Experimental data on compound multiplicity distribution produced in 24Mg-AgBr and 12C-AgBr interactions, both at 4.5A GeV, have been analysed in terms of scaled factorial moments. We have calculated the values of generalized dimension Dq for q = 2, 3, 4, 5, 6 and 7 with the help of intermittency exponent αq. From the knowledge of Dq, the multifractal specific heat is calculated for different multiplicity bin ranges. The parameter λq (λq = (αq + 1)/q) is calculated from αq to look for possible non-thermal phase transition. The analysis reveals no evidence of nonthermal phase transition. Instead, different specific heat in different multiplicity bins has been observed.Analizirali smo eksperimentalne podatke o raspodjelama složene višestrukosti u sudarima 24Mg-AgBr i 12C-AgBr, oba na 4.5A GeV, primjenom prilagodnih faktorijalnih momenata. Izračunali smo vrijednosti poopćene dimenzije Dq za q = 2, 3, 4, 5, 6 i 7 pomoću eksponenta prekidanja αq. Te vrijednosti Dq primijenili smo za izračunavanje višefraktalne specifične topline za niz pretinaca. Parametar λq (λq = (αq + 1)/q) smo izračunali na osnovi αq tražeći netermički fazni prijelaz. Analiza nije pokazala prisutnost netermičkog faznog prijelaza, ali smo našli različite specifične topline u pretincima

Biswas, Subrata

Deb, Argha

Ghosh, Dipak

Haldar, Prabir Kumar

Mandal, Pasupati

English

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ISSN1330–0016CODEN FIZBE7COMPOUND MULTIPLICITY DISTRIBUTION IN NUCLEUS-NUCLEUSINTERACTIONS – PHASE TRANSITION STUDYDIPAK GHOSH, ARGHA DEB, SUBRATA BISWAS, PASUPATI MANDAL andPRABIR KUMAR HALDARNuclear and Particle Physics Research Centre, Department of Physics, JadavpurUniversity, Kolkata - 700 032, IndiaReceived 12 March 2004; Accepted 25 April 2005Online 13 February 2006Experimental data on compound multiplicity distribution produced in 24Mg-AgBrand 12C-AgBr interactions, both at 4.5A GeV, have been analysed in terms ofscaled factorial moments. We have calculated the values of generalized dimensionDq for q = 2, 3, 4, 5, 6 and 7 with the help of intermittency exponent αq. From theknowledge of Dq, the multifractal specific heat is calculated for different multiplic-ity bin ranges. The parameter λq (λq = (αq + 1)/q) is calculated from αq to lookfor possible non-thermal phase transition. The analysis reveals no evidence of non-thermal phase transition. Instead, different specific heat in different multiplicitybins has been observed.PACS numbers: 25.75-q, 24.60 Ky UDC 539.129Keywords: 24Mg-AgBr and 12C-AgBr interactions, compound multiplicity distribution,scaled factorial moments, search for non-thermal phase transition, multifractal specificheat1. IntroductionIn the field of multiparticle production process in high-energy hadronic or nu-clear interactions, the concept of intermittency, a new methodology, was first intro-duced by Bialas and Peschanski [1 – 2]. Intermittency is signaled by the power-lawbehaviour of the scaled factorial moments (SFM) with increasing spatial resolutionof the particle detection. The pioneers suggested a method which involves the sta-tistical counting variable called the scaled factorial moment Fq that is the maintool for studying intermittency. The definition of SFM is obtained from turbulencetheory [3] where it gives a measure of the amount of intermittency in a turbulentsystem. It has been shown that the average scaled factorial moment [1] is equalto the moment of a true probability distribution of particle density, without anystatistical bias.FIZIKA B 14 (2005) 4, 317–326 317ghosh et al.: compound multiplicity distribution in nucleus-nucleus . . .According to Bialas and Peschanski, power-law growth of the factorial momentswith decreasing phase space interval size signals the onset of intermittency in thecontext of high-energy interactions. Intermittency differs from Gaussian fluctuationwith the self-similarity at various scales. It has been conjectured that intermittencyis a general mechanism for producing large local fluctuations, may be a manifes-tation of phase transition [4 – 5]. But no unambiguous experimental evidence hasyet been obtained, which has shown that phase transition and similar mechanismsoccur in particle production process. The nature of such phase transition is anothercrucial problem to be addressed for a meaningful understanding of the dynamics ofparticle production.In most of previous works on intermittency, linear best fits were drawn in thetotal bin range from some pre-conceived ideas. The properties of intermittencyhave been studied [6] to investigate the structure of different phases of a self-similarmultiparticle system using the random cascade model. It is a matter of fact thata self-similar cascade is not consistent with particle production during one phasebut instead requires a non-thermal phase transition [7]. The signals of non-thermalphase transitions can be studied with the help of the parameter λq = (αq + 1)/qwhere αq is the intermittency exponent of q-th order. The condition for non-thermalphase transition may occur when the function λq has a minimum value at q = qc[5 – 7]. The regions q < qc and q > qc are dominated by numerous small fluctuationand rare large fluctuations, respectively. There is a co-existence of the liquid phaseof the many small fluctuations and the dust phase of a few grains of very highdensity depending on whether we probe the system by a moment of order q < qcor q > qc, respectively.Some simple thermodynamic approximations could be applicable to the mul-tiparticle production process. Recently Bershadskii [8] showed that the constantspecific heat approximation (CSH), which is widely applicable in the ordinary ther-modynamics, is also applicable to the multifractal data. According to Bershadskii,we have also calculated the multifractal specific heats for both the interactions inemission angle space individually from the knowledge of generalized dimensions Dqobtained from the scaled factorial moment (Fq). We have divided the full bin rangeinto several sub-intervals to observe the existence of non-thermal phase transition.Investigations, generally in high-energy nuclear collisions, are carried out on theproduced pions with a common belief that these particles are the most informativeas they are produced immediately after the collisions. Very few attempts have beenmade on the medium energy (30 – 400 MeV) knocked-out target protons, whichare supposed to carry some information about the interaction dynamics, becausethe time scale of emission of these particles is of the same order (≈ 10−22 s) asthat of the produced particles. These target protons, which manifest themselves asgrey tracks in nuclear emulsion, are the low energy part of the internuclear cascadeformed in high-energy interactions. If the number of fast target fragments of anevent generally known as the grey particles in emulsion media, is combined withproduced pions of the same event, known as the shower tracks in the same media,a new parameter named as “compound multiplicity” (nc = ng + ns where nc =compound multiplicity, ng =number of grey tracks and ns =number of shower318 FIZIKA B 14 (2005) 4, 317–326ghosh et al.: compound multiplicity distribution in nucleus-nucleus . . .tracks ) is formed which can play an important role in understanding the reactiondynamics in high-energy nuclear interactions. So far, very few attempts have beenmade to work with this parameter [9]. The behaviour of the compound multiplicityspectra thoroughly is to be probed with the available sophisticated tools.In this paper, we have analyzed the fluctuation of compound multiplicity pro-duced in 24Mg-AgBr and 12C-AgBr interactions, both at 4.5 A GeV, in emissionangle space in the framework of scaled factorial moment. Also, the multifractalspecific heat has also been studied in both data sets.2. Experimental detailsThe data set used in this present analysis was obtained by exposing NIKFI-BR2emulsion plates by 24Mg and 12C beam, both at incident energy of 4.5 A GeV atJINR Dubna, Russia. The scanning of the plates is carried out with the help of ahigh-resolution Leitz metalloplan microscope provided with semi-automatic scan-ning and measuring system. The scanning is done using objective 10× in conjunc-tion with a 25× ocular lens. To increase the scanning efficiency, two independentobservers scanned the plates independently. For measurement, 100× oil immersionobjective is used in conjunction with 25× ocular lens. The measuring system fittedwith it had 1 µm resolution along the X and Y axes and 0.5 µm resolution alongthe Z-axis. From the scanned events only those events are chosen which satisfy thefollowing criteria:a) The beam track should lie within the angle 3◦ to the mean beam directionof the pellicle.b) The events, which are within 20 µm thickness from the top or bottom surfaceof the plate, should be rejected.c) The events for which the primary beam track is observed to be a secondarytrack of other interaction should not be analysed and are rejected.According to the emulsion technique, the particles emitted after interactions areclassified as:a. Black particles: They are target fragments with ionization greater or equalto 10I0, I0 being the minimum ionization of a singly-charged particle. Their rangeis less than 3 mm, velocity less than 0.3c and energy less than 30 MeV, where c isthe velocity of light in vacuum.b. Grey particles: They are mainly fast target recoil protons with energy up to400 MeV. They have ionization 1.4I0 ≤ I < 10I0. Their range is greater than 3mmand have velocities 0.3c ≤ V ≤ 0.7c.c. Shower particles: The relativistic shower tracks with ionization less than orequal to 1.4 I0 are namely produced by pions and are not generally confined withinthe emulsion pellicle.d. The projectile fragments are a different class of tracks with constant ionisa-tion, long range and small emission angle.To ensure that the targets in the emulsion are silver or bromine nuclei, we havechosen only the events with at least eight heavy-ionizing tracks of (black+grey). ForFIZIKA B 14 (2005) 4, 317–326 319ghosh et al.: compound multiplicity distribution in nucleus-nucleus . . .our present analysis, we have considered the combination of grey and shower tracksfor studying the characteristics of compound multiplicity distribution. Accordingto the above selection procedure, we have chosen 800 events of 24Mg-AgBr inter-actions and 792 events of 12C-AgBr interactions both at 4.5A GeV. The emissionangle (θ) is measured for each track by taking the coordinates of the interactionpoint (X0, Y0, Z0), coordinates (X1, Y1, Z1) at any point of the linear portion ofeach secondary track and coordinate (Xi, Yi, Zi) of a point on the incident beam.It is worthwhile to mention that emulsion technique possesses very high spatialresolution which makes it a very effective detector for studying the intermittencyphenomenon.3. Method of studyWe here adopt the scaled factorial moment method to investigate the fluctuationpattern of compound multiplicity distribution. The number of particles consideredis finite at the available projectile energy. Hence, the statistical fluctuation is verymuch obvious and the standard moments fail to reveal the dynamical fluctuationof particle density distribution. The scaled factorial moment method is free fromthe hazards of statistical noise pollution.Consider a certain interval in some ‘x’ space, defined as ∆x = xmax−xmin. Theconsidered region is divided into M bins of equal size having width δx = ∆x/M .The normalized scaled factorial moment is defined as [1]〈Fq〉 = 〈Mq−1M∑m=1km(km − 1) · · · (km − q + 1)〈km〉q〉 , (1)where km is the number of particles in the mth bin (m = 1, 2, · · ·M) and angularbracket denotes an average over the whole sample of events, q is the order of momentwhich can assume any positive integral value starting from 2.If the non-statistical fluctuations are self-similar in nature, in the limit of smallbin size, factorial moment obeys the relation〈Fq〉 ∝ Mαq . (2)From (2) we get ln〈Fq〉 = αq lnM + C . (3)In analogy with turbulent fluid dynamics, this property is called “intermittency”. Inconnection with high-energy interactions, αq measures the strength of intermittencyand is called the intermittency exponent. From the intermittency exponent αq, onecan find out the signals of non-thermal phase transition with the help of the relationλq =1 + αqq(4)The generalized dimension (Dq) is calculated from αq by using the relationDq = 1 −αqq − 1. (5)320 FIZIKA B 14 (2005) 4, 317–326ghosh et al.: compound multiplicity distribution in nucleus-nucleus . . .It is well known that in usual thermodynamics, constant specific heat (CSH)approximation is widely applicable in many important cases, e.g., the specific heatof gases and solids is constant, independent of temperature, over a smaller or greatertemperature interval [10]. This approximation is applicable to multifractal data ofmultiparticle production process, as has already been shown by Bershadskii [8].If the q-dependence of Dq satisfies the condition Dq > Dq′ for q < q′, then thespectra are said to exhibit multifractality, and the multifractal specific heat can beobtained from the generalized dimension, Dq by the relation [8].Dq = (a − c) + c ln q/(q − 1) for q > 1 , (6)where c is the specific heat, a is some other constant and q can be interpreted asthe inverse of temperature [10].4. Results and discussionTo analyze and measure the factorial moments in the emission angle (cos θ)-space in compound multiplicity, the two type of data sets (24Mg-AgBr and 12C-AgBr interactions, both at 4.5A GeV), have been divided into 2 to 35 subintervals(M = 2 to 35). The normalized scaled factorial moments of order 2 to 7 are cal-culated for each interval with Eq. (1). Figure 1a shows the nature of the variationof normalized scaled factorial moment of different orders (q = 2 to 7) with loga-rithm of the number of multiplicity bins in the emission-angle space for 24Mg-AgBrinteractions at 4.5A GeV. Figure 2a shows the same plot for 12C-AgBr interac-tions at 4.5A GeV. Both plots do not show perfect linear behaviour in the full binrange. Therefore, we divide the full bin range into parts to probe whether the lin-ear behaviour is prominent in different subintervals of M . Variation of the factorialmoment with lnM in selected intervals (17 ≤ M ≤ 35, 3 ≤ M ≤ 17, 2 ≤ M ≤ 21and 9 ≤ M ≤ 18) gives a clear picture of the intermittent pattern of particle pro-duction. Those are represented in Figs. 1b – e and 2b – e, respectively, for the24Mg-AgBr and 12C-AgBr interactions, both at 4.5A GeV.The intermittency exponents αq’s are evaluated by performing best linear fitsaccording to Eq. (5). Table 1 shows the value of the intermittency exponents αqwith different orders in cos θ-space for different bin ranges in 24Mg-AgBr inter-actions at 4.5A GeV. Table 2 shows the same thing for 12C-AgBr interactions,also at 4.5A GeV. It is evident from the tables that the value of the intermit-tency exponents gradually increases with the increase of orders for all multiplicitybin ranges in both data sets. Here we have analysed the scaled factorial momentsin cos θ-space to search for the signals of non-thermal phase transition. λq-valuesare calculated from the value of the intermittency exponents according to the re-lation (4). The results of our study on the variation of λq with q for24Mg-AgBrFIZIKA B 14 (2005) 4, 317–326 321ghosh et al.: compound multiplicity distribution in nucleus-nucleus . . .Fig. 1. Dependence ofln〈Fq〉 on lnM for differ-ent bin ranges: (a) 2 ≤M ≤ 35, (b) 17 ≤ M ≤35, (c) 3 ≤ M ≤ 17, (d)2 ≤ M ≤ 21 and (e) 9 ≤M ≤ 18, for different or-der of moments (q = 2,3, 4, 5, 6 and 7) in 24Mg-AgBr interactions at 4.5AGeV in cos θ-space.322 FIZIKA B 14 (2005) 4, 317–326ghosh et al.: compound multiplicity distribution in nucleus-nucleus . . .Fig. 2. Dependence ofln〈Fq〉 on lnM for differ-ent bin ranges: (a) 2 ≤M ≤ 35, (b) 17 ≤ M ≤35, (c) 3 ≤ M ≤ 17, (d)2 ≤ M ≤ 21 and (e) 9 ≤M ≤ 18, for different or-der of moments (q = 2, 3,4, 5, 6 and 7) in 24C-AgBrinteractions at 4.5A GeVin cos θ-space.FIZIKA B 14 (2005) 4, 317–326 323ghosh et al.: compound multiplicity distribution in nucleus-nucleus . . .TABLE 1. Intermittency exponents αq for different order of moments (q = 2, 3, 4, 5,6 and 7) for different bin ranges in compound multiplicity spectrum of 24Mg-AgBrinteractions at 4.5A GeV in cos θ-space.Bin range q = 2 q = 3 q = 4 q = 5 q = 6 q = 72≤M≤35 0.584±0.011 1.323±0.017 2.068±0.024 2.804±0.034 3.525±0.045 4.231±0.05817≤M≤35 0.485±0.050 1.233±0.070 2.018±0.074 2.790±0.075 3.530±0.084 4.236±0.1043≤M≤17 0.589±0.014 1.332±0.026 2.101±0.042 2.877±0.060 3.653±0.079 4.425±0.0982≤M≤21 0.614±0.013 1.365±0.024 2.128±0.038 2.888±0.056 3.642±0.077 4.387±0.1009≤M≤18 0.503±0.035 1.121±0.046 1.733±0.055 2.341±0.075 2.942±0.107 3.537±0.146TABLE 2. Intermittency exponents αq for different order of moments (q = 2, 3, 4,5, 6 and 7) for different bin ranges in compound multiplicity spectrum of 24C-AgBrinteractions at 4.5A GeV in cos θ-space.Bin range q = 2 q = 3 q = 4 q = 5 q = 6 q = 72≤M≤35 0.383±0.015 0.947±0.036 1.553±0.061 2.170±0.094 2.788±0.132 3.401±0.17317≤M≤35 0.180±0.039 0.488±0.074 0.811±0.124 1.090±0.208 1.322±0.317 1.511±0.4353≤M≤17 0.457±0.014 1.162±0.026 1.978±0.039 2.841±0.055 3.748±0.076 4.671±0.1002≤M≤21 0.455±0.013 1.126±0.030 1.871±0.052 2.661±0.078 3.476±0.107 4.302±0.1389≤M≤18 0.346±0.019 0.933±0.049 1.639±0.096 2.420±0.153 3.220±0.215 3.996±0.079TABLE 3. Specific heats for different bin ranges in compound multiplicity spectrumof 24Mg-AgBr and 12C-AgBr interactions, both at 4.5A GeV in cos θ-space.Bin range 24Mg-AgBr 12C-AgBr2 ≤ M ≤ 35 0.329 0.49817 ≤ M ≤ 35 0.608 0.2073 ≤ M ≤ 17 0.400 0.6292 ≤ M ≤ 21 0.316 0.7099 ≤ M ≤ 18 0.229 0.884and 12C-AgBr interactions for different intervals are shown in Figs. 3a and b. Fromthe figure, one sees that λq gradually decreases with the increase of orders for324 FIZIKA B 14 (2005) 4, 317–326ghosh et al.: compound multiplicity distribution in nucleus-nucleus . . .all multiplicity bin ranges for both interactions and no such minimum value ofλq is obtained at any value of q. The non-occurrence of minimum of λq speaksagainst the existence of non-thermal phase transition in the compound multiplicityspectrum. To obtain the multifractal specific heat, we have plotted the generalizeddimensions Dq against ln q/(q−1). For both interactions, multifractal specific heatsare calculated from the slope values of the best fitted straight lines which aretabulated in Table 3. The specific heats are different for different bin ranges for thetwo sets of interactions.Thus, we may conclude that our present study reveals no evidence for non-thermal phase transition in compound multiplicity spectrum of 24Mg-AgBr and12CAgBr interactions at 4.5A GeV in the emission angle space. Higher-dimensionalanalysis may help in getting better results.Fig. 3. Variation of λq against q for different orders and for different bin ranges in(a) 24Mg-AgBr and (b) 12C-AgBr interactions, both at 4.5A GeV in cos θ-space.References[1] A. Bialas and R. Peschanski, Nucl. Phys. B 273 (1986) 703.[2] A. Bialas and R. Peschanski, Nucl. Phys. B 308 (1988) 857.[3] B. Mandelbrot, J. Fluid Mech. 62 (1974) 331.[4] A. Bialas and R. Hwa, Phys. Lett B 253 (1991) 436.[5] Peschanski R., Nucl. Phys. B 327 (1989) 144.[6] A. Bialas and K. Zaeeswski, Phys. Lett. B 238 (1990) 413.[7] R. Peschanski, Int. J. Mod. Phys. A 6 (1991) 3681.FIZIKA B 14 (2005) 4, 317–326 325ghosh et al.: compound multiplicity distribution in nucleus-nucleus . . .[8] A. Bershadskii, Eur. Phys. J. A 2 (1998) 223.[9] D. Ghosh et al., Nucl. Phys. A 499 (1989) 850.[10] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part I, Pergamon Press (1980).RASPODJELE SLOŽENE VIŠESTRUKOSTI U SUDARIMAJEZGRA-JEZGRA – PROUČAVANJE FAZNOG PRIJELAZAAnalizirali smo eksperimentalne podatke o raspodjelama složene vǐsestrukosti usudarima 24Mg-AgBr i 12C-AgBr, oba na 4.5A GeV, primjenom prilagodnih fak-torijalnih momenata. Izračunali smo vrijednosti poopćene dimenzije Dq za q =2, 3, 4, 5, 6 i 7 pomoću eksponenta prekidanja αq. Te vrijednosti Dq primijenili smoza izračunavanje vǐsefraktalne specifične topline za niz pretinaca. Parametar λq(λq = (αq + 1)/q) smo izračunali na osnovi αq tražeći netermički fazni prijelaz.Analiza nije pokazala prisutnost netermičkog faznog prijelaza, ali smo našli različitespecifične topline u pretincima.326 FIZIKA B 14 (2005) 4, 317–326

Raspodjele složene višestrukosti u sudarima jezgra-jezgra – proučavanje faznog prijelaza

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