On the temperature of the pionic sources in nucleus - nucleus collisions at 4.5 A GeV/ c

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J. Phys. G: Nucl. Part. Phys.22 (1996) 231–242. Printed in the UK

On the temperature of the pionic sources innucleus–nucleus collisions at 4.5A GeV/c

Alexandru JipaAtomic and Nuclear Physics Department, Faculty of Physics, University of Bucharest, PO BoxMG-11, R-76900 Bucharest-Magurele, Romania

Received 8 November 1994, in final form 18 September 1995

Abstract. In this paper the temperature and density of the particle source in some collisionsat 4.5A GeV/c momentum are discussed. The experimental values for these quantities do notoffer conditions for a phase transition from the hadronic matter to the quark–gluon plasma. Theexperiments have been performed in the framework of the SKM 200 Collaboration from JINRDubna.

1. Introduction

One of the methods of establishing the behaviour of nuclear matter in high temperatureand density conditions is the study of relativistic nuclear collisions. In these collisionsdifferent phase transitions can be realized (figure 1) [1]. To make evident any transition itis necessary to know the temperature and density of the particle source.

Figure 1. The possible phase transi-tions in nucleus–nucleus collisions athigh energies [1].

In the study of relativistic nuclear collisions at energies of a few GeV per nucleon thecollision geometry is important. At these energies the participant–spectator picture of thecollision is frequently used because the following conditions are satisfied:λB � d andλ < RT (figure 2); hereλB is the de Broglie wavelength associated with the nucleon in

0954-3899/96/020231+12$19.50c© 1996 IOP Publishing Ltd 231

232 A Jipa

the projectile nucleus,λ is the mean free path of the nucleon inside the nucleus,d is themean distance between nucleons in the target nucleus andRT is the radius of the targetnucleus. The participant spectator picture for these collisions suggests the creation of twodistinct regions: the participant region and the spectator region. The participant regionis created through the overlapping of the two colliding nuclei; here sequential nucleon–nucleon collisions appear, new particles are created and many new physical phenomenaare generated. The non-overlapping parts of the two colliding nuclei form the spectatorregion(s). Many experimental results suggest the importance of the phenomena at thecontact surface of the two regions. In the participant region the nuclear matter can reachhigh temperature and high density. Here different phase transitions in the nuclear mattercan appear. This region is frequently called the ‘fireball’ region.

Figure 2. The participant–spectator picture of therelativistic nuclear collisions.

To find the state of the participant region it is necessary to study different quantitiescharacterizing the emitted particles. One of these quantities is the temperature of theemitting source. This paper tries to take into account some possibilities so as to obtainthe temperature of the fireball formed in inelastic and central nucleus–nucleus collisions athigh energy, as well as the connections of this quantity with different interesting physicalquantities (participants, size of the emitting source, density, etc).

The experimental results presented in this paper have been performed at the JointInstitute for Nuclear Research, Dubna (Russia), in the framework of the SKM 200Collaboration. Many of the experimental results on different physical quantities used inthis paper have been published elsewhere by different members of this Collaboration [2–14].

2. Experiments

At JINR Dubna the nuclei (4He, 12C, 16O, 20Ne) are accelerated in the Syncrophasotronat 4.5A GeV/c momentum. The detection device is the SKM 200 Spectrometer (figure 3)[2]. This spectrometer has a streamer chamber with the dimensions 2 m× 1 m× 0.6 m.For these experiments the chamber was filled with neon and placed in a magnetic fieldof 0.8 T. Solid targets in the form of thin disks have been mounted inside the chamber;in some experiments the chamber filling gas has been used as a target. The high voltage

Nucleus–nucleus collisions at 4.5A GeV/c 233

(500 kV per pulse, 10.5 ns length of pulse) is supplied by a Marx generator. A stereo-photographic system with three cameras allows one to record the experimental informationon high-sensitivity films.

Figure 3. The SKM 200 spectrometer of JINR Dubna (Russia) [2].

The experimental data are obtained through scanning, measuring and geometricalreconstruction. The charges of the secondary particles were determined by the visualexamination of the ionization degree of the film tracks. Only negative pions with momentahigher than 50 MeV/c have been measured. A geometrical reconstruction of the measuredparticles was performed [3, 4, 6, 9–11, 15]. For these experiments the absolute error in angleis about 2.9◦ for all angles and the relative error in momentum is about 8% for all momenta[9–11].

The streamer chamber is triggered by two systems of scintillation detectors placed infront of and behind it. There are two triggering modes: inelastic(T (θch = 0, θn = 0)) andcentral (T (θch > 0, θn > 0)). Here, θch and θn are the minimum values of the emissionangles accepted for the charged particles and fragments and neutral particles and fragments,respectively, of the projectile nucleus with momenta higher than 3.5A GeV/c (for thisexperiment), which are known as stripping particles.

This detection device does not allow one to identify correctly all particles yielded inthese collisions; the negative pions can be identified with errors less than a few per cent[10, 16]. For this reason the experimental results presented in this paper refer to this kindof particle.

3. Experimental results on temperature

To obtain the temperature of the pionic source in different nucleus–nucleus collisions at4.5A GeV/c the total momentum spectra and the transverse momentum spectra of thenegative pions have been used (figure 4(a)–(d)) [10, 14].

It is important to emphasize the fact that at these energies the contribution of the different

234 A Jipa

Figure 4. The transverse momentum spectrum of the negative pions and the energy spectrum inthe centre-of-mass system for negative pions in: (a) and (b), O–Pb collisions at 4.5A GeV/c;(c) and (d), He–Li collisions at 4.5A GeV/c.

kinds of resonances to the pion yield is around 10% (figure 5 and figure 6) [17–19]. Theseresonances can cool the fireball.

To fit the experimental momentum spectra of the negative pions and to obtain theemission temperature of this kind of particle the following functions have been used [20–23]:

〈pT〉 = c1 exp

(c2

T

T0

)T0 = 150 MeV, c1 = 382.2, c2 = −0.999 (1)

〈pT〉 =(

πmπT

2

)1/2K5/2(mπ/T )

K2(mπ/T )(2)

dσ

dpT= constant× pT(T ET)1/2 exp

(−ET

T

)(3)

Edσ

dE= constant× pE2 1

eE/T − 1. (4)

HereK5/2(mπ/T ), K2(mπ/T ) are MacDonald functions,ET = [m2π +p2

T]1/2, p is the total

Nucleus–nucleus collisions at 4.5A GeV/c 235

Figure 5. The excitation probabilities for different resonances inhigh-energy nucleus–nucleus collisions.

Figure 6. The number of resonances expected inhigh-energy nucleus–nucleus collisions.

momentum of the negative pion andpT is the transverse momentum of the negative pion.The choice of these functions is related to the thermodynamic and hydrodynamic models

for relativistic nuclear collisions [20–22]. Some fits with equations (3) and (4) are presentedin figure 4.

The experimental results on the temperature of the pionic source into a nucleus–nucleuscollision obtained with the above functions are the same, in the limit of the experimentalerrors. For example, in O–Pb inelastic collisions the following results are obtained:T (1) =104.8±6.3 MeV, T (2) = 114±8 MeV, T (3) = 106.8±6.7 MeV, T (4) = 112±9 MeV [8];and in He–Li central collisions the values areT ′(1) = 98.6±2.4 MeV, T ′(2) = 92±4 MeV,T ′(3) = 96 ± 5 MeV, T ′(4) = 101± 6 MeV [8, 9]. Because of this only one set ofexperimental results is presented in the tables for each triggering mode, inelastic or central.

In table 1 the temperatures of the pionic source in 11 inelastic nucleus–nucleus collisionsat 4.5A GeV/c are presented, and in table 2 the results for the same collisions, but in thecentral triggering modeT (2, 0), are presented. The results have been obtained using relation(1). The temperatures increase with the mass number of the target nucleus and, for the sametarget nucleus, increase with the mass number of the projectile nucleus. For central collisionsthe temperatures are higher than the temperatures for inelastic collisions.

It is useful to discuss the connections of the temperature with the participants and with

236 A Jipa

Table 1. Temperatures, participants, radii and densities for inelastic nucleus–nucleus collisionsat 4.5 GeV/c per nucleon.

pT

AP–AT (MeV/c) T (MeV) 〈Qexp〉 〈QN〉 rπ (fm) ρ (fm−3)

He–Li 241± 3 69.0 ± 2.0 2.0 ± 0.5 4.0 ± 1.0 1.80 0.163± 0.041He–C 238± 4 71.0 ± 3.0 2.9 ± 0.3 5.8 ± 0.6 2.11 0.149± 0.015He–Ne 230± 5 76.0 ± 3.3 3.6 ± 0.3 7.2 ± 0.6 2.12 0.183± 0.015He–Cu 227± 6 78.0 ± 4.0 5.7 ± 0.5 12.5 ± 1.1 2.72 0.149± 0.013He–Pb 204± 4 94.0 ± 3.0 9.9 ± 1.0 24.9 ± 2.5 3.06 0.208± 0.021C–C 236± 6 72.2 ± 3.8 4.2 ± 0.4 8.4 ± 0.4 2.60 0.113± 0.005C–Cu 220± 4 82.8 ± 2.7 9.0 ± 0.8 19.6 ± 1.7 3.05 0.165± 0.014Ne–Ne 225± 9 79.4 ± 4.7 7.9 ± 0.7 15.8 ± 1.4 3.09 0.127± 0.011Ne–Zr 195± 5 100.8 ± 3.8 12.3 ± 0.4 27.3 ± 0.9 3.62 0.138± 0.005O–Ne 229± 9 76.7 ± 5.9 6.1 ± 0.7 12.2 ± 1.4 2.86 0.123± 0.014O–Pb 190± 8 104.8 ± 6.3 19.0 ± 0.9 47.3 ± 2.2 4.23 0.149± 0.007

Table 2. Temperatures, participants, radii and densities for central nucleus–nucleus collisions(T (2, 0)) at 4.5 GeV/c per nucleon.

pT

AP–AT (MeV/c) T (MeV) 〈Qexp〉 〈QN〉 rπ (fm) ρ (fm−3)

He–Li 198± 3 98.6 ± 2.4 2.8 ± 0.3 5.6 ± 0.6 1.50 0.396± 0.042He–C 195± 5 100.5 ± 3.7 4.7 ± 0.2 9.4 ± 0.4 1.78 0.398± 0.017He–Ne 189± 5 105.6 ± 4.0 6.1 ± 0.8 12.2 ± 1.6 1.93 0.405± 0.053He–Cu 186± 6 107.6 ± 4.9 8.2 ± 0.5 18.0 ± 1.1 2.20 0.403± 0.024He–Pb 167± 4 123.6 ± 3.7 14.7 ± 1.2 37.1 ± 3.0 2.82 0.399± 0.032C–C 194± 6 101.7 ± 4.6 7.8 ± 0.3 15.6 ± 0.6 2.10 0.402± 0.015C–Cu 181± 4 112.3 ± 3.3 19.7 ± 1.0 42.8 ± 2.2 2.94 0.402± 0.021Ne–Ne 185± 9 108.9 ± 7.3 9.8 ± 0.7 19.6 ± 1.4 2.28 0.395± 0.028Ne–Zr 160± 5 130.4 ± 4.6 26.2 ± 2.1 58.2 ± 4.7 3.27 0.395± 0.032O–Ne 188± 9 106.3 ± 7.2 9.6 ± 0.3 19.2 ± 0.6 2.25 0.399± 0.013O–Pb 156± 8 134.3 ± 7.7 39.6 ± 0.5 98.6 ± 1.2 3.89 0.401± 0.005

the density.One of the quantities that offers information about the participant region is the number

of participants [10]. The number of participant protons can be established from theexperimental results using the following relation [1, 9–12]:

Q = nch − 2nπ − (nsP + ns

T) (5)

whereQ is the number of participant protons,nch is the multiplicity of the charged fragmentsand particles,nπ is the multiplicity of the negative pions,ns

P is the number of spectatorfragments from the projectile nucleus andns

T is the number of spectator fragments from thetarget nucleus.

For the SKM 200 Collaboration experiments the number of participant protons isestablished using the following relation:

Q = nch − 2nπ − (ns1 + n+r + n+

R + np<pF) (5′)

wherens1 is the multiplicity of the particles with momenta higher than 3.5 GeV/c, generatedin the angular range according to the triggering mode of the streamer chamber,n+

r is thenumber of positive fragments with ionization greater than 1 which have length of the track

Nucleus–nucleus collisions at 4.5A GeV/c 237

chord smaller thanr, n+R is the number of positive fragments with ionization greater than 1,

which have length of the track chord betweenr andR with r < R andnp<pF is the numberof positive fragments with ionization greater than 1, which are emitted and have momentap smaller than the Fermi momentum,pF [9–12]. In these experimentsr, R and pF havethe following values: 9.24 cm, 12.58 cm and 240 MeV/c respectively.

To obtain the total number of participant nucleons,QN, the following relationship issuggested [9–12]:

QN = AP + AT

ZP + ZTQ (6)

where AP and AT, ZP and ZT are the mass number and the atomic number of thetwo colliding nuclei (projectile and target), respectively. In other studies the followingrelationship is used:QN = (2–2.5)Q [5]. Equation (6) considers the collision geometryand the collision symmetry better [10, 12]. In tables 1 and 2 the total numbers of participantnucleons have been calculated with relation (6) using the experimental values of theparticipant proton numbers.

The density of the nuclear matter inside the overlapping region of the two collidingnuclei can be estimated using the following relation [5]:

ρ = QN

V= QN

4πr3/3(7)

wherer is the size of the particle source (fireball) at the emission of the pion. This sizecan be determined experimentally or by using phenomenological models.

The interferometry of the identical particles offers the possibility of establishing thesize of the particle source from experimental results on momentum, energy and multiplicity[24, 25].

Another way, used in this paper, considers the following picture of the collision betweennuclei at energies of a few GeV/A: in the overlapping region of the two colliding nucleia very hot central region is created, which flows through colder peripheral regions; theseregions slow down the flow and can absorb some particles created in the hot region. Whathappens at the contact area between the two kinds of region during the flow seems to beimportant [9–11].

Different physical quantities can be calculated in the following working assumptions:(i) the nucleons are spheres of radiir0 and the nuclei are spheres of radiiR = r0A

1/3;(ii) initially, in the target nucleus a geometrical spherical zone occurs, the volume of thegeometrical spherical zone depends on the impact parameterb and on the beam energy;(iii) the ratio (ZP + ZT)/(AP + AT) remains constant for the very hot region; and (iv) thegeometrical spherical zone evolves in a very hot sphere (fireball) and the volume of thesphere is equal to the volume of the geometrical spherical zone [9–11].

In these assumptions the radius of the very hot sphere is

r = c(γ )

2h1/3(3r2

1 + 3r22 + h2)1/3 (8)

where r21,2 = |R2

T − (b ∓ RP)2| and h = 2RP; c(γ ) is a quantity depending on the time

evolution of the fireball. This evolution is related to the contraction Lorentz factor,γ .Because the pion emission is supposed to be a surface emissionc(γ ) = (1/γ )2/3. The resultsof the two methods are similar [5, 10, 11, 26, 27]. In the two tables mentioned previouslythe calculated values of the radii are presented.

Therefore, the two tables contain the experimental results on the participant protons, thecalculated values for the participant nucleons, the calculated values of the size of the pionicsource and the estimated densities.

238 A Jipa

Figure 7. The dependence of the temperature of the pionic source on the number of averageparticipant nucleons in: (a) inelastic nucleus–nucleus collisions at 4.5A GeV/c fit with alinear function; (b) inelastic nucleus–nucleus collisions at 4.5A GeV/c fit with a saturation-like function; (c) central nucleus–nucleus collisions at 4.5A GeV/c fit with a linear function;and (d) central nucleus–nucleus collisions at 4.5A GeV/c fit with a saturation-like function.

It is important to study the dependences of the temperature on the participant protonnumber and on the participant nucleon number, for inelastic and central collisions(figure 7(a)–(d)). To fit the experimental dependences two kinds of function are used,

y1 = a1x + a2 (9)

y2 = b1[1 − exp(−b2x)]. (10)

These functions try to take into account the possibility that the rest mass of theparticipant nucleons represents—together with the compression of the nuclear matter inthe overlapping region of the colliding nuclei and the interaction processes inside thisregion—an energy source which increases the temperature of the fireball. The first functionsuggests a linear dependence of the temperature on the number of participant nucleons. Sucha dependence will favour the central collisions of very heavy nuclei at high and ultrahighenergies in the attempt to obtain the phase transition to the quark–gluon plasma. The secondfunction suggests a saturation-like behaviour of the temperature on the number of participantnucleons. In this case the very large size of the fireball can be related to a smaller bindingof the fireball surface nucleons, as well as to the smaller temperature of the fireball surface.

Nucleus–nucleus collisions at 4.5A GeV/c 239

Figure 8. The dependence of the temperature of the pionic source on the beam energy pernucleon in the centre-of-mass system for central ((a), linear fit; (b) saturation-like fit) andinelastic ((c) linear fit; (d) saturation-like fit) nucleus–nucleus collisions.

The effects of the spectator regions can also contribute to such a behaviour.The fit of the temperature–participant nucleon number dependence suggests a linear

trend (table 3). Therefore, it is possible that there is an increase of the temperature ofthe particle source through the collision of two nuclei with equal high mass numbers (forexample, Au–Au, Pb–Pb, U–U).

Table 3. The values ofχ2 for the fits of the dependencesT = f (〈Qexp〉) andT = f (〈QN〉) ininelastic and central triggering modes.

y = a1x + a2 y = a1x + a2 y = a1(1 − exp(−a2x)) y = a1(1 − exp(−a2x))

T (θch, θn) χ2ndf(T , Q) χ2

ndf(T , QN) χ2ndf(T , Q) χ2

ndf(T , QN)

T (0, 0) 0.97 0.93 3.76 4.52T (2, 0) 1.25 1.26 3.15 4.39

Another important dependence of the pionic temperature is on the beam energy forinelastic and central collisions (figure 8(a)–(d)) [6, 28, 29]. In this case the dependenceseems to be saturation-like (table 4). In figure 9 the pionic temperatures obtained indifferent nucleus–nucleus collisions at kinetic energies up to 3.6A GeV are presented [29].

240 A Jipa

Figure 9. The freeze-out temperature of thepions calculated from the pion multiplicity data pernucleon [29].

Figure 10. The influence of the resonances from thefireball on the dependence of the temperature on the beamenergy [32].

This behaviour of the temperature relative to the beam energy in the centre-of-mass systemcan be an explanation of the absence of significant signals of the transition to the quark–gluon plasma at higher energies (15–200A GeV) [30, 31]. The presence of the resonancesdecreases the temperature of the source (figure 10) [32].

Table 4. The values ofχ2 for the fits of the dependenceT = f (E∗beam) in inelastic nucleus–

nucleus collisions.

y = a1x + a2 y = a1(1 − exp(−a2x))

χ2 4.54 0.39

In each triggering mode the density in all 11 collisions seems to be constant, namely〈ρin〉 = 0.89ρ0 in inelastic collisions and〈ρcen〉 = 2.35ρ0 in central collisions, respectively.ρ0 is the density of the normal nuclear matter (ρ0 = 0.17 Fm−3) [33].

These experimental results are in agreement with some calculations using thehydrodynamical model [29, 33] involving the multiplicity of the negative pions (figure 11).

4. Conclusions

The experimental results presented in this paper suggest the following conclusions.

Nucleus–nucleus collisions at 4.5A GeV/c 241

Figure 11. The dependence of the rationπ/A on thetemperature, for different densities, in relativistic nuclearcollisions.

(i) The temperature in nucleus–nucleus collisions at 4.5A GeV/c depends on thecollision geometry, the number of participants and the beam energy.

(ii) The temperature in inelastic nucleus–nucleus collisions is smaller than in centralnucleus–nucleus collisions.

(iii) At this energy there are no conditions for the phase transition at the quark–gluonplasma.

(iv) It is possible to create a state in which nucleons, pions and a few quarks coexist, orother new forms of nuclear matter (for example, resonance matter [34] or di-quarks plasma[35]) can appear.

Acknowledgments

The author wishes to thank Professor Dr Calin Besliu (Bucharest University) for very usefuldiscussions, Professor Dr Ken Peach (University of Edinburgh) for encouragement, as wellas the members of the SKM 200 Collaboration from JINR Dubna for the efforts in obtainingexperimental data.

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