Evidence of Critical Behavior in the Evidence of Critical Behavior in the Disassembly of Light Nuclei with A ~ 36Disassembly of Light Nuclei with A ~ 36

Yu-Gang MaTexas A&M University, Cyclotron Institute, USA

Shanghai Institute of Nuclear Research, CAS, CHINA(for NIMROD Collaboration)

• Experimental Set-up: NIMROD• Reaction System: 47MeV/u 40Ar+58Ni• 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

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 stud

y 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 );

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 (ZQP

tot12) comprise ~4.3% of violent events.

4 Excitation Energy with 4 Excitation Energy with Calorimetry Method:Calorimetry Method:

E* = (ECP+En) + 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.

Charge Distribution of QP

• Zqp

The minimum The minimum effeff ~ 2.31, close to the Critical Exponent of ~ 2.31, close to the Critical Exponent of

liquid gas phase transition universal class (~2.23) preliquid gas phase transition universal class (~2.23) predicted by the Fisher droplet model!dicted by the Fisher droplet model!

1 2 3 4 5 6 7 8 92

3

5.85.96.06.16.26.36.46.5

eff

E*/A (MeV)

lines: Fisher Droplet Power-Law fit: dN/dZ ~Z-eff

Ref: Fisher, Rep. Prog. Phys. 30, 615 (1969).

The largest fluctuation: Campi Plots

Ref : Campi, J Phys A19 (1988) L917

Campi plot:

ln(Zmax) vs ln(Sln(Zmax) vs ln(S22)) (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

The GAS Branch

Transition Region

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 by

Chomaz, Gulminelli, D’Agostino et al., PRL, PLB

Caloric CurveCaloric Curve

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(Ecm

kin-Vcoul)

T0 = 8.3±0.5MeV at E*/A = 5.6 MeV

No obvious plateau was observed at the largNo obvious plateau was observed at the largest fluctuation point, in comparison with est fluctuation point, in comparison with the heavier system! different physthe heavier system! different phys.

Ref: J. Natowitz et al., Phys Rev C65, 034618 (2002)

Model ComparisonsModel Calculation (A=36, Z=16)• Statistical Evaporation Model: GEM

INI (Pink dotted lines) NO PHASE TRANSITION

Ref: R. Charity et al., ,NPA

• Lattice Gas Model (LGM) (Black lines)• Classical Molecular Dynamics Mode

l (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)

Evaporation model fails to fit the Data; Phase Transition Models reproduce the Data well!

Fragment Topological Structure: Zipf plot

Nuclear Zipf-Plot Rank sorted Fragment Size distribution, where Rank = 1 if the heaviest fragment = 2 if 2nd heaviest fragment, = 3 if 3rd heaviest fragment and so on, in each event. Accumulating all events, we can plot rank sorted mean size <Zrank> vs rank, i.e., Zipf-type plot.

Nucl Zipf-Law if ~1 using Zrank ~ rank- fit when liquid gas phase transition, see Y.G. Ma, Phys. Rev. Lett. 83, 3617(1999)

Zipf law fit:

Zrank ~ rank-

Our Data: Zipf-law (~1 ) is satisfied around E*/A ~ 5.6 MeV/u

Zipf-plots

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:

τeff =2.31 0.03 for distribution of Z – close to Critical Exponent of LGPT =0.33 0.01, =1.150.06; =0.680.04 ==> 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

ThanksThanks !COLLABORATORS

R. Wada, K. Hagel, J. S. Wang, T. Keutgen,

Z. Majka, M. Murray, L. J. Qin, P. Smith,

J. B. Natowitz

R. Alfaro, J. Cibor, M. Cinausero, Y. El Masri,

D. Fabris, E. Fioretto, A. Keksis, M. Lunardon,

A. Makeev, N. Marie, E. Martin, A. Martinez-Davalos,

A. Menchaca-Rocha, G. Nebbia,

G. Prete, V. Rizzi, A. Ruangma, D. V. Shetty,

G. Souliotis, P. Staszel, M. Veselsky, G. Viesti,

E. M. Winchester, S. J. Yennello

Extra Slide 1: Δ-Scaling Analysis of ZmaxDefinition: <Zmax> PN[Zmax] (z()) [(Zmax-Zmax*)/<Zmax>]

where

PN[Zmax] the probability distribution of Zmax <Zmax> the mean value of Zmax Zmax* the most probable value of ZmaxIf Δ-Scaling holds, all probability distributions collapse t

o 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

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IN OUR DATA, Δ-SCALING OF ZMAX 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!)

Extra Slide 2: Critical Exponents

Ref: Gilkes et al., PRL73, 1590(1994) Elliott et al., PRC49, 3185(1994).