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On the Isospin Dependenceof the EoS of Nuclear Matter
D. Cozma, M. Petrovici (NIPNE, Bucharest, Romania)
Heavy-Ion Collisions from the Coulomb Barrier to the Quark-Gluon Plasma
Erice
September 22nd, 2008
– p.1
OVERVIEW
• Introduction & Motivation
• HIC ModelQMD transport model
Essential ingredients
• In-medium EffectsIn-medium NN scattering
Isospin dependence of EoS
• HIC ObservablesObservables
Case Study: Zr+Ru
• Summary and Outlook
– p.2
Introduction
Nuclear Equation of State: E/A=E(ρ)
sources: finite nuclei ρ/ρ0 ≤ 1
heavy-ions ρ/ρ0 ≤ 3
neutron stars ρ/ρ0 ≤ 10
Fuchs PRL86, 1974
– p.3
Motivation
Equation of State of asymmetric nuclear matter:Symmetry energy:
E(ρ, β) = E(ρ) + Esym(ρ) β2 + · · · β =ρn − ρp
ρn + ρp
Esym(ρ) =1
2
∂E(ρ, β)
∂β2|β=0 = a4 +
p0
ρ20
(ρ − ρ0)
- phenomenological models constrained in the low ρ region diverge at high density
– p.4
Motivation
FOPI Collaboration: RuZr and AuAu @ 400 AMeVRami et al., PRL 84,1120; Hong et al., PRC 66, 034901
– p.5
Transport Model
Transport model: Quantum Molecular DynamicsMonte Carlo cascade + Mean field + Pauli-blocking+ in medium cross sectionall 4∗ resonances below 2 GeV - 10 ∆∗ and 11 N∗
• included baryon-baryon collisions:
all elastic channels
inelastic channels NN → NN⋆, NN → N∆⋆,
NN → ∆N⋆, NN → ∆∆⋆, NR → NR′
• included pion-absorption resonance-decay channels:
∆ Nπ, ∆⋆ ∆π, ∆⋆
N1440π, N⋆ Nπ,
N⋆ Nππ, (N⋆
∆π, N⋆ N1440)
– p.6
QMD: Essential Ingredients
Inclusion of collisions:
- binary collisions: geometric criterion
- one needs consistent cross-section ↔potential parameters (mean field)
- use fit to available experimental data; if not available use detailed balance
and isospin symmetry
σ1,2→3,4(√
s) ∼ (2S3 + 1)(2S4 + 1)〈p3,4〉〈p1,2〉
1
s|M(m3,m4)|2
σf→i =p2
i
p2f
gi
gfσi→f
- Pauli blocking due to Fermi statistics: collision allowed with probability
(1 − f ′
1)(1 − f ′
2)
- angular distributions of two-body scattering: same as NN → NN
(determined from an effective model)
– p.7
Nucleon-Nucleon Interaction
Vacuum NN Interaction: - microscopical OBE model (Bonn)
T (~q′, ~q) = V (~q′, ~q) + P
Z
d3k
(2π)3V (~q′, ~k)
m2
E2k
1
2Eq − 2EkT (~k, ~q)
In-Medium NN interaction: - Dirac-Brueckner approach
G(~q′, ~q| ~P , z) = V ∗(~q′, ~q) + P
Z
d3k
(2π)3V ∗(~q′, ~k)
m2∗
E21/2~P+~k
Q(~k, ~P )
z − 2E1/2~P+~k
G(~k, ~q| ~P , z)
Li, Machleidt PRC 48, 1702 Li, Machleidt PRC 49, 566– p.8
Nucleon-Nucleon Interaction
Li, Machleidt PRC 48, 1702; C. Fuchs, PRC 64, 024003
0 30 60 90 120 150 180θ
cm(deg)
0
0.05
0.1
0.15
0.2
dσ/d
Ω (
rel)
0.050 GeV0.100 GeV0.200 GeV0.400 GeV0.600 GeV0.800 GeV1.000 GeV
NPdifferential cross-sections in vacuum
0 30 60 90 120 150 180θ
(cm) (deg)
0
0.05
0.1
0.15
0.2
dσ/d
Ω (
rel)
0.050 GeV0.100 GeV0.200 GeV0.400 GeV0.600 GeV0.800 GeV1.000 GeV
PP differential cross-section in vacuum
0 30 60 90 120 150 180θ
(cm)
0
0.05
0.1
0.15
0.2
dσ/d
Ω
vacuum0.5 ρ
0
1.0 ρ0
2.0 ρ0
3.0 ρ0
4.0 ρ0
PN differential cross-sections@ 400 MeV
0 30 60 90 120 150 180θ
(cm)
0.06
0.07
0.08
0.09
0.1
dσ/d
Ω
vacuum0.5 ρ
0
1.0 ρ0
2.0 ρ0
3.0 ρ0
4.0 ρ0
PP differential cross-sections@ 400 MeV
– p.9
Isospin dependence
EoS of isospin asymmetric nuclear mater:
V n(p)(ρ, β) = a u + b uγ + Vmdi + V pc + V
n(p)asym(ρ, β)
Ea(ρ, β) = ea ρ F (u) β2 Vn(p)asym = ∂Ea(ρ, β)/∂ ρn(p)
F1(u) =2u2
1 + uF2(u) = u F3(u) = u1/2
nucleons and resonances propagatein an isospin dependent mean field
Vasym(n∗) = Vasym(∆0) = V nasym
Vasym(p∗) = Vasym(∆+) = V pasym
Vasym(∆++) = 2V pasym − V n
asym
Vasym(∆−) = 2V nasym − V p
asym
Li, Ko, Ren PRL78, 1644 – p.10
Observables
double neutron to proton ratio (n/p)AB/(p/n)BA
Li,Li, Stocker PRC 73, 051601
neutron/proton ratio at midrapidity
Yong, Li, Chen PLB650, 344
– p.11
Observables
Elliptic flow and Differential Elliptic Flow
dN
dφ= a0 (1 + a1cos(φ) + a2cos(φ))
a2 =1
N
X
i
pi2x − pi2
y
pi2t
– p.12
Elliptic flow (EoS and in-medium NN dep)
dN
dφ= a0 (1 + a1cos(φ) + a2cos(φ))
constraints: |y|<0.50, b=5 fm, Ru+Zr @ 400 MeV
EOS + Cross-sections n+p n p
Isospin indep + Free c.s. -0.040 -0.042 -0.036
Isospin indep + Dens. Dep. c.s. -0.038 -0.038 -0.038
Isospin indep + Dens. Dep. diff. c.s. -0.038 -0.040 -0.036
Isospin dep soft + Free c.s. -0.018 -0.019 -0.017
Isospin dep soft + Dens. Dep. c.s. -0.014 -0.015 -0.013
Isospin dep soft + Dens. Dep. diff. c.s. -0.016 -0.018 -0.014
Isospin dep stiff + Free c.s. -0.014 -0.017 -0.014
Isospin dep stiff + Dens. Dep. c.s. -0.017 -0.020 -0.014
Isospin dep stiff + Dens. Dep. diff. c.s. -0.020 -0.022 -0.018
Isospin dep linear + Free c.s. -0.013 -0.017 -0.008
Isospin dep linear + Dens. Dep. c.s. -0.017 -0.022 -0.010
Isospin dep linear + Dens. Dep. diff. c.s. -0.016 -0.022 -0.009– p.13
Differential elliptic flowsensitivity to in-medium NN interaction
0 0.2 0.4 0.6 0.8 1p
t
-0.02
-0.015
-0.01
-0.005
0
v 2
free cross-sectionsin-medium cross-sectionsin-medium diff cross-sections
Isospin independent EOS
0 0.2 0.4 0.6 0.8p
t
-0.008
-0.006
-0.004
-0.002
0
v 2 (p
roto
ns)
free cross-sectionsin-medium cross-sectionsin-medium diff cross-sections
Isospin dependent (soft) EOSRuZr @ 400 MeV
sensitivity to EOS
0 0.2 0.4 0.6 0.8p
t
-0.02
-0.015
-0.01
-0.005
0
v 2 (p
roto
ns)
isospin indepisospin dep softisospin dep stiffisospin dep linear
Various EOS (free NN cs)RuZr @ 400 MeV
0 0.2 0.4 0.6 0.8p
t
-0.02
-0.015
-0.01
-0.005
0
v 2 (p
roto
ns)
isospin indepisospin dep softisospin dep stiffisospin dep linear
Various EOS (in-medium NN diff cs)RuZr @ 400 MeV
– p.14
Differential elliptic flow
splitting of the n vs. p values
0 0.2 0.4 0.6 0.8p
t
-0.02
-0.015
-0.01
-0.005
0
v 2 (p
roto
ns,n
eutr
ons)
isospin indep protonsisospin indep neutronssoft EOS protonssoft EOS neutronsstiff EOS protonsstiff EOS neutronslinear EOS protonslinear EOS neutrons
Various EOS (free NN cs)RuZr @ 400 MeV
0 0.2 0.4 0.6 0.8p
t
-0.02
-0.015
-0.01
-0.005
0
v 2 (p
roto
ns,n
eutr
ons)
isospin indep protonsisospin indep neutronssoft EOS protonssoft EOS neutronsstiff EOS protonsstiff EOS neutronslinear EOS protonslinear EOS neutrons
Various EOS (in-medium NN diff cs)RuZr @ 400 MeV
Problem ! differential elliptic flow at high pT
– p.15
Summary
• Message: - a2 sensitive to isospin dependent part of EoS; density
dependent NN cross-section of secondary importance
- no clear preference for the isospin dependent part of the equation of
state
• Consistency: - vacuum isospin dependent NN interaction → in-medium NN
cross-sections, equation of state
• Improvements: - determine the origin of the differential elliptic flow at
large pT ;
- introduce momentum dependence in the symmetry energy terms and
account for neutron-proton mass splitting
• To Be Done: - implement in transport code explicit production channels for
deuterium and compare with results from coalescence models;
- study the emission of 3H and 3He from a coalescence model
– p.16