Evidence of phase transition in the break up of Kr-projectile nuclei at 0.95A GeV

Nuclear Physics A 829 (2009) 210–233 www.elsevier.com/locate/nuclphysa

Evidence of phase transition in the break up of Kr-projectile nuclei at 0.95A GeV B. Bhattacharjee ∗ , B. Debnath Department of Physics, Gauhati University, Guwahati-781014, India Received 27 January 2009; received in revised form 24 July 2009; accepted 24 July 2009 Available online 22 August 2009

Abstract Projectile fragmentation mechanism and the possible liquid gas phase transition have been studied by extracting the critical exponents using cluster approximation technique. A ‘toy model’, schematically accounting for pre-equilibrium, has been developed and various moments as well as conditional moments have been evaluated with the data obtained from the toy model to see the effect of mass conservation constraint. An exponent δ, related to third order moment, has been evaluated to obtain a set of values of γ , β and τ that follow the corresponding scaling relation. The values of γ , β and τ so obtained are found to be 1.34 ± 0.19, 0.54 ± 0.15 and 2.31 ± 0.06 respectively. © 2009 Elsevier B.V. All rights reserved. PACS: 25.75.-q; 25.70.Pq; 13.85.Hd Keywords: Nucleus–nucleus collision; Multi-fragmentation; Power law; Phase transition; Charge moment; Critical exponents

1. Introduction It is believed that in the complex scheme of high energy reactions, nuclear fragmentation is relatively a well isolated phenomenon. Nuclear fragmentation in which excited nuclei break up into several pieces of smaller masses, each more massive than an alpha particle, is termed as nuclear multi-fragmentation. It is opined that one of the most important challenges of heavy ion physics is the identification and characterization of nuclear liquid–gas phase transition believed to be * Corresponding author. Tel.: +913612570531; fax: +913612700133.

E-mail address: [email protected] (B. Bhattacharjee). 0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2009.07.015

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the underlying mechanism of nuclear multi-fragmentation process [1,2]. In normal state, nuclear matter shows the properties of a liquid, but the power law behavior followed by mass (charge) yield distribution of the fragments produced from the remnant parts of the colliding nuclei gives the exponent value which lies within the range expected for a system near its critical point. Such observations led different groups [3–13] to suggest that there may be a continuous phase transition associated with a critical point. On the other hand, it has been proposed that a back bending in the caloric curve that leads to a negative specific heat can be signed through the occurrence of abnormally large kinetic energy fluctuations. This signature of 1st order phase transition has been applied to a number of multi-fragmentation data and a liquid–gas phase transition has been tentatively identified, particularly at few hundred MeV/A [1,14–18]. Also, considering the size of the heaviest fragment produced in each collision event as order parameter, a number of other groups [19–25] have studied the distribution of this order parameter for various systems at few to few hundred MeV/A. From the observation of a characteristic bi-modal behavior, a first order type of phase transition is associated with the studied nuclear multi-fragmentation processes. In the last two decades, intense theoretical and experimental investigations have been carried out on relativistic heavy ion collisions to gather information about liquid gas phase transition of low density nuclear matter in hot nuclei and hence the fragmentation mechanism [26–31]. Amongst many phenomenological models, the statistical model has been extensively used in describing the critical nature of nuclear multi-fragmentation. It is known that, in statistical physics, systems consisting of mutually interacting units, in general, cannot be evaluated in exact terms, whereas most systems without interaction are easy to characterize. Thus, an often used approximation is the cluster approximation which tries to transform the problem of interacting units into the approximation of non-interacting clusters. Stauffer [9] has pointed out that percolation, besides being perhaps the simplest statistical tool to study phase transition, also serves as an introduction to cluster approximation of collective phenomena. Percolation problem on a large lattice displays the features of a system undergoing a second-order phase transition [32]. A continuous phase transition in nuclear matter, that might have taken place in the final stage of fragmentation of heavy ion collisions, is generally characterized by the presence of some characteristic features. Certain experimentally observable quantities might undergo fluctuations or diverge or may even tend to vanish near some critical value of the control parameter giving the signatures of continuous phase transition. This critical value of the control parameter corresponds to the critical point in the phase diagram. In cluster distribution technique of studying phase transition, these experimental observables are described by a finite number of critical exponents namely, σ , β, τ , γ , etc. These exponents are universal and depend neither on the structural details of the lattice nor on the type of percolation, but only on the dimension of the lattice. The main characteristic of these exponents is that they obey the scaling relation and all of them can be evaluated from the knowledge of just two. In an attempt to get the signatures of critical behavior in the case of break up of nuclei, several workers have analyzed their experimental data on various systems at different energies [2,29–31, 33–41]. The experimentally evaluated values of σ , β, τ and γ are then compared with the values obtained for different three-dimensional known systems exhibiting critical behavior. A striking agreement in the values of these exponents indicates towards the possibility of existence of criticality in the experimental system under consideration. However, though there is a broad agreement between the values of different exponents as estimated by various groups [34,39], in a number of cases, a careful cross check reveals that the cited values of the exponents suffer from internal inconsistencies and do not follow the corresponding scaling relation [37,38]. There are also reports that the value obtained for an exponent is sensi-

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tive to the range of values chosen for the control parameter and to the system under investigation [37,39]. It may therefore be inferred that the values of the critical exponents, considered as the representative of critical phenomenon, as cited by various groups are not unambiguous. Moreover, Elliott et al. (EOS Collaboration) [35] have shown that out of several techniques adopted by different workers in estimating percolation exponents, only a few of these may serve as true sensitive tools to study the criticality. From a study of 1A GeV Au–C data and its comparison with two theoretical models, such as one that undergoes phase transition and the other that does not, they categorized the traditional signals of critical behavior into a number of insensitive and sensitive tools. Further, EOS and KLMM Collaborations [34,37] have studied Au multi-fragmentation at two different energies, namely at 1.0A GeV and 10.6A GeV respectively. Remembering the results of ALADIN Collaboration on the target mass independence nature of projectile multi-fragmentation [42], a comparison of EOS and KLMM Collaborations’ results suggests that the exponent values calculated at two different energies for the same system differ significantly. KLMM Collaboration on the basis of their analysis concluded that percolation theory becomes a less satisfactory representation of the break-up of the excited nuclei at high energy interactions then it is at lower energies. However, EMU01 Collaboration from the study of Au–Em interactions at KLMM energy has reported a clear evidence of the critical behavior of the fragmentation process with universal features of second order phase transition. Clearly, the findings of one group differ significantly from that of the other indicating that the studies on break up of the nuclei in high energy A + A collisions are far from complete. At the same time, many different measurements of the nuclear caloric curve [43] performed on various systems to realize 1st order type of liquid–gas phase transition also show quite diverse behavior [1]. The EOS Collaboration has studied three systems of different masses, namely, Au, La and Kr interactions on C at 1A GeV. They considered thermal excitation energy E∗th of the remnants as the control parameter and performed analyses on fragment properties and critical exponents. Looking at the differences in the results of these preliminary analyses it is claimed that while Au and La indicate towards a 2nd order phase transition, the Kr fragmentation is possibly governed by 1st order type of phase transition. But the observed results are not insulated from the surface effect, as is seen in three-dimensional (3d) percolation studies and therefore the observed differences in the results cannot exclusively be attributed to the differences in the order of phase transition. Their cited results reveal that various properties of the fragments and values of the exponents τ , β, γ and β/γ vary systematically with mass or otherwise size of the fragmenting nuclei. Moreover, the authors themselves have pointed out that their results on the analysis of heat capacity [22,23] for all the three systems are inconclusive in identifying the exact order of phase transition. Nevertheless, a back bending in the caloric curve in Kr data is taken as a signature of first order phase transition in that system. But the observed back bending is found to be very much sensitive to the particular mass cut for the studied systems of Au and Kr and therefore may not be unambiguously considered as a sensitive tool to realize order of phase transition involved in nuclear multi-fragmentation process. The problem of identification and characterization of nuclear liquid–gas phase transition therefore still remains to be solved. Though some progresses have already been made in understanding the nuclear multifragmentation process, there are still a few more queries that need to be addressed for a better understanding of the process. A few of these are identified as — can those systems that contain critical behavior be readily distinguished from those which do not? Which exactly are insensitive and sensitive tools for studying criticality? Is the list put forwarded by EOS Collaboration final and complete? What is the nature of production of clusters — is it a sequential or simultane-

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ous decay? It is opined that a single existing theoretical model cannot satisfactorily describe all the characteristics of nuclear multi-fragmentation and demands more results on such process for several other systems at different energies. In the present investigation we have analyzed Kr–Em interactions at 0.95A GeV using NIKFI BR2 photo-nuclear emulsion pellicles exposed at SIS, GSI Dramstadt. Details of the experimental technique are given in our earlier publications where we have studied spatial distribution of target fragments in the light of intermittency and multifractality [44–46], universal scaling behavior of fast helium fragments [47] and emission characteristics of intermediate mass projectile fragments [48,49]. In this work we have developed a ‘toy model’ of nuclear multi-fragmentation and examined, using cluster approximation technique, the effect of mass conservation constraint on the traditional signals of critical behavior. Possible signatures of phase transitions have been investigated by using Campi’s technique of finding critical exponents. The estimated values of the critical exponents τ , β and γ of the present investigation are then compared with the values reported for various other systems. A higher order moment M3 , related to skewness [21,50], is also evaluated using the experimental data and the corresponding exponent δ is calculated. An attempt has been made to establish the scaling relationship between these experimentally evaluated exponents τ , β, γ and δ. 2. Results We started our analysis by formulating a ‘toy model’ of nuclear multi-fragmentation that is different from the EOS Collaboration [35] and is based on the following algorithm: first the total charge QPF of a fragmenting system is determined randomly from 36 total system constituents which is the charge of the projectile for present investigation. The number of prompt protons emitted from a particular event is then considered as Nprompt = 36 − QPF . Each QPF is then allowed to disintegrate randomly to give the total number n of PFs in an event from a uniform distribution on (1, QPF ). The assignment of maximum charge Zmaxi of a cluster which may be emitted from an event having n number of PFs is guided by two factors, namely, QPF and n. Following Elliott, we then randomly choose the first cluster with maximum charge Zmax1 from uniform distribution on (1, Zmax ), where Zmax = QPF − (n − 1). Obviously, the subsequent (n − 1) clusters are then to be generated from the remaining (QPF − Zmax1 ) constituents. Thus, Zmax2 , the maximum charge of the second cluster is then chosen from random partitioning of  ), where Z  Zmax on (1, Zmax max = [(QPF − Zmax1 ) − (n − 2)]. For each event this process is is disintegrated into respective (n − 1) number of PFs and ends with repeated until total Q PF  Zmaxn = (QPF − Zmaxk ), where k = 1, 2, . . . , (n − 1). Our algorithm of random partitioning differs from that of the EOS one [35] in the sense that we considered the charge of the remnant part of the projectile nucleus as the charge of the fragmenting system, not the total charge of the projectile. Our technique rests on the random selection of total charge of the fragmenting system from a uniform distribution of (1, Zproj ), while EOS’s technique relies on the random selection on (1, Zproj ) of multiplicity in which the incident projectile will disintegrate. Thus the major difference between the two toy models is that while in our algorithm the mass conservation is violated, as it should be due to pre-equilibrium emission, the EOS model ignores this aspect. In order to introduce the two toy models, their physical meaning and their differences more sensibly, we plot in Figs. 1(a) and (b) the distributions of pre-equilibrium particles and cluster multiplicity, respectively. It could be readily seen from these plots that the two toy models assume respectively a flat distribution of pre-equilibrium particles (this work) and a flat distribution of cluster multiplicity (Elliott et al.). For both distributions, the experimental data lay in between and in this sense in-

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(a)

(b) Fig. 1. (a) Frequency distribution of the prompt particles and (b) multiplicity distribution of projectile fragments for Kr–Em interactions and randomly generated data.

deed the two toy models can be considered as two extreme limits to describe data with minimal hypothesis and without any critical behavior or phase transition. As we shall see, when tested for a total 36 system constituents, both the toy models are found to be capable of producing signals traditionally associated to criticality. The data generated with our toy model may therefore be used as a tool to evaluate the effect of mass conservation constraint on traditional signals of critical behavior. All total 542 minimum biased Kr–Em interactions are considered for the present work. The event statistics for different targets of emulsion are shown in Table 1. Frequency distribution of various charged projectile fragments with Zf  1 emitted from Kr–H, CNO, Em and AgBr interactions are plotted in Fig. 2(a). Similar distribution fitted with

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Table 1 Event statistics of 84 Kr–Em interactions for different targets of nuclear emulsion. Interaction

Number

84 Kr–H

105 215 222

84 Kr–CNO 84 Kr–AgBr

a power law has also been reported in nuclear fragmentation of 238 U at 0.96A GeV, 84 Kr at 1.25A GeV and 131 Xe at 1.22A GeV in nuclear emulsion [51–54]. For the experimental and randomly generated data, considering total number of system constituents as 36 for both the toy models, the yields of the fragments charge distribution, lying between Z = 1–10, have been re-plotted in Fig. 2(b) in log–log scale and a straight line fit to the respective data points gives the values of the exponent τ as 2.12 ± 0.15, 0.66 ± 0.03 (for our) and 2.06 ± 0.21 (for EOS) respectively. The value of Pearson’s correlation coefficient R, whose values lie between −1 to +1 and which measures the strength of linear relationship between two variables [46,55], is found to be not less than 0.981 in any of these cases. 2.1. Sensitivity test of various observables Assuming that the temperature T of the system is linearly dependent on the total multiplicity m, in a number of nuclear experiments, multiplicity is taken as the control parameter. This control parameter could not be finely tuned explicitly in order to convert the system into the critical state. However, in a number of events, a favorable situation is spontaneously developed where the system itself evolves to a critical one. The total charged projectile fragment multiplicity m is defined as [10,11,38] m = Nf + Nα + Nprot ,

(1)

where Nf , Nα and Nprot denote the number of heavy PFs with charge Zf  3, alpha particles with Zf = 2 and the number of emitted protons with Zf = 1 respectively. Here Nprot is determined using charge balance of the PFs. The distance ε of a given event with multiplicity m from the critical point mc is defined as [34,38] ε = mc − m.

(2)

2.1.1. Fluctuations in Zmax It is known that a system exhibits significant fluctuations in the neighborhood of the critical point in a small range of the control parameter and appears at increasingly large scale as ε → 0. It is also known that the most readily observed fluctuations in the cluster distribution are those in the size of the largest cluster [35,50,56]. In Fig. 3(a), the standard deviation of Zmax normalized with respect to the charge of the projectile are shown as a function of multiplicity for our experimental set of data. Large fluctuations in the multiplicity range between 11–19 are readily seen from this plot with a peak at m = 17 ± 1. As in the case of Ref. [35], peaks have also been observed for both the sets of randomly generated data (plots are not shown here). But at this point one has to remember the fact that the point with σZmax /Zproj = 0 and m = 1 corresponds to a single projectile fragment moving with total charge of the projectile itself and therefore is not associated with the fragmentation of the nucleus and hence may be ignored. With this point

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(a)

(b) Fig. 2. (a) Frequency distribution of various charged projectile fragments with Zf  1 for the collisions of krypton beam with various targets of nuclear emulsion at 0.95A GeV. The distributions are plotted on the same X and Y scale. (b) Frequency distribution for experimental as well as randomly generated events in log–log scale upto Zf = 10. Open circles and open triangles are for randomly generated data following our and EOS algorithms respectively for a total 36 system constituents and open squares are for our experimental data. Solid (experimental data), dot (our random events) and dash (EOS’s random event) lines are the best fitted lines.

being excluded, no well defined peak could be observed in σZmax /Zproj vs. m plot [Fig. 3(b)], particularly with our set of generated data, and the behavior of this distribution differs significantly from that of the experimental one. However, recently, Gulminelli et al. [56] have pointed out that there exists a major problem in connecting such fluctuation peak to a phase transition or critical behavior in the analysis of nuclear multi-fragmentation data. A number of effects such as finite-

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(a)

(b) Fig. 3. Standard deviation of normalized Zmax as a function of multiplicity m for (a) experimental data, (b) randomly generated data.

ness of the system under investigation, use of different event sorting procedure, etc., smooth the fluctuation effect to such an extent that not only the transition point is loosely defined and shifted, but also the signal is qualitatively the same for a critical point, a first order phase transition or even a continuous change or cross over. Thus the heap like structure that has been observed in Fig. 3(a) with experimental data may not carry as much information as one would expect for cluster approximation of critical behavior analysis for an infinite system. 2.1.2. Charge moments and conditional moment For a single event, Campi [10] defined the kth moment of charge distribution as

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Mk =



nZf (ε)Zfk

(3)

Zf

and for a collection of data, Mk (ε) in the small bins of multiplicity m as    1  i  1   k nZf (ε)Zf . Mk (ε) = Mk (ε) = N N i

(4)

Zf

Here nZf is the normalized charge distribution and is defined as nZf = NZf /QPF , QPF is the sum of charges of all the projectile spectator protons, fast alpha particles and heavy projectile fragments with charges Zf  3. N denotes the total number of events in a given small range of ε, and Mki is the kth order charge distribution moment for ith event. It may be noted here that in the calculation of various charge moments in the gas phase, the contribution of prompt protons and in liquid phase, the contribution of fragments with largest charge Zmax as well as the prompt protons are excluded. A variable γ2 , related to variance of charge σ 2 , is defined as [11] γ2 =

M2 M0 σ2 = 1 + . Z2 M12

(5)

Here M0 , M1 , M2 are the zeroth, first and second order moments of charge distribution respectively. While M0 and M1 correspond to mean number and mean size of the clusters, M2 is a quantity that is related to the fluctuation in the size of the fragments. In Fig. 4(a), γ2 , used to describe fluctuations in the average cluster size, is plotted as a function of multiplicity m for Kr–Em interactions and compared with the result of EOS Collaboration for Au–C and Kr–C collisions at 1.0A GeV. The error bars show the standard deviation in γ2 . Fig. 4(b) represents the same plot for the randomly generated data. A peak in γ2 is expected from random partitions only if the fragmenting system is of constant size. If this is not the case because of pre-equilibrium (as in the present data-set), no pronounced peak should be expected in the absence of a phase transition or a critical phenomenon. This is exactly what is seen in Fig. 4(b), no distinct rise and fall in γ2  is seen with our set of generated data. However, from Fig. 4(a) it is clearly seen for all the three colliding systems that γ2  increases monotonically with m, attains a maximum and then gradually decreases. Thus the appearance of a crest in γ2  value might have been resulted from a phase transition occurring at m = mc which for the present experiment is found to be 18 ± 1. For gold it was reported to be at m = mc = 32. Further, it is interesting to note from this figure that the emulsion data of this work confirms EOS finding that the position and height of the γ2 peak is essentially determined by the source size [29,31]. To have further insight into our sample of data, the mean values of 2nd moment of charge distribution, M2 , is plotted against m in Fig. 5(a) and compared with the result of EOS Collaboration. Fig. 5(b) represents the same plot for randomly generated events. A similar rise and fall of second order charge moment is clearly evident in Fig. 5(a) for both Kr and Au projectiles. Strong fluctuations in M2  values are found for m = 8–24 for Kr–Em interactions. Here again it is seen that the distribution of 2nd charge moments for experimental set of data points differs significantly from our generated data. Once again, a rise and fall in M2  values with non-critical EOS generated data suggests that M2  can be considered as an observable of critical behavior if the experimental values differ with the values obtained from both the sets of simulated data, differing from only one of the two may not be sufficient. The EMU-01 Collaboration, from their studies on 10.6A GeV Au–Em interactions [39], has pointed out that the distributions of different moments as well as conditional moment are not

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(a)

(b) Fig. 4. Variation of γ2  with total charged fragment multiplicity m for (a) experimental data of 84 Kr–Em (this work) and 84 Kr–C and 197 Au–C (EOS Collaboration); (b) randomly generated data.

influenced by the target masses. The differences in conditional moment of Fig. 4(a) and 2nd charge moment of Fig. 5(a) may therefore be attributed to the difference in the projectile masses of various systems (Au–C and Kr–Em/C). Figs. 6(a) and (b) show the scattered and contour plots of 2nd charge moments M2i versus multiplicity m for individual events of Kr–Em interactions of the present investigation. From the experimental plot it is seen that M2i values fluctuate strongly for multiplicities 10  m  24. Clearly, the experimental data points are grouped in two distinct categories: one consists of events with multiplicity less than 19 with M2i < 0.9 and the other with multiplicity m > 13 and largest M2 values for a given m. While small multiplicity and small M2i values of former group are

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(a)

(b) Fig. 5. Average of second charge moments as a function of multiplicity m for (a) 84 Kr–Em (solid circles) and 197 Au–C (open circles) data and (b) generated data.

believed to be characteristic features of liquid phase, large multiplicities and large M2i values of the later group of events are expected features of gas phase. No such grouping of the data points could be seen [Figs. 6(c) and (d)] with both the sets of random numbers and therefore strongly support liquid gas type of phase transition in Kr multi-fragmentation. 2.1.3. Multiplicity distribution of Intermediate mass fragments Another predicted consequence of the liquid–gas phase transition is the abundant emission of intermediate mass fragments. We, therefore, have plotted in Fig. 7 the average number of intermediate mass fragments whose charges lie within 3–14 (in our case) against the multiplicity

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m for experimental and generated set of data. All these plots look similar and may not serve as a sensitive signature of phase transition study. 2.1.4. Variation of mean fragment size Fig. 8(a) shows the variation of M1  with the maximum charge of the fragments for m < mc and as expected, it monotonically decreases with the increase of Zmax value. However, a similar result with randomly generated events [Fig. 8(b)] suggests that such variation of M1  with Zmax cannot be taken as a genuine signature of percolation type of phase transition in the system under consideration. From the above results one therefore gets enough indications that the system under present investigation shows some definite evidences of critical behavior suggesting the relevance of fur-

(a)

(b) Fig. 6. Plots of second charge moments for individual events as a function of m. (a) Scattered plot for our experimental data, (b) contour plot of same, (c) contour plot for randomly generated events following our algorithm and (d) following EOS algorithm.

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(c)

(d) Fig. 6. (continued)

ther analysis of the data to estimate the various exponents. However, one at the same time cannot deny the fact that these evidences, particularly the scattered plot of 2nd charge moments M2i and the fluctuations in the size of the largest cluster, are compatible with first order phase transition as well thereby suggesting the relevance of studying bi-modality signals [57,58] with the present set of experimental data. 2.2. Critical exponents and testing of scaling laws It is believed that in nuclear fragmentation process the system is at coexistence in the neighborhood of the critical point and cluster distribution follows a pure power law with Zf as nZf (ε) ∼ Zf−τ

at ε = 0

(6)

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(a)

(b) Fig. 7. Distribution of average number of IMFs as a function of multiplicity for (a) experimental events and (b) randomly generated events.

and near the critical point, M2 (ε) and Zmax are related with ε as M2 (ε) ∼ |ε|−γ Zmax (ε) ∼ ε

β

for ε → 0, for ε > 0.

(7) (8)

Here τ , γ , and β are critical exponents. Zmax (ε) is the average charge of the largest fragment for a given distance from the critical m. Far away from the critical point, the behavior of the system is dominated by the mean field regime and these relations are not followed [38]. In scaling theory, kth moment of the cluster number density is related with the values of critical exponents as Mk (ε) ∝ |ε|−(1+k−τ )/σ

for ε → 0,

(9)

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(a)

(b) Fig. 8. Variation of mean size with Zmax ; (a) for experimental events and (b) for randomly generated events.

with the restriction 1 + k − τ > 0 for dimension d > 1 and k  0 for d = 1 [59]. Here σ is a critical exponent related to the characteristic cluster size. Using relation (9) one can easily find out the scaling relation among any three critical exponents and for β, γ and τ , it can be given as 3β + 2γ = τ (γ + β).

(10)

2.2.1. Estimation of critical exponents To find out the different critical exponents, on the basis of Fig. 6(a), we have considered the selected data instead of the whole data set. To estimate the critical exponent γ and the critical multiplicity mc , the “γ -matching” technique is adopted and fitting boundaries were determined following the works of Kudzia [38].

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(a)

(b) Fig. 9. Mean values of second charge moments M2  for (a) liquid and (b) gas phases as a function of ε.

Figs. 9(a) and (b) show the variation of M2  with ε in log–log scale for liquid and gas phases. As mentioned earlier, due to mean field and finite size effects, far away from the critical point cluster distributions do not follow the above power laws. It is therefore reasonable to ignore the extreme points of Fig. 9(a) and (b) and do the straight line fitting with the remaining data points. Also following the suggestion of Fisher Drop Model (FDM), a model that relates various moments to relevant thermodynamical quantities, γ -value in gas phase has been calculated including Zmax . Different ranges were selected for liquid and gas phase events and γ s were estimated for several trial values of the critical point mc . The different values of γ for liquid and gas phases are tabulated in Table 2.

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Table 2 Critical exponent γ for gas and liquid phases for different trial values of mc . Sl. No.

mc

γliquid

R

γgas (inc. Zmax )

R

1 2 3 4 5 6 7 8

15 16 17 18 19 20 21 22

1.96 ± 0.24 1.69 ± 0.24 1.68 ± 0.20 1.34 ± 0.14 1.39 ± 0.12 1.44 ± 0.13 1.61 ± 0.21 1.61 ± 0.16

0.936 0.932 0.940 0.944 0.952 0.950 0.918 0.944

1.54 ± 0.12 1.44 ± 0.15 1.47 ± 0.12 1.33 ± 0.05 1.35 ± 0.05 1.22 ± 0.10 1.25 ± 0.05 1.20 ± 0.05

0.978 0.963 0.975 0.995 0.994 0.972 0.990 0.992

Fig. 10. |γliquid − γgas | as a function of mc .

It is expected that at m = mc , |γliquid − γgas | attains a minimum value and γgas , within the statistical error, should agree with the value of γliquid . A plot of |γliquid − γgas | against various hypothetical mc values [Fig. 10] clearly shows a minimum in |γliquid − γgas | value at mc = 18 ± 1 indicating that critical multiplicity for Kr–Em interactions at 0.95A GeV, as determined by the γ -matching procedure, is 18 ± 1 and the critical exponent γ = 1.34 ± 0.19, the average of γliquid and γgas for which |γliquid − γgas | is minimum. The critical exponent β is calculated using Eq. (8). The value of lnZmax  is plotted as a function of ln |mc − m| in Fig. 11. From the linear fit carried out within the fitting boundaries, determined during the γ -matching procedure, β is estimated to be 0.47 ± 0.05 with R = 0.980. Next, we have made an attempt to estimate the critical exponent τ . In estimating τ it is to be remembered that the assumptions of FDM are not valid for the charges less than 6 and in our case most of the PF’s charge lies within 1–6. Therefore for the calculation of critical exponent τ , rather than Eq. (6), we have used the relation [6,34,35] μ=

ln M3 τ − 4 . = ln M2 τ − 3

(11)

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Fig. 11. Variation of lnZmax  with ln |mc − m|.

The variations of ln M3 as a function of ln M2 for each event and lnM3  against lnM2  are shown in Figs. 12(a) and (b) respectively for the gas phase only for the reason discussed in Ref. [60]. The straight line drawn is the best fitted line for the experimental data points and from the slope of this line the value of τ is found to be 2.31 ± 0.06 which, within statistical error, agrees with the value obtained from the charge distribution of this work. Experimental values of γ , β, and τ for different systems, including this work, are listed in Table 3. β values estimated using the scaling relation (10) for these systems are also listed in this table. From Table 3, it is seen that, for all but Ref. [39], the experimental β values differ significantly from that estimated using scaling relation (10) involving τ and γ as other two parameters, thereby raising doubt about the correctness of the values of different exponents. 2.2.2. Testing of scaling laws In an attempt to get a better set of exponents with our experimental data, we have evaluated another critical exponent δ related to the third order moment, M3 as M3 (ε) ∼ |ε|−δ ,

(12)

where δ bears a scaling relation with γ and τ as 3δ − 4γ = τ (δ − γ ).

(13)

In Fig. 13, variation of lnM3 (ε) with ln |m − mc | is plotted at critical multiplicity mc = 18 to get the value of δ. A linear fit of the data points for ln |m − mc | lying between 2.10–2.70 gives the value of critical exponent δ as 2.95 ± 0.14 with R value 0.997. The above relation (12) is valid for both the phases but we have considered only the gas phase again due to the finite size effect. Using this value of δ in scaling relation (13), exponent γ is found to be 1.20 ± 0.17 which, within the statistical errors, lies close to the graphically obtained value. Thus the experimental values of the set of three exponents δ, γ , and τ obey the scaling relation (13). Having a set of three experimental exponents that obey the scaling relationship, we

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(a)

(b) Fig. 12. (a) Event-by-event distribution of ln M3 vs ln M2 and (b) variation of lnM3  with lnM2 .

re-evaluate the β value considering τ and δ as the other two parameters from the corresponding scaling relation 4β + 2δ = τ (β + δ).

(14)

The value of β thus calculated is found to be 0.54 ± 0.15 which is in good agreement with the value reported by Srivastva et al. [29] and also nearer to the value obtained from relation (10) using our experimental values of τ and γ . Taking β as 0.54 ± 0.15 the successive estimations of τ and γ from the scaling relation (10) give their values respectively as 2.29 and 1.20. It is readily seen that these values are in good agreement with our experimental values. Thus, finally

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229

Table 3 Different values of γ , τ and β for various systems. Exp.

Energy (A GeV)

γ

τ

β

β (calculated)

Ref.

Kr–Em (norm) Kr–C La–C Au–Em Au–Em (norm) Au–Em (norm) Au–Em Au–Em Au–C Au–Em Au–Em d = 3 Ising (liquid–gas) d = 3 percolation

0.95 1 1 1 4 10.6 4 10.6 1 10.6 10.6 –

1.34 ± 0.19 – – – 1.15 ± 0.09 1.17 ± 0.09 1.23 ± 0.09 1.11 ± 0.09 1.40 ± 0.10 0.86 ± 0.05 – 1.23

2.31 ± 0.06 1.88 ± 0.08 2.10 ± 0.06 2.16 ± 0.08 2.12 ± 0.04 2.11 ± 0.05 2.15 ± 0.04 2.16 ± 0.05 2.14 ± 0.06 2.23 ± 0.05 1.88 ± 0.06 2.21

0.47 ± 0.05 0.53 ± 0.05 0.34 ± 0.02 0.32 ± 0.02 0.34 ± 0.01 0.33 ± 0.01 0.36 ± 0.02 0.33 ± 0.02 0.29 ± 0.02 0.25 ± 0.02 0.19 ± 0.02 0.33

0.60 ± 0.25 – – – 0.16 ± 0.07 0.14 ± 0.08 0.22 ± 0.09 0.21 ± 0.10 0.23 ± 0.13 0.26 ± 0.09 – 0.33

Our [29] [29] [29] [38] [38] [38] [38] [34] [39] [37]

–

1.80

2.18

0.41

0.40

Fig. 13. Variation of lnM3  as a function of ln |mc − m| for gas phase.

for Kr–Em interactions at 0.95A GeV, we obtain a set of values of γ , β and τ that obey the scaling relationship as 1.34 ± 0.19, 0.54 ± 0.15 and 2.31 ± 0.06, respectively. 3. Summary From the present investigation on Kr–Em interactions at 0.95A GeV and a comparison of the results with Au–C at about same energy it is seen that the distribution of 2nd charge moments and conditional moment depend on the sizes of the projectiles. The maximum values of these moments are found to be large for the larger projectile system as is the case for a lattice with larger volume. It is further seen from our and EOS generated data that, depending upon the different algorithm followed in regards to the random partitioning the total number of system’s constituents,

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