Equation of State for nuclear matter: research at CHARMS

PART I:

Generalities about the Equation of State (EOS) for ordinary matter and for nuclear matter

PART II:

Our research with the FRS connected to EOS

PART I: Equation of State for ordinary matter and for nuclear matter

Fundamental interactions and residual forces

e.m. interaction

residual e.m. interaction

(e.g. covalent bond)

residual e.m. interaction (molecular

force)

ordinary matter

strong interaction

residual strong interaction (nucleon-

nucleon force)

nuclear matter

ATOM MOLECULE LIQUID

NUCLEON NUCLEUS

Range dependence of the residual force

u

r

u

r

molecule-molecule

(Lennard-Jones)

nucleon-nucleon (Skyrme)

Nuclear matter in normal condition (nuclei) behaves as a liquid!

The scales are very different:

Ordinary matter Nuclear matter

Density: 1 g/cm3 3 1014 g/cm3

Typical distance: 10-10 m 10-15 m

How do microscopic properties translate into macroscopic

properties?

EOS for the ordinary matter

How does molecular force change when we have a great number of molecules? How strong is the molecular bond?

Macroscopic quantities that are observable when I heat a liquid: - volume, V - pressure, P - temperature, T

Relation among V, P, T: the equation of state

IDEAL GAS: .

REAL GAS (Van der Waals): .

nRT)nbV)(V

naP(

2

2

nRTPV

strenght intramolecular

force

volume molecule

Solution of the EOS (van der Waals)

for a given material (a and b

given)

V

P

Isothermes

spinodal region

liquid gas

coexistence

measuring P, V, T a and b intramolecular force

high T

low T

)()(11 322

2

TBTBRT

ab

V

nP

V

na

nbV

nRTP VV

drreNRT

ab

V

nTB kTrU

V2

0

)(2 )1(2)(

2nd virial

coefficient

How can I explore experimentally the P,V diagram? 1) I increase T at constant pressure

V

P

Isothermes

gas

coexistence

liquid

100°C

1 atm

T

E

P constantV increasing

Caloric curve

(liquid-gas phase transition)

liquid-gas coexistence

How can I explore experimentally the P,V diagram? 2) I increase P at constant temperature increase

(compression)

V

P

Isothermes

gas

coexistence

liquid

heat bath

nRT

PVZ compressibility

EOS2

2

V

na

nbV

nRTP

RTV

na

nbV

VVZ

constT

)(

Study of the EOS for the nuclear matter

“Exploring the nuclear-matter phase-diagram and identifying the different phases of nuclear matter is one of the main challenges of modern nuclear

physics.” NUPECC

1) Measuring the phase transitions

2) Measuring the compressibility

How can I explore experimentally the P,V diagram?

projectile

target spectator

spectatorparticipant„fireball“

Nucleus-nucleus collisions at relativistic energies

2) this part is compressed

1) these parts get excitation energy E*

Liquid-gas phase transition

Liquid phase Fragmentation

Transition(coexistence)

Multifragmentation

Gas phase Vaporisation

Phase transition superfluid liquid

superfluid

liquid

coexistence

gas

E/MeV

5

7010 300

A25

0.5

T/M

eV

Superfluid phase revealed by structural effects e.g. even-odd staggering

Liquid-gas phase transition

1)What is T? How to measure T?

2)What is E*? How to measure E*?

3)What is P? Is it constant?

4)What is V? Is it measurable?

Classical Temperature

Temperature is a macroscopic observable that rules the exchange of energy between bodies.

Correlation between the temperature and the energy of the molecules of the ideal gas AT EQUILIBRIUM

< x > = < y > = < z > = 1/2 k T

kTe)(p

T

E tot = n < >

T high T low

p 1

EF

erm

i

Zero Temperature

Nuclear Temperature: zero temperature

The nucleus is:a mesoscopic systema fermionic quantum system

the nucleons inside the nucleus do not have the same degrees of freedom: they have increasing energy

T

E tot

p 1

EF

erm

i

This part hereis approximatelyan exponential

Te1

1)(p

ETOT = a T2

aAE/A T2

Nuclear Temperature: non-zero temperature

Thermometers

  

p ln p

p = number of nucleons with energy

-1/T = slope

SLOPE THERMOMETER: Energy spectrum for nucleons from evaporation

T

p

p1

p2

1 2

ISOMER THERMOMETER: Nuclei in a heat bath at T>0.The energy of different isomers will be different

ISOTOPE THERMOMETER: Nuclei in a heat bath at T>0.The mass (or binding energy) of different isomers will be different

TT

T

T

1

2

1

2 ee

e

e

Y

Y

p

p 12

2

1

Excitation energy

AA*E

Excitation energy : quantity related to the individual energy of the nucleons

Pressure and VolumePressure : pressure done by the nucleons

Volume: volume occupied by the nucleons?????

Problems behind the liquid-gas phase transition

V

P

Isothermes

gas

coexistence

liquid

100°C

1 atm

1) Costant pressure? Operational definition of the volume?2) Quantum system3) Mesoscopic system4) Fast heating (no thermalisation – no equilibrium)5) Mixture of two liquids (proton and neutron subsystems)

„Isospin dependence of the EOS“

Compressibility of nuclear matter

Nuclear compressibility is directly related to the nuclear force

u

r

(E/A) = Internal energy per nucleon = internal energy stored in compression

/0 = normalised density

= nuclear compression modulus = curvature at =0

large = hard EOS

small = soft EOS

Methods to investigate the compressibility1) From fireball (Flow, Kaon production)2) From scattering (Giant resonance)

Flow

The „squeeze out“ or „flow“ is directly related to the gradiente pressure in the fireball nuclear compression modulus

Kaon productionThe bulk of K+ mesons is produced in secondary or multiple reactions of nucleons in the fireball:N1+N2 N1N2 N3 K+ + Such secondary reactions occur predominantly at high nuclear density kaon production yields are sensitive to the compression nuclear compression modulus

Giant monopole resonance

The isoscalar giant monopole resonance (GMR) is a compressional mode of excitation. It is of particular interest because its energy is directly related to compressibility. By measuring the inelastically scattered alpha particles at forward angles, including 0° degrees, one can deduce the energy.

Problems behind the compressibility

1) Short time span (dynamical picture hydrodynamical models)

2) Momentum dependence interaction (MDI)3) Mesoscopic system (finite size)4) In-medium effects5) Mixture of two liquids (proton and neutron subsystems)

„Isospin dependence of the EOS“

The study of the EOS at GSI (Germany)

ALADIN

                                             

KAOS

The kaon spectrometer is capable of determining the momentum and charge

of the particles, their emission angle, the centrality of the reaction including the total number of participating nucleons,

and the orientation of the reaction plane. The momentum is measured via the

deflection angle of the particle in the magnetic field and its recorded hit

position in the focal plane. The velocity is deduced by reconstructing the flight path and measuring the time of flight. With these quantities known, the rest

mass and thus the particle species can be unambiguously determined.

FOPI

The charged particles produced by a nickel-nickel collision at an energy of 1.93 GeV per nucleon leave tracks in the central drift chamber. The individual signals in the detector (squares) are automatically connected to form the track. Unambiguous identification of the particle is possible from the curvature of the track and additional information from other sections of the FOPI detector. In the example shown, two strange particles (K0 and ) arise simultaneously and decay after a short flight.

  

PART II: Our research with the FRS connected to EOS

4 detectors „landscape“FRS „a microscope“

... a different approach!

x2, x4 B

t2, t4 velocity

flight path

x2, t2

x4, t4

beam monitor

target

scintillator

scintillator

ionisationchamber

ionisationchamber

beam

1 AGeV 238U Ti

2ZΔE

A/Z from time and position:

Z from IC:

cβ

Bρ

m

e

Z

A

0

x2, x4 Bt2, t4 velocity

flight path

x2, t2

x4, t4

beam monitor

target

scintillator

scintillator

ionisationchamber

ionisationchamber

beam

1 AGeV 238U Ti

2ZΔE

A/Z from time and position:

Z from IC:

cβ

Bρ

m

e

Z

A

0

velocity is calculated from B:very precise evaluation

0mA

eZBρv

Z=26

longitudinal velocity

1 A GeV 238U on H2+Ti

Our observables: velocity spectra and cross sections

v long

B e a m f r a m e v t r a nsv

L a b or a t or y f r a m ev t r a nsv

f r a gm e nt a t ion

fi s s ion

v long

B e a m f r a m e v t r a nsv

L a b or a t or y f r a m ev t r a nsv

f r a gm e nt a t ion

fi s s ion

The integral of these spectra gives us the fission cross-section and the fragmentation cross-section

Fission

Our observables

Phases

1 - superfluid

2 - liquid

3 - coexistence

4 - gas

E/MeV

5

7010 300

A25

0.5

T/M

eV

Results from e.m.-induced fission of 70 different secondary projectiles (Steinhäuser et al., Nuc. Phys.A 634 (1998) 89 )

Structural properties survive at low energy

1 - Superfluid phase

Structural effects are restored in the end

products of hot decaying nuclei

Z

Fissioning nucleus: 226Th

2 - Liquid phase: an example: 1 GeV p on 238U

proton 1 GeV

fissionfragments

Intra-nuclear Cascade Sequential Evaporation / Fission

2 - Liquid phase: the cross sections of spallation and fragmentation residues

"evaporation corridor"

or"attractor line"

IDEA BEHIND LIMITING FRAGMENTATION

2 - Liquid phase: the velocity of spallation and fragmentation residues

Morryssey systematics is found to be valid:1) for small A in spallation / fragmentation reactions2) for compound nuclei which fission

3 - Liquid-gas coexistence: an example: 238U + Pb

fission

sequential evaporation

sequential evaporation

238U

Pb

238U

Pb

break-up pre-fragment

3 - Liquid-gas coexistence: an example: 238U + Pb

3 - Liquid-gas coexistence: indications in the cross sections of "light" residues

“attractor line”

average positionof the finalresidues of 238U

possible paths of theevaporation chain

pre- fragmentsafter abrasion

experimentaldata

For more violent collisions the evaporation starts at lower excitation energies !!!

238U 1.59 break-up abrasion

evaporation

3 - ISOSPIN THERMOMETER

3 - Liquid-gas coexistence : indications in the cross sections of "light" residues

abrasion

break- up

evaporation

“attractor line”

experimentaldata

E*=ATFO2

E*=27A MeV

abrasionevaporation

break- up

Liquid-gas coexistence:

indications in the velocity of "light" residues

Liquid-gas coexistence : indications in the velocity of "light" residues

break-up

this is due to a dynamical process!

ABRABLA

fission

sequential evaporation

sequential evaporation

238U

Pb

238U

Pb

break-up pre-fragment

break-up

event Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Zbound Zb3 1 41 20 6 4 3 2 2 0 0 0 0 0 78 74 2 33 21 7 3 2 2 2 0 0 0 0 0 70 64 3 8 4 3 3 3 3 2 2 2 2 0 0 32 24 4 32 13 12 4 2 2 2 2 2 0 0 0 71 61 5 64 4 2 2 0 0 0 0 0 0 0 0 72 68 6 17 12 7 4 3 2 2 2 2 0 0 0 51 43 7 26 10 6 6 3 2 2 2 2 0 0 0 59 51

ALADIN data

ALADIN data Au+Au at 1 A GeV

ABRABLA data Au+Au at 1 A GeV