B-2 Multifragmentation – 0Introduction

• 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

B-2 Multifragmentation – 1Generalities

J. B

on

do

rf et a

l., Ph

ys

. Re

p. 2

57

(19

95

)13

3

At freeze-out: density ~ 0/3temperature T ~ 5 MeVexcitation energy E* ~ 4-6 AMeV

preequilibriumemission

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).

B-2 Multifragmentation – 2From evaporation to vaporisation

E

E

towards vaporisation

multifragmentationquasi-projectile

evaporation

INDRA Au+Au at 60 AMeV

ALADIN multiplicity of IMF’s

peripheral central

peripheralcentral

Zbound = Zwith Z 2

multifragmentation

evaporationvaporisation

A.Schüttauf et al., Nucl. Phys. A 607 (1996) 457

AMeV

B-2 Multifragmentation – 3Caloric curve of nuclear matter

Caloric curve of nucleus Caloric curve of water

Excitation energy per Nucleon (MeV)

Tem

per

atu

re (

MeV

)

J. Pochodzalla et al., Phys. Rev. Lett. 75(1995)1040

liquid

gas

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 , T~150 MeV)

B-2 Multifragmentation – 4Phase diagram

B-2 Multifragmentation – 5Equation of state

Generally, the equation of state of a system is a relation between three thermodynamical variables.

For the nuclear matter:

0( , ) ( , 0) ( , )C THE T E T E T E

internal energy

compression energy at T=0

thermal energy

binding energyof the infinite

nuclear matter in its ground state

density temperature

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

( R=r0.A1/3 with r0=1.2fm )

20

20

( )( )

18C

KE

0

220 2

9 Cd EK

d

Compression

energyCompressibility

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.

B-2 Multifragmentation – 6Equation of state

B-2 Multifragmentation – 7Spinodal region and multifragmentation

isotherms

Coexistence zone of liquid-gas phases for T<Tc = 17.9 MeV with a spinodal region characterized by a mechanically instable regime with a negative compressibility K = -1/V.dP/dV

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.

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

1 ln( ( , ))S E N

T E E

B-2 Multifragmentation – 8Nuclear temperature

Experimental thermometers

yiel

d

E

Maxwell-Boltzmann distribution:

Isotopic temperatures

( / )( ) . kinE Tkin kinN E E e

6 7

3 4

13.33.

ln(2.18. ).

HeLiLi Li

He He

MeVT

Y Y

Y Y

1 2 3 4((( ) ( )) / )1 2

3 4

..

.B B B B TY Y

R a eY Y

yields of the species

binding energies

constant containing the spins and A’s

Kinetic temperatures

B-2 Multifragmentation – 9Detectors for multifragmentation

ALADIN INDRA

MINIBALLEOS

Spectrometers 4detectors

B-2 Multifragmentation – 10How to reach multifragmentation

A.Schüttauf et al., Nucl. Phys. A 607 (1996) 457

maximum fragment production in central

collisions

B-2 Multifragmentation – 11Kinetic temperatures

Au+Au at 600 AMeV, mid-peripheral collisions

T. Odeh, PhD thesis, University Frankfurt (1999)

( / )( ) . kinE Tkin kinN E E e

Maxwell-Boltzmann fit:

B-2 Multifragmentation – 12Isotopic temperatures

Au+X at 600 AMeV

T. Odeh, PhD thesis, University Frankfurt (1999)

6 7

3 4

13.33.

ln(2.18. ).

HeLiLi Li

He He

MeVT

Y Y

Y Y

+

B-2 Multifragmentation – 13Statistical 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)

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.

J. Bondorf et al., Phys. Rep. 257(1995)133

B-2 Multifragmentation – 14Statistical models

T. Odeh, PhD thesis, University Frankfurt (1999)

Experimental results and statistical model

TH

eLi

Temperature

Multiplicities

Good agreement for the fragments but not for the light particles!

B-2 Multifragmentation – 15Dynamical 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)

Nucleus-nucleus version!

J. Cugnon, Phys. Rev. C 22 (1980) 1885D. Doré et al., Phys. Rev. C 63 (2001) 034612

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.

B-2 Multifragmentation – 16Dynamical and statistical models

Combination of dynamical and statistical models

multifragmentation

cascade

cascade+

multifragmentation

E

yiel

d

B-2 Multifragmentation – 17Isospin tracer

Ru+Zr and Zr+Ru at 400 AMeV

40Zr and 44Ru have stable isotopes with the same mass A = 96.

2 Zr Zr Ru Ru

Z Zr Zr Ru Ru

Z Z ZR

Z Z

Zr+Ru or Ru+Zr

relative abundance of protons

RZ (Zr+Zr) = +1 and RZ (Ru+Ru) = -1RZ = 0 full mixing

B-2 Multifragmentation – 18Isospin 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