Spectra of various particle species have been calculated with the Quantum Molecular Dynamics (QMD) model for very central collisions of Au+Au. They are compatible with the idea of a fully stopped thermal source which exhibits a transversal expansion besides the thermal distribution of an ideal gas. How- ever, the microscopic analyses of the local flow velocities and temperatures indicate much lower temperatures at densities associated with the freeze-out. The results express the overall impossibility of a model-independent determi- nation of nuclear temperatures from heavy ion spectral data, also at other energies (e.g. CERN) or for other species (i.e. pions, kaons, hyperons

Greiner, Walter

Konopka, Jens

Stöcker, Horst

English

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:nucl-th/9505018 v1   12 May 1995On the impossibility of temperature extraction from heavy ioninduced particle spectraJens Konopka, Horst St¨ ocker, and Walter GreinerInstitut f¨ ur Theoretische Physik, Postfach 11 19 32, Johann Wolfgang Goethe-Universit¨ at,D–60054 Frankfurt am Main, Germany(February 8, 2006)AbstractSpectra of various particle species have been calculated with the QuantumMolecular Dynamics (QMD) model for very central collisions of Au+Au. Theyare compatible with the idea of a fully stopped thermal source which exhibitsa transversal expansion besides the thermal distribution of an ideal gas. How-ever, the microscopic analyses of the local ﬂow velocities and temperaturesindicate much lower temperatures at densities associated with the freeze-out.The results express the overall impossibility of a model-independent determi-nation of nuclear temperatures from heavy ion spectral data, also at otherenergies (e.g. CERN) or for other species (i.e. pions, kaons, hyperons)!Typeset using REVTEX1The study of nuclear matter under extreme conditions is the basic motivation for heavyion experiments [1,2]. The aim is the creation of a large bulk of heated and compressednuclear matter [3]. However, the temperature and the density are not straightforwardlydeterminated from these experiments. It is even not clear from the beginning, whether itmakes sense to speak about a temperature, since the degree of equilibration in a single eventis not a measurable quantity either. In this letter the relation between spectral information,which is measurable in experiments, and the density and the degree of random motion duringthe intermediate reaction stages is investigated using transport theory. The latter aspect isclosely related to the question of thermalization and the possible analogy to thermostaticproperties of heated and/or compressed matter.Single particle and N-body transport models have been used extensively to gain insightinto the stiﬀness of the equation of state, the strength of the momentum dependence and in-medium nucleon-nucleon scattering cross-sections [4–8]. These calculations predict variousobservables in terms of diﬀerent underlying assumptions. The question is, which of theseaspects is dominating the dynamics of heavy ion collisions. This procedure is in fact veryindirect. It would be more convenient to have a direct measure of thermodynamical quan-tities, like pressure or temperature, which could be connected to the properties of excitednuclear matter.As an idealized test case, we performed QMD calculations [4,5,9] of a heavy systemfor b=0, namely Au (150 MeV/nucleon) + Au. Such reactions are supposed to yield thelargest accessible participant region, where the nucleons involved undergo a rapid sequenceof binary collisions. Consequently, equilibrium, with a large number of nucleons involved,is more likely established in central collisions than in peripheral reactions or collisions oflighter partners.In the present analysis no information other than mass, charge and four-momentum ofthe emitted products will be used for extraction of further information on the propertiesof the hot and dense nuclear matter formed in the transient state. Thus such an analysiswould be applicable to experimental data as well [10,11].2After a full QMD propagation for 300 fm/c, involving all nuclear interactions and hardbinary scatterings, fragments are calculated via a 3 fm conﬁguration space coalescence.Then, in addition, all charged products are propagated on their mutual Coulomb-trajectoriesfor another 106 fm/c ≈ 3 · 10−18 s.Having obtained the full triple-diﬀerential spectra in such a manner for each chargeindependently, the spectra were ﬁtted by a transversally expanding, thermally equilibratedsource [12].This implies that the transverse spectrum is composed of a directed ﬂow and a randomcomponent, which can be associated with a temperature. The strength of the collectivecomponent is assumed to increase linearly with the transverse distance; the shape of thesource is a homogeneous sphere. Nonrelativistically the double-diﬀerential distribution readsd2f(T,pmax)dpl dpt= N2pmaxZ pmax0dp0p0pmax￿ηπ￿322π pt×exp{−η(p2t + p2l + p20)} I0(2ηptp0) (1)Here η is an abbreviation for A2mNT. With these formulas the average ﬂow momentum pernucleon is23pmax and the averaged ﬂow kinetic energy amounts to pmax/4mN.In order to get information on the thermal collective energy sharing, we restrict our anal-ysis for the moment to the transverse momentum spectra only. This procedure is motivatedby the fact that transverse momenta are newly created and are not directly connected tothe initial (longitudinal) beam momentum. A possible collective expansion in longitudinaldirection cannot unambiguously be associated to the properties of the hot and dense reac-tion zone because the system may have memory about its history, in particular the incidentmomentum.Fig. 1 shows transverse momentum spectra of various fragment charges obtained withQMD for the system Au (150 MeV/nucleon, b=0) + Au (symbols) together with ﬁts to thesecalculated data, which are based on the assumptions from above. In fact, the correspondingcount rates have been ﬁtted, rather than the invariant distributions, which are displayed. Allspectra are compatible with temperatures between 20 and 25 MeV and averaged collective3ﬂow velocities of 0.1–0.13 c. Fig. 2 shows the minimum of the χ2 distributions associatedwith the various ﬁts, which exhibit a valley over a wide range of temperatures. For heavyfragments such as α’s or even heavier species, the minimum is well deﬁned. It is representedin the by the bullet: For this class of probes the collective component can be extracted for thismodel in an unambiguous way: In the valleys of the χ2 distributions the corresponding ﬂowmomentum depends only weakly on the temperatures. On the contrary, for light particleslike free protons and deuterons one observes that1) the minimum is not as pronounced as in the case of the heavy composite fragments,particularly the ﬁt for p’s is of considerable minor quality and is not unique2) the valley extends for p’s like a banana from low temperatures with large ﬂow to hightemperatures with almost no ﬂow with only a slight gradient. This implies that the heavyfragments clearly exhibit collective ﬂow in these ultracentral collisions, while the ﬂow cannotunambiguously be determined with light fragments. Collective ﬂow is here associated withazimuthally symmetric collective expansion.Fig. 2 shows the line corresponding to energy conservation When the ﬁt procedure workswell, the minimum of the χ2 distribution is in line with energy conservation. However, this isnot the case for protons, which is a ﬁrst indication that the proton emission pattern cannotbe understood in terms of the simple ansatz of collective ﬂow + thermal motion, althoughthe proton spectrum in Fig. 1 seems to be properly ﬁtted.Fig. 2 indicates that the heavier fragments exhibit smaller averaged transverse collectiveexpansion velocities: In the model these fragments form via the conﬁguration space coa-lescence mechanism in the interior of the expanding nuclear matter. On the surface of theexploding system, where the local expansion velocity is highest, it is less probable to formheavy fragments, since the number density of nucleons around is much smaller than in theinterior.Let us summarize the analysis of the ﬁnal spectra, apparently the momentum spacepopulation of all fragment species is reasonably well described by two parameters, the tem-perature T and an averaged collective velocity vc. The spectral information suggests tem-4peratures larger than 20 MeV. Since this depends only on the ﬁnal state of the collision,when all interactions on the nuclear length scale have ceased, it is concluded that the ashesof the exploding nuclear matter seem to indicate these high temperatures. The associatedaverage collective velocity amounts to 10-15% of the speed of light. Quantitatively similarconclusions have been obtained from measured transverse momentum and global kineticenergy spectra in central collisions of Au+Au at 150 MeV/nucleon [10,13,14]. However, adetailed analyses of the nuclear chemistry, i.e. the fragment abundances, in the very sameexperiment yields low temperatures of the order of 8 MeV to account for the large fragmentabundances [15]. This has lead to speculations on diﬀerent time scales for the chemical andthe dynamical equilibration processes. The temperatures determined with the two methods,chemical determination from isotope ratios or fragment abundances on the one hand side anddetermination from the fragment spectral slopes on the other hand can not be reconciled inthis scenario. We demonstrate below that this is due to misconceptions of the interpretationof yields and spectra in an instantaneous break-up of a homogeneous, globally equilibratedsystem.QMD allows for a microscopic calculation of all quantities relevant to the reaction dy-namics and the nuclear thermodynamics. The time evolution of the composite system isfollowed in more detail [12] by dividing conﬁguration space in various cells on a cylindricalgrid. For each of these cells the mean and the spread of the local velocity distribution iscalculated for a superposition of many events, obtained under the same macroscopic condi-tions. The mean velocity illustrates the collective velocity of each individual cell, whereasthe spread around the mean reveals the thermal, i.e. random, motion inside the cell. Thedirectional dependences of the temperature parameters (longitudinal, transversal, and az-imuthal temperatures T , T⊥, Tϕ) have also been extracted. They correspond to the widthsof the local velocity distribution in the directions parallel to the beam, perpendicular to thebeam, and in azimuthal direction. These values are equal in each cell in the case of localequilibrium. If the temperatures were, in addition, constant over the full volume, this wouldbe global thermal equilibrium. The complete information on the local collective velocities,5and on the degree of randomness is available. It tests our initial assumption of a linearlyincreasing ﬂow velocity proﬁle and we can analyze to which extend global or local thermalequilibrium is established in the model.In the following we characterize the central reaction zone in terms of the global param-eters density and temperature, which are important for the (multi-)fragmentation of thesource. The central reaction zone is deﬁned here as the volume of all cells with density atleast half of the maximum density at this instant. This deﬁnition implies a time-dependentvolume, which is considered here.A considerable fraction of the nucleons remains in the central volume (Fig. 3 a)): evenafter 100 fm/c about one third of the system is still in the central zone. Matter outsidethis zone has experienced the higher densities earlier on. In terms of a hydrodynamicalpoint of view, matter in the QMD model freezes out at diﬀerent times. The temperaturedistributions indicate rapid cooling and prove that the source of nucleons and clusters is notglobally equilibrated, even for the most central collisions in microscopic models (Fig. 3 b)).However, in the longitudinal direction the spread of the velocity distribution is higher by50% in the intermediate reaction stages (t=40–60 fm/c), while the system is transversallyequilibrated. This cooling is associated with a decrease of the density too (Fig. 3 c)).Fig. 4 shows that at low densities (< 0.5ρ0), where the fragment freeze-out is commonlyassumed to occur, the microscopically determined true (local) temperatures (symbols) havedropped below 10 MeV. Thus they strongly deviate from the slope temperatures extractedfrom the simple macroscopic ﬁt to the (microscopically calculated) fragment spectra (lightgrey shaded area). The values compare reasonably well with curves of constant entropyobtained with the quantum statistical model (QSM) [16] and with a QSM analysis of exper-imental fragment yields [15] (dark shaded area).This explanation of the large discrepancy between the temperatures extracted from thechemical analysis and the temperatures extracted from the fragment spectra puts into ques-tion all temperature extractions from slopes of particle spectra, even if an additional collec-tive motion is taken into account. This was done, for example, in our macroscopic analysis6above. The values are 2–3 times higher than the correctly extracted microscopic values.This ﬁnding concerns also temperatures extracted from slopes of pion spectra at highestincident energies, e.g. at the AGS and CERN [17].What is wrong with the assumptions underlying the global ﬁt? A thermally equilibratedsource at some temperature T can disintegrate not only due to the thermal pressure, but alsodue to some additional collective expansion. The probability of ﬁnding a particle which hasreceived a collective velocity vcoll., i.e. dN/dvcoll., is essentially unknown. It can be expressedasdNdvcoll.=dNdr·drdvcoll.=dNdr· dvcoll.dr!−1. (2)Hence, the density distribution as well as the ﬂow velocity proﬁle enters as the ﬁrst andsecond term on the right hand side of the preceding equation. For a ﬁt to the spectra, ahomogeneous density with a sharp cut-oﬀ and a linearly increasing velocity proﬁle have beenused [18].Therefore, the high transverse momenta components of the particle spectra do not cor-respond to the high momentum tails of hot source, but to the collective motion of fast cellswith low internal temperature. The microscopic analysis suggests that the combinations ofdensity and temperatures, which are traversed in the course of the reaction, are in agreementwith the expectations from the quantum statistical analysis of the fragment distributions inthe ﬁnal state [15].In summary, temperatures obtained from slopes of particle spectra can only serve asan upper estimate for the conditions at freeze-out, even if collective ﬂow has been takeninto account. The main origin of this uncertainty is the fact that the conﬁguration spacedistribution is essentially unknown. In view of this analysis, model estimates of temperaturesin high-energy and ultrarelativistic heavy ion collisions [17] are expected to overestimate thethermal energy reached in such reactions considerably. Our analysis suggests that chemicalthermometers, e.g. π/p ratios, which take into account feeding from unstable states andabsorption may be more appropriate.7ACKNOWLEDGEMENTThis work was supported by BMFT, DFG, and GSI.8REFERENCES[1] H. St¨ ocker and W. Greiner, Phys. Rep. 137, 277 (1986).[2] R. B. Clare, D. Strottmann, Phys. Rep. 141, 177 (1986).[3] Proceedings on the NATO Advanced Study Institute Programme on Hot and DenseNuclear Matter, eds. W. Greiner, H. St¨ ocker, and A. Gallmann, NATO ASI Series BVol. 335, Plenum, N.Y. (1994).[4] J. Aichelin et al., Phys. Rev. C 37, 2451 (1988).[5] G. Peilert et al., Phys. Rev. C 39, 1402 (1989).[6] C. Gale et al., Phys. Rev. C 35, 1666 (1987).[7] B. Bl¨ attel et al., Rep. Prog. Phys. 56, 1 (1993).[8] A. Faessler, Prog. Part. Nucl. Phys. 30, 229 (1993).[9] G. Peilert et al., Phys. Rev. C 46, 1457 (1992).[10] J.P. Coﬃn et al, Proc. of Nucleus-Nucleus Collisions V, Taormina, Italy, 1994, eds. M.di Toro, E. Migneco, and P. Piattelli, Nucl. Phys. A 583 567c (1995).[11] M. A. Lisa et al., LBL-Preprint 35504 UC-414 (1994).[12] J. Konopka et al., Proc. of Nucleus-Nucleus Collisions V, Taormina, Italy, 1994, eds. M.di Toro, E. Migneco, and P. Piattelli, Nucl. Phys. A 583 357c (1995).[13] S.C. Jeong et al., Phys. Rev. Lett. 72, 3468 (1994).[14] W. Reisdorf for the FOPI-Collaboration, Proceedings of the XXII Workshop on GrossProperties of Nuclei and Nuclear Excitations, Hirschegg 1994, p. 93.[15] C. Kuhn et al., Phys. Rev. C 48, 1232 (1993).[16] D. Hahn and H. St¨ ocker, Nucl. Phys. A476 718 (1988).9[17] E. Schnedermann and U. Heinz, Phys. Rev. Lett. 69, 2908 (1992).[18] H. St¨ ocker et al., Z. Phys. A 303, 259 (1981).10FIGURESFIG. 1. Invariant transverse momentum distributions of various charged fragments emergingfrom Au (150 MeV/nucleon, b=0) + Au collisions. The lines indicate ﬁts to the calculated pointsinvolving a azimuthally symmetric, transversally expanding, and thermally equilibrated source.The ﬁts have been performed for each fragment specie independently. The associated count ratesrather than the displayed invariant distributions have been ﬁtted. All spectra are compatible witha temperature between 20 and 25 MeV and averaged collective ﬂow velocities between 0.10 and0.13 c.FIG. 2. χ2 distributions for the two parameter ﬁt to the spectra in Fig. 1. Contourlines areat 2, 3, 4, and 5 times the minimum value. The minima are indicated as black circles. The thicksolid lines correspond to those pairs of parameters where the total kinetic energy is conserved. Forthe case of protons, the quality of the ﬁt in terms of χ2 is poor. The energy constraint tends topredict lower temperature and larger ﬂow.FIG. 3. Thermodynamics in the central reaction zone, i.e. the volume where the local densityis at least half of the maximum value. Mass content a), spreads of the local velocity distributionsb), and maximum as well as averaged density c) are displayed at a function of time.FIG. 4. Temperatures versus densities for central Au+Au collisions at 150 MeV/nucleon. Themicroscopically determined local temperatures (symbols as in Fig. 3) are considerably lower thanthe simple macroscopic ﬁt to the spectra (light grey shaded area). However, they compare reason-ably well with curves of constant entropy from QSM and with experimental data from a quantumstatistical analysis of the fragment distributions (dark grey shaded area) [15].110.0 0.2 0.4 0.6 0.8pt/A [GeV/c]10-12510025101251022510325A2/ptdN/dpt[1/(GeV/c)2event]Au(150 MeV/nucleon, b=0)+AuZ=7-9Z=4-6Z=3 ( 2)Z=2 ( 2)d ( 2)p0.0 0.2 0.4 0.6 0.8pt/A [GeV/c]10-12510025101251022510325A2/ptdN/dpt[1/(GeV/c)2event]QMDJ. Konopka et al., Fig. 1120.050.100.150.200.25p0[GeV/c] pbad2d Z=215 20 25 30T [MeV]0.050.100.150.20p0[GeV/c] Z=315 20 25 30T [MeV]Z=4-615 20 25 30 35T [MeV]Z=7-9J. Konopka et al., Fig. 2130100200300400AAu+Au 150 AMeV, QMDa)010203040T[MeV]TTT b)40 60 80 100t [fm/c]0.00.51.01.52.0/0max< >c)J. Konopka et al., Fig. 3140.0 0.2 0.4 0.6 0.8 1.0/ 0051015202530T[MeV]05101520253080 70 60 50 40time [fm/c]11.522.5S/A=3J. Konopka et al., Fig. 415

On the impossibility of temperature extraction from heavy ion induced particle spectra

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