We investigated the multifractality of target fragments of 24Mg-AgBr interaction at low energy (4.5 AGeV) in emission angle phase space, using a new method as proposed by Takagi. The analysis involves the step of measuring the generalised dimension Dq, which in turn deducts the multifractal behaviour of target fragments. We also determine the multifractal specific heat.Istražujemo mnogofraktalnost izlaznih sporih čestica u sudarima 24Mg–AgBr na niskoj energiji (4.5 AGeV) u faznom prostoru kuta emisije, primjenom nove metode koju je predložio Takagi. U analizi se određuje poopćena dimenzija Dq iz koje se izvode mnogofraktalna svojstva izlaznih čestica. Također izvodimo mnogofraktalnu specifičnu toplinu

Deb, Argha

Dutta (Chattopadhyay), Keya

Dutta, Ishita

Ghosh, Dipak

Sarkar, Rinku

English

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Printed ISSN 1330–0016Online ISSN 1333–9133CD ISSN 1333–8390CODEN FIZBE7MULTIFRACTALITY AND SLOW PARTICLE PRODUCTION IN 24Mg–AgBrINTERACTION AT 4.5 AGeV ENERGYDIPAK GHOSH, ARGHA DEB, KEYA DUTTA (CHATTOPADHYAY),RINKU SARKAR and ISHITA DUTTANuclear And Particle Physics Research Center, Department Of Physics,Jadavpur University, Kolkata 700 032, IndiaReceived 28 June 2004; Accepted 13 December 2006Online 23 February 2007We investigated the multifractality of target fragments of 24Mg–AgBr interactionat low energy (4.5 AGeV) in emission angle phase space, using a new method asproposed by Takagi. The analysis involves the step of measuring the generaliseddimension Dq, which in turn deducts the multifractal behaviour of target fragments.We also determine the multifractal specific heat.PACS numbers: 25.70.Np, 25.70.Pq UDC 539.172Keywords: 24Mg–AgBr interaction, energy 4.5 AGeV, generalised dimension Dq, targetfragments, multifractal behaviour, specific heat1. IntroductionTo characterize mathematically at-all-scales intrinsically-irregular systems, theconcept of fractality is needed. The word fractal, coined by B. Mandelbrot [1],means to create irregular fragments. A fractal structure has the property that ifone magnifies a small portion of it, the same complexity is shown as for the entiresystem. This feature establishes the absence of regularity in the system. Accordingto Mandelbrot, there is a fractal face to the geometry of nature. The power-lawbehaviour of normalised scaled factorial moments also suggests the existence offractal properties in multiparticle production [2 – 4]. The nonuniform fractality, i.e.the multifractality, has been the focal point of a number of theoretical investiga-tions [5, 6] on dynamics of multiparticle production. The earlier G-moment methodto study the multifractality is faulty in this respect that experimental data sets donot show exact linearity in a log–log plot of moment against bin size, as expectedfrom the mathematical formula. This problem may arise due to the nonvalidity ofthe assumed mathematical limit (number of points tends to infinity) for the real ex-perimental data where number of particles in each event is always finite. To removethis difficulty, Takagi proposed a method [7] which reveals that the fractal prop-FIZIKA B (Zagreb) 15 (2006) 4, 183–188 183ghosh et al.: Multifractality and slow particle production in24Mg–AgBr . . .erties are associated with multiparticle production process at high energies. Thismethod has been properly applied to study the fractal characteristics in electron–positron annihilation [8, 9] and UA5 data on proton-antiproton collisions [10]. Itis well known that pions produced in relativistic interactions are very informativeabout the dynamics of particle production process. But to get a deep insight, itis essential to study target fragments, which can be a good source of information.On the other hand, Bershadskii [11] reported that the constant specific heat (CSH)approximation, widely used in the ordinary thermodynamics, is applicable to themultifractal data. Moreover, the constant specific heats, calculated from a few setsof data corresponding to different types of interactions, are approximately the sameand, therefore, may turn out to be a universal characteristic of the multiparticleproduction process at high energies. This creates a new path to carry out study onconstant specific heat approximation in multiparticle production in high-energy re-actions. This report represents a work on the fractal nature of particles evaporatedfrom a target in 24Mg–AgBr interactions at 4.5 AGeV using Takagi’s procedure,and also to study multifractal specific heat in emission-angle phase space.It is interesting to note that this analysis reveals also multifractal behaviour oftarget fragments.2. Experimental detailsIn order to probe the fractality in target fragmentation, we precisely study theinteractions initiated by 24Mg in AgBr at 4.5 AGeV. The data set used in thepresent analysis was obtained by irradiating stacks of G5 nuclear emulsion platesby a 24Mg beam of incident energy 4.5 AGeV at JINR Dubna. The scanning ofthe plates was carried out with the help of a high resolution Leitz Metalloplanmicroscope provided with an online computer system. The details of scanning andmeasurement are given in our previous works [12].For this analysis, we have considered only the slow-target fragments known asblack particles. According to the emulsion terminology, black tracks are identifiedby ionisation lying between 1.4 I0 < I < 10 I0, where I0 is the minimum ionisationof a singly charged particle. These black tracks have the range < 3 mm.The spatial angle of emission (θ) of each black track with respect to thebeam axis is calculated by noting the space coordinates of the interaction center(x0, y0, z0), one point on the incident beam track and one point on the respectivesecondary track, and applying simple coordinate geometry. The average multiplicityof produced black particles of the sample is 9.6 ± 0.47.3. Method of studyWe consider the target fragmentation process at some incident energy. In sucha process, the particle distribution is considered in a phase space x. A single eventcontains n particles. The multiplicity n changes from event to event according to184 FIZIKA B (Zagreb) 15 (2006) 4, 183–188ghosh et al.: Multifractality and slow particle production in24Mg–AgBr . . .the distribution Pn(x). The selected phase space interval of length x is divided intoM bins of equal size, the width of each bin being δx = x/M . Then the multiplicitydistribution for a single bin is denoted as Pn(δx) for n = 0, 1, 2, 3 . . ., where weassume that the inclusive particle distribution dn/dx is constant, and Pn(δx) isindependent of the location of the bin. n hadrons contained in a single event aredistributed in the interval x = xmax − xmin. The multiplicity n changes from eventto event according to the distribution Pn(x). If the number of independent eventsis Ω, then the particles produced in those events are distributed in ΩM bins of sizeδx. Let N be the total number of target-associated slow particles produced in theseΩ events and naj the multiplicity of black particles in the jth bin of the ath event.The theory of multifractals [13] has been motivated to consider the normalizeddensity Paj defined byPaj = naj/N . (1)This is of course also true when N → ∞. Then one has to consider the Takagi’smoment of order q asTq(δx) = lnΩ∑a=1M∑j=1P qaj for q > 0 , (2)which behaves like a linear function of the logarithm of the resolution R(δx),Tq(δx) = Aq + Bq lnR(δx) ,where Aq and Bq are constants independent of δx. If such a behaviour is observedfor a considerable range of R(δx), a generalised dimension may be determined asDq = Bq/(q − 1) . (3)Now, evaluating the double sum of P qaj , Takagi showed [7] that for a sufficientlylarge Ω,ln〈nq〉 = Aq + (Bq + 1) lnR(δx) .While analyzing real data [8 – 10], it was observed [14] that a plot of ln〈nq〉 againstδx saturates for large x regions. This deviation may be due to the non-flat behaviourof dn/dx in the large x region. Takagi suggested that 〈n〉 would be a better choicefor the resolution R(δx) because dn/d〈x〉 is flat by definition [13, 16]. ChoosingR(δx) = 〈n〉, one hasln〈nq〉 = Aq + (Bq + 1) ln〈n〉 (4) ,a simple linear relation between ln〈nq〉 and ln〈n〉. The generalized dimension Dqcan be obtained from the slope values using the relation (3).FIZIKA B (Zagreb) 15 (2006) 4, 183–188 185ghosh et al.: Multifractality and slow particle production in24Mg–AgBr . . .The case with q = 1 is obtained by taking an appropriate limit [13]. D1 can beobtained from the relation〈n ln n〉/〈n〉 = C1 + D1 ln〈n〉 , (5)where C1 is a constant. So far the methodology is developed for non-overlapping xbins, but it can be applicable for overlapping bins [15].It is well known that in classical thermodynamics, the constant specific heatapproximation (CSH) is widely applicable in many important cases, e.g. the spe-cific heat of gases and solids is constant, independent of temperature over a greateror smaller temperature interval [16]. This approximation is also applicable to mul-tifractal data of target fragmentation process, which has already been shown byBershadskii [11].One can get the generalised dimension Dq from CSH approximation asDq = (a − c) + c ln q/(q − 1) . (6)Here c is the specific heat, a is some other constant and q can be treated as theinverse of temperature, q = 1/T [16].If one plots Dq versus ln q/(q − 1), the slope of the graph will give the value ofspecific heat. In this paper, we investigate the multifractal nature of 24Mg-AgBrinteraction at 4.5 AGeV and from the knowledge of Dq, we calculate the specificheat. It is observed that the generalised dimension Dq decreases with the increaseof order number q. This reflects the multifractal geometry in the case of particlesevaporated from the target.4. Analysis and discussionTo analyze our data, it is considered that a single event contains n black particlesdistributed in a particular interval in emission-angle phase space. This interval isdivided into overlapping bins whose size is varied in steps of 0.1.The central value is0. In Fig. 1, ln〈nq〉 is plotted against ln〈n〉 for q = 1, 2, 3 and 4. A linear behaviouris obtained in all of the above log-log plot. The generalized dimensions D2, D3,D4, D5 are obtained from the slopes of such linear plots by using Eq. (3). D1 isobtained from Eq. (5). In Fig. 2, Dq is plotted against q. This plot shows thatDq decreases gradually with the increase of q, which suggests multifractality in thecase of target-evaporated particles. We also plot Dq against ln q/(q − 1) (Fig. 3),which is also a linear plot. From the slope of its linearity, we get the specific heat.The generalized dimensions are given in Table 1. The calculated specific heat forthese interaction data is0.19 ± 0.017 .This analysis shows that even at very low energy (4.5 AGeV) the target frag-ments from the 24Mg–AgBr interaction show multifractal behaviour.186 FIZIKA B (Zagreb) 15 (2006) 4, 183–188ghosh et al.: Multifractality and slow particle production in24Mg–AgBr . . .Fig. 1 (left). Variation of ln〈nq〉 with ln〈n〉.Fig. 2. Variation of Dq with q.Fig. 3. Variation of Dq with ln q/(q − 1).TABLE 1. Values of Dq for different q values of24Mg–AgBr interaction.Data D1 D2 D3 D4 D524Mg - AgBr at 4.5 AGeV 0.750 0.730 0.709 0.690 0.675FIZIKA B (Zagreb) 15 (2006) 4, 183–188 187ghosh et al.: Multifractality and slow particle production in24Mg–AgBr . . .AcknowledgementsWe thank Prof. K. D. Tolstov of JINR, Dubna, for giving us the exposed anddeveloped emulsion plates. We also gratefully acknowledge the financial help of theUniversity Grant Commission (India) under COSIST programme.References[1] B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York (1982).[2] P. Lipa and B. Buschbeek, Phys. Lett. B 223 (1989) 465.[3] I. M. Dremin, Mod. Phys. Lett. B 251 (1990) 455.[4] P. Carruthers, Int. J. Mod. Phys. A 4 (1989) 5587.[5] R. C. Hwa, Phys. Rev. D 41 (1990) 1456.[6] C. B. Chiu and R. C. Hwa, Phys. Rev. D 43 (1991) 100.[7] F. Takagi, Phys. Rev. Let. 72 (1994) 32.[8] W. Braunschweig et al., Tasso Collab., Phys. Lett. B 231 (1989) 548.[9] W. Braunschweig et al., Tasso Collab., Z. Phys. C 45 (1989) 159; P. Abreu et al.,DELPHI Collab., Z. Phys. C 52 (1991) 271.[10] R. E. Ansorge et al., UA5 Collab., Phys. C 43 (1989) 357.[11] A. Bershadski, Euro Phys. J. A 2 (1998) 223.[12] D. Ghosh, Euro Phys. Lett. 23 (1993) 91.[13] H. G. E. Hentschel and I. Procaccia, Physica (Amsterdam) D 8 (1983) 435.[14] F. Takagi, Phys. Rev. Lett. 53 (1984) 427; Phys. Rev. C 32 (1985) 1799.[15] V. Simak et al., Phys.Lett. B. 206 (1988) 159.[16] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 Pergamon Press, New York(1980).MNOGOFRAKTALNOST I TVORBA SPORIH ČESTICA U SUDARIMA24Mg–AgBr NA ENERGIJI 4.5 AGeVIstražujemo mnogofraktalnost izlaznih sporih čestica u sudarima 24Mg–AgBr naniskoj energiji (4.5 AGeV) u faznom prostoru kuta emisije, primjenom nove metodekoju je predložio Takagi. U analizi se odred–uje poopćena dimenzija Dq iz koje seizvode mnogofraktalna svojstva izlaznih čestica. Takod–er izvodimo mnogofraktalnuspecifičnu toplinu.188 FIZIKA B (Zagreb) 15 (2006) 4, 183–188

Mnogofraktalnost i tvorba sporih čestica u sudarima 24Mg–AgBr na energiji 4.5 AGeV

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