1. Evidence of Critical Behavior in the Disassembly of Light Nuclei with A ~ 36 Yu-Gang Ma Texas A&M University, Cyclotron Institute, USA Shanghai Institute of Nuclear Research, CAS, CHINA (for NIMROD Collaboration) Experimental Set-up: NIMROD Reaction System: 47MeV/u 40 Ar+ 58 Ni Events selection: good Quasi-Projectile (QP) events in violent collisions Evidence of Critical Behavior + Model Comparisons the Largest Fluctuations the Critical Exponent analysis the Fragment Topological Structure the Caloric Curve -------- HIC03 Conference, Montreal, June 25-28, 2003

2. 4  -NIMROD Array NIMROD = Neutron Ion Multidetector for Reaction Oriented Dynamics The NIMROD multidetector -- a new 4  array of detectors build at Texas A&M to study reactions mechanisms in heavy ion reactions. The charged particle detectors are composed of silicon telescopes and CsI(Tl) scintillators covering angles between 3º and 170º. These charged particle detectors are placed in a cavity inside the revamped TAMU neutron ball. 166 CsI; 2 Si-Si-CsI telescopes + 3 Si-CsI telescopes in each forward ring (Ring2-9 );

3. Event Selection and QP Reconstruction 1 Violent Collisions Bin1+Bin2 were selected by Mcp-Mn correlation. 2 A new method to reconstruct the QP source was developed. 3 28000 Good QP events (Z QP tot  12) comprise ~4.3% of violent events. 4 Excitation Energy with Calorimetry Method: E* =  (E CP +E n ) + Q, the highest E*/A ~ 9MeV/u First, 3 source fits to LCPs Second, employ the parameters of fits to control the EVENT-BY-EVENT assignment of individual LCP to one of the source (QP, or NN, or QT) using Monte Carlo sampling techniques. The probability of QP’s LCP is We associate IMFs (Z>3) with the QP source if they have rapidity >0.65Yproj.  

4. Charge Distribution of QP Zqp The minimum  eff ~ 2.31, close to the Critical Exponent of liquid gas phase transition universal class (~2.23) predicted by the Fisher droplet model!  lines: Fisher Droplet Power- Law fit: dN/dZ ~Z -  eff Ref: Fisher, Rep. Prog. Phys. 30, 615 (1969).

5. The largest fluctuation: Campi Plots Ref : Campi, J Phys A19 (1988) L917 Campi plot: ln(Zmax) vs ln(S 2 ) (event-by-event) can explore the critical behavior, where Zmax is the charge number of the heaviest fragment and S 2 is normalized second moment Features: The LIQUID Branch is dominated by the large Zmax The GAS Branch is dominated by the small Zmax Critical point occurs as the nearly equal Liquid and Gas branch. The LIQUID Branch The GAS Branch Transition Region

6. Fluctuation of Zmax and Ek tot Zmax (order paramter) Fluctuation: Normalized Variance of Zmax/Z QP : NVZ =  2 / There exists the maximum fluctuation of NVZ around phase transition point by CMD and Percolation model, see : Dorso et al., Phys Rev C 60 (1999) 034606 Total Kinetic Energy Fluctuation: Normalized Variance of Ek/A: NVE =  2 (Ek/A) / The maximum fluctuation of NVE exists in the same E*/A point! A possible relation of Cv to kinetic energy fluctuation was proposed by Chomaz, Gulminelli, D’Agostino et al., PRL, PLB

7. Caloric Curve 1. Sequential Decay Dominated Region (LIQUID-dominated PHASE): T ini = (M 2 T 2 –M 1 T 1 )/(M 2 -M 1 ) where M 1, T 1 and M 2, T 2 is apparent slope temperature and multiplicity in a given neighboring E*/A window. Ref: K. Hagel et al., Nucl. Phys. A 486 (1988) 429; R. Wada et al., Phys. Rev. C 39 (1989) 497 2. Vapor Phase (Quantum Statistical Model correction): feed-correction for isotopic temperature T iso Ref: Z. Majka et al., Phys. Rev. C 55 (1997) 2991 3. Assuming vapor phase as an ideal gas of clusters: T kin = 2/3E th kin = 2/3(E cm kin -V coul ) T 0 = 8.3±0.5MeV at E*/A = 5.6 MeV No obvious plateau was observed at the largest fluctuation point, in comparison with the heavier system! different phys No obvious plateau was observed at the largest fluctuation point, in comparison with the heavier system! different phys. Ref: J. Natowitz et al., Phys Rev C65, 034618 (2002)

8. Model Comparisons Model Calculation (A=36, Z=16) Statistical Evaporation Model: GEMINI (Pink dotted lines)  NO PHASE TRANSITION Ref: R. Charity et al.,,NPA Lattice Gas Model (LGM) (Black lines)  Classical Molecular Dynamics Model (CMD) (  LGM+Coulomb) (Red dashed lines)  Both with PHASE TRANSITION! Ref: Das Gupta and Pan, PRL Observables vs T scaled by T 0: T 0 (Exp)=8.3 ±0.5MeV (Black Points)  T 0 (GEMINI) = 8.3 MeV T 0 (LGM) = 5.0MeV  T(PhaseTran) T 0 (CMD) = 4.5MeV  T(PhaseTran) Fig.(e) 2 nd Zmax; Fig.(f) Evaporation model fails to fit the Data; Phase Transition Models reproduce the Data well!

9. Fragment Topological Structure: Zipf plot Nuclear Zipf-Plot  Rank sorted Fragment Size distribution, where Rank = 1 if the heaviest fragment = 2 if 2 nd heaviest fragment, = 3 if 3 rd heaviest fragment and so on, in each event. Accumulating all events, we can plot rank sorted mean size vs rank, i.e., Zipf-type plot. Nucl Zipf-Law if  ~1 u sing Z rank ~ rank -  fit when liquid gas phase transition, see Y.G. Ma, Phys. Rev. Lett. 83, 3617(1999) Zipf law fit: Z rank ~ rank -  Our Data: Zipf-law (  ~1 ) is satisfied around E*/A ~ 5.6 MeV/u Zipf-plots

10. CONCLUSIONS (1) The Maximum Fluctuation Shows around E*/A~5.6MeV/u via: near equal Liquid branch and Gas branch coexists in Campi Plots fluctuation of order parameter (Zmax) fluctuation of total kinetical energy Δ-scaling changes from Δ=1/2-scaling to Δ=1-scaling (Pls see Extra Slide) (2) Caloric Curve has no plateau, in comparison with heavier system : E*/A|crit ~ 5.6 ±0.5MeV, T|crit ~ 8.3 ±0.5MeV (3) Fisher Droplet Model and Critical Exponent Analysis: τ eff =2.31  0.03 for distribution of Z – close to Critical Exponent of LGPT  =0.33  0.01,  =1.15  0.06;  =0.68  0.04 ==> Liquid-Gas Universal Class! (Pls see Extra Slide) (4) Fragment Topological Structures: Zipf’s law, fragment hierarchy, is satisfied around E*/A|crit (5) Overall good agreements with Phase Transition Model calc. were attained This body of evidence suggests a phase change in an equilibrated system at, or extremely close to, the critical point for such light nuclei

11. Thanks Thanks ! COLLABORATORS R. Wada, K. Hagel, J. S. Wang, T. Keutgen, R. WadaK. HagelJ. S. WangT. Keutgen Z. Majka, M. Murray, L. J. Qin, P. Smith,Z. MajkaM. MurrayL. J. QinP. Smith J. B. Natowitz R. Alfaro, J. Cibor, M. Cinausero, Y. El Masri,R. AlfaroJ. CiborM. CinauseroY. El Masri D. Fabris, E. Fioretto, A. Keksis, M. Lunardon,D. FabrisE. FiorettoA. KeksisM. Lunardon A. Makeev, N. Marie, E. Martin, A. Martinez-Davalos,A. MakeevN. MarieE. MartinA. Martinez-Davalos A. Menchaca-RochaA. Menchaca-Rocha, G. Nebbia,G. Nebbia G. Prete, V. Rizzi, A. Ruangma, D. V. Shetty,G. PreteV. RizziA. RuangmaD. V. Shetty G. Souliotis, P. Staszel, M. Veselsky, G. Viesti,G. SouliotisP. StaszelM. VeselskyG. Viesti E. M. Winchester, S. J. YennelloE. M. WinchesterS. J. Yennello

12. Extra Slide 1: Δ-Scaling Analysis of Zmax Definition:  P N [Zmax]   (z(  ))   [(Zmax-Zmax*)/  ] where P N [Zmax]  the probability distribution of Zmax  the mean value of Zmax Zmax*  the most probable value of Zmax If Δ-Scaling holds, all probability distributions collapse to a single universal scaling function  for a given value of the scaling exponent Δ Ref: Botet and Ploszajczak et al., Phys. Rev. Lett. 86 (2001) 3514 ------------------------------------------------------------------------- IN OUR DATA, Δ-SCALING OF Z MAX SHOWS CHANGE FROM Δ=1/2-SCALING BELOW 5.6 MeV/u TO Δ=1-SCALING ABOVE 5.6 MeV/u, which relates to phase change  (NOTE: in arXiv:nucl-ex/0303016 v1, we use scaling of Zmax/Zqp(tot) rather than Zmax as here!)

13. Extra Slide 2: Critical Exponents Ref: Gilkes et al., PRL73, 1590(1994)Elliott et al., PRC49, 3185(1994).