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
MAHNAZ Q. HASEEB
Department of Physics, COMSATS Institute of Information Technology,
Park Road, Islamabad, Pakistan
SAYYED A. HADI
Karakoram International University, Gilgit-Baltistan, Pakistan
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.
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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
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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
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K. K. Olimov, M. Q. Haseeb & S. A. Hadi
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
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Rapidity and Angular Dependences of Spectral Temperatures of Negative Pions
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.
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K. K. Olimov, M. Q. Haseeb & S. A. Hadi
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.
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Rapidity and Angular Dependences of Spectral Temperatures of Negative Pions
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
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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
1.2 ≤ |y| ≤ 1.6 Experiment 2923 67± 2 1.01 127± 3 2.41FRITIOF 5915 61± 1 8.30 118± 1 8.49
1.8 ≤ |y| ≤ 2.8 Experiment 628 36± 2 1.19 148± 5 1.10FRITIOF 1038 37± 1 0.46 152± 3 3.99
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
20–40 Experiment 3375 68± 2 2.16 115± 2 0.98FRITIOF 7307 67± 1 5.11 114± 1 3.25
40–60 Experiment 3874 98± 2 2.44 106± 2 1.17FRITIOF 9634 98± 1 5.42 105± 1 2.08
60–80 Experiment 3893 120± 2 2.43 102± 2 2.09FRITIOF 10323 116± 1 6.03 99± 1 1.78
80–100 Experiment 3916 126± 2 2.94 100± 2 2.31FRITIOF 10218 121± 1 7.37 95± 1 1.64
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.
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Rapidity and Angular Dependences of Spectral Temperatures of Negative Pions
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
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K. K. Olimov, M. Q. Haseeb & S. A. Hadi
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.
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Rapidity and Angular Dependences of Spectral Temperatures of Negative Pions
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
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K. K. Olimov, M. Q. Haseeb & S. A. Hadi
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
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April 19, 2013 16:58 WSPC/143-IJMPE S0218301313500201
Rapidity and Angular Dependences of Spectral Temperatures of Negative Pions
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
April 19, 2013 16:58 WSPC/143-IJMPE S0218301313500201
K. K. Olimov, M. Q. Haseeb & S. A. Hadi
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