On the Isospin Dependence of the EoS of Nuclear...
Transcript of On the Isospin Dependence of the EoS of Nuclear...
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