Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Nuclear Equation of State
from Reactions and Structure
P. Danielewicz
Natl Superconducting Cyclotron Lab, Michigan State U
Hirschegg 2009: Nuclear Matter at High Density
January 18 - 24, 2009
Session in honor of Jörn Knoll
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Outline
1
Jörn Knoll
2
EOS Intro
3
High-ρ EOS from Reactions
Basics of Central Reactions
Flow Anisotropies
4
S(ρ) from Structure
Binding Formula
Half-∞ Nuclear-Matter Calculations
5
aa (A) from IAS Data
Use of Charge Invariance
Data Analysis & Interpretation
New Developments
6
Conclusions
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
J. Knoll: Looking for Deep Understanding
Freeze-out with nonequilibrium
Green’s functions, ’08-’09
Extreme processes in nuclear
collisions, ’84-’88
Vector mesons in matter, ’01-’07
Nuclear sideward flow, ’82
Beyond Markov approximation
in transport, with emphasis on
broad resonances, ’99-’03
Deuteron yields and entropy
generated in central collisions,
’82
Bremsstrahlung in matter,
Landau-Pomeranchuk-Migdal
effect, ’89-’96
Event-by-event analysis of
collision events, ’82
Aftermath of quark-gluon
plasma: strangeness and vector
mesons, ’88-’93
Finite particle-number effects on
emission from central collisions,
’79-’81
Rows on rows model, ’77
Instabilities in expanding nuclear
systems, ’83-’90
Semiclassical scattering with
complex trajectories, ’74-’77
Merger and decay of strings in
ultrarelativistic collisions, ’84-’88
DWBA and eikonal approaches
to electron scattering, ’73-’74
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
My Personal Overlap
Freeze-out with nonequilibrium
Green’s functions, ’08-’09
Extreme processes in nuclear
collisions, ’84-’88
Vector mesons in matter, ’01-’07
Nuclear sideward flow, ’82
Beyond Markov approximation
in transport, with emphasis on
broad resonances, ’99-’03
Deuteron yields and entropy
generated in central collisions,
’82
Bremsstrahlung in matter,
Landau-Pomeranchuk-Migdal
effect, ’89-’96
Event-by-event analysis of
collision events, ’82
Aftermath of quark-gluon
plasma: strangeness and vector
mesons, ’88-’93
Finite particle-number effects on
emission from central collisions,
’79-’81
Rows on rows model, ’77
Instabilities in expanding nuclear
systems, ’83-’90
Semiclassical scattering with
complex trajectories, ’74-’77
Merger and decay of strings in
ultrarelativistic collisions, ’84-’88
DWBA and eikonal approaches
to electron scattering, ’73-’74
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
This Talk
Freeze-out with nonequilibrium
Green’s functions, ’08-’09
Extreme processes in nuclear
collisions, ’84-’88
Vector mesons in matter, ’01-’07
Nuclear sideward flow, ’82
Beyond Markov approximation
in transport, with emphasis on
broad resonances, ’99-’03
Deuteron yields and entropy
generated in central collisions,
’82
Bremsstrahlung in matter,
Landau-Pomeranchuk-Migdal
effect, ’89-’96
Event-by-event analysis of
collision events, ’82
Aftermath of quark-gluon
plasma: strangeness and vector
mesons, ’88-’93
Finite particle-number effects on
emission from central collisions,
’79-’81
Rows on rows model, ’77
Instabilities in expanding nuclear
systems, ’83-’90
Semiclassical scattering with
complex trajectories, ’74-’77
Merger and decay of strings in
ultrarelativistic collisions, ’84-’88
DWBA and eikonal approaches
to electron scattering, ’73-’74
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Equation of State from Reactions & Structure
EOS: any nontrivial relation between thermodynamic vbles
characterizing the matter, e.g. p(ρ, T ) or EA (ρp , ρn , T ).
Central Reactions: ρ changed due to compression.
Structure: ρ changes in the surface.
Energy breakdown in uniform matter, due to charge symmetry:
ρn −ρp 2
E0
E
(ρ
,
ρ
)
=
(ρ)
+
S(ρ)
A n p
A
ρ
0
S(ρ) ' aaV + L3 ρ−ρ
ρ0
aaV =?
L =?
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Central Reactions in Terms of Boltzmann Eq
Central reactions of heavy nuclei described statistically in terms
of Boltzmann equation for the Wigner function f - density of
particles in space and momentum:
velocity
force
∂p ∂f
∂p ∂f
∂f
+
−
=
∂t
∂p ∂r
∂r ∂p
Z
Z
dp2
dΩ0 v12
dσ
dΩ0
× (1 − f1 )(1 − f2 )f10 f20 − (1 − f10 )(1 − f20 )f1 f2
gain
Here, p - single-particle energy and
scattering cross-section.
loss
dσ
dΩ
- nucleon-nucleon
System energy specified in terms of the Wigner functions,
allowing to consider nonequilibrium situations, while
constraining the equilibrium, E = E{f }. Single-ptcle energy:
δE
(p) =
δf (p)
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Central Reactions
Reaction plane: plane in which the centers of initial nuclei lie.
Spectators: nucleons in the reaction periphery, little disturbed
by the reaction.
Participants: nucleons that dive into compressed excited
matter.
Nuclear EOS deduced from the features of collective flow in
reactions of heavy nuclei.
Collective flow: motion characterized by significant
space-momentum correlations, deduced from momentum
distributions of particles emitted in the reactions.
Euler eq. in ~v = 0 frame:
mN ρ
Nuclear Equation of State
∂
~
~v = −∇p
∂t
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
EOS and Flow Anisotropies
EOS assessed through reaction plane anisotropies
characterizing particle collective motion.
∂ ~
~
v = −∇p
Hydro? Euler eq. in ~v = 0 frame: mN ρ ∂t
where p - pressure. From features of v , knowing ∆t, we may
learn about p in relation to ρ. ∆t fixed by spectator motion.
For high p, expansion
rapid and much
affected by spectators.
net baryon
density
excitation
energy
For low p, expansion
sluggish and
completes after
spectators gone.
Simulation by L. Shi
Nuclear Equation of State
spectator
density
t=0
5
10
15
20 fm/c
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Sideward Flow Systematics
Deflection of forwards and backwards moving particles away
from the beam axis, within the reaction plane.
Au + Au Flow
Excitation Function
Note: K used as a label
PD, Lacey & Lynch
Science298(02)1592
The sideward-flow
observable results from
dynamics that spans
a ρ-range varying with
the incident energy.
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
2nd -Order or Elliptic Flow
Another anisotropy, studied at midrapidity:
v2 = hcos 2φi, where φ is azimuthal angle
relative to reaction plane.
Au+Au v2
Excitation Function
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Constraints on EOS
Au+Au flow anisotropies:
ρ ' (2 − 4.6)ρ0 .
No one EOS yields both
flows right. Discrepancies:
inaccuracy of theory.
Most extreme models for
EOS can be eliminated.
PD, Lacey & Lynch
Science298(02)1592
Neutron Matter:
Uncertainty in
symmetry energy
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Symmetry Energy in the Binding Formula
Bethe-Weizsäcker formula:
Z2
(N − Z )2
+a
(A)
+Emic
a
A
A1/3
In the standard formula aa (A) ≡ aaV ' 21 MeV, the symmetry
term has purely volume character.
E = −aV A + aS A2/3 + aC
A-dependent symmetry coefficient?? Capacitor analogy:
aa
Nuclear:
E = −av A + as A2/3 +
(N − Z )2
A
aa
(N − Z )2
= E0 (A) +
A

Q ≡ N − Z
2
Q
Electrostatic:
E = E0 +
⇒
A
C ≡
2C
2aa
Note: For coupled capacitors, capacitances add up.
Contributions to C with different A-dependence??
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Symmetry Energy in the Binding Formula
Bethe-Weizsäcker formula:
Z2
(N − Z )2
+a
(A)
+Emic
a
A
A1/3
In the standard formula aa (A) ≡ aaV ' 21 MeV, the symmetry
term has purely volume character.
E = −aV A + aS A2/3 + aC
A-dependent symmetry coefficient?? Capacitor analogy:
aa
Nuclear:
E = −av A + as A2/3 +
(N − Z )2
A
aa
(N − Z )2
= E0 (A) +
A

Q ≡ N − Z
2
Q
Electrostatic:
E = E0 +
⇒
A
C ≡
2C
2aa
Note: For coupled capacitors, capacitances add up.
Contributions to C with different A-dependence??
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Symmetry Energy in the Binding Formula
Bethe-Weizsäcker formula:
Z2
(N − Z )2
+a
(A)
+Emic
a
A
A1/3
In the standard formula aa (A) ≡ aaV ' 21 MeV, the symmetry
term has purely volume character.
E = −aV A + aS A2/3 + aC
A-dependent symmetry coefficient?? Capacitor analogy:
aa
Nuclear:
E = −av A + as A2/3 +
(N − Z )2
A
aa
(N − Z )2
= E0 (A) +
A

Q ≡ N − Z
2
Q
Electrostatic:
E = E0 +
⇒
A
C ≡
2C
2aa
Note: For coupled capacitors, capacitances add up.
Contributions to C with different A-dependence??
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Symmetry Energy in the Binding Formula
Bethe-Weizsäcker formula:
Z2
(N − Z )2
+a
(A)
+Emic
a
A
A1/3
In the standard formula aa (A) ≡ aaV ' 21 MeV, the symmetry
term has purely volume character.
E = −aV A + aS A2/3 + aC
A-dependent symmetry coefficient?? Capacitor analogy:
aa
Nuclear:
E = −av A + as A2/3 +
(N − Z )2
A
aa
(N − Z )2
= E0 (A) +
A

Q ≡ N − Z
2
Q
Electrostatic:
E = E0 +
⇒
A
C ≡
2C
2aa
Note: For coupled capacitors, capacitances add up.
Contributions to C with different A-dependence??
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
A-dependence of Symmetry Coefficient
E.g. volume-surface breakdown of energy & asymmetry:
E = ES + EV
N − Z = (NS − ZS ) + (NV − ZV )
(NV − ZV )2
(N − Z )2
ES = aS A2/3 + aaS S 2/3 S
A
A
under charge symmetry, i.e. N ↔ Z invariance.
EV = aV A + aaV
Minimization of the energy E with respect to (N − Z ) partition
between volume and surface yields:
(N − Z )2
E = E0 + EA = E0 +
2/3
A
+ AaS
aV
a
a
Capacitance for asymmetry:
2C ≡
A
A
= V + S
aa (A)
aa
aa
volume capacitance
Nuclear Equation of State
A2/3
E.g. droplet model
different radii for n&p
Myers&Swiatecki
AnnPhys55(69)395
surface
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
A-dependence of Symmetry Coefficient
E.g. volume-surface breakdown of energy & asymmetry:
E = ES + EV
N − Z = (NS − ZS ) + (NV − ZV )
(NV − ZV )2
(N − Z )2
ES = aS A2/3 + aaS S 2/3 S
A
A
under charge symmetry, i.e. N ↔ Z invariance.
EV = aV A + aaV
Minimization of the energy E with respect to (N − Z ) partition
between volume and surface yields:
(N − Z )2
E = E0 + EA = E0 +
2/3
A
+ AaS
aV
a
a
Capacitance for asymmetry:
2C ≡
A
A
= V + S
aa (A)
aa
aa
volume capacitance
Nuclear Equation of State
A2/3
E.g. droplet model
different radii for n&p
Myers&Swiatecki
AnnPhys55(69)395
surface
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
A-dependence of Symmetry Coefficient
E.g. volume-surface breakdown of energy & asymmetry:
E = ES + EV
N − Z = (NS − ZS ) + (NV − ZV )
(NV − ZV )2
(N − Z )2
ES = aS A2/3 + aaS S 2/3 S
A
A
under charge symmetry, i.e. N ↔ Z invariance.
EV = aV A + aaV
Minimization of the energy E with respect to (N − Z ) partition
between volume and surface yields:
(N − Z )2
E = E0 + EA = E0 +
2/3
A
+ AaS
aV
a
a
Capacitance for asymmetry:
2C ≡
A
A
= V + S
aa (A)
aa
aa
volume capacitance
Nuclear Equation of State
A2/3
E.g. droplet model
different radii for n&p
Myers&Swiatecki
AnnPhys55(69)395
surface
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
A-dependence of Symmetry Coefficient
E.g. volume-surface breakdown of energy & asymmetry:
E = ES + EV
N − Z = (NS − ZS ) + (NV − ZV )
(NV − ZV )2
(N − Z )2
ES = aS A2/3 + aaS S 2/3 S
A
A
under charge symmetry, i.e. N ↔ Z invariance.
EV = aV A + aaV
Minimization of the energy E with respect to (N − Z ) partition
between volume and surface yields:
(N − Z )2
E = E0 + EA = E0 +
2/3
A
+ AaS
aV
a
a
Capacitance for asymmetry:
2C ≡
A
A
= V + S
aa (A)
aa
aa
volume capacitance
Nuclear Equation of State
A2/3
E.g. droplet model
different radii for n&p
Myers&Swiatecki
AnnPhys55(69)395
surface
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Half-Infinite Matter in Skyrme-Hartree-Fock
To one side infinite uniform matter & vacuum to the other
Wavefunctions: Φ(rr ) = φ(z) eikk ⊥ ·rr ⊥
matter interior/exterior:
φ(z) ∝ sin (kz z + δ(kk ))
φ(z) ∝ e−κ(kk )z
Discretization in k -space. Set of 1D HF eqs solved using
Numerov’s method until self-consistency:
2 2
~ k⊥
d
~2
d
−
φ(z) +
+ U(z) φ(z) = (kk ) φ(z)
dz 2m∗ (z) dz
2m∗ (z)
PD&Lee, NP(in press). Before: Farine et al, NPA338(80)86
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Isoscalar (Net) & Isovector Densities from SHF
Results for different Skyrme
interactions in half-∞ matter.
If net & difference densities have the
same shape, the surface does not
contribute to capacitance for asymmetry.
Greater difference in shapes, and
greater surface contribution to capacitance, for faster drop of S(ρ)
with decease in ρ in uniform matter
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Symmetry Coefficient
With iso-density ρa = (ρn − ρp ) 2aaV /µa and µa = ∂E/∂(N − Z ),
the symmetry coefficient is rigorously given by:
Z
Z
Z surface region
A
1
1
1
3
3
= V
d r ρa (r ) = V
d r ρ(r )+ V
d3 r (ρa − ρ)(r )
aa (A)
aa
aa
aa
A
A2/3
' V + S
aaS − L Correlation f/SHF
aa
aa
A−1/3
1
1
' V +
aa (A)
aa
aaS
for large-A nuclei
⇒
L - slope parameter of
symmetry energy S
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Charge Invariance
?aa (A)?
Conclusions on sym-energy details, following
E-formula fits, interrelated with conclusions on other terms in
the formula: asymmetry-dependent Coulomb, Wigner & pairing
+ asymmetry-independent, due to (N − Z )/A - A correlations
along stability line (PD NPA727(03)233)!
Best would be to study the symmetry energy in isolation from
the rest of E-formula! Absurd?!
Charge invariance to rescue: lowest nuclear states
characterized by different isospin values (T ,Tz ),
Tz = (Z − N)/2. Nuclear energy scalar in isospin space:
sym energy
Ea = aa (A)
T2
(N − Z )2
= 4 aa (A) z
A
A
→ Ea = 4 aa (A)
Nuclear Equation of State
T (T + 1)
T2
= 4 aa (A)
A
A
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Charge Invariance
?aa (A)?
Conclusions on sym-energy details, following
E-formula fits, interrelated with conclusions on other terms in
the formula: asymmetry-dependent Coulomb, Wigner & pairing
+ asymmetry-independent, due to (N − Z )/A - A correlations
along stability line (PD NPA727(03)233)!
Best would be to study the symmetry energy in isolation from
the rest of E-formula! Absurd?!
Charge invariance to rescue: lowest nuclear states
characterized by different isospin values (T ,Tz ),
Tz = (Z − N)/2. Nuclear energy scalar in isospin space:
sym energy
Ea = aa (A)
(N − Z )2
T2
= 4 aa (A) z
A
A
→ Ea = 4 aa (A)
Nuclear Equation of State
T (T + 1)
T2
= 4 aa (A)
A
A
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Charge Invariance
?aa (A)?
Conclusions on sym-energy details, following
E-formula fits, interrelated with conclusions on other terms in
the formula: asymmetry-dependent Coulomb, Wigner & pairing
+ asymmetry-independent, due to (N − Z )/A - A correlations
along stability line (PD NPA727(03)233)!
Best would be to study the symmetry energy in isolation from
the rest of E-formula! Absurd?!
Charge invariance to rescue: lowest nuclear states
characterized by different isospin values (T ,Tz ),
Tz = (Z − N)/2. Nuclear energy scalar in isospin space:
sym energy
Ea = aa (A)
(N − Z )2
T2
= 4 aa (A) z
A
A
→ Ea = 4 aa (A)
Nuclear Equation of State
T (T + 1)
T2
= 4 aa (A)
A
A
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
aa (A) Nucleus-by-Nucleus
T (T + 1)
A
In the ground state T takes on the lowest possible value
T = |Tz | = |N − Z |/2. Through ’+1’ most of the Wigner term absorbed.
→ Ea = 4 aa (A)
Formula generalized to the lowest state of a given T (e.g.
Jänecke et al., NPA728(03)23). Pairing depends on evenness of T .
?Lowest state of a given T : isobaric analogue state (IAS) of
some neighboring nucleus ground-state.
Tz=-1
Tz=0
Tz=1
Study of changes in the
symmetry term possible
nucleus by nucleus
T=1
T=0
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
IAS Data Analysis
In the same nucleus:
E2 (T2 ) − E1 (T1 ) =
aa−1 (A)
4 ∆T 2
=
A ∆E
4 aa T2 (T2 + 1) − T1 (T1 + 1)
A
+Emic (T2 , Tz ) − Emic (T2 , Tz )
?
= (aaV )−1 + (aaS )−1 A−1/3
Data: Antony et al. ADNDT66(97)1
Emic :
Koura et al., ProTheoPhys113(05)305
v Groote et al., AtDatNucDatTab17(76)418
Moller et al., AtDatNucDatTab59(95)185
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
aa (A) from IAS
Symbol size proportional to relative significance.
∼Linear dependance from A & 20 on.
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Different Shell Corrections
Line: best fit to Koura at A > 20.
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
aa vs A
Line: best fit at A > 20.
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Best a(A)-Descriptions
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Symmetry-Energy Parameters
aaV = (31.5 − 33.5) MeV, aaS = (9.5 − 12) MeV, L ∼ 95 MeV
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Recent Developments: More SHF Calculations
Coefficient-extraction from spherical SHF results for
realistic (blue) and humongous (green) (up to A ∼ 106 ) nuclei,
using codes of P.-G. Reinhard
Nuclei in Nature too small for a clean surface/volume
separation! Consequence: aaV bit smaller, aaS bit larger
and L bit lower than first deduced.
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Conclusions
The nuclear EOS is pursued from different sides: structure,
reactions, astrophysics, lattice QCD, effective theory
Collective flow in central reactions constrains nuclear
pressure at densities ρ = (2 − 4.5)ρ0 . Most extreme model
EOS eliminated.
Systematic of isobaric analogue states constrains the
symmetry energy at ρ . ρ0 .
Outlook:
– Analysis of rapidity distributions from 1-10 GeV/nucleon
regime, to improve transport-model assumptions.
– Use of asymmetry skins & charge radii.
Nuclear Equation of State
P. Danielewicz
Jörn Knoll
EOS Intro
High-ρ EOS from Reactions
S(ρ) from Structure
aa (A) from IAS Data
Conclusions
Conclusions
The nuclear EOS is pursued from different sides: structure,
reactions, astrophysics, lattice QCD, effective theory
Collective flow in central reactions constrains nuclear
pressure at densities ρ = (2 − 4.5)ρ0 . Most extreme model
EOS eliminated.
Systematic of isobaric analogue states constrains the
symmetry energy at ρ . ρ0 .
Outlook:
– Analysis of rapidity distributions from 1-10 GeV/nucleon
regime, to improve transport-model assumptions.
– Use of asymmetry skins & charge radii.
Nuclear Equation of State
P. Danielewicz