Impact parameter dependence of collective flows and

J. Phy. G: Nucl. Pan. Phys. IS (1992) 681-705. Primed in the

UK

Impact parameter dependence of collective flows and particle multiplicities in heavy-ion reactions Guo-qiang Lit, Y LotfyL S W Huang, Tbmoyuki Maruyamas, Dao T Khoa and h a n d Faessler Instirut fur 'Iheorelische Physik der Univenilal Tiibingen, D-7400 Tiibingen, Federal Republic of Germany Received 25 July 1991, in final form 3 Oclober 1991 AbslracL We sludy the impact parameter dependence of mllective flows and panicle multiplicities in h e a y i o n reactions from 84-2100 MeV A - ' . W e consider mainly the 'Oca C a system. For mmparison with experimental data other synems are also considered. When the bombarding energies arc M o w 400 MeV A-', we use the quantum molecular dynamics (QMD) with the nucleon-nucleon moss seclion obtained from Ihe G-matrix. At bombarding energies in the GeV A-' region, we use relativislic quantum molecular dynamics (RQMD) with the Cugnon paramelrilalion for lhe taryonbaryon cross seclions. We calculate the degree of equilibrium, the directed transveme m ~ x e n ! f i mmr! the ~lpidi!;, dir!rih!!tioE. >,E mc!!ip!icitirs of photons, pions, e 0 5 ezd hons are also calculated and analysed. T h e i r dependence on lhe nuclear equation of slate Vr studied in detail. lhere Is fair agreement between Ihe theoretical mulls and experimental data for panicle produclion c m s seclions and mulliplicities.

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1. Introduction

One of the main purposes of hedvy-ion (HI) reactions is to study the properties of nuclear matter at various densities and temperatures. During a HI reaction at intermediate bombarding energy, a piece of nuclear matter with density up to 3 p 0 ( p o RZ 0.17 f r C 3 is the saturation density of normal nuclear matter) can be formed which lasts only for a very short period. Ewperimentdlly one knows the initial conditions of the reaction, and one measures its final observables. In order to leam the properties of the intermediate compressed stage, one needs a transport theory which can follow the time evolution of this reaction from its initial and intermediate stages to the final observables. One also needs a set of experimental observables for this reaction which are sensitive to the properties of the compressed nuclear matter such as its equation of state (EOS) and which is not seriously affected by final-state interactions. The development of the transport theory and the identification of the observables which are sensitive to the EOS is one of the challenging tasks for theorists in this field. t Permanenl address: Physics Depanmeni. Hangzhou Univenily. Hangrhou. Peoples Republic of China. $ Permanenl address: Physics Depanmenl. El-Minia Universily, El-Minia, Egypt. 5 Presenl address: lnstitul fiir ?heoretische Physik, Univenitht Giessen, D-6300 Giessen, Federal Republic of Germany. ~54-3899/92/040681+25Sfl4.5fl @ 1992 IOP Publishing L i d

681

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The Boltzmann-Uehling-Uhlenbeck (BUu) approach [l-31 and quantum molecular dynamics (QMD) [MI,together with their relativistic extensions, the so-called relativistic Boltzmann-Uehling-Uhlenbeck (RBUU)approach [7-91 and relativistic quantum molecular dynamics (RQMD)[lo, 111, are at present the most successful transport theories for the intermediate energy HI reactions. In these theories, both the effects of the mean field and stochastic two-body collisions are taken into account. The mean field is related to the EOS of the nuclear matter and by using different mean fields in the simulation of the HI reactions one might obtain useful information about the nuclear EOS. Collective flows such as t h e directed transverse momentum and the flow angle, and the particle production cross section in the HI reactions, are often advocated as suitable experimental observables for the determination of the nuclear EOS [12151. The observed collective flow and the pion multiplicity are explained by the BUU approach [l]. Because of the complication of the HI reactions, such as its nonequiiiiirium features ana the finite-size effects, an unambiguous conciusion has yet to be made concerning the EOS of nuclear matter or, more specifically, its incompressibility. More accurate and exclusive experimental data need to be measured and more detailed theoretical investigations are necessary for the determination of the nuclear EOS from the HI reactions. One of the advantages of the theoretical simulation over the experiment for the HI reactions is the possibility of studying the impact parameter dependence of the observables. Experimentally, the impact parameter of a HI collision is usually estimated from the multiplicities of charged particles such as protons [1&19]. In this paper we study the impact parameter dependence of the collective flows and the particle multiplicities in the intermediate energy H I reactions from 84-2100 MeV A-'. At bombarding energies below 400 MeV A-' we use QMO with the in-medium I" _^^^

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approach for the particle production in HI reactions, we also compare our theoretical results with experimental data for some observables. In section 2 we give a description of our theoretical formalism. The results are presented, compared with the data and discussed in section 3. A short summary Of our paper is presented in section 4. 2. Theoretical framework

2.1. QMD and RQMD

The detailed description of QMD and RQMD has been given in [a, 10, 111. We present here only a short outline or these approaches. In the QMD,each nucleon is represented by a Gaussian wavepacket with a fixed width. Their centroids (ri0,p;,) are propagated according to the classical Hamilton equations of motion

683

Impaci parameter dependence in heavy-ion reaciions

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The single-particle Hamiltonian H i( H = H i ) appearing in equation (1) is constructed from a Skyrme-type zero-range force [Z],supplemented by a finite-range Yukawa force and an effective Coulomb interaction. The Skyrme-type mean field potential in the QMD has the form

The momentum dependence of the in-medium NN interaction (MDI) is introduced through the following phenomenological form, with its parameters (t4 = 1.57 MeV, 1, = 5 x M e V 2 ) fitted to the proton-nucleus optical potential 161.

ine parameters of the Skyrme force are adjusted so tnat we nave the right saturation density and binding energy per nucleon for normal nuclear matter but have different incompressibilities (IC = 200 and 380 MeV) for the nuclear EOS. In table 1 we list the parameters used in the QMD. Another important ingredient in the QMD calculation is the in-medium NN cross section which allows us to simulate NN collisions during the time evolution of the HI reaction. The in-medium NN cross section used in the present QMD calculation is derived from the G-matrix which is the solution of the Bethe-Goldstone equation for two colliding nuclear matters [20-221. I

Table 1. Parameiers o i the mean deid p i e n i i a i . equation [Ti. lo ihe soft and hard EDS rvllh the MDI. respectively.

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",*'.," -6 "L the transport theories, since the experimental observables must be independent of the reference frame. BUU approach was extended to RBUU by connecting it with the Walecka model [7-9, 241, whereas QMD was extended to RQMD in the light of the constraint Hamiltonian dynamics [lo, 11, 25, 261. A covariant Hamiltonian dynamics for an A-particle system is expressed in terms of 8 A variables: 4 A position coordinates qi, and 4 A momentum coordinates pi,. Since physical events are described as world lines in the liA-dimensional phase space, 2 A constraints (4i = 0, i = 1,. , , ,2.4) have to be introduced in order to eliminate the extra 2 A - I degrees of freedom and to define a global time par"ter r . We first choose A constraints as the on-mass-shell conditions. The remaining A constraints = 0 defines the are taken to be time fixations. Among them we assume that olohal time parame!e.r r 2nd that ~ ! h e i mns!rain6 are not explicitly dependent on r: D----The Hamiltonian is then constructed as a linear combination of 2 A - 1 constraints: A.

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reached, and hence the nuclear EOS does not play a significant role. One interesting point to emphasize is the fact that at subthreshold energies, the multiplicities of etas and kaons show clear sensitivity to the nuclear EOS, especially at Small impact parameters. These exclusive data, together with their inclusive cross sections, are promising observabies to fuc the incompressibility of the nuclear matter. Acknowledgment This work was supported by the GSI Darmstadt under the contract no 99262.

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