Size matters: Origin of binomial scaling in nuclear fragmentation experimentsWolfgang Bauer and Scott PrattDepartment of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University,East Lansing, Michigan 48824~Received 27 August 1998!The relationship between measured transverse energy, total charge recovered in the detector, and size of theemitting system is investigated. Using only very simple assumptions, we are able to reproduce the observedbinomial emission probabilities and their dependences on the transverse energy. Our results show that theobserved scaling can arise due to a combination of finite-size effects and detector acceptance effects.@S0556-2813~99!07705-5#PACS number~s!: 25.70.Pq, 24.10.Pa, 24.60.KyDuring the last decade, evidence has been mounting thatnuclear matter undergoes a phase transition in the nuclearfragmentation process. From general considerations regard-ing the elementary nucleon-nucleon interaction ~repulsive atshort and attractive at intermediate distances!, we expect thenuclear phase diagram to show a van der Waals ‘‘liquid-gas’’ phase transition of first order, terminating in a second-order transition at the critical point.Recent observations point to evidence of first- andsecond-order transitions. In experiments studying Au-Aucollisions conducted at the GSI, a measurement of the tem-perature as a function of excitation energy found possibleevidence for a two-phase coexistence regime @1#, not unlikethe scenarios predicted by statistical multifragmentationmodels with excluded volume @2,3#. Other experiments con-ducted at the Bevalac focused on the extraction of criticalexponents from ~almost! completely reconstructed Au frag-mentation events on C targets, studying the dependence ofthe second moment of the charge distribution and size of thelargest fragment as a function of the total charged particlemultiplicity @4–6#. It was shown @7,8# that these data areconsistent with the second-order phase transition predictedby the nuclear percolation model @9–12#.If one wants to gain a fundamental understanding of thefragmentation process that goes beyond simple equilibriummodel descriptions of the phenomena, then a proper descrip-tion of the origin and time evolution of fluctuations is essen-tial @13–16#, in particular if one wants to understand whyparticular molecular dynamics codes produce fragments ~ornot! and what their connections to the fundamental processesof nuclear fragmentation are @17–27#.In this light, the recent findings of Moretto et al. and oth-ers are very interesting @28–32#. This group found that theprobability Pnof emitting n intermediate mass fragments~IMFs! follows a binomial distributionPn~m,p!5m!n!~m2 n!!pn~12 p!m2n. ~1!The parameters m and p are related to the average and vari-ance of the distribution:^n&5(n5 0`nPn~m,p!5m•p, ~2!s25(n50`~n2^n&!2Pn~m,p!5mp~12p!. ~3!This result suggests that one may interpret the parameter p asthe elementary probability for the emission of one fragmentand the parameter m as the total number of tries. This wouldindicate that the problem of multifragment emission is reduc-ible to that of multiple one-fragment emission. The claim forreducibility and its interpretation as the consequence of asimple barrier penetration phenomenon was further strength-ened by the observation that ln(p21) has a linear dependenceon 1/AEt, where Etis the total transverse energy, Et5 (lEklsin2ul. Finally, the same scaling was found for dif-ferent beam energies and different projectile-target combina-tions.Other authors have criticized the above work, pointingour that there are different emission probabilities for differ-ent size IMFs, and that there are problems in the transforma-tion between the total transverse energy and a true thermalenergy @33#, focusing on autocorrelations between the num-ber of IMFs and the transverse energy @34#. ~See also thereplies to these criticisms in Refs. @32,35#.!In the present paper we add to this discussion by showinghow binomial distributions arise naturally from finite-sizeeffects. In particular, we focus on the dependence of theexperimentally recovered charge as a function of the mea-sured transverse energy. We then demonstrate one way inwhich the dependence of the binomial parameter p on thetotal transverse energy could arise. To be fair, one shouldalso note that the possibility of ‘‘spacelike’’ interpretations,i.e., finite source size, was discussed in Ref. @31# as an alter-native to the ‘‘timelike’’ interpretation outlined above.We begin our study by generating power-law-distributedrandom fragmentation events. This is accomplished by deter-mining the charge of individual fragments with a probabilitydistribution proportional to Z2t, where Z is the fragmentcharge andtis the power-law exponent. For definiteness, wewish to generate events with exactly Zsyscharges. If an eventhas less than Zsyscharges, we add another fragment; if it hasmore than Zsyscharges, we throw it out. For an infinite sys-tem, we would expect the multiplicity distributions for indi-vidual fragments of a given Z to follow a Poisson distribu-tion,PHYSICAL REVIEW C MAY 1999VOLUME 59, NUMBER 5PRC 590556-2813/99/59~5!/2695~4!/$15.00 2695 ©1999 The American Physical SocietyQn~l!5lnexp~2 l!n!with l5^n&5s2. ~4!And since the combined probability distribution of twoPoisson-distributed variables is again a Poissonian,Qn~l1!^ Qn~l2![(i5 0nQi~l1!Qn2 i~l2!5 Qn~l11 l2!,~5!we would expect that the multiplicity distribution of the totalnumber of IMFs is also Poissonian.The individual probability distributions for IMFs, how-ever, cannot be exactly Poissonian, because the tails of thedistributions are cut off due to the finite size of the emittingsystem. Thus the probability distributions in our simulationare closer to a binomial distribution with rather large valuesof m and small values of p. ~When m→`, p→ 0 such thatmp5const, we obtain a Poissonian as the limit of a binomialdistribution.! Typical values of p we find for the probabilitydistributions of our individual fragments are <33 102 2for asystem of 100 total charges. These small values of p implythat the probability distributions are very close to a Poissondistribution. However, for a system of only ten charges thereare significant deviation from the Poisson limit due to thefinite-size corrections.We now ask what the combined probability distributionfor fragments charges in the interval 3 to k, k5