The calculation of the critical exponent τ is crucial in the determination of critical phenomena in heavy ion reactions. This, however, is obscured by the unavoidable mixing of critical and noncritical events that results in nonclean signals. Here we report on a method to extract critical events from a set of mixed ones. In comparing to the traditional one, based on the so-called Campi plot, a distinct advantage is found.Fil: Dorso, Claudio Oscar. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires; ArgentinaFil: Lopez, J. A.. University of Texas at El Paso; Estados Unido

Dorso, Claudio Oscar

Lopez, J. A.

Physical Review C

English

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

PHYSICAL REVIEW C, VOLUME 64, 027602Selection of critical events in nuclear fragmentationC. O. Dorso1 and J. A. López21Universidad de Buenos Aires, Nun˜ez 1428, Buenos Aires, Argentina2University of Texas at El Paso, El Paso, Texas 79968~Received 27 March 2001; published 23 July 2001!The calculation of the critical exponentt is crucial in the determination of critical phenomena in heavy ionreactions. This, however, is obscured by the unavoidable mixing of critical and noncritical events that resultsin nonclean signals. Here we report on a method to extract critical events from a set of mixed ones. Incomparing to the traditional one, based on the so-called Campi plot, a distinct advantage is found.DOI: 10.1103/PhysRevC.64.027602 PACS number~s!: 25.70.Pq, 24.10.Lx, 25.70.Mn, 65.20.1wichexasarticeacitomi--ethabs.leeneveidr-asdall-nsoweisiti-ntsbil-ing.e.,ingaif-mu-ot.i-tnea abeicalymdbybe-isSince the pioneering work of the Purdue group, in whthe mass spectra resulting from multifragmentation ofcited nuclei was first fitted to a power law, a lot of work hbeen devoted to the characterization of this phenomenonphase transition. This task has been undertaken both theocally and experimentally~for a recent review see Refs.@1,2#!via the calculation of critical exponents@3#, caloric curves@4,5#, thermal response functions@6–9#, etc.In particular, the critical exponentt can be extracted fromthe mass spectrum of events produced exactly at the cripoint, which should have a power-law decay such asnA'A2t, wherenA is the number of fragments of sizeA perunit mass. At nearby points, the yield is modified tonA'A2t f (z) by a scaling factorf (z) which obscures the purpower law. In the scaling factorz5As«, « is the relativedistance to the critical point«5(p2pc)/pc , and s is an-other critical exponent. As the experimental data containsassortment of events at different impact parameters, extion energies, etc., the resulting spectra will unavoidable ctain a mix of critical and noncritical events.Originally, the critical exponent was estimated fromixed data by means of a fitting parameter~‘‘apparent expo-nent’’! resulting from a power-law fit to critical and noncritcal data. Since the scaling function attains the value of 1the critical pointf (0)51, the minimum of this apparent exponent was identified with the true critical exponent, wherpure power law would apply~see Ref. @1#!. Nowadays,whenever possible, critical events must extracted fromdata and used to calculatedirectly from them. Unfortu-nately, the most commonly used technique that presumselects critical and near-critical events, based on thecalled Campi plot@7,8,10#, does not live up to its promisesA Campi plot is a representation of data from multipevents in which the log of the size of the maximum fragmin event i ln(Amaxi ) is plotted against the logarithm of thsecond moment of the mass distribution in the same eln(M2i ). ~The kth moment is defined asMk5(AknA , withthe biggest fragment excluded from the sum in the ‘‘liquside’’ @11#.! The resulting graph displays a typical ‘‘boomeang’’ shape with a down-pointing branch believed to besociated to overcritical events~i.e., an exponential mass yielwith large multiplicity and small biggest fragment!, an hori-zontal branch presumably related to undercritical events~i.e.,0556-2813/2001/64~2!/027602~4!/$20.00 64 0276-s aeti-alna-n-ataelyo-tnt-a ‘‘U-shaped’’ distribution of large biggest cluster and smmultiplicity!, and an ‘‘elbow’’ region usually taken as connected to critical events with power-law mass distributio@12#.In previous works, ratherad libitum criteria have beenused to select events, presumably critical, from the elbregion and to extract from them @12#. Unfortunately, thisresults in a wide mix of noncritical and critical events. Wnow use percolation to critically examine the effects of thcriterion on the resulting value of the critical exponent.Bond percolation is a simple procedure that displays crcal behavior@11,13–17#. A three-dimensional grid~of say,L3L3L nodes! can be fragmented into connected segmeby breaking each of the internode links with some probaity pb . Repeating this process for many grids and combinthe results, produces a smooth distribution of ‘‘masses,’’ iumber of connected nodes. At some critical bond breakprobability pc(L), the resulting mass distribution followscharacteristic power-law decay, but mixing results from dferentpc’s produces a scaled power-law decay different frothe critical one. Critical and noncritical events tend to poplate different, but overlapping, segments of the Campi plWe now examine the wide mixing of critical and noncritcal events produced by thead libitumcriteria used in the pasto select events from the elbow region. This will be dousing anL56 grid which will be broken 20 000 times witheach of the probabilitiesp50.01,0.02, . . . ,0.99 for a total ofalmost 2 million events. Using the resulting mass spectrCampi plot will be populated, and a subset of events willselected from the elbow region, and used to obtain a critexponent.The critical bond activation probabilitypc(L), and thecorresponding exponentt can be calculated as explained bElliot et al. @3#. The critical exponent must be obtained froa one-parameter fit (t) as the overall normalizationq0 isdetermined via the Riemannz functionq051/(AA12t, withthe mass distribution A normalized to the system size anM1 to 1.The critical mass spectra is selected as that best fittedthis single-parameter power law according to a minimumx2.This fitting procedures are performed in the mass rangetween 0.02Atot and 0.15Atot to avoid finite size effects. Thebond probability value for which the best fit occurs©2001 The American Physical Society02-1nstoweldtantsfndhtaderynt-ndse.heheon-f anicalfom-ieldtoined,t 2s.eioe eBRIEF REPORTS PHYSICAL REVIEW C 64 027602pc(6)>0.3260.01 and the corresponding critical exponeis t>2.1960.02, both quite consistent with previoucalculations.We now proceed to study the effects of the event selecThe Campi plot resulting from the percolation data is shoin Fig. 1 along with a rectangular region used as an evselector in the references mentioned before. The left paneFig. 2 shows the mass distribution of the events containethe central rectangle. The corresponding critical exponent52.360.02, quite a distance away from the true value cculated before. The origin of this discrepancy can be fouby inspecting the bond probability distribution of the evenselected by the cut.The bond probability distribution can defined asPi5n(pi)/( in(pi) with n(pi) representing the number oevents with pb5pi present in the sample selected, aFIG. 1. Campi plot of percolation data produced with almosmillion breakups of a 63636 grid at several breakup probabilitieThe central rectangular region shows the cut used to select ncritical events. The inset shows the subdivision of the central regused in the text.FIG. 2. Mass distribution~left panel! and distribution of breakupprobabilities ~right panel! of percolation events contained in thcentral rectangle of previous figure.02760tr.nntofinisl-d( in(pi) the total number of events in the rectangle. The rigpanel of Fig. 2 shows that this distribution is very broindeed, and that the mix contains events coming from vdifferent bond activation probabilities.To see if thepb variation is related to the relative positioof the event in the Campi plot, we further divide the recangle in 12 regions, as shown in the insert of Fig. 1, acalculate the bond probability distribution for each of theregions. Table I shows the values oft thus obtained, and Fig3 some typical probability distributions. As we can see tmixing is always high and the values oft vary substantiallyindependent of the relative position in the Campi plot. Tselection rule used appears not to reduce the mixing of ncritical events.But the effect of the cut is even more troublesome. ICampi plot is produced usingonly pc ~i.e., with the probabil-ity that produces the critical power-law mass distributio!,the cut introduces large deviations in the expected critmass distribution. The value oft obtained in such a case ocritical but cut-selectedevents is again too big:t52.3. Ap-parently, the cut produces a biased exclusion of certain cponents of the mass distributions which are needed to ythe correct power-law distribution.In summary, the usage of a cut not only does not helpreduce the existing~and unavoidable! mixing of critical andnoncritical events, but it also disturbs the mass distributionan uncontrollable way. Many other cuts were also explorand although some cases led to better’s, specially thoseincreasing the upper limit of the rectangle, none yieldedpbTABLE I. The values oft vary substantially for all 12 regionsof Fig. 1.Region A B C D E Ft 2.8 2.6 2.2 2.1 2.9 2.5Region G H I J K Lt 2.3 2.2 2.6 2.75 2.3 2.3ar-nFIG. 3. Typical bond probability distribution in some of thregions of the central rectangle.2-2hbtsare-seeohrity-editlufod-ky,loter-tsofd-fitther ofane-nlyuchellosa-.D.rvmo-leic-.BRIEF REPORTS PHYSICAL REVIEW C 64 02760202760ee,ss-rs-e-s-,erat pc , the bin that produced the closest critical multiplicitcontains a narrower distribution ofPi with a much reducedmixing of noncritical events than other bins.To underline this last result, Fig. 5 shows a Campi pwith all the critical events produced in the percolation excise ~dark circles! at pb5pc , along with those eventssampled in by the bin-and-fit procedure~white circles! forpb5pc . Just as in the wide distribution of the critical evenexplains the failure of the cut selection, the wide coveragethe sampled events explains the goodness of the bin-anrule.In conclusion, the selection of events using cuts onCampi scatter plot does not reduce the relative numbenoncritical events and perturbs the mass distribution inundesirable way. Sorting events by multiplicity bins and slecting the one that gives the best one-parameter fit, not oseems to give much better results, i.e., events with a mhigher concentration of critical events, but it is also a wdefined methodology.This work was supported by the Universidad de BuenAires ~Grant No. TW98! and the National Science Foundtion ~Grant No. PHY-96-00038!. J.A.L. acknowledges thehospitality of the Universidad de Buenos Aires, and C.Othat of the University of Texas at El Paso.FIG. 5. Campi plot with all the critical events from the perclation data~dark circles!, and the events in the critical multiplicitybin ~white circles!. Critical events are distributed over the whoregion, including outside the elbow region. Events in the multiplity bin sampled by the bin-and-fit rule also have a wide spreadl-@1# J. A. López and C. O. Dorso,Phase Transitions in NucleaMatter ~World Scientific, Singapore, 2000!.@2# M. Bruno, A. Bonasera, C. O. Dorso, and J. P. Mastinu, NueCimento2, 1 ~2000!.@3# J.B. Elliot et al., Phys. Rev. C62, 064603~2000!.@4# J. Pochodzallaet al., Phys. Rev. Lett.75, 1040~1995!.@5# A. Strachan and C.O. Dorso, Phys. Rev. C59, 285 ~1999!.o@6# F. Gulminelli and P. Chomaz, Phys. Rev. Lett.82, 1402~1999!.@7# M. D’Agostino et al., Nucl. Phys.A650, 329 ~1999!.@8# M. D’Agostino et al., Phys. Lett. B473, 219 ~2000!.@9# A. Chernomoretz, M. Ison, S. Ortiz, and C. O. Dorso, Criticaity in Nuclear Fragmentation~unpublished!.@10# X. Campi, J. Phys. C48, 2 ~1987!.@11# D. Stauffer and A. Aharony,Introduction to Percolationdistributions withpc higher than 20%.~For other deficien-cies of the Campi plot see Ref.@18#.!In view of this, an event selection method other than tused cuts must be implemented. The ultimate goal is toable to extract critical breakups from a mixture of evenand since these critical events produce a power-law mdistribution at a preferredcritical multiplicity, an approachbased on multiplicity selection, as proposed in Ref.@3#,seems to be more promising. The underlying idea is toduce the volume of data~mixed events! with a minimum lossof critical information by classifying the events in multiplicity bins, and then finding the multiplicity bin which producethe best one-parameter fit, i.e., the critical multiplicity. Thprocedure can be expected to work as Elliot and co-workhave managed to show that the total multiplicity is montonically connected to the bond-activation probabilities in tcase of percolation@19#. ~We have also verified this for oucalculations and have observed a small multiplicdeviation.!The results of this multiplicity binning and fitting procedure is shown in Fig. 4 for the percolation calculations prsented before. The left panel of Fig. 4 shows the masstribution from the bin that produced the best power law fiand the power-law curve that resulted from the fit. The vaobtained for the critical exponents ist52.1960.02 in exactagreement with the true value.To understand why this procedure yields better resultst, the right panel of Fig. 4 shows the distribution of bonactivation probabilitiesPi . As evidenced by the sharp peaFIG. 4. Left panel: mass distribution and the power-law fit frothe bin with the best fit. The critical exponent ist52.1960.02.Right panel: distribution of bond-activation probabilitiesPi .2-3nBRIEF REPORTS PHYSICAL REVIEW C 64 027602Theory~Taylor & Francis, London, 1992!.@12# M. Belkacem, V. Latora, and A. Bonasera, Phys. Rev. C52,271 ~1995!.@13# W. Baueret al., Proceedings of the 7th High Energy Heavy IoStudy~GSI, Darmstadt, Germany, 1984!, p. 701.@14# W. Baueret al., Phys. Lett.150B, 53 ~1985!.@15# W. Baueret al., Nucl. Phys.A452, 699 ~1986!.02760@16# W. Baueret al., Phys. Rev. C38, 1297~1988!.@17# X. Campi and J. Desbois,Proceedings of the 23rd BormioConference~Ric. Sci. Educ. Perm., Milano, 1985!, p. 497.@18# A. Barrañón, C. O. Dorso, and J. A. Lo´pez, Rev. Mex. Fis.~tobe published!.@19# J.B. Elliot et al., Phys. Rev. C55, 1319~1997!.2-4

Selection of critical events in nuclear fragmentation

CONICET Digital

PHYSICAL REVIEW C, VOLUME 64, 027602Selection of critical events in nuclear fragmentationC. O. Dorso1 and J. A. Lo´pez21Universidad de Buenos Aires, Nun˜ez 1428, Buenos Aires, Argentina2University of Texas at El Paso, El Paso, Texas 79968~Received 27 March 2001; published 23 July 2001!The calculation of the critical exponent t is crucial in the determination of critical phenomena in heavy ionreactions. This, however, is obscured by the unavoidable mixing of critical and noncritical events that resultsin nonclean signals. Here we report on a method to extract critical events from a set of mixed ones. Incomparing to the traditional one, based on the so-called Campi plot, a distinct advantage is found.DOI: 10.1103/PhysRevC.64.027602 PACS number~s!: 25.70.Pq, 24.10.Lx, 25.70.Mn, 65.20.1wSince the pioneering work of the Purdue group, in whichthe mass spectra resulting from multifragmentation of ex-cited nuclei was first fitted to a power law, a lot of work hasbeen devoted to the characterization of this phenomenon as aphase transition. This task has been undertaken both theoreti-cally and experimentally ~for a recent review see Refs. @1,2#!via the calculation of critical exponents @3#, caloric curves@4,5#, thermal response functions @6–9#, etc.In particular, the critical exponent t can be extracted fromthe mass spectrum of events produced exactly at the criticalpoint, which should have a power-law decay such as nA’A2t, where nA is the number of fragments of size A perunit mass. At nearby points, the yield is modified to nA’A2t f (z) by a scaling factor f (z) which obscures the purepower law. In the scaling factor z5As« , « is the relativedistance to the critical point «5(p2pc)/pc , and s is an-other critical exponent. As the experimental data contains anassortment of events at different impact parameters, excita-tion energies, etc., the resulting spectra will unavoidable con-tain a mix of critical and noncritical events.Originally, the critical exponent was estimated frommixed data by means of a fitting parameter ~‘‘apparent expo-nent’’! resulting from a power-law fit to critical and noncriti-cal data. Since the scaling function attains the value of 1 atthe critical point f (0)51, the minimum of this apparent ex-ponent was identified with the true critical exponent, where apure power law would apply ~see Ref. @1#!. Nowadays,whenever possible, critical events must extracted from thedata and used to calculate t directly from them. Unfortu-nately, the most commonly used technique that presumablyselects critical and near-critical events, based on the so-called Campi plot @7,8,10#, does not live up to its promises.A Campi plot is a representation of data from multipleevents in which the log of the size of the maximum fragmentin event i ln(Amaxi ) is plotted against the logarithm of thesecond moment of the mass distribution in the same eventln(M2i ). ~The kth moment is defined as M k5(AknA , withthe biggest fragment excluded from the sum in the ‘‘liquidside’’ @11#.! The resulting graph displays a typical ‘‘boomer-ang’’ shape with a down-pointing branch believed to be as-sociated to overcritical events ~i.e., an exponential mass yieldwith large multiplicity and small biggest fragment!, an hori-zontal branch presumably related to undercritical events ~i.e.,0556-2813/2001/64~2!/027602~4!/$20.00 64 0276a ‘‘U-shaped’’ distribution of large biggest cluster and smallmultiplicity!, and an ‘‘elbow’’ region usually taken as con-nected to critical events with power-law mass distributions@12#.In previous works, rather ad libitum criteria have beenused to select events, presumably critical, from the elbowregion and to extract t from them @12#. Unfortunately, thisresults in a wide mix of noncritical and critical events. Wenow use percolation to critically examine the effects of thiscriterion on the resulting value of the critical exponent.Bond percolation is a simple procedure that displays criti-cal behavior @11,13–17#. A three-dimensional grid ~of say,L3L3L nodes! can be fragmented into connected segmentsby breaking each of the internode links with some probabil-ity pb . Repeating this process for many grids and combiningthe results, produces a smooth distribution of ‘‘masses,’’ i.e.,number of connected nodes. At some critical bond breakingprobability pc(L), the resulting mass distribution follows acharacteristic power-law decay, but mixing results from dif-ferent pc’s produces a scaled power-law decay different fromthe critical one. Critical and noncritical events tend to popu-late different, but overlapping, segments of the Campi plot.We now examine the wide mixing of critical and noncriti-cal events produced by the ad libitum criteria used in the pastto select events from the elbow region. This will be doneusing an L56 grid which will be broken 20 000 times witheach of the probabilities p50.01,0.02, . . . ,0.99 for a total ofalmost 2 million events. Using the resulting mass spectra aCampi plot will be populated, and a subset of events will beselected from the elbow region, and used to obtain a criticalexponent.The critical bond activation probability pc(L), and thecorresponding exponent t can be calculated as explained byElliot et al. @3#. The critical exponent must be obtained froma one-parameter fit (t) as the overall normalization q0 isdetermined via the Riemann z function q051/(AA12t, withthe mass distribution nA normalized to the system size andM 1 to 1.The critical mass spectra is selected as that best fitted bythis single-parameter power law according to a minimum x2.This fitting procedures are performed in the mass range be-tween 0.02A tot and 0.15A tot to avoid finite size effects. Thebond probability value for which the best fit occurs is©2001 The American Physical Society02-1BRIEF REPORTS PHYSICAL REVIEW C 64 027602pc(6)>0.3260.01 and the corresponding critical exponentis t>2.1960.02, both quite consistent with previouscalculations.We now proceed to study the effects of the event selector.The Campi plot resulting from the percolation data is shownin Fig. 1 along with a rectangular region used as an eventselector in the references mentioned before. The left panel ofFig. 2 shows the mass distribution of the events contained inthe central rectangle. The corresponding critical exponent ist52.360.02, quite a distance away from the true value cal-culated before. The origin of this discrepancy can be foundby inspecting the bond probability distribution of the eventsselected by the cut.The bond probability distribution can defined as Pi5n(pi)/( in(pi) with n(pi) representing the number ofevents with pb5pi present in the sample selected, andFIG. 1. Campi plot of percolation data produced with almost 2million breakups of a 63636 grid at several breakup probabilities.The central rectangular region shows the cut used to select near-critical events. The inset shows the subdivision of the central regionused in the text.FIG. 2. Mass distribution ~left panel! and distribution of breakupprobabilities ~right panel! of percolation events contained in thecentral rectangle of previous figure.02760( in(pi) the total number of events in the rectangle. The rightpanel of Fig. 2 shows that this distribution is very broadindeed, and that the mix contains events coming from verydifferent bond activation probabilities.To see if the pb variation is related to the relative positionof the event in the Campi plot, we further divide the rect-angle in 12 regions, as shown in the insert of Fig. 1, andcalculate the bond probability distribution for each of theseregions. Table I shows the values of t thus obtained, and Fig.3 some typical probability distributions. As we can see themixing is always high and the values of t vary substantiallyindependent of the relative position in the Campi plot. Theselection rule used appears not to reduce the mixing of non-critical events.But the effect of the cut is even more troublesome. If aCampi plot is produced using only pc ~i.e., with the probabil-ity that produces the critical power-law mass distribution!,the cut introduces large deviations in the expected criticalmass distribution. The value of t obtained in such a case ofcritical but cut-selected events is again too big: t52.3. Ap-parently, the cut produces a biased exclusion of certain com-ponents of the mass distributions which are needed to yieldthe correct power-law distribution.In summary, the usage of a cut not only does not help toreduce the existing ~and unavoidable! mixing of critical andnoncritical events, but it also disturbs the mass distribution inan uncontrollable way. Many other cuts were also explored,and although some cases led to better t’s, specially thoseincreasing the upper limit of the rectangle, none yielded pbTABLE I. The values of t vary substantially for all 12 regionsof Fig. 1.Region A B C D E Ft 2.8 2.6 2.2 2.1 2.9 2.5Region G H I J K Lt 2.3 2.2 2.6 2.75 2.3 2.3FIG. 3. Typical bond probability distribution in some of theregions of the central rectangle.2-2BRIEF REPORTS PHYSICAL REVIEW C 64 02760202760at pc , the bin that produced the closest critical multiplicity,contains a narrower distribution of Pi with a much reducedmixing of noncritical events than other bins.To underline this last result, Fig. 5 shows a Campi plotwith all the critical events produced in the percolation exer-cise ~dark circles! at pb5pc , along with those eventssampled in by the bin-and-fit procedure ~white circles! forpb5pc . Just as in the wide distribution of the critical eventsexplains the failure of the cut selection, the wide coverage ofthe sampled events explains the goodness of the bin-and-fitrule.In conclusion, the selection of events using cuts on theCampi scatter plot does not reduce the relative number ofnoncritical events and perturbs the mass distribution in anundesirable way. Sorting events by multiplicity bins and se-lecting the one that gives the best one-parameter fit, not onlyseems to give much better results, i.e., events with a muchhigher concentration of critical events, but it is also a welldefined methodology.This work was supported by the Universidad de BuenosAires ~Grant No. TW98! and the National Science Founda-tion ~Grant No. PHY-96-00038!. J.A.L. acknowledges thehospitality of the Universidad de Buenos Aires, and C.O.D.that of the University of Texas at El Paso.FIG. 5. Campi plot with all the critical events from the perco-lation data ~dark circles!, and the events in the critical multiplicitybin ~white circles!. Critical events are distributed over the wholeregion, including outside the elbow region. Events in the multiplic-ity bin sampled by the bin-and-fit rule also have a wide spread.@1# J. A. Lo´pez and C. O. Dorso, Phase Transitions in NuclearMatter ~World Scientific, Singapore, 2000!.@2# M. Bruno, A. Bonasera, C. O. Dorso, and J. P. Mastinu, NuevoCimento 2, 1 ~2000!.@3# J.B. Elliot et al., Phys. Rev. C 62, 064603 ~2000!.@4# J. Pochodzalla et al., Phys. Rev. Lett. 75, 1040 ~1995!.@5# A. Strachan and C.O. Dorso, Phys. Rev. C 59, 285 ~1999!.@6# F. Gulminelli and P. Chomaz, Phys. Rev. Lett. 82, 1402 ~1999!.@7# M. D’Agostino et al., Nucl. Phys. A650, 329 ~1999!.@8# M. D’Agostino et al., Phys. Lett. B 473, 219 ~2000!.@9# A. Chernomoretz, M. Ison, S. Ortiz, and C. O. Dorso, Critical-ity in Nuclear Fragmentation ~unpublished!.@10# X. Campi, J. Phys. C 48, 2 ~1987!.@11# D. Stauffer and A. Aharony, Introduction to Percolationdistributions with pc higher than 20%. ~For other deficien-cies of the Campi plot see Ref. @18#.!In view of this, an event selection method other than theused cuts must be implemented. The ultimate goal is to beable to extract critical breakups from a mixture of events,and since these critical events produce a power-law massdistribution at a preferred critical multiplicity, an approachbased on multiplicity selection, as proposed in Ref. @3#,seems to be more promising. The underlying idea is to re-duce the volume of data ~mixed events! with a minimum lossof critical information by classifying the events in multiplic-ity bins, and then finding the multiplicity bin which producesthe best one-parameter fit, i.e., the critical multiplicity. Theprocedure can be expected to work as Elliot and co-workershave managed to show that the total multiplicity is mono-tonically connected to the bond-activation probabilities in thecase of percolation @19#. ~We have also verified this for ourcalculations and have observed a small multiplicitydeviation.!The results of this multiplicity binning and fitting proce-dure is shown in Fig. 4 for the percolation calculations pre-sented before. The left panel of Fig. 4 shows the mass dis-tribution from the bin that produced the best power law fit,and the power-law curve that resulted from the fit. The valueobtained for the critical exponents is t52.1960.02 in exactagreement with the true value.To understand why this procedure yields better results fort , the right panel of Fig. 4 shows the distribution of bond-activation probabilities Pi . As evidenced by the sharp peakFIG. 4. Left panel: mass distribution and the power-law fit fromthe bin with the best fit. The critical exponent is t52.1960.02.Right panel: distribution of bond-activation probabilities Pi .2-3BRIEF REPORTS PHYSICAL REVIEW C 64 027602Theory ~Taylor & Francis, London, 1992!.@12# M. Belkacem, V. Latora, and A. Bonasera, Phys. Rev. C 52,271 ~1995!.@13# W. Bauer et al., Proceedings of the 7th High Energy Heavy IonStudy ~GSI, Darmstadt, Germany, 1984!, p. 701.@14# W. Bauer et al., Phys. Lett. 150B, 53 ~1985!.@15# W. Bauer et al., Nucl. Phys. A452, 699 ~1986!.02760@16# W. Bauer et al., Phys. Rev. C 38, 1297 ~1988!.@17# X. Campi and J. Desbois, Proceedings of the 23rd BormioConference ~Ric. Sci. Educ. Perm., Milano, 1985!, p. 497.@18# A. Barran˜o´n, C. O. Dorso, and J. A. Lo´pez, Rev. Mex. Fis. ~tobe published!.@19# J.B. Elliot et al., Phys. Rev. C 55, 1319 ~1997!.2-4

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Selection of critical events in nuclear fragmentation

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