rapidity and angular dependences of spectral temperatures of negative pions produced in 12 c 12 c...

April 19, 2013 16:58 WSPC/143-IJMPE S0218301313500201

International Journal of Modern Physics EVol. 22, No. 4 (2013) 1350020 (14 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0218301313500201

RAPIDITY AND ANGULAR DEPENDENCES OF SPECTRAL

TEMPERATURES OF NEGATIVE PIONS PRODUCED

IN 12C12C COLLISIONS AT 4.2A GeV/c

KHUSNIDDIN K. OLIMOV∗

Department of Physics, COMSATS Institute of Information Technology,

Park Road, Islamabad, Pakistan

Physical-Technical Institute of Uzbek Academy of Sciences, Tashkent, Uzbekistan

[email protected]

MAHNAZ Q. HASEEB

Department of Physics, COMSATS Institute of Information Technology,

Park Road, Islamabad, Pakistan

[email protected]

SAYYED A. HADI

Karakoram International University, Gilgit-Baltistan, Pakistan

[email protected]

Received 19 October 2012Revised 11 February 2013Accepted 21 February 2013Published 15 April 2013

The average spectral temperatures of negative pions in 12C12C collisions at 4.2A GeV/cwere extracted from transverse momentum as well as scaled center-of-mass kinetic energydistributions for different intervals of rapidity and emission angle of π− in the center-of-mass system (cms) of 12C12C collisions. The temperatures extracted from experimentalspectra of negative pions were compared systematically with the corresponding tem-peratures deduced from spectra of π− calculated using Modified FRITIOF model. Thedependence of the average spectral temperatures of negative pions on the rapidity andemission angle of π− in the cms of 12C12C collisions at 4.2A GeV/c was analyzed.

Keywords: Relativistic nucleus–nucleus collisions; spectral temperatures of hadrons;transverse momentum distribution; kinetic energy distribution; Hagedorn thermody-namic model; collective flow; modified FRITIOF model.

PACS Number(s): 25.10.+ s, 25.75.Dw

∗Corresponding author.

1350020-1

http://dx.doi.org/10.1142/S0218301313500201

mailto:[email protected]




K. K. Olimov, M. Q. Haseeb & S. A. Hadi

1. Introduction

The analysis of transverse momentum as well as energy spectra of pions produced

in relativistic nucleus–nucleus collisions can give us information on the condition of

the system at freeze-out, and particularly about the temperature. It is important to

mention that the temperature and density are the main parameters of the nuclear

equation of the state (EOS) that determine the phase transition of the nuclear

matter. Pions are used as a thermometer and incident energy as a measure of the

pressure to derive the useful information on EOS.1–5 The negatively charged pions

can be separated easily from the other products of nuclear collisions. It is necessary

to mention that pions are most abundantly produced particles at the energies of

the Dubna synchrophasotron. The significant part of produced pions comes from

decay of baryon resonances, excited dominantly in relativistic nuclear collisions. In

particular, it was shown in Refs. 6–14 that a significant fraction of pions produced

in experiments on 2 m propane (C3H8) and 1 m hydrogen bubble chambers of

Joint Institute for Nuclear Research (JINR, Dubna, Russia) originated from decay

of ∆(1232) resonances.

In Ref. 15 the spectral temperatures of negative pions in d12C, 4He12C and12C12C collisions at 4.2A GeV/c were extracted from fitting noninvariant center-

of-mass (cm) energy spectra of π− mesons with Maxwell–Boltzmann distribution

function. The temperatures of π− mesons were analyzed in Refs. 15–20 for colli-

sions of different sets of nuclei at various energies in the past. Analysis of transverse

momentum as well as transverse mass distributions is preferred for estimating the

hadron temperatures due to the Lorentz invariance of such spectra with respect

to longitudinal boosts.21–23 In our recent work,24 we extracted the spectral tem-

peratures of negative pions in d12C, 4He12C and 12C12C collisions at 4.2A GeV/c

from fitting transverse momentum distributions of π− mesons in the framework of

Hagedorn thermodynamic model.22,25 The spectral temperatures of π− extracted

in Ref. 24 from transverse momentum spectra proved to be noticeably lower than

the corresponding temperatures deduced from noninvariant cms energy spectra of

π− mesons in Ref. 15.

The collective flow, which can be described as a movement of a large number

of ejectiles either in a common direction or at a common magnitude of velocities,

has become a dominant observable in all high energy heavy ion experiments.26,27

Depending on the direction of collective motion, the collective flow is further subdi-

vided into longitudinal, radial, transverse, elliptic flows as well as sideflow.26,27 For

the first time the collective expanding was deduced in Ref. 28, where the experimen-

tal curved cross-section spectra of protons and pions in nucleus–nucleus collisions

at 0.8 GeV/nucleon were described by the blast wave model. In this model, it was

assumed that the available energy is equally divided between translational energy

of the blast and the thermal motion of the particles in the exploding matter.28 The

same method was used for central collisions of heavy nuclei at cms kinetic energies

of a few hundred MeV per nucleon29 to estimate the entropy of fireball formed in

1350020-2


Rapidity and Angular Dependences of Spectral Temperatures of Negative Pions

these reactions. The collective flow was also observed experimentally in azimuthal

distributions of particles by Gesellschaft fur Schwerionenforschung (GSI)/Lawrence

Berkeley National Laboratory (LBNL) collaborations at the Bevalac accelerator.30

The experimental data on collective flow effects gave rise to a series of theoretical

works attempting to extract the EOS.27 In Ref. 31, the transverse mass spectra

of pions, kaons and protons from heavy-ion reactions as measured by the NA44

collaboration were fitted with simple Boltzmann distributions, from which T , the

inverse slope parameter, often interpreted as the apparent temperature of the emit-

ting source, was extracted for each particle species analyzed. It was deduced that

T had two components: a thermal part, Tthermal, and a second part resembling

the collective expansion with an average transverse velocity 〈βt〉.31 Thus collective

flow has become an important feature and is a base of analysis of all heavy-ion

reactions,26–35 and it should also show up as an averaged feature in experimental

samples of minimum biased nucleus–nucleus collisions.

In the present work, we shall extract the average spectral temperatures of π−

mesons in 12C12C collisions at 4.2A GeV/c from their transverse momentum (pt)

as well as the scaled cms kinetic energy (Ek) distributions for different intervals

of rapidity and emission angle of π− in the center-of-mass system (cms) of 12C12C

collisions. The temperatures will be extracted from pt distributions and scaled cm

Ek spectra of π− in the framework of Hagedorn thermodynamic model22,25 and the

Standard Thermal Model of freeze-out with the cms,1,3,36 respectively. The depen-

dence of the average spectral temperatures on the rapidity and emission angle of

negative pions in the cms of 12C12C collisions will be investigated using the above

two methods of temperature extraction. Comparison of the average temperatures

of π− deduced from the pt and scaled cm Ek spectra will be made. The tempera-

tures extracted from experimental spectra of negative pions will also be compared

systematically with the corresponding temperatures obtained from spectra of π−

calculated using Modified FRITIOF model37–40 adapted to intermediate energies.

2. Experimental Procedures and Analysis

The experimental data were collected using 2 m propane (C3H8) bubble chamber

of the Laboratory of High Energy of JINR (Dubna, Russia) placed in a magnetic

field of strength 1.5 T and irradiated with a beam of 12C nuclei accelerated to a

momentum of 4.2 GeV/c per nucleon at the JINR synchrophasotron. This initial

momentum of projectile 12C nuclei corresponds to incident kinetic energy≈ 3.4 GeV

per nucleon. To select events of inelastic 12C12C interactions in the total set of in-

teractions with propane, we used criteria based on the determination of the total

charge of secondary particles, the presence of protons emitted into backward hemi-

sphere, the number of π− mesons produced, etc., as given in Refs. 41–44. These

criteria separated about 70% of the total number of inelastic 12C12C events that

was estimated by using the known cross-sections for 12Cp and 12C12C interactions

and the proton-carbon ratio in the C3H8 molecule. The remaining 30% interactions

1350020-3



were extracted statistically from 12Cp and 12C12C collision events in propane by in-

troducing the relevant weights. The weights were determined in such a way that the

numbers of events occurring on carbon and hydrogen corresponded to the numbers

expected on the basis of the known cross-sections for inelastic interactions.43–46 The

corrections to account for the loss of particles emitted under a large angle to the

object plane of the camera were introduced. All the negatively charged particles

were identified as π− mesons, since at our incident energies π− mesons constitute

the main fraction (> 95%) among the negatively charged particles, and admixture

of fast electrons and negative strange particles among them does not exceed 5%.

Most of the pions with momentum lower than 70 MeV/c were not registered be-

cause of their short range in the propane bubble chamber. The average error in

measuring angles of the negative pions was ≈ 0.8◦, while the mean relative error in

determining momenta of π− mesons was about 6%. Additional information about

the experimental procedures can be found in Refs. 41–46.

The experimental statistics analyzed in the present work consists of 20,52812C12C collision events at 4.2A GeV/c, measured under conditions of 4π geom-

etry. For the purpose of comparison, we simulated 50,000 events using Modified

FRITIOF model37–40 for 12C12C collision events at 4.2A GeV/c. The model pa-

rameters used for simulation were the same as given in Ref. 38. In experiment all

the collision events regardless of the impact parameter of a collision were recorded.

Therefore, the collision events in the Modified FRITIOF model were simulated

for minimum bias events without restrictions on the impact parameter of a colli-

sion. This model could describe satisfactorily many results obtained in relativistic

hadron–nucleus and nucleus–nucleus collisions at JINR experiments.38–40,45,46

The Hagedorn thermodynamic model22,25 predicts that the normalized trans-

verse momentum (pt) distribution of hadrons can be described using the expression

dN

Nptdpt= A · (mtT )

1/2 exp

(

−mt

T

)

, (1)

where N (depending on the choice of normalization) is either the total number of

inelastic events or the total number of respective hadrons, mt =√

m2 + p2t is the

transverse mass, T is the spectral temperature and A is the fitting constant. This

relation (1) will be referred to as one-temperature Hagedorn function throughout

the present paper. Correspondingly, in case of two temperatures, T1 and T2, the

above formula is modified as

dN

Nptdpt= A1 · (mtT1)

1/2 exp

(

−mt

T1

)

+A2 · (mtT2)1/2 exp

(

−mt

T2

)

, (2)

referred to as two-temperature Hagedorn function in this work. The above expres-

sions (1) and (2) were derived assuming that mt ≫ T .

The standard thermal model of freeze-out with the cms1,3,36 predicts that

the scaled cm kinetic energy spectra of hadrons in the limit Ek ≫ T can be

1350020-4



described as

dN

NpEdEk= A · exp

(

−Ek

T

)

, (3)

where p, E and Ek are the momentum, total energy and kinetic energy of a hadron

in the cms of colliding nuclei. The above relation (3) will be referred to as one-

temperature simple exponential function throughout the present paper. In case of

two temperatures, T1 and T2, the above expression becomes

dN

NpEdEk= A1 · exp

(

−Ek

T1

)

+A2 · exp

(

−Ek

T2

)

, (4)

referred to as two-temperature simple exponential function in the present work.

In Refs. 15 and 24, it was shown that noninvariant cms energy and the ptspectra of negative pions in d12C, 4He12C and 12C12C collisions at 4.2 A GeV/c

were characterized by two-temperature shapes and described very well using two-

temperature function fits. However, the fits of pion spectra with one-temperature

Hagedorn (1) and simple exponential function (3) are also important since they

allow one to extract the average spectral temperature of pions.

The experimental rapidity distribution of negative pions in the cms of 12C12C

collisions at 4.2A GeV/c are shown in Fig. 1 along with the corresponding spectrum

calculated using the Modified FRITIOF model. As seen from Fig. 1, the experimen-

tal rapidity spectrum is almost symmetric relative to ycm = 0 as is expected for

the symmetric system with identical colliding 12C nuclei. It is seen from this figure

-3 -2 -1 0 1 2 3

0.0

0.2

0.4

0.6

(1/N

ev)(dN/dycm)

ycm

Fig. 1. Rapidity distribution of negative pions in the cms of 12C12C collisions at 4.2A GeV/c inexperiment (•) and Modified FRITIOF model (◦). The distributions are normalized by the totalnumber of inelastic events.

1350020-5



0.0 0.2 0.4 0.6

10-4

10-3

10-2

10-1

100

101

102

0.0 0.2 0.4 0.6

10-5

10-4

10-3

10-2

10-1

100

101

102

0.0 0.4 0.8 1.2 1.6

10-5

10-4

10-3

10-2

10-1

100

101

102

0.0 0.5 1.0 1.5

10-4

10-3

10-2

10-1

100

101

pt, GeV/c

b

(1/N

)(dN

/(p

tdp

t)), (

GeV

/c

)-2

pt, GeV/c

c

pt, GeV/c

d

(1/N

)(dN

/(p

tdp

t)), (

GeV

/c

)-2

pt, GeV/c

a

Fig. 2. Transverse momentum distributions of negative pions in experiment (•) and ModifiedFRITIOF model (◦) in 12C12C collisions at 4.2A GeV/c for the following intervals of rapidityand emission angle of pions in the cms of 12C12C collisions: (a) |y| ≤ 0.1; (b) 1.8 ≤ |y| ≤ 2.8;(c) 0◦ ≤ θ ≤ 20◦; (d) 80◦ ≤ θ ≤ 100◦. Fits of pt spectra by the one-temperature Hagedornfunction in experiment (solid line) and Modified FRITIOF model (dashed line). All distributionsare normalized by the total number of negative pions.

that the experimental rapidity distribution is described quite satisfactorily by the

model spectrum.

In the present work, we shall explore how the average spectral temperatures of

negative pions change with the rapidity when one goes from midrapidity (central

region) towards fragmentation region of colliding 12C nuclei. The change of the

average spectral temperatures of π− mesons with increase of emission angle of π−

in the cms of 12C12C collisions from 0◦ to 90◦±10◦ will also be studied. The average

spectral temperatures of π− will be extracted from both the pt and scaled cm Ek

spectra of negative pions using fits with the functions (1) and (3), respectively.

1350020-6



0.4 0.8 1.2 1.6

10-4

10-3

10-2

10-1

100

101

0.0 0.4 0.8 1.2 1.6

10-5

10-4

10-3

10-2

10-1

100

101

102

0.0 0.4 0.8 1.2 1.610

-5

10-4

10-3

10-2

10-1

100

101

0.0 0.4 0.8 1.2

10-3

10-2

10-1

100

101

102

Ek, GeV

b

Ek, GeV

d

(1/N

)(dN

/(pEdE

k))

, G

eV

-3c

Ek, GeV

c

(1/N

)(dN

/(pEdE

k))

, G

eV

-3c

Ek, GeV

a

Fig. 3. Scaled cm kinetic energy distributions of negative pions in experiment (•) and ModifiedFRITIOF model (◦) in 12C12C collisions at 4.2A GeV/c for the following intervals of rapidityand emission angle of pions in the cms of 12C12C collisions: (a) |y| ≤ 0.1; (b) 1.8 ≤ |y| ≤ 2.8;(c) 0◦ ≤ θ ≤ 20◦; (d) 80◦ ≤ θ ≤ 100◦. Fits of the scaled Ek spectra by the one-temperaturesimple exponential function: experiment (solid line) and Modified FRITIOF model (dashed line).All distributions are normalized by the total number of negative pions.

The experimental and model pt distributions of π− mesons along with the fits

by the one-temperature Hagedorn function for the intervals of rapidity |y| ≤ 0.1

and 1.8 ≤ |y| ≤ 2.8 and emission angles 0◦ ≤ θ ≤ 20◦ and 80◦ ≤ θ ≤ 100◦

in the cms of 12C12C collisions are presented in Figs. 2(a)–2(d). As seen from

Figs. 2(a)–2(d), the experimental pt distributions of π− are described satisfactorily

by the Modified FRITIOF model calculations. As seen from these figures, both

experimental and model pt spectra are fitted satisfactorily by the one-temperature

1350020-7



Table 1. Dependence of the average spectral temperatures of negative pions extracted from trans-

verse momentum (pt) and scaled cm kinetic energy (Ek) spectra on the interval of rapidity in thecms of 12C12C collisions at 4.2A GeV/c.

pt spectra Scaled cm Ek spectra

Interval of y in cms Type Nπ T , MeV χ2/n.d.f. T , MeV χ2/n.d.f.

|y| ≤ 0.1 Experiment 2859 108 ± 2 1.29 95± 2 2.99FRITIOF 7246 107 ± 1 2.54 86± 1 5.24

0.3 ≤ |y| ≤ 0.5 Experiment 5233 103 ± 2 1.41 97± 2 2.69FRITIOF 12759 103 ± 1 5.50 92± 1 2.54

0.7 ≤ |y| ≤ 0.9 Experiment 3575 94± 2 2.01 119± 2 3.31FRITIOF 8814 90± 1 5.83 111± 1 9.38



Table 2. Dependence of the average spectral temperatures of negative pions extracted fromtransverse momentum (pt) and scaled cm kinetic energy (Ek) spectra on the interval of emissionangle in the cms of 12C12C collisions at 4.2A GeV/c.

Interval of θ in pt spectra Scaled cm Ek spectra

cms degrees Type Nπ T , MeV χ2/n.d.f. T , MeV χ2/n.d.f.

0–20 Experiment 1547 27± 1 0.90 123± 3 1.11FRITIOF 2802 25± 1 4.88 116± 1 1.52





Hagedorn function. The experimental and model scaled cm Ek distributions of π−

mesons with the fits by the one-temperature simple exponential function for the

intervals of rapidity |y| ≤ 0.1 and 1.8 ≤ |y| ≤ 2.8 and emission angles 0◦ ≤ θ ≤ 20◦

and 80◦ ≤ θ ≤ 100◦ in the cms of 12C12C collisions are shown in Figs. 3(a)–3(d).

As seen from Figs. 3(a)–3(d), the experimental scaled cm Ek distributions of π−

are reproduced satisfactorily by the model spectra. As seen from these figures,

both experimental and model scaled cm Ek spectra are fitted satisfactorily by the

one-temperature simple exponential function.

The corresponding average spectral temperatures extracted from pt and scaled

cm Ek spectra for different intervals of cm rapidity and emission angle of π− mesons

in 12C12C collisions at 4.2A GeV/c are presented in Tables 1 and 2. As seen from

Table 1, the average spectral temperature of π− extracted from pt spectra decreases

from the maximal value, 108 ± 2 MeV, to the minimal value, 36 ± 2 MeV, as one

goes from cm midrapidity towards fragmentation region of colliding 12C nuclei.

1350020-8



This result is expected since pions in central rapidity region are produced mostly

in central hard 12C12C collisions, and hence at higher temperatures, as compared

to pions in region of fragmentation of colliding nuclei originated predominantly in

peripheral soft 12C12C interactions, and hence at lower temperatures. This is also

confirmed by the cm angular dependence of the average spectral temperature of π−,

extracted from pt spectra, presented in Table 2. As seen from Table 2, the average

spectral temperature of π− extracted from pt spectra increases from 27 ± 1 MeV

to 126 ± 2 MeV as the cm emission angle of π− increases from 0◦ to 90◦ ± 10◦.

This is because pions emitted at cm angles around 0◦ and 90◦ are originated at

around 12C fragmentation and central rapidity region, respectively. It is interesting

to mention that in order to select central collisions some authors16,47,48 restricted

particle spectra to cms emission angles 90◦ ± 10◦ or 90◦ ± 20◦ since this approach

was less affected by nonequilibrium processes.17

On the other hand, as seen from Tables 1 and 2, the average spectral temper-

atures extracted from the scaled cm Ek spectra of π− show the totally opposite

behavior as compared to the temperatures extracted from pt spectra of π−. As seen

from these tables, the temperatures extracted from the scaled cm Ek spectra of

π− increase as one goes from cm midrapidity towards fragmentation region of col-

liding 12C nuclei. Correspondingly, the average temperature of π− extracted from

the scaled Ek spectra decreases as the cm emission angle of π− increases from 0◦

to 90◦ ± 10◦. Such behavior of spectral temperatures extracted from cm kinetic

energy spectra could be explained by the influence of the longitudinal boosts on

Table 3. Average spectral temperatures of negative pionsin 12C12C collisions at 4.2A GeV/c extracted from total ptand total scaled cm Ek spectra by fitting with the one-tem-perature Hagedorn and one-temperature simple exponen-tial function, respectively.

Totalspectra Type T , MeV χ2/n.d.f.

Experiment 103± 2 4.46pt FRITIOF 94± 1 2.68

scaled cm Ek Experiment 119± 2 3.78FRITIOF 106± 1 4.17

Table 4. Spectral temperatures, T1 and T2, and their relative contributions, R1 and R2,for negative pions in 12C12C collisions at 4.2A GeV/c extracted from total pt and totalscaled cm Ek spectra by fitting with two-temperature Hagedorn and two-temperaturesimple exponential function, respectively.

Total spectra T1, MeV R1 T2, MeV R2 χ2/n.d.f.

pt 78 ± 4 (87 ± 14)% 146± 9 (13 ± 6)% 0.69

scaledcm Ek 94 ± 5 (91 ± 8)% 168± 12 (9 ± 6)% 0.20

1350020-9



0.0 0.4 0.8 1.2 1.6

10-3

10-2

10-1

100

101

102

0.0 0.4 0.8 1.2 1.6

10-4

10-3

10-2

10-1

100

101

0.0 0.4 0.8 1.2 1.6

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

0.0 0.4 0.8 1.2 1.6

10-4

10-3

10-2

10-1

100

101

102

b

(1/N

ev)(dN

/(pEdE

k))

, G

eV

-3c

Ek, GeV

(1/N

ev)(dN

/(p

tdp

t)), (G

eV

/c)-2

pt, GeV/c

a

c

(1/N

ev)(dN

/(p

tdp

t)), (G

eV

/c)-2

pt, GeV/c

d

(1/N

ev)(dN

/(pEdE

k))

, G

eV

-3c

Ek, GeV

Fig. 4. Total transverse momentum distribution of negative pions in experiment (a) and ModifiedFRITIOF model (c); total scaled kinetic energy distribution of negative pions in the cms of 12C12Ccollisions at 4.2A GeV/c in experiment (b) and Modified FRITIOF model (d); Fits by the one-temperature (dashed line) and two-temperature (solid line) Hagedorn functions (a) and simpleexponential function (b); Fits (solid line) by the one-temperature Hagedorn function (c) and one-temperature simple exponential function (d). All distributions are normalized by the total numberof inelastic events.

such spectra. It is important to mention that the average spectral temperatures

extracted from the Modified FRITIOF model spectra of π− mesons, as seen from

Tables 1 and 2, reproduce quite satisfactorily the corresponding temperatures ex-

tracted from experimental spectra of negative pions. It follows from above results

that pt spectra of π− are preferable for estimating adequately the temperatures of

π− as compared to the scaled cm Ek spectra. This result agrees with the analo-

gous statements of some authors21–23 that transverse momentum distributions of

hadrons are preferable for estimating the hadron temperatures due to the Lorentz

invariance of such spectra with respect to longitudinal boosts.

1350020-10



We also compared the average spectral temperatures of π− extracted from the

total pt and total scaled cm Ek spectra of π− in 12C12C collisions at 4.2A GeV/c

using fits by the one-temperature functions (1) and (3), respectively. The corre-

sponding results are shown in Table 3. As seen from this table, the average temper-

ature of π− extracted from cm kinetic energy spectrum is significantly higher than

the temperature extracted from the pt spectrum both in the experiment and Mod-

ified FRITIOF model. This confirms that the temperature extracted from cm Ek

spectra of π− is affected markedly by the longitudinal boosts along the collision

axis.

As shown in our recent paper24 and earlier works,15–19 the transverse momen-

tum as well as energy spectra of pions, produced in relativistic nuclear collisions,

are characterized by two-temperature shapes. For the sake of comparison, in the

present work we also fitted the total pt and total scaled cm Ek spectra of π− in12C12C collisions at 4.2A GeV/c by two-temperature functions (2) and (4), re-

spectively. The corresponding extracted temperatures T1 and T2 along with their

relative contributions R1 and R2 to the total multiplicity of π− mesons are given

in Table 4. As seen from Figs. 4(a) and 4(b), the total pt and total scaled cm Ek

spectra of π− in 12C12C collisions are fitted very well by two-temperature functions

(2) and (4), respectively, as compared to the one-temperature fits. This is also seen

from comparison of the values of χ2/the number of degrees of freedom (n.d.f.) given

in Tables 3 and 4. However, as mentioned and seen above, the one-temperature fits

of pion spectra are important for extracting the average spectral temperatures of

pions. It is also seen from Table 4 that the temperatures T1 and T2 extracted from

the total scaled cm Ek spectra of π− in 12C12C collisions proved to be noticeably

higher than the corresponding temperatures extracted from the total pt spectra

of π−, as was also expected. As to the model spectra presented in Figs. 4(c) and

4(d), they are reproduced quite well using the fits with the one-temperature func-

tions (1) and (3), and the fitting of these spectra with two-temperature functions

(2) and (4) leads to almost coinciding temperatures T1 ≈ T2 suggesting that the

one-temperature fit is sufficient for the Modified FRITIOF model spectra.

3. Summary and Conclusions

The average spectral temperatures of negative pions were extracted from the ptand scaled cm Ek spectra of π− in 12C12C collisions at 4.2A GeV/c for differ-

ent intervals of cm rapidity and cm emission angle of π− using fits with the one-

temperature Hagedorn and one-temperature simple exponential function, respec-

tively. The average spectral temperature of π− extracted from pt spectra decreased

from 108±2 MeV to 36±2 MeV when going from cm midrapidity towards fragmen-

tation region of colliding 12C nuclei. This result was expected since pions in central

rapidity region are produced predominantly in central hard 12C12C collisions, and

hence at higher temperatures, as compared to pions in region of fragmentation of

colliding nuclei originated mostly in peripheral soft 12C12C interactions, and hence

1350020-11



at lower temperatures. This finding was confirmed by the increase of the aver-

age spectral temperature of π−, extracted from pt spectra, from 27 ± 1 MeV to

126 ± 2 MeV as the cm emission angle of π− increased from 0◦ to 90◦ ± 10◦. On

the other hand, the average spectral temperatures extracted from the scaled cm Ek

spectra of π− showed the totally opposite behavior, which could be explained by

the influence of the longitudinal boosts on the kinetic energy spectra of negative

pions. It was concluded that pt spectra of π− were preferable for adequate estima-

tion of the spectral temperatures of π− as compared to the scaled cm kinetic energy

spectra. This result supports the statements of several authors21–23 that transverse

momentum distributions of hadrons are preferred for estimating the hadron tem-

peratures due to the Lorentz invariance of pt spectra with respect to longitudinal

boosts.

The average spectral temperatures extracted from the Modified FRITIOF model

spectra of π− mesons reproduced quite satisfactorily the corresponding tempera-

tures extracted from experimental spectra of negative pions.

The transverse momentum as well as energy spectra of negative pions produced

in minimum bias 12C12C collisions at 4.2AGeV/c were fitted noticeably better using

the two-temperature thermal model as compared to the one-temperature model. It

would be oversimplified to believe that the origin of pions in a minimum bias sample

of 12C12C interactions could be described by a two thermal sources. As mentioned

earlier, the phenomenon of collective flow has become the well established and an

important feature of relativistic heavy ion collisions. Inverse slope parameter, T ,

or an apparent temperature of the emitting source, of transverse mass spectra of

hadrons was shown to consist of two components: a thermal part, Tthermal, and a

second part resembling the collective expansion with an average transverse velocity

〈βt〉.31 Hence, the observed two-temperature shape of transverse momentum and

energy spectra of negative pions produced in 12C12C collisions at 4.2A GeV/c can

likely be explained by the collective flow effects.

Acknowledgments

We express our gratitude to the staff of Laboratory of High Energies of JINR

(Dubna, Russia) and of Laboratory of Multiple Processes of Physical-Technical In-

stitute of Uzbek Academy of Sciences (Tashkent, Uzbekistan), who took part in the

processing of stereophotographs from 2 m propane (C3H8) bubble chamber of JINR.

The authors are thankful to V. V. Uzhinskii and A. S. Galoyan (JINR, Dubna), who

kindly provided the code of Modified FRITIOF model. Kh. K. Olimov is grateful to

the Higher Education Commission (HEC) of the Government of Pakistan for sup-

port under Foreign Faculty Hiring Program (FFHP) and Startup Research Grant.

We thank the officials of COMSATS Institute of Information Technology (Islam-

abad, Pakistan) for hospitality and providing necessary facilities for the fruitful

work. S. A. Hadi thanks the management of Karakoram International University

(Gilgit-Baltistan, Pakistan), especially Vice-Chancellor and Dean of Sciences, for

1350020-12



granting him the study leave to continue his graduate studies at the Department of

Physics of COMSATS Institute of Information Technology (Islamabad, Pakistan).

References

1. J. Kapusta, Phys. Rev. C 16 (1977) 1493.2. J. W. Harris et al., Phys. Lett. B 153 (1985) 377.3. R. Stock, Phys. Rep. 135 (1986) 259.4. R. Stock et al., Phys. Rev. Lett. 49 (1982) 1236.5. P. Braun-Munzinger and J. Stachel, Annu. Rev. Nucl. Part. Sci. 37 (1987) 97.6. D. Krpic, G. Skoro, I. Picuric, S. Backovic and S. Drndarevic, Phys. Rev. C 65 (2002)

034909.7. Kh. K. Olimov, Phys. Rev. C 76 (2007) 055202.8. Kh. K. Olimov, S. L. Lutpullaev, B. S. Yuldashev, Y. H. Huseynaliyev and A. K.

Olimov, Eur. Phys. J. A 44 (2010) 43.9. Kh. K. Olimov, Phys. Atom. Nucl. 73 (2010) 433.

10. Kh. K. Olimov et al., Phys. Rev. C 75 (2007) 067901.11. Kh. K. Olimov and M. Q. Haseeb, Eur. Phys. J. A 47 (2011) 79.12. K. K. Olimov, M. Q. Haseeb, A. K. Olimov and I. Khan, Cent. Eur. J. Phys. 9 (2011)

1393.13. Kh. K. Olimov, M. Q. Haseeb, I. Khan, A. K. Olimov and V. V. Glagolev, Phys. Rev.

C 85 (2012) 014907.14. Kh. K. Olimov, M. Q. Haseeb and I. Khan, Phys. Atom. Nucl. 75 (2012) 479.15. S. Backovic et al., Phys. Rev. C 46 (1992) 1501.16. R. Brockmann et al., Phys. Rev. Lett. 53 (1984) 2012.17. L. Chkhaidze et al., Z. Phys. C 54 (1992) 179.18. B. Li and W. Bauer, Phys. Rev. C 44 (1991) 450.19. L. Chkhaidze, T. Djobava and L. Kharkhelauri, Bull. Georg. Natl. Acad. Sci. 4 (2010)

41.20. L. Chkhaidze et al., Nucl. Phys. A 831 (2009) 22.21. V. D. Gudima et al., Phys. Elem. Part. Atom. Nucl. 17 (1986) 1093.22. R. Hagedorn and J. Rafelski, Phys. Lett. B 97 (1980) 136.23. R. Hagedorn, Multiplicities, Pt distributions and the expected hadron–quark-gluon

phase transition, CERN Preprint TH-3684 (CERN, Geneva, 1984).24. Kh. K. Olimov and M. Q. Haseeb, Phys. Atom. Nucl. 76 (2013) 595.25. R. Hagedorn and J. Ranft, Suppl. Nuovo Cimento 6 (1968) 169.26. W. Reisdorf and H. G. Ritter, Annu. Rev. Nucl. Part. Sci. 47 (1997) 663.27. N. Herrmann, J. P. Wessels and T. Wienold, Annu. Rev. Nucl. Part. Sci. 49 (1999)

581.28. P. J. Siemens and J. O. Rasmussen, Phys. Rev. Lett. 42 (1979) 880.29. P. J. Siemens and J. I. Kapusta, Phys. Rev. Lett. 43 (1979) 1486.30. H. A. Gustaffsson et al., Phys. Rev. Lett. 52 (1984) 1590.31. NA44 Collaboration (I. Bearden et al.), Phys. Rev. Lett. 78 (1997) 2080.32. EOS Collaboration (J. C. Kintner et al.), Phys. Rev. Lett. 78 (1997) 4165.33. J. Gosset et al., Phys. Rev. Lett. 62 (1989) 1251.34. E877 Collaboration (J. Barrette et al.), Phys. Rev. C 56 (1997) 3254.35. NA49 Collaboration (H. Appelshauser et al.), Phys. Rev. Lett. 80 (1997) 4136.36. S. Nagamiya et al., Phys. Rev. Lett. 49 (1982) 1383.37. B. Gankhuyag and V. V. Uzhinskii, Modified FRITIOF code: Negative charged parti-

cle production in high-energy nucleus–nucleus interactions [in Russian]. JINR PreprintNo. P2-96-419 (JINR, Dubna, 1996).

1350020-13



38. A. S. Galoyan, G. L. Melkumov and V. V. Uzhinskii, Phys. Atom. Nucl. 65 (2002)1722.

39. A. I. Bondarenko et al., Phys. Atom. Nucl. 65 (2002) 90.40. A. S. Galoyan et al., Phys. Atom. Nucl. 66 (2003) 836.41. H. N. Agakishiyev et al., Z. Phys. C 27 (1985) 177.42. N. Akhababian et al., Sov. J. Nucl. Phys. 38 (1983) 90.43. D. Armutlisky et al., Z. Phys. A 328 (1987) 455.44. A. I. Bondarenko et al., The Ensemble of interactions on carbon and hydrogen nuclei

obtained using the 2 m propane bubble chamber exposed to the beams of protonsand H-2, He-4, C-12 relativistic nuclei at the Dubna Synchrophasotron [in Russian],JINR Communications P1-98-292 (JINR, Dubna, 1998).

45. A. I. Bondarenko et al., Phys. Atom. Nucl. 60 (1997) 1833.46. Ts. Baatar et al., Phys. Atom. Nucl. 63 (2000) 839.47. S. Nagamiya et al., Phys. Rev. C 24 (1987) 971.48. V. G. Grishin et al., Temperature and density of nuclear matter in CC interactions

at P = 4.2 GeV/c per nucleon [in Russian], JINR Preprint P1-86-639 (JINR, Dubna,1986).

1350020-14

rapidity and angular dependences of spectral temperatures of negative pions produced in 12 c 12 c...

Documents