equation of state for nuclear matter: research at charms part i: generalities about the equation of...
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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