter has density A 0. 16fm and binding energytained by assuming that the excited system expands isentropecally(see Fig. 2). This is an idealization from Ref [211a16 MeviA. Usually the Equation of State(EOS)ofthe idealized nuclear matter is used to examine if ation and finite size effect are taken into account, theliquid-gas phase tranboiling"temperature (i.e. 15.64 Mev from Fig. 1)what temperature and densitywill come down. 26, 27 For instance, in mean-fieldThe Skyrme parametrization for the interactionpotential energy has been demonstrated 2324l to be a 150Sinteraction a peak in specific heat at 10 Mev forgood approximation for Hartree-Fock calculations.A150Sm was found. Without the Coulomb interaction. infor an equation of statbulk matter with the same isospin asymmetry as 5Smversus density) corresponding to nuclear forcesthe peak is located at 13 Me V. As will be described(Skyrme effective interaction)is shown in Fig.1.Inlater, both experimental data and more realistic mod-this figure isotherms are drawn for various temperaels point to much lower temperature when phasetures(10, 12, 14, 15, 15.64 and 17 MeV). The preschange occurs.re contributed by kinetic energy was calculatedthe finite temperature Fermi-gas model. The similarity2.2 The percolation modelwith Van der Waals EOS is obvious. With the pa-The percolation has been extensively studied inrameters chosen here the critical temperature is 15.64 condensed matter physics. 2 The applications in nuMev. The spinodal region(ap/app2that to the molecular dynamics calculation. The start2m(1) ing point of CMd is a thermally equilibrated sourcewhich has been produced by the above l- LGM: i.e. theThe interaction constant E. is related to the bindingnucleons are initialized at their lattice sites with menergy of the nuclei. In order to incorporate the iso-tropolis sampling and have their initial momenta withthe LGM(thereafter. we will called Maxwell-Boltzmann sampling. From this startingI-LGM"). the short-range interaction constant E. ispoint we switch the calculation to CMD evolutionchosen to be different between the nearest neighboring under the influence of a chosen force. Note that in thislike nucleons and unlike nucleonscase p is, strictly speaking, not a'freeze-out'densityEnE=0 Mevfor molecular dynamics calculation but merely definesE=-533 Mev(2) the starting point for time evolution. However, sincewhich indicates the repulsion between the near- classical evolution of a many particle system is en-est-neighboring like nucleons and attraction between tirely deterministic and the initialization does have inthe nearest-neighboring unlike nucleons. This kind of it all the information of the asymptotic cluster distriisospin-dependent interaction incorporates, to a cer- bution, we continue to call A the freeze-out'densitytain extent, the Pauli exclusion principle and effecThe form of the force in the cmd is also chosetively avoids producing unr asonable clusters, such as to be isospin dependent in order to compare with theof the system is calculated at an assumed'freeze-out' expressed asdu The potential for unlike nucleons isdi-proton and di-neutron clusters etc. the disassemblresults of I-LGMdensity p=(A/L)A, beyond A nucleons are too farapart to interact.Un (r)=CIB( /ry"-(r /r)"lexp(r/a)tropolis algorithm. 49 Various observables based on where ro=1. 842 fm is the distance between thephase space can be calculated in a straightforward of two adjacent cubes so that A=/rn=0.16 fm .Thefashion for each event. The cluster distribution in parameters of the potentials are p=2, q=l, a=1.3.BLGM can be obtained by using the rule that two nu- =0.924 and C=1966 Mev. With these parameters thecleons are part of the same cluster if their relative ki- potential is minimum at ro with the value.33 MeVnetic energy is insufficient to overcome the attractive and zero when the nucleons are more than 1.3 ro apartand strongly repulsive when r is significantly less than(P/2)-E、ss,<0(3) Ph. We now turn to the nuclear potential between likenucleons. Although we take Epp=Enn=0 in I-LGM,theThis method has been proved to be similar to the fact that we do not put two like nucleons in the sameio-Klein prescription in condensed cube suggests that there is short-range repulsion be-matter physics and was shown to be valid in LGM. tween them. We have taken the nuclear force betweenSince LGM is a model of the nearest-neighboring two like nucleons to be the same expressions as aboveinteraction, a long-range Coulomb force is not amena- +5.33 Mev up to r= 1. 842 fm and zero afterwards oveble to lattice gas type calculation. Pan and Dasprovide a prescription, based on simpleln()=Um(r)-U2(5,(<)Ing neighboring sites form part of the same cluster Fig. 2 shows the above potential Vmp or yp, -a/ r, Sphysical reasoning, to decide if two nucleons, occuor not. They first tried to map the LGM calculation potential form automatically cuts off at riro= a(Eq 4)to a classical molecular dynamics type prediction,or中国煤化工山 es In any"rdeboth first carried out without any Coulomb interaction. rivatCNMHntage in any mo-If the calculations match quite faithfully, then they can lecNUCLEAR SCIENCE AND TECHNIQUESol.15was2≤?≤3, which is in a reasonable range for critimulti-fra10analogous to a continuous liquid to gas phase transi-tion observed in more common fluidIn the Fisher droplet model the fragment massdistribution may be representedY(A)=YoA5where Yo, T, X and Y are parameters. However at the1.0critical point X=l and y=l and the cluster distributionFig2 Molecular dynamics potential for like nucleon pair(Upp)given by a pure power lawand unlike nucleon puir( Uhp). From Ref [53Y(A)=Y,AThe system evolves with the above potential. The The model predicts a critical exponent t-2.21time evolution equations for each nucleon are, asIn this paper, we will use some results whiclusual glvwere recently obtained in Cyclotron Institute, Texas0p/r=∑V(),A&M University to show some signatures of nuclearar /at=p,/mLGPT. Of course, many other works are also in-(6) cluded in this review article. For the former work, weNumerically, the particles are propagated in the phase used Ar +Ni reaction at 47 Me v/nucleon and reconspace by a well-known Verlet algorithm ($4) one of the structed the quasI-projectile(QP)using a new methodfinite-difference methods in molecular dynamics with which is based on three source fits and the Montecontinuous potentials. At asymptotic times, for in- Carlo sampling for the assignment of QP. Well destance. the original blob of matter expands to 64 times fined QP sources have been obtained and their disas-its volume in the initialization, the clusters are easily sembly features have been analyzed. For simplicityrecognized: nucleons which stay together after an ar- we call experimental data of this work as TAMUbitrarily long time are part of the same cluster. The data 1581observables based on cluster distribution in both modFirstly, in Fig 3 we present, for the QP from theby switching on/off the Coulomb interaction within servei hs of aAr+Ni, yield distributions, dN/dz, ob-or nine different excitatthe molecular dynamicsnergy(E/A)intervals from tamu data3 Signatures of the nuclear liquid gasAt low excitation energy a large Z residue alwaysphase transitionremains, i.e. the nucleus is basically in the liquidphase accompanied by some evaporated light particles3.1 Fisher droplet model: power lawWhen E*/A reaches -6.0 MeV/nucleon. thischarge/mass distributionmuch less prominent. As E*/A continues to increasethe charge distributions become steeper, which indiThe Fisher droplet model has been extensively cates that the system tends to vaporize. To quantitaapplied to the analysis of multifragmentation since the tively pin down the possible phase transition point. wepioneering experiments on high energy proton-nucleus use a power law fit to the QP charge distribution in thecollisions by the Purdue group [3.55. 6l Relative yields range of Z-2-7 to extract the effective Fisher-law pa-of fragments with 3 increases with increasing. In this energy zone, the fragmentation is basically dominated by evaporation and sequential decayis important. But above 6 Me v/nucleon excitation()EA=6575(h)/A=75-840e/A8.4-94energy, decreases with increasing . Inplots in nine different excitation energy bins. Thedots are data and the lines are power-law fits(Eq. 10). FromTAMU35300.1AFig 14 The phase separation parameter as a function of etation energy in TAMU data.3.025(Mev)Fig13 Zipf parameter as a function of excitation energy in登丘are produced in the relativistic heavy ion collisionSome preliminary results are also obtained. OIIn a related observation which is consistent withthe formulation of Zipf's law, percolation model calculations"suggest that the ratio Sp0.5 around the phase separation point. Here Zamox isthe atomic number of the second heaviest fragment ineach event. Fig 14 shows Sp versus E*A for TAMUdata. Sp=0.5 near 5.5 Me V/nucleon and exhibits es-sentially linear behavior (with two different slopes) Fig. iH中国煤化工CN Gin TAMU data SITheabove and below that point, which also supports the meNUCLEAR SCIENCE AND TECHNIQUEvol 15this region of excitation, the nucleus is essentially parameter is bimodalfully vaporized and each cluster shows a similar be-In analyses of INDRA data, the bihaviorparameter is definedIn relating theoretical calculation with I-CMD forA36 (581 we obtain the very similar pictures as the3≤Z.≤data. Fig. 16 shows the scattering plots for Z2max andZmax for temperature T=3 to8 MeV. An obvious tran- P was chosen as a sorting parameter which may besition of the behavior occurs around 5.0 MeV, namely connected with the density difference of the twotrom the negative correlation to positive correlation, phases (a-pg), which is the order parameter for thehich inflects the phase transition point. Of course, liquid gas phase transition. Fig 17 shows the eventin I-CMD simulation, the input parameter is tempera- distributions observed at different incident energiesture instead of the excitation energy which can be de- These results show that none of the distributions istermined from the data. However, the qualitative re- Gaussian, which would indicate a pure phase, but thegas phase part is dominant above 45 Mev/nucleonT=3. 0 MeVT:3. 5 MeVwhile the liquid part is more important below 32MeV/nucleon. Two components are roughly equalaround 39 Mevinucleon. The observed trends are5relevant to the liquid gas phase transitionXe+ Sn single source4.5Me1=5.0 MeVT=5.5 MeV1400 ETTTWTTTTTTEA=45 MeVT8. 0 MevT=B, 5 Mev075050250050.751Fig 17 Event distributions observed at different incident erles-l are dividedtext for the variable used for event distribution. From Indratemperature windows in I-CMD simulation. The y-axis is forZ2mas and x-axis for Zmux. From Ref[58]3.5 BimodalityAnother proposed test of phase separation is bimodality which was suggested in [73]. As has beenon curvature anomalies of any thermodynamic poteral as a function of an observable which can then beseen as an order parameter. It interprets a bimodalityof the event distribution as coexistence, each compenent representing a different phase. It provides a defi-nition of an order parameter as the best variableseparate the two maxima of the distribution. In this中国煤化工州framework when a nuclear system is in the coexis. figCN MH Gtence region, the probability distribution of the order tiogy, rrom IAMU aataMA Yu-Gang et al. Recent progress of nuclear liguid gas phase transitionFor our tamu data for very light system if we rather than the phase space. As shown below this kindconsider the clusters with Z<3 as a gas and the clus- of information entropy can be taken as a method toters with Z34 as a liquid, a parameter characterizing determine nuclear liquid gas phase transitionthe bimodal nature of the distribution can be definedFig 19 shows the information entropy for disassembly of Xe. 6I The information entropy exhibits arise and fall with temperature, which is similar to the(12) behaviors of Nimr and S2(see Fig. 11). The tercures extracted from thealues of hFig 18 shows the mean value of P as a function of tent with the transition temperatures in Fig. 1, indicat-E/A. Here again, the slope shows a distinct change at ing that information entropy ought to be a good diagE*/A=5-6 Me v where P=0, i.e. the point of equal nosis of phase transition. Physically, the maximum ofdistribution of Z in the two phasesH reflects the largest fluctuation of the multiplicityprobability distribution in the phase transition point3.6 Multiplicity information entropyIn this case it is the most difficult to predict how manyThe concept of multiplicity information entropy clusters will be produced in each event. i.e. the disorhas been also introduced into the diagnosis of nuclear der(entropy) of information is the largest. Generallyliquid gas phase transition recently. o Originally, the speaking, the larger the dispersal of multiplicity probinformation entropy was defined by Shannon inin- ability distribution, the higher the information entropyformation theory and it measures the"amount of and then the disorder of system in the event topologyinformation" which is containemessages sent One should make a careful distinction between thisalong a transmission line. It can be expressed asinformation entropy, on the one hand and the originalH=∑P(13)where P; is a normalized probability, and 2,p=1. mentum space rather than event space and it alwaysJaynes proposed that a very general technique forincreases with temperaturediscovering the least biased distribution of the p:con-More recently, we have appliedhe same multisists in the maximization of the Shannon H entropy, plicity information entropy to the relativistic heavysubject to whatever constraints on p are appropriate to lon collision already. [77Ithe particular situation. The maximization of H wasthus put forward as a general principle of statistical:"nference-one which could be applied to a wide variety of problems in economics, engineering and manyother fields, such as quantum phenomena. In highenergy hadron collisions, multiparticle productionshould obemaximum entropy principle. ThisFig19 Information entropy H as a function of temperaturekind of stochasticity can be also quantified via the The symbols are the same as in Fig. 11. From Ref [161information entropy which has been shown to be agood tool to measure chaoticity in hadron decaying 3.7 Campi plotbranching process. b In different physical conditionsinformation entropy can be expressed with differentOne of the well known characteristics of the sys-stochastic variables. In this work on HIC, we define pi tems undergoing a continuous phase transition is theas the event probability of having"I" particles pro- occurrence of the largest fluctuations. These largeduced, i.e. (pi l is the normalized probability distribu- flucty中国煤化工 sity of the systertion of total multiplicity, the sum is taken over whole ariseIP/ distribution. This emphasis is on the event space the cuci nmh g the latent heat atin macroscopIc systems such be-NUCLEAR SCIENCE AND TECHNIQUESVoL 15havior gives rise to the phenomenon of critical opalCampi suggested the use of event-by-event scatter plots of the natural log of the size of the largestcluster(here InZmux) versus the natural log of the nor-7=3.0Mev7=3.5MeT=4.0 MeVmalized second moment(InS,)of the cluster distribu-tion with the heaviest fragment removed,Z'n(z)S2xz2(2(14)I T4, 5 MeV1 T=5. 0 MeV1 755 MeVwhere Zi is the charge number of QP clusters and n z)is the multiplicity of the cluster Z,. As a means to oberve the region of the largest fluctuations, such plotsvery instructive In previoussearches for critical behavior 32)!rs6oMey·0 7-6.5 Mvp.叩r7oMvIn Fig. 20 we present such plots for the nine selected excitation energy bins in our TAMU data. Fig.2I The Campr'sin nine ditferent temperatureAgain, in the low excitation energy bins of E*/As3.7 r-axis for S2. From Ref [58]The y-axis is for Zmux andMeV/nucleon, only the upper(liquid phase)branch tion, 58 Fig. 21 shows such plot. Similarly, the sameexists, while at EW/A=7.5 Mev/nucleon, only thelower Zax(gas phase) branch is strongly dominant.transition is observed around 5.0 Mev indicating theHowever, in the region of intermediate E*/A of 4.6nset of phase transition6.5 MeV/nucleon, there appears to be a significaUsing the general definition of the kth moment asfrom the liquid dominated branch to the∑Zn(Zpor branch, indicating that the region of maximalfluctuations signaling a transition between the two Campi also suggested that the quantity, h, defined asphases is to be found in that range.As the Zmax Zomax correlation shown above weMmOobtain the similar Campi's plot with I-CMD simula-where Mo, M, and M,, the zeroth moment, first mo-ment and second moment of the charge distributioncould be employed to search for the critical region. Insuch an analysis, the position of the maximumvalue is expected to define the critical point, i.e. the1[(ae4-13200Ee42028(E/A2837critical excitation energy E, at which the fluctua-tions in fragment sizes are the largestThe excitation energy dependence of the averagevalues of n obtained in an event-by-event analysis of1EA=37-46E/46-56L(EA=58-65our data is shown in Fig 22. n reaches its maximum inthe 5-6 Mev excitation energy range. In contrast toobservations for heavier systems, there is no welldefined peak in n for our very light system and n is1aEA65750)EA75840/A8494elatively constant at higher excitation energies. Weo that s value is lower than 2 which is the中国煤化工 tical behavior ifFig.20 The Campi plot for different excitation energy winCN MH Gver, 3D percolationdows. From TAMUstudies indicate that finite size effects can lead to aMA Yu-Gang et al. Recent progress of nuclear liquid gas phase transitiondecrease of n with system size. 80.8 For a percolation systems of the same size employing a finite latticesystem with 64 sites, peaks in n under two are ob- bond percolation model. Such a case is known to dis-served. Therefore, the lone criterion 1>2 is not suffi- play true critical behavior. They found that NVZ peakscient to discriminate whether or not the critical point close to the critical point in the percolation modelis reached. To carry out further quantitative explora- calculation but shows no such peak in the random par-tions of maximal fluctuations we have investigated tition model calculation. This indicates that the massseveral other proposed observables expected to be conservation criterion, by itself, can not induce thelated to fluctuations and to signal critical behavior. peak of NVZ. The details can be found in(821These are discussed belowFor our tamu daof Zux as a functionFig 23. A clear maximum characterizing the largestfluctuation of this order parameter, is seen, which islocated around e/A=5-6 Mev/nucleon(Mevg.22 y of the QP system formed in Ar+ Ni as a function ofexcitation cnergy From TAMU data. I001.8 Fluctuations in the distribution of ZIt is supposed that the cluster size distributions[MevFig 23 NVZ of the QPformed in Ar+Ni as a tunctionshould manifest the maximum fluctuations around the of excitation energy in our TAMU data IS8/critical point where the correlation length divergesAs a result of constraints placed by mass conservation, 3.9 Fluctuations in the distributation of total ki.the size of the largest cluster should then also showlarge fluctuations. Thus, it has been suggested thatnetic energy and negative heat capacitya possible signal of critical behavior is the fluctuationThe system we have studied is a hot system. Ifin the size of the maximum fragment. 32) Recently, critical behavior occurs, it should also be reflected inDorso et al. employed a molecular dynamics model to large thermal fluctuations. Using a definition similarinvestigate fluctuations in the normalized variance to that of the normalized variance of Zmax, we can de-(Nva.fine the normalized variance of total kinetic energyNVZper nucleon(17)NVE=(Zmax) 182 They found that this quantity can indeed where Ein /A is the total kinetic energy per nucIGaof the atomic number of the heaviest fragmentEkin /a>display a maximum in the critical regionand aEmia is its width. Fig 24 shows the NVE as aIn their work, Dorso et al. performed calculations function of excitation energy in TAMU data. The ob.of the NVZ on two simple systems, one of which served behavior is very similar to that of NVZ. Again,should not exhibit critical behavior and the other does. the maximal fluctuation was found at E/A=5-6For the first they used a random partition model, in Me v/u. The maximal thermal fluctuations are foundwhich the population of different mass numbers is in the same region as the maximal fluctuations in theobtained by randomly choosing values of A following largest cluster sizes.a previously prescribed mass distribution. 8.In thisThe use of kinetic energy fluctuations as a tool tocase the fluctuations in the populations are of statisti- meascanonical heat capacities has also beencal origin or are related to the fact that the total mass prop中国煤化工 that for a givenAnot is fixed. No signal of criticality is to be expectedCN MH Gy stored in a sub-In the second case they explored the disassembly of system of the microcanonical ensemble is a goodNUCLEAR SCIENCE AND TECHNIQUESkinetic heat capacity for QP events (grey contoursand central Au+C(black dots). Au+Cu(squares, trian-S 0.06gles), Au+Au reactions before(open stars) and aftersubtraction of I A Mev radial flow(black stars).3.10 Universal fluctuations: Ascaling24 NVE of the QP system formed in Ar+Ni as a functionThe recent developed theory of universal scalingl energy in our TAMU data.laws of order-parameter fluctuations provides athermometer while the fluctuations associated to the method to select order parameters and characterizepartial energy can be used to evaluate the heat capac- critical and off-critical behavior, without any assumpity. An example of such a decomposition is given by tion of equilibrium. In this framework, universalthe kinetic energy Ek and the interaction energy Ej. A-scaling laws of the normalized probability distribu-The interaction energy fluctuation can then be studied tion PIm of the order parameter m for different"sys-as a function of the total energy and the heat capacity. tem size", should be observedand can be evaluated according toCT P[m]=p(Z))=p(m'CT2-σWith0≤A≤1, whereandm*where a=0, is the fluctuation of the interaction and the most probable values of m, respectively, andenergy El, T is the temperature, and CK is the kineticgAza )is the(positive)defined scaling function whichheat capacity that can be evaluated by taking the nu- depends only on a single scaled variable Zay. If themerical derivative of =E*- with respect to T.scaling framework holds, the scaling relation is validEq- (19)shows that a negative heat capacity correindependently of any phenomenological reasons forsponds to partial energy fluctuations in the microcan- changing . I The A-scaling analysis is very ro-onical ensemble that exceed the corresponding fluc-bust and can be studied even in small systems if thetuations in the canonical ensemble(an=CkT)In particular, first order phase transitions arprobability distributions P[m] are known with a sufficient precisionmarked by singularities and negative heat capacibotet et al. introduced this universal scalities. 35 4I corresponding to fluctuations anomalously method to the indra data for Xe+ Sn collisionslarger than the canonical expectation. If the system isin statistical equilibrium, a measurement of anomalousthe range of bombarding energies between 25Me v/nucleon and 50 Me V/nucleon. They chose thefluctuations at a given energy is an unambiguouslargest fragment charge, Zmax, as the order parameterproof of a thermal first order phase transition. Fig. 25 It was found that at Eub 32 MeV/nucleon, there is ashows the normalized partial energy fluctuations andtransition in the fluctuation regime of Zma which iscompatible with a transition from the ordered phasewith 4= 1/2 scaling to the disordered phase with AI scaling of excited nuclear matter. 5 From this study,they attributed the fragment production scenario to bein the family of aggregation scenarios which contains2557.502.5575both equilibrium models such as the Fisher dropletE/Ao(A Mev)model, the Ising model, or the percolation model andFig. 25el normalized partial energy fluctuations and off-equilibrium models such as the SmoluchowskiAu+C (black doteiry for QP events (grey contours)and central model of gels For such scenarios the average size of) Au+Cu (squares, triangles), Au+Au reac.tions before(open stars)and after subtraction of I A Mev ra中国煤化工 order parameter anddial flow(black stars). Right panel: heat capacity per nucleon theof the source for QP events and central reactions. FromCNMHGitical point obeys aRef[871No. 1LA Yu-Gang et al Recent progress of nuclear liquid gas phase transitionWe have applied the same technique to the znatdistribution from our TAMU data. Fig. 26(a)shows theA-scaling features of PIZnaxI distributions for030505Ar+Ni collisions in different excitation energywindows0.10The upper panel shows that A-scaling of PIZuxldistributions for all E*/A windows above 2.0 Mev Fig. 27 of. From TAMUwith A=l Essentially for this A=l scaling the curves data.at higher excitation energy above 5.6 Mev/nucleon with the mean value, ie. ocam--constant(seento a sincurves of lower EW/A deviate from this first-scalingFig. 27). The saturation of the reduced fluctuations of(i.e. A=1). This situation of lower E*/A curves disap- Zmax (i e.ima. ) observed above corresponds to thepears when A=l/2 scaling was used. Fig. 26(b)shows transition to the regime of maximal fluctuations. 891the curves with A=l/2. Now these curves collapseMore recently, we have applied the A-scaling toonto one curve, i. e. they obey A=l/2 scaling, but in multiplicity distribution, strange particle multiplicitythis case, the scaling curves for higher E*/A which and the number of binary nucleon-nucleon collision inbeys A=I scaling as shown in Fig. 26(a)becomes CERN-SPS energy for p+p, C+C and Pb+Pb fromparison, Fig. 26(c)only plot the higher E*/A scaling even in such high energy hadron transpon a 3worse and deviate from lower E*/A curves. For com- 20 to 200 GeV/nucleon. It looks the scaling holdcurves with A=l and the lower E*/A scaling curveswith A=ln. In this way, a transition from A=1/ 3.11 Caloric curvescaling to A=l scaling was found in theofhe caloric curve which relates the internal enE*/A-5.6 MeV for our light system. The latter scaling ergy of an excited system at thermodynamic equilib-corresponds to the fluctuations of the Zmax growing rium to its temperature is a priori the simplest experi10mental tool to look for the existence of a phase transi-DIN collaboration I collected the outcome of 97Au or197 Au collisions at 600 Me V/nucleon with data ob-lined by means of less energetic collisions. THtation energy was determined by following the procedure prescribed in Ref. [901 and the correspondingtemperature fixed by means of arguments relating thisquantity to the so called double ratio procedure inhe present case the ratios of He/He and Li/'li iso.topes. The corresponding curve showed the features ofa rather strong first order transition, with a characteristic close to constant temperature T over a large energy interval lying between 3 and 10 MeV excitationenergy per nucleon, which may be interpreted as asign for the generation of latent heat and followed bya strong increase of T with excitation energy above10 MeV. However, a critical discussion followed thisFig. 26 A-scaling for different EvA window: all E*/A winlows above 2.0 Me/nucleon are shown together with A=l(a);which the hypotheses and simplifica-ame as(a) but with A=ln2 scaling(b); for four ENA windowtions中国煤化工 temperature, theabove 5. 6 Mevinucleon with A-l scaling and four E*A win- freezeCN MHGT at high excita-ows below 5.6 MeVInucleon with A=l/2 scaling(c). From tion energy were examined. %)TaMU data.NUCLEAR SCIENCE AND TECHNIQUESVoL 15Ma et al. measured Ar on Ni collisions at 52 and temperature as shown in the following section95 Mevinucleon with the 4r multi-detector (INDRA)in order to investigate this point. 93 Several double 3.12 Critical temperatureisotopic yield ratios were used in order to define thele collected the experimental data for calorictemperature. We led to different apparent slope tem- curve in terms of different mass windows and founderature increase as a function of the excitation energy. that the onset temperature of the temperature plateauThe issue of the experiment and its interpretation have so-called the limiting temperature(Tm), shows thebeen performed by means of statistical models. 94-9 dropping trends with the system mass. 102 See Fig. 29A temperature plateau is absent for such a light sys- This reflects the influence of the Coulomb interactionon nuclear instability. in a recent theoretical calcula-In our TaMU data the caloric curve is also con- tion, Li et al. also shows the limiting temperaturesstructed, Quasi-projectile has been reconstructed decrease with the increasing of mass number for nuwith a new method. The determination of initial clei along the B-stability line, 103 which is in agree-temperature has been deduced by taking the cascade ment with the earlier results of Song and Su ermethod for lower excitation energies 9 or quantum al. 10-4. 105tstatistical model correction for higher excitation ener-Mean values of Tin/T for five different massesinto account. Fig. 28 shows the resultant ca- which result from averaging the results of differentloric curve of quasi-projectile Ar. Based upon the calculations/04-l13are shown in Fig.30.The estimatedfluctuation data and Fisher power law analysis of our uncertainties are relatively small, 6%. For com-TAMU data, the corresponding initial temperature parison, the figure also presents ratios of Tim/. whicharound the critical point is 8.3+0.5 Mev in the region are expected to result assuming only finite size effectsof 5.5 Me v/nucleon excitation energyas derived from a lattice calculation 4 and the ratioMore measurements for the caloric curves have of the nuclear binding energy per nucleon along thebeen performed in various groups. For instance, for line of beta stability to the bulk binding energy per100 Me V/nucleon Au+C from EOS group, 4.8 GeV nucleon, 16 MeV. We have employed the mean varia-He+Ag/Au reactions from ISIS group, the excited tion of Tm/T with A. determined from commonlysystems with A$110 particles produced by means of used microscopic theoretical calculations, togetherdifferent projectiles and targets with a bombarding with the five experimental limiting temperatures re-energy of 47 Me v/nucleon"0o etc. Nuclear expansion ported in Ref [7], to extract the critical temperature ofdenstyyol A collection of the experimental data for variation as if it were an experimental uncertainty.density was also extracted from the analysis of caloric nuclear matter. In doing so we treat the theoreticalthe caloric curves has been used to deduce the critical Since the various interactions employed have beenuned"to other nuclear properties, we consider this aTuen cascade correctedreasonable approach. The results are presentedT OSM correctedFig. 29 Limiting temperatures vs mass. Limiting temperaturesderived from double isotope yield ratio measurements are rep)n of the distance from critical temperature(b). See textfor details. From Tamu data 5Here, Y( AZ. ) is the yield of selected events withprecise, is in good agreement with the values of 2.31 and A Z values and M is the fragment multiplicity+0.03 obtained from the charge distribution around The denominator r(Az, )which represents thepoint of maximal fluctuations and 2.15+0.1 ex- uncorrelated yield is built, for each fragment multtracted from the correlation of InS, vs InS.plicity, by taking fragments in different events of theTo summarize this section, we report in Table I a selected sample. With such a correlation methodcomparison of our results with the values expected for events with nearly equal-sized fragments are produced,the 3D percolation and liquid gas system universality we expect to see peaks appearing at AZ values closeclasses and with the results obtained by Elliott et al. to zero. Taking into account secondary decay of fragfor a heavier system. Obviously, our values for this ments, the bin AZ=0-1 was only considered At 32 Alight system with Ag36 are consistent with the values Mev incident energy peaks were observed in this binof the liquid gas phase transition universality class for each fragment multiplicity. 812>See Fig.36rather than the 3D percolation class.Background"has been suitably constructed inorder to estimate whether the3.15 Spinodal instabilitywith equal-sized fragments is statistically significantRecently the multifragmentation of very heavy Fig.36 shows the higher order correlation functionsfused systems formed in central collisions between for the first bin in AZ with their statistical errors; theXe andn sn at 32, 39. 45 and 50 A MeV with IN- full line corresponds to the extrapolated"background"DRA has been studied. 4. 2These fused systems All events corresponding to the points whose error barcan be identified to well defined pieces of nuclear is fully located above this line correspond to a statis-matter and eventually reveal fragmentation properties tically significant enhancement of equal-sized frto be compared to models in which bulk instabilities ment partitions. The number of significant eventsare present.amounts to 0. 1% of the selected fusion eventsMany theories have been developed to explainobserved weak but unambiguous enhancedmultifragmentation(see for example Ref [20], [21]中国煤化工 ed fragments at32[124] for a general review of models). In particular,CN MHGas a signature ofthe concept of multifragmentation resulting from thees as the origin of multifragmentaMA Yu-Gang et al Recent progress of nuclear liquid gas phase transitionIn our calculation, we use [-LGM and I-CMd toinvestigate the isospin dependence of the apparentcal temperature for Xe-isotopes. Ain Section 3. 1, a minimum of power-law parameterTmin exists if the critical behavior takes place. Fig. 37 (a)81displays T parameter as a function of temperatureforXe nuclei with the different isospin. The I9parameters in Fig. 37(a) locate closely at 5.5for all the systems, which illustrates its minor de-ata Xe+Snpendence on the isospin. However, r parameters showi different values outside the critical region for nucleii with different isospin, e. g, r decreases with isospinwhen 755.5 Mev(multifragmentation region)Fig. 37(b) plots the information entropy H as afunction of temperature for Xe isotopes. The informa3o i so tion entropy h has peaks close to 5. 5 Mev for all iso-topes. These peaks indicateof theFig-36 Higher-order charge correlations: quantitative results phase space in the critical point is the largest. In otherfor experimental data at different incident energieIs words, the system at the critical temperature has theindicate the events where A=0-l curves show the(see text). Vertical bars correspond to statisticald largest fluctuation/stochasticity which leads tohorizontal bars define Zuo bins From Ref. [123]largest disorder and particle prouction ratetion in the FermI energythe critical point, the information entropy H increasesis consistent with the measurement of negative mi. with the isospinical heat capacities for the same selectedSimilarly, Fig. 37(c)gives the IMF multiplicity asfused events at 32A and 39A Mevll30 which are pre- a function of temperature for Xe isotopes. Again,thedicted to sign a first order phase transition. 3Hitical behavior occurs at the similar temperature, ithas weak sensitivity to the isospin, as reveals3.16 Isospin effectsIn Fig. 37(d) we give the temperature dependences of Campi, s second moment of the fragmeSince nuclei are cd of neutrons and pro- mass distribution. At the percolation point S2 ditons, isospin effects may be very important for the verges in an infinite system and is at maximum in anuclear liquid-gas phase transition. 32)As the asym- finite system. Fig. 37(d)gives the maximums of S2metry between neutron and proton densities becomes around 5.5 Mev for different isotopes, respectivelya local property in the system, calculations predict Again, the critical behavior occurs at the same tem-neutrons and protons to be inhomogenously distrib- perature, independent of the isospin, as t revealsnin the system resulting in a relativelythe ap-tron-rich gas and relatively neutron-poor liq parent critical temperature is very weak sensitive touid.46. 13. 34) The critical temperature may also be re- the isospin of sources in the limited isospin rangeduced with increasing neutron excess reflecting the This conclusion is not contradict with the previousfact that a pure neutron liquid probably does not ex- studies on the isospin dependence of critical temperaist. 3While our recent calculations suggest that the ture like in [46] where the span of isospin is fromrather narrow range of isospin values available in the symmetrical nuclear matter to pure neIfboratory might not allow us to observe the decrease we onlyin critical temperature, 35, 136 efforts are underway to mentally中国煤化工nopsstudy the fractionation of the isospin in the 2-4xCN MH Gre is neglect.co-existence region.This conclusion might indicate that it will be difficultNUCLEAR SCIENCE AND TECHNIQUESVol ISand(B, C)d isodiscussed4 Conclusionsliquid gas phase tratensive observables relating this kind of phase transi-the frasobeys the power law which was predicted by Fisherdroplet model. For intermediate mass fragments, theirdmultiplicity rises to a maximum and then falls with thetemperature or excitation. The emission rate of lightparticles and charged particles shows a turning point35404550556.0657,075around the phase transition point. From fragment sizestructure,we found that there exists a particular hier-Fig-37 Critical observables: rparameter from the power law archical arrangement, so-called the nuclear Zipt-typeof IMF ()and Campi's second moment(d)as functions of law, which was supported by both the calculation oftemperature and isospin of disassembling sources. From lattice gas model and Texas A&M data and EMU13Ref. 436cern data. The scattering plots of the correlationto search for the isospin dependence of critical tem- between zmax and Zomax shows a significant transitionperature which signals the liquid gas phase transition around the phase transition point. Bimodality paramefor medium size nuclei in the experimental point of ter also gives a phase separation betview. In addition, values of power law parameter of liquid phase and dominant gas phase when phase tran-cluster mass distribution, mean multiplicity of inter- sition takes placemediate mass fragments (IMF), information entropyBesides, many observables demonstrate the exis-(H) and Camp i's second moment(S2)also show minor tence of maximal fluctuations when phase transitiondependence on the isospin of Xe isotopes at the criti- occurs. These fluctuation observables include thevalues of T, H, NIMF and S2 will reveal outside the of the distributions of order parameters, zma and erya point. In contrary, some isospin dependences of the Campi scattering plots and the normalized variancescritical regionkinetic energy and related heat capacity. A-scalingAs the isospin effects are not large, the influence analysis also shows a universal behavior at higher ex-of sequential decays becomes important and may ob- citation energy where the saturation of the reducedscure the isospin fractionation effect one wishes tofluctuations of Zmux(i.e.study. To minimize such problems, isobar pairs, suchas(t, He), which have the same number of internal corresponds to the transition to the regime of largeexcited states, have been used. Some indications for fluctuations from the ordered phase at lower excitationisospin fractionation are provided by the sensitivity of energyr(t/y He)distributions to the overall Ni/Z ratio of theCaloric curves, critical temperature, critical exsystem. 37 The ratios of Y()Y(He)have also been ponents, phase co-existence diagram and spinodalobserved to decrease with incident energies, in quali- instability are also discussed. All of them can providetative agreement with the predictions from the isospin some useful information on liquid gas phase transipendent lattice gas model 146.53 134 1361 Light isobarstionsuch as (t, He) pair may suffer from contamination ofBV凵中国煤化工 should keep inpre-equilibrium processes, Attempts have been made mind thCNMH Gen done in thisto use additional mirror isobar pairs such as (Li, Be) field by groups spread world-wide. The literature onMA Yu-Gang et al Recent progress of nuclear liquid gas phase transitionthis subject is enormous. Our review report had to be 27 Raduta Al H. Raduta Ad R. Phys Rev Lett. 2(X)1. 87necessarily selective. Some interesting works must202701have left out. we apologize for all the omissions that 28 De J N, Das Gupta S, Shlomo Set ul. Phys Rev C. 1997occurredAcknowledgments29 Stauffer D, Aharony A. Introduction to percolation theory.Taylor and Francis, London. 1992We appreciate many helpful discussions with 30 Bauer W. Dean D R Mosel U et al. Phys Lett B 198.5colleagues and/or friends, especially Joe Natowitz150:53Jean Peter, Bernard Tamain, Subal Das Gupta, Jicai 31 Bauer W Post U. Dean D R et al. Nucl Phys. 1986, A 452Pan, Manyee Betty Tsang and Bao-an Li, Roy Wada,Kris Hagel, Sherry Yennello et al.32 Campi X, Desbois J. Invited contribution to the TopicalReferencesMeeting on"Phase Space Approach to Nuclear Dynam-ics". Trieste, 1985Lynden-Bell B Physica A. 1999. 263: 29333 Campi X J Phys A: Math Gen. 1988. 19: L9172 Bertsch G F Science. 1997. 277:161934 Bondorf J P, Botvina A S. Iljinov A S et al. Phys Rep,3 Finn JEet al. Phys Rev Lett. 1982. 49: 13211995,257:1334 Rivet M Fet al. Phys Lett B. 1996. 388: 21935 Gross D HE Phys Rep, 1997, 279: 1195 Gross DH E. Rep Prog Phys, 1990, 53: 605 and refer- 36 Randrup J, Koonin S E Nucl Phys, 1987,A471:355cences therein37 Sa B H, Zheng Y M. Zhang X Z Phys Rev C,1989, 406 Pochodzalla J et al. Phys Rev Lett. 1995, 75: 104026807 Natowitz J B et al. Phys Rev C, 2002, 62: 03461838 Raduta A H. Raduta A R. Phys Rev, 1997, C55: 13448 Bondorf J Pet al. Phys Rev C. 1998, 58: R2739 Mekjian A Z. Phys Rev Lett. 1977. 38: 6409 Gilkes M Let al. Phys Rev Lett. 1994. 73: 159040 Bondorf J P. Donangelo R, Mishustin I M et al. Nucl Phys,10 D'Agostino M et al. Phys Lett B, 2000, 473: 2191985,A444:46012 Schmidt M er al. Phys Rev Lett. 2001. 86: 119142 Kuo TTS, Ray S. Sharmanna J et al. Int J Mod Phys E.13 Labastie M. Whetten R L. Phys Rev Lett. 1990. 65: 15671996,5:30314 Borderie B et al. Phys Rev Lett, 2001, 86:325243 Pan J C. Das Gupta S. Phys Rev C, 1996. 53: 131915 Botet R ef al. Phys Rev Lett, 2001, 86: 351444 Das Gupta S et al. Nucl Phys, 1997, A621: 89716 Ma Y G Phys Rev Lett, 1999. 83: 361745 Carmona J M. Richert J. Tarancon A Nucl Phys, 199817 Ma Y G Chin Phys Lett. 2000, 17: 34018 Elliott JB et al. Phys Rev Lett, 2002, 88: 042701; 46 Chomaz Ph, Gulminelli F Phys Lett B. 1999. 447: 22147 Ma G et al. Eur Phys I A, 1999, 4: 21719 Elliott J B et al. Phys Rev C, 2003, 67: 2460948 Qian W L Su R K J Phys G, 2003. 29: 1023o Richert J, wagner P Phys Rep, 2001, 350: 149 Metropolis M et al. J Chem Phys, 1953, 21: 108721 Das Gupta S, Mekjian A. Tsang M B. Adv Nucl Phys, 50 Coniglio A, Klein E J Phys A: Math Gien. 1980. 13: 27752001, 26: 91 and references therein51 Pan J, Das Gupta S. Phys Rev C. 1993. 53: 131922 Ma Y G Zhang H Y, Shen wQ Prog Phys (in Chinese), 52 Stillinger F H, Weber T A Phys Rev. 1985, 31: 52622002,22:9953 Ma Y G j Phys G, 2001. 27:23 Ring P, Schuck P. The nuclear many-body problem, New 54 Verlet L Phys Rev, 1967. 159:98York: Springer-Verlag, 1980. 555 Hirsch A S et al. Phys Rev C. 1984, 29: 50824 Jaqainan H Mekjian A Z. Zamick L. Phys Rev, 1983. 56 Minich R W et al. Phys Lett B, 1982,118:458C27:278257 Fisher M E. Rep Prog Phys, 1969, 30: 61525 Reif F. Fundamentals of statistical and thermal physics,中国煤化工 IMROD CollaboratioNew York: McGraw HilL, 1965, Chap. 8CNMHGKcutgen T Majka Z,26 Gulminelli G et al. Phys Rev Lett, 2003, 91: 202701uray il L, otu r, IvawwIz J B, Alfaro R. CiborNUCLEAR SCIENCE AND TECHNIQUESVol 15J, Cinausero M.El Masri Y Fabris D, Fioretto E, Kiekies 88 BotetR Ploszajczak M. Phys Rev E, 2000. 62: 1825A. Lunardon M. Makeev A. Marie N. Martin E, Marti- 89 Frankland I D et al. ArXiv nucl-ex/0201nezDavalos A, Menchaca-Rocha A Nebbia Gi Prete G 90 Campi X, Krivine H, Plagnol E Phys Rev C,1994.50Ruangma A, Shetty D V, Souliotis G Staszel Pselsky M, Viesti Gi winchester E M, Yennello S J)Albergo S et al. Nuovo Cimento A, 1985. 89Arxiv/nucl-ex: 030301 6 and long papers, to be published 92 Campi x, Krivine H, Plagnol E. Phys Lett B. 1996. 385: I59 Ma Y G Wada R, Hagel K et al.(NIMROD Collabora- 93 Ma Y G et al.(INDRA Collaboration). Phys Lett B, 1997.tion). Ann Report of Cyclotron Institute, Texas A&M390:41University P. I-18, 200194 Borderie B er al. Phys Lett B. 1996. 38822460 Gulminelli F, Chomaz Ph Int J Mod Phys E, 1999. 8: 527 95 Borderie B et aL. Eur Phys J A 1999, 6: 19761 Peaslee G F et al. Phys Rey C. 1994. 49: R227196 Gulminelli F, Durand D Nucl Phys A, 1997, 615:11762 Ogilvie C A et al. Phys Rev Lett. 1991.67:121497 Hagel K et al. Nucl Phys. 1988.A486: 42963 Tsang M Bet al. Phys Rev Lett. 1993. 71: 150298 Majka Z et al. Phys Rev C,1997.55: 299164 Ma Y G Shen w Q Phys Rev C, 1995, 51: 7109 Kwiatkowski K et al. Phys Lett B. 1998. 423: 2165 Ma Y G Eur Phys J A, 1999. 6: 367100 Cibor Jet aL. Phys Lett B, 2000, 473: 2966 Zipf GK. Human behavior and the principle of least effort, 101 Natowitz J B, Hagel K, Ma Y Get al. Phys Rev C, 2002.Addisson- Wesley Press. Cambridge, MA66:031601(R)67 Turcotte D L. Rep Prog Phys, 1999, 62: 1377Hagel K. Ma Y G et al. Phys Rev Lett.68 Pan J, Das Gupta S Phys Rev C, 1998. 57: 18392002、89:212701al. Acta Phys Polo B, 2001, 32: 3099l03 Liu M, Liz X, Liu JF Ch70 Ma Gong H Q Su R K Phys Rev C, 1991. 44: 25057I Cole A J Phys Rev C, 2002, 65: 031601R105 Zhang LL, Song H Q. Wang P et al. Phys Rev C.199972 Sugawa Y Horiuch H Prog The Phys, 2001, 105: 13159:329273 Chomaz Ph, Gulminelli F, Duflot V. Phys Rev E, 2001, 106 Levit S, Bonche P Nucl Phys, 1985. A437: 42664:046114107 Besprovany J, Levitt S. Phys Lett B, 1989, 21774 Borderie B J Phys G, 2002, 28: R217108 Zhang Y J, Su R K, Song H Q et al. Phys Rev C, 1996. 575 Denbigh K G Denbigh J S. Entropy in relation to uncom-plete knowledge, Cambridge University Press, 1995109 Das A, Nayak R, Satpathy L J Phys G. 1992. 18: 86976 Brogueira Per al. Phys Rev D, 1996. 53: 5283110 Song H Q, Qian Z X Su R K. Phys Rev C, 1993. 477 Ma GL Ma Y G Wang K et al. Chin Phys Lett. 2003, 202001III Song H Q, Qian Z X. Su RK. Phys Rev C. 1994. 4978 Stanley H E. Introduction to phase transitions and criticalphenomena. Oxford University Press, Cambridge, Eng1 12 Baldo M, Ferreira L S. Phys Rev C. 1999. 59: 682land, 1992113 Bonche P. Levit S, Vautherin D Nucl Phys. 1985. A43679 Hauger J Aet al. Phys Rev C, 2000. 62: 02461626580 Campi X, Krivine H Z Phys, 1992, A344: 81114 Elliott JB et aL. Phys Rev C. 2000. 62: 06460381 Campi X. Krivine H Nucl Phys, 1992. A545:161c115 Youngblood D H, Clark H L, Lui Y w. Phys Rev Lett.82 Dorso C O. Latora V C, Bonasera A Phys re60:034606116 Ogul R, Botivina A S. Phys Rev C, 2002, 66: 051601(R)83 Elattari B, Richert J, Wagner P. Phys Rev Lett 1992, 69[7 Karnaukhoy V et al. Phys Rev C, 2003, 67: 01160I(R)45: Nucl Phys,1993.A560:60118 Guggenheim E A. J Chem Phys, 1945. 13: 25384 Chomaz Ph. Gulminelli F Nucl Phys. 1999, A647: 1531 19 D' Agostino M et aL. Nucl Phys. 1999. A650: 3298.5 Chomaz. Ph, Duflot V, Gulminelli F Phys Rev Lett, 2000. 120 Elliott J B et al. Phys Rev C. 1997. 55: 131985:358中国煤化工块3886 D'Agostino M er al. Phys Lett B, 2000, 473: 21987 D'Agostino M et al. ar Xiv nucl-cx/0310013CNMH Gon). Nuc! Instr MethMA Yu-Gang et aL. Recent progress of nuclear liquid gas phase transition123 Borderie B ef al. ar Xiv: nucI-ex/0106007XXXVIII Int Winter Meeting on Nuclear Physics, ed lori[24 Moretto L G Wozniak G J. Ann Rey Nucl Part Sci. 1993.L, Moroni A. Ricerca scientifica ed educazione perma43: 379 and references thereinnente, Bormio, Italy, 2000. 404125 Colonna M. Chomaz Ph Ayik S Phys Rev Lett, 2002, 88: 131 Chomaz Ph, Gulminelli F Nucl Phys, 1999. A647: 153132 Li B A. Ko C M. Ren Z Phys Rev Lett. 1997. 78: 1644126 Ayik S Colonna M. Chomaz Ph Phys Lett B, 1995. 353: 13.3 Muller H Serot B D. Phys Rev C. 1995. 52: 2007234 Samaddar S K. Das Gupta S. Phys Rev C, 2000. 61Moretto L G al. Phys Rey Lett. 1996. 77: 2634abacaru G et aL.(INDRA Collaboration), Proc of the 135 Ma Y G et al. Chin Phys Lett. 1999. 16: 256XXXVIll Int Winter Meeting on Nuclear Physics, ed lori 136 Ma Y G et al. Phys Rev. 1999, C60: 24607L, Moront A, Ricerca scientifica ed educazione perma- 137 Dempsey J Fet aL. Phys Rev. 1996. C54: 171ente, Bormio, Italy, 2000, 433138 Xu H S ef al. Phys Rev Lett. 2000. 85: 716129 Borderie B er aL. (INDRA Collaboration), Phys Rev Lett, 139 Shetty D V et al.(NIMROD Collaboration), Phys Rev C.2001,86:32522003,68:05460130 L Neindre n et al.(INDRA Collaboration Proc of the中国煤化工CNMHG