A simple geometrical model (often quoted in literature as the Glauber model)
of heavy ion collisions is recapitulated. It is shown that the transverse
energy distribution of heavy ion collisions follow the geometry of the
collision. An extension of the model to include rapidity and transverse mass
particle spectra is discussed.Comment: 7 page

Lietava, R.

English

arXiv

A simple geometrical model (often quoted in literature as the Glauber model) of heavy ion collisions is recapitulated. It is shown that the transverse energy distribution of heavy ion collisions follow the geometry of the collision. An extension of the model to include rapidity and transverse mass particle spectra is discussed.A simple geometrical model (often quoted in literature as the Glauber model) of heavy ion collisions is recapitulated. It is shown that the transverse energy distribution of heavy ion collisions follow the geometry of the collision. An extension of the model to include rapidity and transverse mass particle spectra is discussed

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submitted to acta physica slovacaA SIMPLE MODEL FOR TRANSVERSE ENERGYDISTRIBUTION IN HEAVY ION COLLISIONSDedicated to Ja´n Piˇsu´t on the occasion of his 60th birthdayR. LietavaDept. of Theoretical Physics, Comenius University, Bratislava, Slovakia,CERN, Geneva, SwitzerlandA simple geometrical model (often quoted in literature as the Glaubermodel) of heavy ion collisions is recapitulated. It is shown that the trans-verse energy distribution of heavy ion collisions follow the geometry of thecollision. An extension of the model to include rapidity and transversemass particle spectra is discussed.1 IntroductionDetailed studies of nucleus-nucleus interactions require the use of detailed eventgenerators such as QGSM, RQMD or VENUS. These models incorporate manyfeatures, and it can be difficult to understand the origin of relations betweenvariables. In the following, we describe a simple geometrical model relating thenumber of collisions to the cross section.We will assume a Glauber model. This model is widely used in heavy ioncommunity [1]-[10]. Present estimates of the collision centrality by the WA97and NA50 collaborations are based on this kind of model. The physical ideasbehind the model are found in [11].2 ModelA good introduction to the model is given in [12]. Here we recapitulate the basicassumptions of the model. The nucleus-nucleus collision is described in termsof nucleon-nucleon collisions. The nucleons of the nucleus with mass number Aare distributed in nucleus with probability density ρ(~b, z) normalised to unity.It is convenient to introduce the thickness function for a nucleus A asTA(~b) =Zρ(~b, z)dz12 R. LietavaIf the nucleus A collide with a nucleus B at impact parameter b the probabilityof having at least one inelastic nucleon-nucleon collision isp = σinZTA(~s)TB(~b − ~s)d2~sThe probability for the occurrence of n inelastic baryon-baryon collisions inthe A-B collision is thenP (n,~b) =ABnpn(1 − p)AB−nIt is easy to represent above formulas by a Monte-Carlo program: the nucleons in nucleus A are generated according the probability densityρA(~b, z) the nucleons in nucleus B are generated according the probability densityρB(~b, z) nucleons from nucleus A are “collided” with nucleons from nucleus B. Thecollision of nucleon i in nucleus A at coordinates (~biA, ziA) with nucleon ofnucleus B at coordinates (~bjB, zjB) takes place if j~biA −~bjBj <p(σin/pi)The number of participants Npart (sometimes they are called “wounded nu-cleons”) is defined as the number of nucleons in the target and projectile whichhave at least one inelastic collision. The number of participants at impact pa-rameter ~b isNpart = AZd2~sTA(~s)1 − (1 − σinTB(~s −~b))B+BZd2~sTB(~s −~b)(1 − (1 − σinTA(~s))A.The number of collisions is the number of binary nucleon-nucleon collisions.The number of collisions at impact parameter ~b isNc = ABσinZTA(~s)TB(~b − ~s)d2~sBoth the number of participants and the number of collisions are easilycalculated using a Monte-Carlo code.Simple model 33 Relation with Transverse EnergyIt is not possible to relate the geometrical model described above to physicallyobservable quantities without further assumptions. To proceed we shall maketwo assumptions both based on empirical observations:The number of produced particles is proportional to the number of par-ticipantsThe number of produced particles is proportional to the transverse energyin a given phase space window.The NA35 collaboration has studied multiplicity distributions in SS collisionsusing detailed simulations in terms of the FRITIOF and VENUS models [13].In both models they find that the average charged multiplicity per participantis approximately constant as a function of the number of participants. Thesimple proportionality of < n− > and < Np > was also assumed by Bia lasin the wounded nucleon model [1] and a similar result is obtained in the DualParton Model [14]. Comparison of several detailed models (FRITIOF, DTNUC1.02, VENUS 4.12) with data is done in [15]. The proportionality between thetransverse energy and number of produced particles was discussed in [16].As an example of measured transverse energy distribution, fig.1 shows thedifferential cross section of the transverse energy produced in Pb-Pb and S-Aucollisions at central rapidity as measured by NA49 collaboration [17].The differential cross section of the number of participants for the Pb-Pband S-Au collisions calculated in our model is shown in fig.2. The density of thenucleons in nucleus was taken to be uniform in a sphere of radius R = 1.2A1/3.The inelastic cross section is σin = 30 mbarn. Comparing fig.1 and fig.2 it isseen that energy per participant is about 1.5 GeV. This number fixes the y-scaleabsolutely. Knowing that the number of generated events in fig.2 was 106 andthat they were generated in a circle with the area 6.35 barn we see that theagreement of the cross section (measured in barn/GeV) is good.We also present a variation of this model based on [7, 10]. We assume thatparticles are produced in each inelastic nucleon-nucleon collision. When travers-ing the nucleus, a nucleon loses its energy. We assume a simple law according towhich the momentum of nucleon in the CMS nucleon-nucleon system is reducedby by a factor 1.7 (pold/pnew = 1.7) in each collision. This corresponds to thelost of about 0.5 units of rapidity per collision. The number of charged particles(normalized to 1. atp(s) = 20.) emitted in each collision is then calculatedaccording to the empirical rule:n(p(s)) = ln(p(s)/2.3)/ ln(8.7)where n(2.3) = 0. and n(20) = 1. , note thatp(s) = 20GeV is the SPS heavyion energy for sulphur-sulphur collision andp(s) = 17GeV is SPS energy forlead-lead collision.4 R. Lietava10-610-510-410-310-210-110 200 400 600 800Fig1 1. Dierential cross section of the transverse energy produced in Pb-Pb and S-Auat central rapidity as measured by NA49 coll. [17]In fig. 3 the differential cross section of the quantityw =Xcollisionsn(s) (1)is plotted. The choice of this quantity is motivated by the fact that if there is nodegradation of the energy the quantity w is equal to the number of collisions.Comparison of fig.3 and fig.2 shows that the modified model is also able toreproduce the data.4 ConclusionsWe show that the transverse energy and charged particle distributions of heavyion collisions are determined by the geometry of nuclei collisions. Assuming thatthe number of produced particles and the transverse energy are proportional tothe number of participants the model is able to describe the data for differentheavy ion systems.This model allows us to estimate the number of participants and impactparameters corresponding to different centrality triggers [18]. Models basedon the same principle were used to quantify the centrality of the collision inthe WA97 [19] and NA50 [20] collaborations. (The NA49 collaboration uses aSimple model 511010 210 310 40 50 100 150 200 250 300 350 400Npartdσ/dN part (arbitrary units)Fig 2. Dierential cross section of the number of participants in Pb-Pb collisions(crosses) and S-Au collisions (histogram)different approach as they are able to measure the number of participants usingeither a zero degree calorimeter or the number of measured protons [21].The phenomenological observation that the number of produced particles isproportional to the number of participants can be obtained in a modified modelwherenucleons loose their energy in each collision when traversing the nucleusparticles are produced in each nucleon-nucleon collision in amount depen-dent on the CMS energy of the nucleon-nucleon collision.The extension of the modified model to include the rapidity and transversemomentum spectra of produced particles is straightforward.5 AcknowledgementI am indebted to Orlando Villalobos Baillie for many useful comments to themanuscript of this paper.References6 R. Lietava11010 210 310 40 50 100 150 200 250 300 350 400wdσ/dw (arbitrary units)Fig1 3. Dierential cross section of the transverse energy in modied model, forexplanation see text.[1] A. Bia las, A. Bleszynski, M.Czyz : Nucl.Phys.B 111 (1976), 461; .[2] N. Pisutova,P.Lichard, J.Pisut: Phys.Lett.B 172 (1986), 451;N. Pisutova, J.Pisut: Acta Physica Polonica B 18 (1987), 177; ,J.Ftacnik, K.Kajantie, N.Pisutova, J. Pisut: Phys. Lett. B 196 (1987), 387;[3] G.Baym, P.Braun-Munziger,V.Ruuskanen: Phys.Lett.B 190 (1987), 29;[4] A.D.Jackson,H.Bogild: Nucl.Phys.A 470 (1987), 669;[5] M.Kutschera, J.Hu¨fner, K.Werener: Physs.Lett. B 192 (1987), 237;[6] R.Shanta,S.K.Gupta: Z.Phys.A 338 (1991), 183;[7] J.Y.Zhang et al.: Phys. Rev. C 46 (1992), 748;[8] D. Kharzeev et al.: Z.Phys.C 74 (1997), 307;[9] S.Jean,J.Kapusta: Phys. Rev. C 56 (1997), 468;[10] R.Lietava, J. Pisut: Eur.Phys.J C 5 (1998), 135;[11] R.J.Glauber, in Lecures in Theoretical Physics, edited by W.E.Brittin et al.,Intersciemce, New York,1959, Vol. I.,p.315,K.C.Chung,C.S.Wang,A.J.Santiago, G.Pech: Phys. Rev. C 57 (1998), 847;[12] C.Y.Wong: Introduction to High-Energy Heavy Ion Collisions, World Scien-tic,1994[13] J. Ba¨chler et al.,: Z. Phys. C 51 (1991), 157;[14] A. Capella and J. Tran Thanh Van,: Phys.Lett.B 93 (1980), 146;[15] I. Kralik, Ph.D. Thesis, Institute of Experimental Physics, SAS, Kosice, 1995.[16] B. Lasiuk at al. Hadronic observanles at NA49 School and Workshop on HeavyIon Collisions, September 1.-5., 1996, Bratislava, Slovakia. NA49 note 117;Simple model 7[17] S.Margetis et al.: Phys.Rev.Lett. 75 (1995), 3814;[18] R.Lietava and O. Villalobos Baillie, A simple model for average multiplicity,WA97 internal note, July,1996;[19] E.Andersen et al.: Phys. Lett. B 433 (1998), 209;[20] C. Baglin et al.: Phys. Lett. B 251 (1990), 472;[21] J. Ba¨chler et al.: Phys.Rev.Lett. 72 (1994), 1419;

A simple model for transverse energy distribution in heavy ion collisions

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