Phase transition in Nuclei
Olivier LOPEZ
Séminaire 1-2-3 – Décembre 2006 - LPC Caen
Phase diagram of NM
Big Bang
QGP
200 MeV
200 AGeV
Temperature
20
100 AMeV
Gas
Hadron
50 AMeV
Liquid
20 AMeV
LG Coexistence
Density 
1
5?
Underlying Physics
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DFT approach to nuclear physics: towards an universal functional
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Nuclear matter phase diagram and finite nuclei phase transitions
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Study the energy functional for asymmetric nuclear matter
Constrain the isovector part of the energy (symmetry energy)
Produce sub- and super-saturation density matter through HI-induced reactions
Scan the low-temperature region of the nuclear matter phase diagram
Characterize the phase transition (location, order, critical points,…)
Evidence finite size effects (anomalies in thermodynamical potentials)
Complementary to the ALICE Physics program at higher energy (QGP)
From finite nuclei to dense nuclear matter
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Constrain Mean-Field models for Astrophysics
Study the structure and pahse properties of Neutron Star crusts
Understand the dynamics of supernova type II explosion (EOS)
Density Functional Theory
Self-consistent mean field calculations (and extensions) are
probably the only possible framework in order to understand the
structure of medium-heavy nuclei.
E = <y | H | y>
H = <f | Heff | f > = E[]
Symmetry energy (basics)
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Standard Bethe-Weisäcker formula for Binding Energy :
E = -avA + asA2/3 + acZ2/A1/3+asym(N-Z)2/A + d
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Symmetry Energy : Esym = asym(N-Z)2/A is therefore the change in
nuclear energy associated to the changing of proton-neutron
asymmetry N-Z
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In nuclear matter (isoscalar+isovector) :
E(n, p) = E0() + E1(n, p) with E1(n, p) = S()(n-p)/2
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Pressure : P = 2E/
Symmetry Energy (questions)
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Little is known at
super and subsaturation density
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Dependence on the
neutron-proton
asymmetry ?
Phase transition and Neutron stars
(Extended) MF theories with a density
functional constraint in a large density
domain are a unique tool to understand
the structure of neutron stars.
Multifragmentation and Phase transition
Multifragmentation
as a possible signature of
the liquid-gas phase
transition
Threshold for Multifragmentation
From G. Bizard et al., Phys. Lett. B 302, 162 (1993)
Hot nuclei and de-excitation
Evaporation
1
Multifragmentation
3
 ~ 0
T < 5 MeV
Vaporization
8
 < 0
T= 5-15 MeV
E*/A (MeV)
 << 0
T>15 MeV
Multifragmentation as a signal of
liquid-gas phase transition?
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Simultaneous emission for fragments :
tff < tn
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Equilibrated system in (,T) plane :
Isotropic emission
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Nuclear system at sub-saturation density :
/0 << 1
Multifragmentation as a simultaneous process
tFF ~ tn
Angular correlation functions :
Ncorr(qFF) - Nuncorr(qFF)
R(qFF) =
From D. Durand, Nucl. Phys. A 630, 52c (1998)
Ncorr(qFF) + Nuncorr(qFF)
Multifragmentation as an equilibrated process…
The “rise and fall” of MF emission
Universality
Mass scaling
From A. Schuttauf et al., Nucl. Phys. A 607, 457 (1996)
Multifragmentation at low density …
Statistical Multifragmentation Model (SMM)
Statistical weight :
58Ni+197Au central collisions
W = eS(V,T)
V=(1+c)V0 with c>0
Volume
From N. Bellaize et al., Nucl. Phys. A 709, 367 (2002)
Multifragmentation and statistical
description
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Reaction dynamics and Fermi
motion is not taken into
account → additional free
parameter Erad (radial flow)
for Statistical Models
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Is explicitly incorporated in
dynamical
(semi-classical)
approaches like HIPSE or
QMD,
(quantal)
like
AMD/FMD…
From N. Bellaize et al., Nucl. Phys. A 709, 367 (2002)
Heavy Ion Phase Space Explorator
D. Lacroix, A. Van lauwe and D. Durand, Phys. Rev. C 69, 054604 (2004)
Signals of Phase
transitions
Signals of phase transition
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Caloric curve: T=f(E*)
SMM
A=100
T
Free nucleons gas
E*  T
coexistence
10
Back-bending
5
Fermi gas
E*  T2
5
10
From INDRA collaboration (1999)
E*/A
From J. Pochodzalla et al.,
Phys. Rev. Lett. 75, 1040 (1995)
Signals of (1st order) Phase
transition
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Abnormal energy fluctuations
Thermodynamical relations :
T-1 = (S E)
S
C = ( E/  T) = -T2(2S/  E2)
If one divides the system in two
independent subsystems (1)+(2) :
Entropy
T
T-1 = (dS/dE)V
And we get for the partial energy
fluctuations of system (1) :
Temperature
C
Specific
heat
E t = E1 + E2
C = dE/dT
s12 = T2 C1C2/(C1+C2)
Energy
C12
C  C1 + C2 = C - s 2/T2
1
1
Latent Heat
(true at all thermodyn. conditions)
Signals of (1st order) Phase
transition
Peripheral Au+Au reactions
Central Xe+Sn reactions
M. D’Agostino et al., Physics Letters B 473, 219 (2000)
N. Le Neindre, PHD Thesis Caen (1999)
Liquid-gas phase transition
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Critical phenomena : power laws, scalings, exponents
Caloric curves : back-bending
Universal scaling : D-scaling (order-disorder)
Disappearance of collective properties : Hot GDR,
Shape transition (Jacobi)
Abnormal fluctuations : negative
capacities/susceptibilities
Charge correlations : spinodal decomposition
Bimodality : order parameter for phase transition
The case of Bimodality
Bimodality : theoretical aspects
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Related to a convex
intruder of the S(X)
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Appearance of a
double-humped distribution for the probability distribution P of
the order parameter X
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Examples :
From Ph. Chomaz, M. Colonna and J. Randrup, Phys. Rep. 389, 263 (2004)
X=E
X=V
Bimodality : experimental results
From M. Pichon, B. Tamain et al., Nucl. Phys. A 779, 267 (2006)
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Peripheral Au+Au
reactions
at
E/A=80 MeV
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Transverse energy
sorting (→ T)
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Bimodality of Zmax,
Zasym is observed in
the third panel
Bimodality : interpretation
Normal density (J) vs dilute (E*) system ?
Same T
From O. Lopez, D. Lacroix and E. Vient, Phys. Rev. Lett. 95, 242701 (2005)
Futures
SPIRAL/SPIRAL2
Isospin dependence of the
level-density parameter for
medium-sized nuclei
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Isospin dependence of the
liquid-gas phase transition
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Limiting temperature for
nuclei
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Mass splitting of p-n in
asymmetric nuclear matter
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Cluster emission threshold
for p-rich nuclei around
A=115 for moderate
E*/A (~1-2 MeV)
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Link to astrophysics and
compact nuclear matter
(NS)
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INDRA-SPIRAL experiments : status
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E494S : Isospin dependence of the level-density parameter
 33,36,40Ar
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+ 58,60,64Ni at E/A=11.1-11.7 MeV => Pd isotopes, E*/A=2-3 MeV
Coupling with VAMOS
Scheduled in March-April 2007 (moving D5-G1 is planned 01/07)
E475S : Emission threshold for complex fragments from compound
nuclei of A=115 and N~Z (p-rich)
 75,78,82Kr
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+ 40Ca at E/A=5.5 MeV
Done in March 2006 (calibration under progress)
Isospin dependence of the level-density
parameter a
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E* dependence :
a=aA
with : a = 1/(K+kE*/A)
K =7 , k =1.3
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N-Z dependence is assumed
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(A) a = a A e-b(N-Z)2
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(B) a = a A e-g(Z-Z0)2
From S. I. Al-Quraishi et al., Phys. Rev. C 63 (2005), 065803
Long-term range
Need for new detectors
MINIBALL/MSU
EOS
ALADIN
4p array (exclusive measurements)
ISIS
Low Energy thresholds (E/A<1 MeV/u)
INDRA
Mass and charge identification (1<A<100)
Very High angular resolution (Dq<0.5°)
Modularity / Flexibility (coupling/transportation)
FAZIA
Four pi A and Z Identification Array
CHIMERA
NIMROD
LHASSA
FAZIA : next generation 4p array
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Compactness of the device
Ebeam from barrier up to 100 A.MeV
Telescopes: Si-ntd/Si-ntd/CsI
Possibility of coupling with other detectors
Complete Z (~70) and A (~50) id.
Low-energy & identification threshold
Digital electronics for energy, timing and pulse-shape id.
FAZIA project
Courtesy of JM Gautier (LPC Caen)
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Visit us at http://fazia.in2p3.fr
FAZIA : next-gen 4p array
E/A= 6.2 MeV
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Digital electronics
Pulse Shape Analysis
E/A= 7.8 MeV
36Ar
Tandem Orsay (2003)
40Ar
CIME / GANIL Sept. 06
Long-term range is: EURISOL
(I) Density dependence of the nuclear symmetry
energy (DDSE)
56Ni - 74Ni, 106Sn -132Sn, E/A = 15 – 50 MeV
 (II) Neutron-Proton effective mass splitting (NPMS)
56Ni - 74Ni, 106Sn - 132Sn , E/A=50-100 MeV
 (III) Isospin-dependent phase transition (IDPT)
56Ni - 74Ni, 106Sn -132Sn, 200Rn - 228Rn, E/A = 30 – 100 MeV
 (IV) Isospin fractionation, Isoscaling (IFI)
56Ni - 74Ni, 106Sn -132Sn, 200Rn - 228Rn, E/A = 30 – 100 MeV
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Key Points are :
 large panoply of beams (light, medium, large A) over the maximal N/Z extension
 Beam energy range around and above the Fermi domain (15-100AMeV)
6
8
 Beam intensity around 10 -10 pps, small emittance, good timing (<1ns)
Phase transition in Nuclei
To be continued…
Nature of Phase transitions
Phase transitions reflect the self-organization of a system and are ruled
by common properties such as predicted by universality classes and
Renormalization Group theory.
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Solid, liquid and gas phases
Plasma (electrons, QGP, ...)
Magnetic properties in solid state matter (para/ferromagnets)
Bose-Einstein condensates
Superfluidity (Cooper pairs)
Fund. symmetries breakings (matter/antimatter, electroweak, …)
Nuclei ! …
T (MeV)
Dynamics of the phase transition
Spinodal decomposition?
Boltzman-Langevin (Stochastic Mean-Field)
Metastable regions
10-15
“GANIL” trajectory
Spinodal
region
0.3
1

A. Guarnera et al, Phys. Lett. B 403, 191 (1997)
R  10 fm
Privileged wavelength are formed : R ~10 fm
Symmetry Energy (future)
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Neutron-proton asymmetry is
different between the bulk and
surface for exotic nuclei
neutron
proton
r
Modified BW formula :
E = -aVA + asA2/3 + ac

(r)
Z2
A1/3
+
aVsym
S
1 + A-1/3aVsym/asym
(N-Z)2
+d
A
For A>>1, aVsym→ asym, for small A → weakening of SE
Multifragmentation as an equilibrated process…
129Xe+natSn
at 50AMeV; Multifragmentation
dN
dcos(qcm)
-1
cos (qcm)
+1
Isotropic emission
in cm frame
From N. Marie et al., Phys. Lett. B 391, 15 (1996)
Phase transition and critical
phenomena
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Power laws and scaling
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Power law of the A-distribution :
P(A) = A-t f(eAs)
e = (T-Tc)Tc
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3D Ising Model :
t = 2.2
s = 0.66
Experimentally :
t = 2.12 ± 0.13
s = 0.64 ± 0.04
From M. D’Agostino et al., Nucl. Phys. A 650, 329 (1999)
Bimodality : exp. results
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Observed whatever
the sorting
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Characteristic of a
1st order phase
transition
Statistical Models and drip lines
Enhancement of
Carbon emission for
p-rich nuclei
4He+116-124Sn
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Hauser-Feshback
calculations (BUSCO)
for Ba isotopes
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E=180 MeV
2
124Ba
Calc.
s c (mb)
1.5
Exp.
1
130Ba
0.5
138Ba
E*/A ≈ 1.5 MeV
0
1.2
1.25
1.3
1.35
1.4
1.45
1.5
N/Z
Figure 1. Carbon emission in 4He + 116,124 Sn. Data from Ref. 5.
From J. Brzychczyk et al., Phys. Rev. C 47, 1553 (1993)
75,78,82Kr
+ 40Ca at E/A=5.5 MeV forming CN 115-122Ba !