1. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 1 Coherent signals of the critical behavior in light nuclear systems Yu-Gang Ma SINAP Shanghai Institute of Applied Physics (SINAP), CAS For NIMROD Collaboration: R. Alfarro, 5 J. Cibor, 4 M. Cinausero, 2 Y. El Masri 6, D. Fabris, 3 E. Fioretto, 2 K. Hagel 1, A. Keksis 1, T. Keutgen, 6 M. Lunardon, 3 Y. G. Ma 1,a, Z. Majka, 4 A.Makeev 1,E. Martin 1, A.Martinez-Davalos, 5 A.Menchaca-Rocha, 5 M. Murray 1, J.B.Natowitz 1, G. Nebbia 3, L. Qin 1, G. Prete, 2 V. Rizzi, 3 A.Ruangma 1, D. V. Shetty 1, P. Smith 1, G. Souliotis 1, P.Staszel, 4 M. Veselsky 1, G. Viesti, 3 R. Wada 1, J. Wang 1, E.Winchester 1, S. J. Yennello 1 1 Texas A&M University, College Station, Texas 2 INFN Laboratori Nazionali di Legnaro, Legnaro, Italy 3 INFN Dipartimento di Fisica, Padova, Italy 4 Jagellonian University, Krakow, Poland 5 UNAM, Mexico City, Mexico 6 UCL, Louvain-la-Neuve, Belgium a Shanghai Insititute of Applied Physics, Shanghai

2. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 2 outline Motivation Experimental set-up: NIMROD and some analysis details Coherent evidence of critical behavior Model comparisons Conclusions

3. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 3 Introduction (I): Phase Transition in Nuclei Liquid Gas Phase Transition from cool nuclei to the full disassembly of nuclei Isis Data: 8-10GeV/c  -, p+Au Labs: MSU-Miniball, TAMU-Nimrod, Indiana U-Isis, GANIL-Indra, GSI-Aladin, Catania-Chimera Quark-Gluon Phase Transition ： from Hadronic Matter to QGP RHIC Detectors: STAR, PHENIX, PHOBOS, BRAHMS Recent results: dAu vs AuAu ISiS STAR Central Collisions 130GeV/c Au+Au

4. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 4 Motivation Ref: J. Elliott et al., Phys. Rev. Lett. 88 (2002) 042701 J.Natowitz, K. Hagel, Y.G. Ma et al., Phys Rev Lett 89, 212701 (2002); arXiv:nucl-ex/0206010 Limiting Temperatures Nucleonic MatterTc Nuclear Matter Tc

5. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 5 Brief Description of the Experimental Set-up: Texas A&M NIMROD

6. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 6 Experimental Set-up: 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 ); CsI Si

7. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 7

8. 8 Reaction systems 47MeV/u Ar + Al, Ti and Ni Complete events of central collisions are chosen Quasi-projectile (QP) was reconstructed on the base of event-by-event by our new method: Monte Carlo sampling based on three source fits for LCP and rapidity cut for IMF

9. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 9 Event Selection: central collisions Mn: total neutron numbers; Mcp: total charge numbers What is central collisions? Bin1+Bin2 are selected by Mcp and Mn Bin1+Bin2 : ~20% all events; Nearly Complete events: Zqp>=12, 4% of all evts Ar+Ni Central collisions Peripheral collisions Nearly Complete Event

10. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 10 A New Method to Reconstruct Quasi-Projectile

11. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 11 Vpar (cm/ns) Vper (cm/ns) Bin5-Peripheral Bin4 Bin3 Bin2 Bin1-Central Velocity contours of protons Vini

12. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 12 A new method to reconstruct QP Our new method to reconstruct QP: 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. deutronstritons 6.4º 18.2º 32.1º 61.2º 120º Previous methods to reconstruct the Quasi- projectile: (1)Selected very peripheral collisions; (2)velocity cut: assuming the particles whose velocities V>Vcm(QP) as the QP particles, and then double the backward hemisphere to obtain E* and T DRAWBACK: (1) low E*/A! (2) fluctuation was washed out! 3 source fits: red: QP, blue:NN, pink:QT

13. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 13 QP Source Reconstruction and Determination of E*/A Mixed QP Vpar p d t 3 He 4 He Li Velocity contour plots and parallel velocity distribution for Ar+Ni at Bin2 window  RED hatch areas are QP component  Energy Balance: E* =  (ECP+En) + Q where ECP,En=kinetic energy of CP and neutron in the source frame; En was obtained assuming a Maxwell-Boltzmann thermal distribution, consistent with volume emission, i.e., En = 3/2MnT = 3/2MnSqrt(E*/a), where a=A/8 is used, Mn was obtained as the difference between the nucleon number (A0) of the QP and the sum of nucleons bound on the detected CP (Mn=A0-  ACP), Q is the mass excess of the QP system

14. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 14 Similarity of Quasi-projectiles

15. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 15 Coherent Experimental Evidence for Critical Behavior a.The Fisher droplet model analysis b.The largest fluctuations c. Fragment hierarchical distribution: nuclear Zipf law d. caloric curve: determination of critical temperature e. critical exponent analysis: universal class of LGPT

16. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 16 Charge Distribution of QP: Fisher Droplet Model 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). Fisher Droplet Model predicts that there exists a minimum of  eff for the charge distributions when the phase transition occurs!

17. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 17 Charge Distribution without Zmax of QP Exponential-law fit: dN/dZ’ ~ exp(- eff Z’), where Z’ ~ Z but Zmax excluded on the event- by-event basis

18. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 18 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 Charge of the largest fragment 2 nd Normalized moment 1. LIQUID 3. GAS 2. Critical points

19. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 19 The Largest 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:

20. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 20 Universal flucutuation: Δ-Scaling Analysis of Zmax Central 25,32,39,45,50MeV/u Semicentral 25,32,39 Semicentral 45,50 + central 39,45,50 INDRA: Xe+Sn KNO scaling ~  =1 variable rescaling normalization rescaling increasing energy Δ-scaling law is observed when two or more probability distributions P[m] of the stochastic observable m collapse onto a single scaling curve Φ(z) if a new scaling observable is defined: z =(m-m*)/ Δ This curve is: ΔP[m] = Φ(z)= Φ[(m-m*)/ Δ] where Δ is a scaling parameter, m* is the most probable value of the variable m, and is the mean of m.

21. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 21 Universal flucutuation: Δ-Scaling Analysis of Zmax

22. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 22 Fragment Topological Structure: Zipf plot Assuming we have M particles in a certain event, we can define Rank n from 1 to M for all particles from Zmax to Zmin. Rank (n) = 1 if the heaviest fragment = 2 if 2 nd heaviest fragment, = 3 if 3 rd heaviest fragment and so on Accumulating all events, we can get the Rank(n) sorted mean atomic number for the each corresponding Rank (n), and plot vs n. We called such plot as Zipf-type plot Nuclear Zipf-type plot reflects the topological structure in fragmentation. Y.G. Ma., Phys. Rev. Lett. 83, 3617 (1999) Ref: Y.G. Ma., Phys. Rev. Lett. 83, 3617 (1999) Original concept was introduced in Language Analysis by G. Zipf. Later on the similar behaviors were found in the various fields, e.g., the distributions of cities, populations, Market structure, and earthquake strength, and DNA sequence length etc. – Related to Self-organized Criticality

23. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 23 Nuclear Zipf’s Law in Lattice Gas Model Ref: Y.G. Ma, Eur. Phys. J. A 6, 367 (1999); 129 Xe,  f =0.38 Zipf’s law ( =1) It’s consistent with other signatures Zipf-type plot:

24. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 24 Pb+Pb/Plastic Zipf-law ( ~1 ) is satisfied

25. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 25 Fragment hierarchical Structure: nuclear Zipf plot Zipf law fit: Z rank ~ rank -  NIMROD Data: Zipf-law (  ~1 ) is satisfied around E*/A ~ 5.6 MeV/u Zipf-plots our data

26. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 26 Z max -Z 2ndMax Correlation (scattering plots) Exc1Exc2Exc3 Exc4 Exc5 Exc6 Exc7Exc8Exc9 Z2ndmax Zmax Transition Region Ref to: Sugawa and Horiuchi, Prog The Phys 105 (2001) 131

27. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 27 Z max -Z 2ndMax Correlation (average values)

28. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 28 caloric curve: apparent Kinetic Energy Spectra in the Source Rest Frame; Sorting QP events by ~ 1 MeV/u E*/A window; apparent kinetic temperature apparent isotopic temperature apparent caloric curve

29. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 29 Caloric Curve: initial 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 physics

30. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 30 Determination of the Critical Exponents: , , ,  Z dN/dZ  =2.31

31. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 31 Determination of the Critical Exponents: , , , 

32. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 32 Model Comparisons: Lattice Gas Model Classical MD SMM Sequential Decay (Gemini)

33. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 33 Model comparison Lattice gas model (LGM) GEMINI: sequential decay model the hot compound nuclei de-excite via binary decay Ref: R. Charity et al.,,NPA483,371(1988) ------------------------------------------------ Classical Molecular Dynamics model Fragment prescription: Congilio-Klein method Ref: Pan, Das Gupta, PRL80, 1182 (1998) P-V phase diagram

34. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 34 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) Evaporation model fails to fit the Data; Phase Transition Models give a correct trends as Datal!

35. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 35 Model comparisons: Campi plot data GEMINI LGM CMD TRANSITION NO TRANSITION 

36. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 36 Model comparison: Zmax-Z2max correlation data GEMINI LGM CMD TRANSITION NO TRANSITION 

37. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 37 SMM calculation (A=36) (Botvina)

38. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 38 Campi Plots Zmax-Z2max Correl. TRANSITION

39. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 39 CONCLUSIONS (1) The lightest system and the most complete studies in nuclear LGPT experimentally (2) 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 (3) Fragment hierarchical Structures: Zipf’s law, fragment hierarchy, is satisfied around E*/A|crit rather than the equal-size fragment distribution which is predicted by spinodal instablity (1st phase transition) (4) Caloric Curve has no plateau, in comparison with heavier system : E*/A|crit ~ 5.6 ±0.5MeV, T|crit ~ 8.3 ±0.5MeV (5) 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! (6) Overall good agreements with Phase Transition Model calc. were attained This body of evidence is coherent and suggests a phase change in an equilibrated system at, or extremely close to, the critical point for such light nuclei rather than 1st order phase transition For details, see Y.G. Ma et al., Phys. Rev. C71, 054606 (2005); PRC69, 031604 ( R ) (2004).

40. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 40 Acknowledgements NIMROD Collaboration: R. Alfarro, 5 J. Cibor, 4 M. Cinausero, 2 Y. El Masri 6, D. Fabris, 3 E. Fioretto, 2 K. Hagel 1, A. Keksis 1, T. Keutgen, 6 M. Lunardon, 3 Y. G. Ma 1,a, Z. Majka, 4 A.Makeev 1,E. Martin 1, A.Martinez-Davalos, 5 A.Menchaca- Rocha, 5 M. Murray 1, J.B.Natowitz 1, G. Nebbia 3, L. Qin 1, G. Prete, 2 V. Rizzi, 3 A.Ruangma 1, D. V. Shetty 1, P. Smith 1, G. Souliotis 1, P.Staszel, 4 M. Veselsky 1, G. Viesti, 3 R. Wada 1, J. Wang 1, E. M. Winchester 1 and S. J. Yennello 1 1 Texas A&M University, College Station, Texas 2 INFN Laboratori Nazionali di Legnaro, Legnaro, Italy 3 INFN Dipartimento di Fisica, Padova, Italy 4 Jagellonian University, Krakow, Poland 5 UNAM, Mexico City, Mexico 6 UCL, Louvain-la-Neuve, Belgium a Shanghai Insititute of Applied Physics, Shanghai

41. Y. G. Ma, CCAST Workshop, Aug 19-21, Beijing 41