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Nuclear Liquid Gas Phase Transition
and Nuclear Zipf Law
Yu-Gang Ma
Shanghai Institute of Applied Physics,
Chinese Academy of Sciences
Fragment Topological Structure: Zipf plot
•
•
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
What is the nuclear Zipf’s law:
• 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 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)
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
Model calculation
Lattice gas model (LGM)
Fragment prescription:
Congilio-Klein method
Ref: Pan, Das Gupta and Grant, PRL80, 1182 (1998)
Classical Molecular Dynamics
model
Nuclear Zipf’s Law in Lattice Gas Model
Zipf-type plot:
129Xe,
•Zipf’s law (=1)
f=0.38
Ref: Y.G. Ma, Eur. Phys. J. A 6, 367 (1999);
It’s consistent with
other signatures
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
Experimental evidence:
(1)EMU13 CERN Exp: Multifragment
emission following Pb+Pb and
Pb+Plastic collision @ 158AGeV
(2) NIMROD: Quasi-projectile
fragmentation from Ar+Ni, Ti and Al
158 AGeV Pb+Pb/Plastic, EMU13 CERN Exp.
Zipf-law (~1 ) is satisfied
λ=
158, 10.6, 0.64 AGeV
Pb+Pb/Plastic
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
Our NIMROD data
Y. G. Ma, J. Natowitz, R. Wada et al. (NIMROD
Collaboration), Nucl. Phys. A 749, 106c (2005)
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);
Evaporation model fails to fit the Data;
E*(Exp)~5.6MeV/A
Phase Transition Models reproduce the Data well!
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
Charge of the largest fragment
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
The LIQUID Branch
Transition Region
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.
1. LIQUID
2. Critical points
The GAS Branch
3. GAS
2nd Normalized moment Ref : Campi, J Phys A19 (1988) L917
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.,
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!
Phys Rev C 60 (1999) 034606
A possible relation of Cv to kinetic
energy fluctuation was proposed:
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
Zmax-Z2ndMax Correlation (average values)
Determination of the Critical Exponents: ,  ,, 