Radial collective flow and thermalization are studied in gold on gold
collisions at 150, 250 and 400 A MeV bombarding energies with a
relativistically covariant formulation of a QMD code. We find that radial flow
and "thermal" energies calculated for all the charged fragments agree
reasonably with the experimental values. The experimental hardware filter at
small angles used in the FOPI experiments at higher energies selects mainly the
thermalized particles.Comment: 4 pages with 4 EPS figures included. Version accepted for publication
  in Phys. Rev. 

Gábor Papp

H. Feldmeier

H. Stöcker

H.A. Gustafsson

H.H. Gutbrod

H.W. Barz

J. Aichelin

J.P. Bondorf

Judit Németh

M.A. Lisa

P.J. Siemens

S.C. Jeong

W. Bauer

W. Reisdorf

W.C. Hsi

English

arXiv

Crossref

Collective flow in central Au-Au collisions at 150, 250, and
                    
                      400

Németh, J

Papp, G

ELTE Digital Institutional Repository (EDIT)

arXiv:nucl-th/9711039v3  17 Dec 1998Collective flow in central Au–Au collisions at 150, 250 and 400 A MeVJudit Németh1 and Gábor Papp1,2∗1 Institute for Theoretical Physics, Eötvös University, H-1088 Budapest, Hungary.2 GSI, Plankstr. 1, D-64220, Darmstadt, Germany.(August 10, 2018)Radial collective flow and thermalization are studied in gold on gold collisions at 150, 250 and 400A MeV bombarding energies with a relativistically covariant formulation of a QMD code. We findthat radial flow and ”thermal” energies calculated for all the charged fragments agree reasonablywith the experimental values. The experimental hardware filter at small angles used in the FOPIexperiments at higher energies selects mainly the thermalized particles.PACS numbers: 25.75.Ld, 25.75.-q, 25.70MnKeywords: Collective flow; Molecular DynamicsIn recent years central-collisions studies became a focus of attention in the intermediate energy domain (100-500A MeV) [1,2]. One of the measurables concentrated on by the experiments is connected to the flow. It is a well-known fact [3], that at larger impact parameters there is a sideward flow and a squeeze-out flow; these quantitieswere measured [4] and calculated [5]. The sideward and squeeze-out flows mostly disappear in very central collisions.However, calculations [5] predicted a large collective energy (radial flow) in central collisions and this was confirmedlater by experiments [6]. This collective energy can be visualized with a blast model [1,7], where the system expandsspherically around the center of mass.0%20%40%60%80%100%0 20 40 60 80time (fm/c)150 AMeV0%20%40%60%80%100%0 20 40 60 80time (fm/c)250 AMeV0%20%40%60%80%100%0 20 40 60 80time (fm/c)400 AMeVFIG. 1. Number of particles that collided less than three times (solid line), and without collisions (dotted line) for threeincident energies, at b = 0.5fm impact parameter, as the function of time. Initially (t = 0) the nuclei are at a distance of 2 fmfrom each other and the total overlap for free evolution would occur at 29, 23 and 19 fm/c for the three energies, respectively.Recently we deduced a momentum dependent, relativistically invariant two-body force [8], which can be appliedto QMD calculations [9]. To check the validity of the force, we made detailed calculations for central gold on goldcollisions in the energy domain 150–400 A MeV [8] and compared the results with experimental data [1]. Theagreement turned out to be highly satisfactory; even the number of the intermediate mass fragments (IMF) is veryclose to the experimentally measured value. It is worth examining what can we learn about the radial flow using thismodel.First we study to what extent one may speak about thermalization of the nucleons in central and semi-centralcollisions. The system is usually assumed to be thermalized if the nucleons collided a few (∼ 3) times. From Fig. 1one can see that for higher energies (250− 400AMeV ) only 20% of the nucleons did not collide more than twice (andmay be considered as unthermalized). At low energy (150 A MeV) this fraction amounts to 60%, due to the largePauli blocking (for 150 AMeV 65% of the possible collisions were blocked, while for 400 AMeV the blocking was only25%). We followed the path of some nucleons in collisions with 400 A MeV energy. Those nucleons which did notcollide at all (less than 3%) are generally positioned in the outer layer of the colliding nuclei.∗Present address: CNR, Dept. of Physics, Kent State University, Kent OH 44242, USA.1FIG. 2. Contour plots averaged over 360 events for 400 A MeV collisions with b=0.5 fm impact parameter. Shown arethe raw proton distribution (far left), the distribution with a θlab > 1.2◦ experimental hardware filter (center left), with anadditional θlab ∼ 21◦ filter (center right), and with an extra θlab < 30◦ filter (far right) in the scaled momentum – rapidityplane. The hardware filter around θlab ∼ 5◦ is not visible. The contours are separated by factors 1.5.Assuming that nucleons having collided at least three times are thermalized, one may conclude from Fig. 1, thatat least for high energies the thermalization rate is ∼ 80%. For a spatially homogeneous system this would mean anisotropic momentum distribution. For this reason it is interesting to examine the momentum distribution by makinga contour plot of the invariant cross section d2σ/pTdpTdy of the outgoing single protons in the two dimensional spaceof the transverse momentum scaled with the projectile momentum, ppT , and the rapidity y scaled with the projectilerapidity yp in the center of mass system. We note that in the energy domain of interest (up to 600MeV ) the scaledrapidity and scaled momentum are equal to each other within 5%. As a consequence, a thermalized distributionshould appear as circles in the y − p plane. First we examined the contour plots without experimental filters atb = 0.5fm and 400 A MeV collision energy after 150 fm/c evolution. One can see from Fig. 2, that in contrast toa thermalized system, the raw data do not show an isotropic distribution (see the most inner – highest multiplicitycontour, having a distortion factor ∼ 1.5). However, applying only the seemingly negligible small laboratory forwardangle filter (excluding particles at θlab < 1.2◦), which means the exclusion of a small number of protons only, anearly isotropic distribution is recovered. The interpretation of this result is the following: the unthermalized (notyet collided) protons leave the collision region with high energy in the forward direction; thus, the remaining protondistribution is already thermalized. Applying further angular filters does not change this behavior.FIG. 3. Contour plots for proton distributions at 150 A MeV, b=1.5 fm (left block of 2 panels) and b=7.5 fm (right block of2 panels) in the scaled momentum – rapidity plain. The left figures of each block show the raw data, while the righthand onesshow the data with the most relevant θlab > 1.2◦ and θlab < 30◦ hardware filters.Figs. 3 and 4 show contour plots for protons at 150 and 400 A MeV, respectively, for central and peripheral events.In order to compare our result with the experimental one, where the effect of the filters in the range 1.2◦ < θlab < 30◦was averaged out, we took only the relevant 1.2◦ < θlab and θlab < 30◦ filters. One can see that for central events thefiltered distributions are almost isotropic (except for the θlab < 30◦ filters), however, the raw, unfiltered values arenot. This fact is very pronounced in the case of 150 A MeV. In the case of large impact parameters the distributionsare always distorted. The same observation was made in connection with the experimental measurements [1].Experimentally the collective radial flow is determined from the kinetic energy distribution of the large fragments.However, since at 400 MeV the total mass of all large fragments for central collisions is less than 3% of all theparticles, such a method produces very poor statistics in our case. In order to determine the radial flow we consideredto possibilities: calculating the flow in the 1. gas scenario, when all the nucleons are considered to be individualobjects contributing to the radial flow; or in the 2. cluster scenario, when all the charged fragments (single protons2and clusters) are taken into account. For comparison we present results for both.FIG. 4. Same as Fig. 3 at 400 AMeV.In order to determine the radial flow of the nucleons we divided the solid angle into 32 equal pieces and calculatedthe flow velocity β(k) within each sector k (k=1,32) asu(k) =∑i∈Ωk~ri~pi|~ri|/∑i∈Ωkmi , β(k) =u(k)√1 + u2(k)(1)where u(k) is the four velocity associated to sector k and the summation is extended for all the charged particleswithin the solid angle Ωk, with mi being the mass of a fragment, while ri is the position of the charged particle at theend of the calculation (150 fm/c). We repeated the same for the gas scenario with each mi being the nucleon mass.The fluctuation of the flow velocity is less than 2% in all the 32 sections for each energy at b = 0.5fm both for thecluster and gas algorithm. Furthermore, the fluctuation of the number of particles within the sectors is at most 3-4%.The extracted values are also stable against changing the end-time of the calculation: from the freeze-out time upto twice the freeze-out time the change of the flow energy is 10% and does not change further for larger end-times.These results show that the radial flow can be determined in a very reliable way. Encouraged by the isotropy of thesystem and by the lack of other collective excitations (side flow or squeeze-out) we define a disoriented (”thermal”)energy as the rest kinetic energy1,Edo =1M∑i=1AiM∑i=1(√m2i + p2i −mi)− Eflow withEflow =1M∑i=1AiM∑i=1√m2i +(~ri~pi)2r2i−mi (2)where Ai is the mass number of a cluster, M is the total multiplicity of the clusters (and M = A, the total numberof particles, for the gas algorithm).In Table I we give the calculated flow velocity, the flow energies, and the ”disoriented/thermal” energies evaluatedboth for the gas and for the fragments (cluster). As a comparison, we give the values extracted from Ref. [1], wherethey use a blast wave model fit to the experimentally measured kinetic energies of heavy fragments. We note thatthese quantities are model dependent, and evaluating them we used the natural way for the QMD model, not theexperimental procedure. We find the agreement surprising.Finally we make a remark comparing our result to the one of the EOS [2] group. In this experiment the selectioncriteria were based on the multiplicity only and the deduced values of the radial flow velocity are considerably1The energy defined here is not exactly the thermal energy, since a thermal energy can be defined for completely thermalizedsystems only in the local rest frame. As the complicated experimental procedure obtaining the thermal energy cannot bereproduced within the given model, we consider the disoriented energy Edo to be close to the thermal one.3lower compared to the flow velocities of the FOPI experiment. In our model we find that the multiplicity trigger isnot sufficient enough for selecting the central collisions; we got a considerable amount of events contributing fromb = 4.5 − 5fm. As a result, using the multiplicity trigger, our flow velocities, β, are reduced by ∼ 15 − 20% inagreement with the EOS result. Consequently the “thermal” energy is increasing by the same amount.In conclusion, we have investigated collective radial flow in the case of Au+Au collision at 150, 250 and 400 AMeV. The results were compared to FOPI experiments [1] and we found a reasonable agreement. Furthermore theexperimental setup automatically filters out the unthermalized particles for higher energies. These results suggestthat the compressibility and the momentum dependence of the used force is highly satisfactory. The application ofour force for higher energies when particle creation is important, is in progress.TABLE I. Average flow energies (MeV), “disoriented/thermal” energy and flow velocity at 400 (upper), 250 (middle) and150 A MeV (lower part) at b=1 fm impact parameter for the gas (g) and cluster (c) assumption.Eflow EexpflowEdo Eexpth β βexpg (400) 63.2±2.5 - 37.2±3 - 0.345±0.006 -c (400) 59.7±2.5 56.8±6.3 32.7±2 32.8±6.3 0.338±0.006 0.334±0.017g (250) 41.2±1.2 - 18.2±0.5 - 0.285±0.002 -c (250) 34.8±1.0 34.0±3.9 16.8±0.6 21.5±3.9 0.264±0.002 0.263±0.014g (150) 21.6±0.5 - 12.9±0.8 - 0.21±0.002 -c (150) 18.3±0.3 19.9±2.3 10.2±0.5 12.6±2.3 0.194±0.002 0.204±0.011AcknowledgmentsWe would like to thank to Prof. H. Feldmeier for the continuous discussions and suggestions and to Prof. G.Fai for carefully reading the manuscript. One of the authors (J.N.) should like to express her thanks to Prof. W.Greiner and the University of Frankfurt and to Prof. W. Nörenberg and the GSI for their kind hospitality, during hervisits to these institutions, where part of this work was done. Discussions with W. Reisdorf and A. Gobbi are highlyacknowledged. We express our thanks to the FOPI group providing the experimental filter program. This work waspartly supported by Hungarian OTKA grant T022931 and FKFP grant 0126/1997.[1] W. Reisdorf et al., Nucl. Phys. A612 (1997) 493.[2] M. A. Lisa et al., Phys. Rev. Lett. 75 (1995) 2662.[3] H. Stöcker, J. A. Maruhn and W. Greiner, Phys. Rev. Lett. 44 (1980) 725.[4] H. A. Gustafsson et al., Phys. Rev. Lett. 52 (1984) 1590; H. H. Gutbrod, A. M. Poskanzer and H. G. Ritter, Rep. Prog.Phys. 52 (1989) 1267.[5] H. W. Barz et al., Nucl. Phys. A531 (1991) 453; W. Bauer et al., Phys. Rev. C47 (1993) R1838; W. Nörenberg and G.Papp, in “Critical Phenomena and Collective Observables”, ed. S. Costa, S. Albergo, A. Insola, C. Tuvé; World Scientific,1996, Singapore, p. 377.[6] S.C. Jeong et al., Phys. Rev. Lett. 72 (1994) 3468; W.C. Hsi et al., Phys. Rev. Lett. 73 (1994) 3367.[7] J. P. Bondorf, S. I. A. Garpman, J. Zimányi, Nucl. Phys. A296 (1978) 320; P. J. Siemens and J. O Rasmussen, Phys. Rev.Lett. 42 (1979) 880.[8] H. Feldmeier, J. Németh and G. Papp, APH Heavy Ion Physics 3 (1996) 71; H. Feldmeier, J. Németh and G. Papp,nucl-th/9804045.[9] J. Aichelin and H. Stöcker, Phys. Lett. B176 (1986) 14; J. Aichelin, G. Peilert, A. Bohnet, A. Rosenhauer, H. Stöcker andW. Greiner, Phys. Rev. C37 (1988) 2451; J. Aichelin, Phys. Rep. 202 (1991) 233.4

Collective flow in central Au--Au collisions at 150, 250 and 400 AMeV

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Collective flow in central Au-Au collisions at 150, 250 and 400 A MeV

Collective flow in central Au-Au collisions at 150, 250 and 400 A MeV

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