Nuclear Liquid Gas Phase Transition and Nuclear Zipf Law

1. Nuclear Liquid Gas Phase Transition and Nuclear Zipf Law Yu-Gang Ma Shanghai Institute of Applied Physics, Chinese Academy of Sciences

2. Fragment Topological Structure: Zipf plot • What is the nuclear Zipf’s law: • Assuming we have M particles in a certain event, we can define Rankn from 1 to M for all particles from Zmax to Zmin. • Rank (n) = 1 if the heaviest fragment • = 2 if 2nd heaviest fragment, • = 3 if 3rd heaviest fragment • and so on • Accumulating all events, we can get the Rank(n) sorted mean atomic number <Zn> for the each corresponding Rank(n), and plot <Zn> vs n. • We called such plot as Zipf-type plot in nuclear fragmentation. • Nuclear Zipf-type plot reflects the topological structure in fragmentation. • 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

3. The ideal of Nuclear Zipf law • The model: Lattice gas model and Classical molecular dynamics model • Nuclear Zipf plot • Nuclear Zipf law in nuclear fragmentation

4. Model calculation Lattice gas model (LGM) Classical Molecular Dynamics model Fragment prescription: Congilio-Klein method Ref: Pan, Das Gupta and Grant, PRL80, 1182 (1998)

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

6. More extensive calc. Other phase transition signatures (left figure) takes place in the same temperature when λ=1 (right figure). Y.G. Ma, Phys. Rev. Lett. 83 (1999) 3617

7. Experimental evidence: • EMU13 CERN Exp: Multifragment emission following Pb+Pb and Pb+Plastic collision @ 158AGeV • NIMROD: Quasi-projectile fragmentation from Ar+Ni, Ti and Al

8. 158, 10.6, 0.64 AGeV Pb+Pb/Plastic 158 AGeV Pb+Pb/Plastic, EMU13 CERN Exp. Zipf-law (~1 ) is satisfied λ=

10. NIMROD QP fragmentation: Ar+Ni, Ni and Al Zipf-plots our data Zipf law fit: Zrank ~ rank- Our Data: Zipf-law (~1 ) is satisfied around E*/A ~ 5.6 MeV/u Y. G. Ma, J. Natowitz, R. Wada et al. (NIMROD Collaboration), Nucl-ex/0410018, submitted to Phys. Rev. C

11. Our NIMROD data Y. G. Ma, J. Natowitz, R. Wada et al. (NIMROD Collaboration), Nucl. Phys. A 749, 106c (2005)

12. Our NIMROD data and 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 T0: T0(Exp)=8.3 ±0.5MeV (Black Points) T0(GEMINI) = 8.3 MeV T0(LGM) = 5.0MeV T(PhaseTran) T0(CMD) = 4.5MeV T(PhaseTran) Fig.(e) 2nd Zmax; Fig.(f) YGMa et al., PRC69, 31604(2004); E*(Exp)~5.6MeV/A Evaporation model fails to fit the Data;Phase Transition Models reproduce the Data well!

13. Backup slides • Experimental data: QP • 47MeV/nucleon 40Ar+Ni, Ti and Al • See ref: YGMa et al., PRC69, 31604(2004); YGMa et al., Nucl-ex/0410018

14. The Largest Fluctuation: Campi Plots • Campi plot: • ln(Zmax) vs ln(S2) • (event-by-event) can explore the critical behavior, where Zmax is the charge number of the heaviest fragment and S2 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 Charge of the largest fragment Transition Region 1. LIQUID 2. Critical points The GAS Branch 3. GAS Ref : Campi, J Phys A19 (1988) L917 2nd Normalized moment

15. The Largest Fluctuation of Zmax and Ektot Zmax (order paramter) Fluctuation: Normalized Variance of Zmax/ZQP: NVZ =  2/<Zmax> 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)/<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:

16. Caloric Curve: initial 1. Sequential Decay Dominated Region (LIQUID-dominated PHASE): Tini = (M2T2 –M1T1)/(M2-M1) where M1, T1 and M2, T2 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 Tiso Ref: Z. Majka et al., Phys. Rev. C 55 (1997) 2991 3. Assuming vapor phase as an ideal gas of clusters: Tkin = 2/3Ethkin = 2/3(Ecmkin-Vcoul) T0 = 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

17. Zmax-Z2ndMax Correlation (average values)

18. Determination of the Critical Exponents: ,  ,, 