B2 Multifragmentation 0

1
B-2 Multifragmentation 0 Introduction
Generalities From evaporation to
vaporisation Caloric curve of nuclear
matter Phase diagram Equation of state
Spinodal region and multifragmentation Nuclear
temperature Detectors for multifragmentation
How to reach multifragmentation Kinetic
temperatures Isotopic temperatures
Statistical models Dynamical models
Dynamical and statistical models Isospin tracer
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B-2 Multifragmentation 1 Generalities
Definition decay of a composite nuclear system
into several heavy fragments (3 ? Z ? 30). It is
a very fast decay mode, the time scales involved
are at most of the order of several hundred fm/c
(1 fm/c 3.10-24 s).
J. Bondorf et al., Phys. Rep. 257(1995)133
At freeze-out density r
r0/3 temperature T 5 MeV excitation
energy E 4-6 AMeV
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B-2 Multifragmentation 2 From evaporation to
vaporisation
INDRA AuAu at 60 AMeV
quasi-projectile
multifragmentation
DE
evaporation
towards vaporisation
peripheral
central
E
ALADIN multiplicity of IMFs
multifragmentation
vaporisation
evaporation
AMeV
Zbound ?Z with Z? 2
A.Schüttauf et al., Nucl. Phys. A 607 (1996) 457
peripheral
central
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B-2 Multifragmentation 3 Caloric curve of
nuclear matter
Caloric curve of nucleus
Caloric curve of water
gas
Temperature (MeV)
liquid
Excitation energy per Nucleon (MeV)
J. Pochodzalla et al., Phys. Rev. Lett.
75(1995)1040
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B-2 Multifragmentation 4 Phase diagram
1.The liquid phase nuclear matter in its ground
state, at low temperatures and densities. 2.The
condensed phase supposed to be cold matter at
high densities where nucleons are organized into
a crystal. 3. The gaseous phase appears at
fairly high temperatures and low densities at
which the nuclei evaporate into a hadron gas. 4.
The plasma phase deconfined mixture of quarks
and gluons coming from the dissociation of
hadrons into their elementary constituents (?
5-10 ?0 , T150 MeV)
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B-2 Multifragmentation 5 Equation of state
Generally, the equation of state of a system is a
relation between three thermodynamical
variables. For the nuclear matter
density
temperature
binding energy of the infinite nuclear matter in
its ground state
internal energy
compression energy at T0
thermal energy
Saturation point For a sufficiently heavy
nucleus, increasing its number of constituents
does not modify the density of nucleons in its
central part. The saturation density ?0 is
independent of the nuclear size. ?0 0.17 ?
0.02 nucleon.fm-3 (? Rr0.A1/3 with r01.2fm )
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B-2 Multifragmentation 6 Equation of state
Compression energy
Compressibility
Low K? ( 200 MeV) ? soft equation of state (one
has to give relatively little
compression energy to reach high densities) High
K? ( 400 MeV) ? hard equation of state Recent
experimental results in heavy-ion collision
studies seem to favor a soft equation of
state. A. Andronic et al., Nucl. Phys. A
661(1999)333c, C. Fuchs et al., Phys. Rev. Lett.
86(2001)1974 Any equation of state is based on
the knowledge of the elementary interactions
between the constituents. ? The nucleon-nucleon
interaction potential has a dominant term that is
repulsive at short range ( ? 0.5 fm ) and
attractive at longer range ( ? 0.8 fm ) ? NN
potential molecule potential ? EoS (infinite
nucleon system) EoS (Van der Waals gas) ?
isotherms, liquid-gas phase transition Problem
the fermionic nature of the nucleons ? simple
real fluid ? approximate theoretical
description from the saturation point as the
balance between the attractive part of the
nuclear interaction potential and the repulsion
between nucleons.
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B-2 Multifragmentation 7 Spinodal region and
multifragmentation
isotherms
spinodal region
Nuclei reaching the spinodal region blow up into
several fragments, undergoing a reaction process
of multifragmentation. This decay mode is a way
to study the transition between the liquid and
gas phases.
Coexistence zone of liquid-gas phases for TltTc
17.9 MeV with a spinodal region characterized by
a mechanically instable regime with a negative
compressibility K -1/V.dP/dV
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B-2 Multifragmentation 8 Nuclear temperature
Definition of the temperature provided by
statistical mechanics This definition is
applicable to any isolated system, like a nuclear
system if one regards the very short range of the
nuclear forces. Requirement full statistical
equilibrium ? Difficult to achieve due to the
short time range of the reaction, the finite
size of the system, the complex dynamics, and the
various collisions that occur in a collision. ?
Experimental results interpreted as a signal of
an equilibrium A. Schüttauf et al., Nucl. Phys.
A 607(1996)457
binding energies
Isotopic temperatures
Experimental thermometers
Maxwell-Boltzmann distribution
yield
constant containing the spins and As
yields of the species
Kinetic temperatures
E
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B-2 Multifragmentation 9 Detectors for
multifragmentation
4p detectors
Spectrometers
ALADIN
INDRA
EOS
MINIBALL
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B-2 Multifragmentation 10 How to reach
multifragmentation
maximum fragment production in central collisions
A.Schüttauf et al., Nucl. Phys. A 607 (1996) 457
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B-2 Multifragmentation 11 Kinetic temperatures
AuAu at 600 AMeV, mid-peripheral collisions
Maxwell-Boltzmann fit
T. Odeh, PhD thesis, University Frankfurt (1999)
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B-2 Multifragmentation 12 Isotopic temperatures

AuX at 600 AMeV
T. Odeh, PhD thesis, University Frankfurt (1999)
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B-2 Multifragmentation 13 Statistical models

• Assumption of an equilibrated source emitting
  fragments in either microcanonical,
• canonical or grand canonical ensembles.
• The break-up process is either spontaneous, all
  fragments are emitted at the same
• time, or, it is a slow process, the
  fragments are emitted sequentially.

Example the SMM code (Statistical
Multifragmentation)
J. Bondorf et al., Phys. Rep. 257(1995)133
It is a mixed approach, based on the
microcanonical assumption (conservation of the
total energy) and using canonical prescriptions
of partitions. It assumes that fragments are
distributes in a certain available volume V
(supposed to be the freeze-out volume) following
Boltzmann statistics. The density of the
freeze-out corresponds to the coexistence region
of the phase diagram. The internal structure of
the fragments is described by means of the liquid
drop model. The mass and charge are exactly
conserved with every single event. The produced
fragments may be excited and may also undergo a
secondary decay. It depends on their mass
fragments up to oxygen can de-excite by breaking
into several single nucleons and light clusters.
Heavier, excited fragments can evaporate light
particles.
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B-2 Multifragmentation 14 Statistical models
Experimental results and statistical model
Multiplicities
Temperature
THeLi
Good agreement for the fragments but not for the
light particles!
T. Odeh, PhD thesis, University Frankfurt (1999)
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B-2 Multifragmentation 15 Dynamical models
The dynamical models follow the time evolution of
the system, from the collision until the
freeze-out.
Example the INC code (Intra-Nuclear Cascade)
J. Cugnon, Phys. Rev. C 22 (1980) 1885 D. Doré et
al., Phys. Rev. C 63 (2001) 034612
Nucleus-nucleus version!
The code does not follow the state of the
ensemble of cascade particles but the state of
each cascade particles as a function of time.
This permits to take into account in a total
explicit way the motion of the nucleons and the
collisions it generates. At the beginning, the
nucleons are randomly positioned in a sphere.
Particles move along straight line trajectories
until two of them reach their minimum distance of
approach dmin. All the particles are followed
in this way until a stopping time tstop. This
time is determined from the excitation energy of
the remnant, the emission anisotropy , and the
saturation of the cumulative numbers of
collisions or escaping particles. In the
nucleus-nucleus case, the stopping time has been
set to 40 fm/c.
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B-2 Multifragmentation 16 Dynamical and
statistical models
Combination of dynamical and statistical models
yield
cascade multifragmentation
cascade
E
multifragmentation
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B-2 Multifragmentation 17 Isospin tracer

• RuZr and ZrRu at 400 AMeV
• 40Zr and 44Ru have stable isotopes with the same
  mass A 96.

ZrRu or RuZr
relative abundance of protons
RZ (ZrZr) 1 and RZ (RuRu) -1 RZ 0 ? full
mixing
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B-2 Multifragmentation 18 Isospin tracer
Relative abundance of protons as a function of
rapidity for central collisions
centrality of the collisions
F. Rami et al., Phys. Rev. Lett. 84(2000)1120