Flow, particle yields, equilibration in heavy ion reactions between 0.1 and 2 A GeV

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A630 (1998) 15c-26c

Flow, particle yields, equilibration in heavy ion reactions between 0.1 and 2 A GeV W. Reisdorf, ~ ~Gesellschaft ffir Schwerionenforschung mbH, Planckstrafie 1, D64291 D a r m s t a d t , Germany 1. I N T R O D U C T I O N The nuclear equation of state (EOS) represents the dependence of pressure P(T, p) or alternatively of the energy per nucleon E/A(T,p) on t e m p e r a t u r e T and density p. It is a bulk property of nuclear matter. When using nucleus-nucleus collisions to infer the EOS several complications arise. First, as in the well known mass formula, the binding energy per nucleon is heavily modified by surface, Coulomb and a s y m m e t r y effects. For the ground state energy we know that these effects modify the bulk or volume term ( - 1 6 MeV) by a factor of two (to - 8 MeV) and hence are not small. Second, one is in a dynamic situation and hence has to cope with phenomena and properties that are of only minor interest to the 'equilibrium physicist': a) nucleons impinging with high velocity on a nucleus experience a repulsive m o m e n t u m dependent force; b) to understand a non-equilibrium situation, knowledge about the transport coefficients, such as viscosity (which in elementary kinetics is inversely proportional to nucleon-nucleon cross sections), is needed. The m o m e n t u m dependent repulsion leads to sidewards deflections in the early nonequilibrium phase of the collisions and therefore limits the m a x i m u m overlap density achieved in a nucleus-nucleus collision. Viscosity can limit the maximal pressure achieved and certainly is relevant in determining the so-called 'fi'eeze-out' conditions when all particles cease to interact and move towards the detectors. A third subject that has lead recently to increasing theoretical and experimental activity is that of quasiparticles moving in the nuclear medium: under the influence of the surrounding baryonic m a t t e r mesons may change their effective mass which modifies the thresholds for their production, and they will experience a mean field which will influence their propagation. The observables (besides the obvious triple-differential m o m e n t u m distributions of all e m i t t e d particles) that we dispose of to try to unravel the complex collisions and extract the 'fundamentals' can be divided roughly as follows: 1. collective particle flow, as a response to pressure and field gradients 2. clusterization as a response to expansion and cooling 3. produced particles; at intermediate energies these are primarily pions, as signals of nucleonic excitations and, much rarer, strange particles which have recently emerged as signals of medium modifications of hadrons 0375-9474/98/$19 © 1998 Elsevier Science B.V. All rights reserved. PII S0375-9474(97)00740-9

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4. dileptons as penetrating probes of the high density stage 5. two-particle correlations as signals of the space-time evolution. It is also for space and time reasons that this talk will be limited primarily to the first two subjects and touch briefly on the third, leaving the last two observables to other contributions to this conference.

2. R a d i a l flow There is a hierarchy of models that have been used to connect heavy ion d a t a with the equation of state. The simplest model is the thermal model in which one assumes optimistically that in head-on collisions a fireball is created which freezes out immediately after having achieved chemical and thermal equilibration. This is a one source model. The hydrodynamical model allows for the introduction of a continuuum of locall 9 equilibrated moving sources which is at the origin of the concept of 'flow'. Topping this simpleminded hierarchy, are 'microscopic' transport models which give up the local equilibrium assumption still present, in at least the one-fluid versions of the hydrodynamic approach. In fluid dynamics [1,2] one follows the evolution of many subsystems, 'cells', that are driven by field and kinetic pressure gradients (hence the direct connection to the EOS) while overall conserving energy, charge and momentum. After a brief compression time the system expands again. At some time the interactions cease ('freeze-out') and, aside fi'om Coulomb acceleration, the particles have acquired the velocities that will be seen in detectors. For non-central collisions the final configuration can be roughly divided into 'spectator' cells plus a central part, the fireball. One can select a central collision by demanding a maximal conversion of the initially longitudinal kinetic energy into transverse energy plus a high degree of axial s y m m e t r y of the event topology. In favourable cases (large systems, such as Au on Au) essentially only the fireball part is left. This fireball is not enclosed in a box (thermal model) but is expanding in much the same way as the universe. If we sit in the center we see those 'cells' moving fastest that are the furthest away, giving rise to the famous red shift characterized by the Hubble constant. If we sit outside (detectors) we see a blue shift. At freeze out the configuration of nucleons is expected to look roughly as shown in Figure 1: there is a Woods-Saxon like density profile with a bulk part, having subsaturation density and an extended even lower-density surface part. Within the bulk part the flow velocity is rising linearly. W i t h such a scenario, and assuming that the local temperatures are not varying much over the configuration, one can describe kinetic energy spectra [3] of e m i t t e d clusterized nucleons (Figure 2). A characteristic feature is that in contrast to a pure Boltzmann scenario (no flow) one sees curved mass dependent spectra. Those of heavier particles show the blueshift in a spectacular way as they are less sensitive to thermal fluctuations that sit on top of the flow patterns and tend to wash out the flow features. Another way of showing evidence for radial flow are the mass-dependent average kinetic energies (Figure 3). In a thermalized system enclosed in a box, rather than expanding, the kinetic energies should be independent on mass. Because of their small sensitivity to thermal smearing heavier particles are well suited to constrain flow profiles. The assumption of realistic Woods-Saxon type density profiles

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scaledrapidity Fig. 4. Comparison of rapidity distributions for fragments with Z=5,6 with various flow profile assumptions (Au+Au at 250 A MeV). (with a diffuseness about twice that of cold nuclei) is in accordance with measured rapidity distributions (restricted to low transverse momenta), see Figure 4, while box like density profiles (limited to the bulk part in Figure 1) predict too broad distributions and the assumption of a shell-like freeze-out (with a single flow velocity [4]) predicts two-humped distributions not supported by the data.

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It is very likely that the flow profiles of single nucleons are even more diffuse than those of elusterized matter. Fragments are formed by coalescence of nucleons and require a minimum density to be formed fast enough in the transient system. They are therefore most likely formed by fluctuations in the bulk part in Figure 1. As a consequence of the large surface tails the averafle flow velocity of nucleons and light particles will be larger. This is suggested in Figure 2 where the spectra for charge one particles are seen to be harder than predicted by a scenario (solid lines) ignoring this effect. The systematics of radial flow expansion is still in a somewhat unsatisfactory state, in part because various au120 © Au+Au Z= 1,2 (EOS) TI thors use different methods to extract - D,u+AuZ , { { the flow from the data, in part because ~" q~ 80 the event topologies are not as sim• Ni+Ni pion,p, d ple (i.e. isotropic) as assumed in most ~ 40 analyses, and in part because the sysl l ,l,I I , I ll,,,I , I,I tem size dependence is not yet sufficiently investigated. Most data in the ~ 0.4 energy range discussed in this talk are for the system Au on Au. ~ 0.2 Figure 5 shows the incident energy dependence of average flow velocities and 10 10 apparent temperatures. At beam energies per nucleon EB/A below 500 MeV beam energy (MoWA) flow velocities deduced including heavy Fig. 5. Expansion systematics: temperaclusters [3] (open squares in the Figure) tures (upper panel) and average flow velociare largest and correspond to about ties. 60% of the available total kinetic energy. At these energies the extraction of radial flow from light charged particles only (Z < 3) [5,6] is difficult as evaporation fi'om heavier primary clusters at late times severely distorts the spectra. At EB/A > 500 MeV [5] heavy clusters become very rare and flow studies have been limited to hydrogen and helium isotopes which are less sensitive to details of flow profiles. On the other hand evaporation should no longer be a problem. Despite the different methods used, it appears from the data obtained so far that the radial flow velocity tends to saturate around 1 GeV somewhat below 0.4c. This could be connected with a changing event topology (anisotropy) and definitely requires further investigations. The temperature, T, profile (upper panel in Figure 5) seems to be smoother, rising monotoneously for EB/A = 0.1 to 2 GeV. Analyses where T is a free parameter (in addition to the average flow velocity flf [5]) generally yield less accurate values because of strong anticorrelations of T with fir. The analysis in Ref. [3] was constrained by energy conservation taking into account flow and particle multiplicity. Recently data for a smaller system (Ni+Ni) have become available [7] (full circles in the Figure) from 1 to 2 GeV. While T continues to rise, it is found (at 1 GeV) that flow, if measured at 90 ° in the c.m., is smaller in the lighter system. The presence of radial flow dearly signals that a certain pressure has been built up J

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W. Reisdorf/Nuclear Physics A630 (1998) 15c-26c

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and that subsequently thermal energy was converted by an at least partially adiabatic expansion (see later) into collective flow. The amount of flow is expected to be primarily dependent on nuclear viscosity which determines both how much pressure can be built up in the highest density stage and later at what density freeze out (and hence the end of the cooling mechanism) occurs. From simple hydrodynamic estimates, neglecting viscosity and using Rankine-Hugoniot equations (see for example [S]) one can however deduce that the 'stiffness' of the EOS, or incompressibility, will influence radial flow only moderately: while a stiff EOS creates a higher pressure gradient at fixed maximal density, this is almost entirely compensated by the fact that a soft EOS will allow for higher densities. Calculations [5] with less restrictive microscopic models confirm this insensitivity. Nevertheless the observation of radial flow represents an important message as its partial or total absence would signal a strong conflict with hydrodynamic thinking and futhermore, as we shall see, as a cooling agent necessary to try to understand the chemical composition of the fireball at 'detection time'.

3. C l u s t e r i z a t i o n

The expansion will lead to cooling which in turn initiates condensation of nucleons into clusters. Figure 6 shows that in the energy regime from 100 to 1000 MeV coalescence of nucleons in central collisions is not an exception but rather the rule: even at 1 GeV about 50% of the protons occur in clusters. Clearly then, coalescence is not just a perturbation but should have a massive back-influence on the spectra of those protons that stay 'single'. At the present time the degree of clusterization in central collisions is not really understood. For Au + Au at 250 A MeV the measured trend of the charge distribution per event, dM/dZ (Fig.6), is severely underestimated by statistical model calculations (in this case the Quantum Statistical Model QSM [9]), despite the fact that the large collective flow energy was subtracted to obtain the energy available for thermalization. For the asymmetric system K r + A u at 400 A MeV [10] dM/dZ could be reproduced only in the framework of the statistical model if it was assumed that fragments come from an equilibrated system involving just one third of the total system mass and in addition only about one third of the excitation energy per nucleon (i.e. 1/9 of the total available energy) was 'thermalized'. In figure 8 we show that quasiclassical molecular dynamics calculations [3,11] also fail to predict the relatively large number of observed intermediate mass fragments (IMF, Z > 2) in central collisions. Further information relevant to this puzzling problem is shown in Fig. 9 where the system size dependence of IMF multiplicities is studied ( A u + A u vesus Ni+Ni). The Ni data have been multiplied by the ratio of the Au to Ni masses. For large rapidities (i.e. spectator dominated fragments) one finds [12] the 'universality' seen by the ALADIN collaboration [13] (right panel in Figure 9), while closer to midrapidity the relative clusterization is decreased in the lighter system, showing the importance of surface to volume ratio in the fireball physics. The failure of the hydrothermal model, Fig. 7, could perhaps be attributed to the fact that QSM does not provide for a vapor and liquid mixture. Two rather different entropies have been extracted from yield analyses restricted to light charged particles (Z = 1,2)

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Physics A630

(1998) 15c-26c

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[14] and analyses including heavy clusters [15]. This could indicate the presence of two phases. We might just be experiencing the making of a quantum fluid. The lack of full quantum features in the molecular dynamics calculation (Fig. 8) might on the other hand explain the failure of the dynamic model. Fully antisymmetrized molecular dynamics codes are now gradually able to handle large systems [16] and will be interesting to follow in the future. A faster bypass to the computation problem might be the use of socalled Quantum-Langevin calculations [17] which try to implement quantum fluctuations into classical dynamical approaches using Monte Carlo techniques. Such calculations predict

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W. Reisdorf/Nuclear Physics A630 (1998) 15c 26c

a much higher degree of clusterization. A very interesting feature coming out of the antisymmetrized theory is the apparent sensitivity of the multifragmentation to the assumed EOS [16] which was already speculated upon in [3]. o

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The possibility of non-equilibrium processes raises the question if there are more direct ways to check whether the necessary conditions for equilibration are fulfilled. For this purpose a so called 'mixing' experiment was performed at GSI. Four systems were studied in which mass symmetry was kept, but isospin asymmetry was changed: 96Zr+96Zr(Z=40), 96Rn+96Rn(Z=44), 96Ru(beam)+96Zr, and 96Zr+96Ru(target). An isospin mixing map of a large portion of phase space was determined using the yield ratio R =3H/3He. The use of all four reactions (nuclear physics-wise the last two are of course identical) allows one to both 'calibrate' isospin distribution, using the symmetric systems, and to eliminate apparatus distortions (such as forward-backward asymmetries), using the asymmetric system in 'normal' and 'inverted' kinematics. We introduce the relative ratio (2R - Rz,. - R n u ) / ( R z r - RR~) which is +1 for Zr+Zr (R = Rz~) and - 1 for R u + R u (R -- RR,). Making a cut at an intermediate scaled rapidity y(0) (which is 4-1 for projectile/target rapidity), we find [18] the result illustrated in Fig.10 where the relative ratio is plotted versus the centrality of the selected events (which is highest in the bin denoted ERAT5). Due to additional low transverse-momentum cuts, we believe that 'spectator' contributions should be negligible in this part of phase space (for exact midrapidity particles, i.e. y(0) = 0, the effect should be, and was, zero). The conclusion is spectacular: within the present statistics (this preliminary result was obtained using only a small sample of the available data), 'participant' matter is found 'fully mixed' only if there is no significant spectator matter around, i.e. for the most central collisions. The potential of such studies to inspect reaction dynamics is very promising.

4. A x i a l l y a s y m m e t r i c flow In non-central collisions the presence of 'spectator' matter (which did not experience first chance collisions because of a lack of geometrical overlap) introduces a forwardbackward, as well as an in-plane and out-of-plane asymmetry in the flow phenomena. While the fireball undergoes a three-dimensional explosion populating large portions of momentum space, the spectators remain confined (except for Fermi motion and relatively small thermal fluctuations) to the reaction plane, thus allowing to determine this plane. The spectators are however deflected to the side by the exploding fireball and by repulsive field gradients at the border between participant and spectator matter.

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Since their experimental discovery [19 21], these flow effects ('sideflow' and 'squeezeout') have been studied in great detail using various observables to quantize the flow. For the precise definition of these quantities we refer to the original literature: the quantity

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fy is the midrapidity slope of the average transverse momentum as a function of rapidity, for the related quantity FDO see [3], the angle 0~ is essentially arctan(Fy). The left panel of Fig. 11 illustrates the rise and fall of sideflow with the impact parameter (selected using the sum of transverse kinetic energies): the drop to almost zero for the most central collisions (as it must for symmetry reasons) testifies the high quality of present-day event selection. Most sideflow data are taken near the maximum. The right panel of the same figure illustrates how the 'hydrodynamic limit' of flow can be reached by observing the flow of heavy fragments, which are subject to small thermal fluctuations [22,3]. Tile left panel of figure 12 shows systematics obtained by the Plastic Ball [23], the EOS [24,25] and the FOPI collaborations. The latter data show more flow because they were obtained using IMF's. There is a clear system-size dependence of the flow which can be associated to increased viscosity due to the higher surface-to-volume ratio of smaller nuclei. Beyond about 600 A MeV incident energy, the data also show a gentle decrease of sideflow, which becomes more spectacular if one switches to logarithmic scaling of the beam energy (right panel Fig. 12) and joins up to a point measured at AGS [26]. (The gap is now being filled [27]). In this figure the particle-size dependence has been removed. The rise and fall of sideflow with beam energy is a remarkable signal of nature telling us that (repulsive) sideflow is primarily a characteristic of this particular energy range (0.1 to 2 A GeV). The very steep rise around 100 A MeV deserves to be studied in more detail in the future. Under certain restrictions [28] ideal non-viscous hydrodynamics predicts that properly scaled sideflow should depend neither on size nor energy. The fast rise around 100 A MeV is probably due to a switch from an attractive to a repulsive regime (see [29]) that is strongly connected also with the liquid-to-vapour transition; in addition we expect that viscosity (which is high at energies below 100 A MeV because of Pauli blocking) will drop allowing the fast buildup of pressure. At the high end of the energy range considered here the 'fall' might again be initiated by a combination of effects: the increasing importance of nucleonic degrees of freedom (resonances) and the shortening of the 'passing time' which diminishes the 'cross-talk' (via Fermi motion) between participants and spectators. It is then interesting to sit on top of the maximum sideflow (around 600 A MeV) and to test the sensitivity to the EOS. Figure 13 shows (for Au on Au) a recent effort [30] to unravel momentum dependent repulsions and static field effects. In the forward hemisphere the ratio of all protons (clusterized and single) emitted towards the 'positive' beam side (in the direction of the impact parameter vector) to those emitted to the opposite side is shown. For F O P I and this energy, this observable is virtually free of any apparatus distortions. By applying two cuts on the polar angle range (see the Fig.) one finds that the data for the more forward angles are sensitive to momentum dependence, while at more backward angles the sensitivity to the static part of the EOS is increased (in the Figure the labels S and H stand for hard or soft EOS with respective incompressibilities of 200 and 380 MeV, M signifies that momentum dependence is included). At present the theory used here (IQMD, see [31]) does not reproduce the data, making conclusions difficult. Clearly, high accuracy of both data and theory is required to draw convincing conclusions on the EOS: the expected effects are at best on the 30% level. Similar observations hold for the 'squeeze-out' flow [21,32] where one compares the flow of energy and particles within the reaction plane with that outside. Because of space limitations we shall not discuss this here.

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W. Reisdorf/Nuclear Physics A630 (1998) 15c-26c PARTICLES

At EB/A > 500 MeV particle production, primarily pions, becomes important. We shall touch here only very briefly on aspects that have come up in recent years. Historically, pion multiplicity has played an important role [33] as it was conNi + Ni - 1.93 AGeV sidered to be a 'primeval' quantity that is delivering its message about the high • A (p±/m > 0.5) 0.2 density phase without distortions by subsequent processes in the expansion phase, 0.1 and hence could be used to try to infer 0 ......................................................... Y ' : : ~ ............ information on the EOS of compressed matter. Transport codes that reproduce "0.1 measured pion yields, show however that E it is not so much the stiffness of the EOS, -0.2 but rather radial flow that influences and A limits strongly the finally observed mul• K* (p±/m > 0.5) 0.2 tiplicity. Suppressing the cooling mechv anism by artificially preventing position0.1 momentum correlations to develop during the expansion, Danielewicz [34] could 0 r~ show that the observed pion multiplici"0.1 ties would actually be higher by a factor ._~ no pot. _, - ] -= vector pot. only 1.7 if there was no local cooling (i.e. no "0.2 ,_: scalar+vector pot. expansion). At SIS/BEVALAC energies , I , I , -2 -1 0 it is accepted that pions are created exclusively thru excitation of nucleonic resy(0) onances and their subsequent very fast (still in the medium) decay, the A(1236) Fig. 14. Upper panel: comparison of sideplaying a dominant role. It seems that flow of A's with that of protons (open trianthe cooling and creation of fi'ee pions in gles). Lower panel: Comparison of sideflow later stages are compensating each other of K + with that of protons (histogram). Calto within about 10% stabilizing the sum culations with various KN potentials are also of A plus free pions at an early (still comshown. pressed) stage. In these theoretical ideas the lifetime of the A in the medium plays a crucial role, as the reabsorption of pions proceeds most likely again thru two steps via an intermediate A. The presence of 'free' pions and 'decay' pions at detection time is suggested by the now well established 'low-pt enhancement' (over a purely thermal spectrum) seen in transverse momentum spectra at SIS, AGS and CERN energies. Recently a direct look at resonances has become possible by identifying the A in the invariant mass peak of pionproton correlations [35,36]. This allows in principle to check if the A mass and lifetime (width) are modified in hot nuclear matter. The EOS collaboration [35] has found that the A mass is shifted down by about 70 MeV in central collisions of Ni on Ni at 2 A GeV, while its width was not dramatically modified from the free A value. For more -

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peripheral collisions the shift was smaller. Simulations with transport codes and thermalmodel considerations suggest that most of this shift is generated by collisional cooling. The thermodynamically consistent shape of the resonance is under debate [37], interesting isospin dependence of the invariant mass distribution has been observed [36] and the role of higher lying resonances on both the pion spectra and the pion proton correlations is under investigation. Systematic studies of in-medium properties of kaons have been carried out using chiral perturbation theory [38]. It was shown that the effective masses of kaons K + (antikaons K - ) increase (decrease) with increasing density of the baryonic medium, massively influencing their production thresholds and hence the observed K + / K - yield ratios (see [39]). Recent theoretical work [40] has suggested that in-plane flow and azimuthal distributions of strange particles may provide useful information on in-medium properties of these particles. Fig. 14 shows measurements [41] of A and K + sideflow and compares it to proton flow. There is a remarkable difference in the flow of these two particles despite their associated production: while the As 'flow with the nucleons', the K + show virtually no flow. Only a realistic KN potential reproduces this fact. Furthermore quasi absence of flow was also seen for K ° and K - mesons in the same reaction [41], although with less statistical significance. Due to the theoretical possibility of K - condensation in dense stellar matter [42], experimental information for the K - is particularly interesting, but difficult to obtain at energies below 2 A GeV because of low production cross sections.

6. C O N C L U S I O N We have seen that radial flow is now well established; however, a consensus on how to extract it fi'om the data and system size systematics must still be obtained. At the end of the evolution, condensation (clusterization) is occurring. We probably see the back-tail end of the liquid-gas transition, a quantum fluid in the making. We have established the rise and fall of sideflow. Concerning pions we are now getting more details about their origin from nucleonic resonances decaying in the medium. Strange particles provide a link between hadron, heavy ion and astrophysics. Perhaps we see the front tail-end of chiral symmetry restoration. The determination of the EOS of nuclear matter or at least of strong constraints on it remains an important and yet unfinished task of the field. Very accurate and systematically complete data are needed, as well as elaborate theories that cope properly with quantum mechanics (antisymmetrization), are relativistically sound (momentum dependence) and consistent with the fundamental symmetries of quantum chromodynamics (in-medium effects) while remaining applicable for practical calculations.

Acknowledgement It is a pleasure to thank my collegues at F O P I for their help and countless discussions, and in particular R. Averbeck, P. Crochet, M. Eskef, N. Herrmann, Y. Leifels, D. Pelte, B. de Schauenburg, for making some of their work available prior to publication.

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