Probes of EOS in Heavy Ion Collisions :results from transport theories
Maria Colonna INFN - Laboratori Nazionali del Sud (Catania)
Nuclear Structure & Astrophysical Applications
July 8-12, 2013ECT*, TRENTO
Heavy Ion Collisions: a way to create Nuclear Matter in several conditions of ρ, T, N/Z… Nuclear effective interaction Equation of State (EOS)
Self-consistent Mean Field calculations (like HF) are a powerful
framework to understand the structure of medium-heavy (exotic)
nuclei.
Source: F.Gulminelli
Widely employed in the astrophysical context (modelization of neutron stars and supernova explosion)
2 21 1'2|(t)]ρ[H,|12t),(1,1'ρt
i
)(1'2'|(t)]ρ[H,|12t),(12,1'2'ρt
i 322 O
)(12,1'2')(2,2')ρ(1,1'ρ)(12,1'2'ρ 112 1,20 vHH
],δK[ρ]K[ρ1'|(t)]ρ,[H|1t),(1,1'ρt
i 11101
Dynamics of many-body systems
Mean-field Residual interaction
)||,( 21 vK F
),( vK F KK
Average effect of the residual interaction
one-body
Fluctuations
TDHF
0 K
Dynamics of many-body systems
-- If statistical fluctuations larger than quantum ones
Main ingredients:
Residual interaction (2-body correlations and fluctuations)In-medium nucleon cross sectionEffective interaction
(self consistent mean-field)
)'()','(),( ttCtpKtpK
ff 1
Transition rate WInterpreted in terms ofNN cross section
see BHF
K
1. Semi-classical approximation
Transport equation for the one-body distribution function f Chomaz,Colonna, Randrup Phys. Rep. 389 (2004) Baran,Colonna,Greco, Di Toro Phys. Rep. 410, 335 (2005)
collcoll IfIhft
tprfdt
tprdf
,,,,, Residual interaction:
Correlations, Fluctuationsk δk
ff 1
(1,2) (3,4)
Two-body Collision Integral
Fluctuations in collision integral Stochastic Mean-Field (SMF) model Fluctuations are projected in
coordinate space(density profile)
(semi-classical analog of Wigner function)
Vlasov
vrel
3
1
2
4
Boltzmann-Langevin (BL) equation
stochastic NN collisions
Collision integral fully stochastic, but approx. description of mean-field effects…
Molecular Dynamics approaches (AMD, ImQMD, QMD,…)2.
Zhang and Li, PRC74,014602(2006)J.Aichelin, Phys.Rep.202,233(1991)
Colonna, Ono, Rizzo, Phys. Rev..C82,054613 (2010)
A.Ono, Phys.Rev.C59,853(1999)
The nuclear interaction, contained in the Hamiltonian H, is represented byeffective interactions (Skyrme, Gogny, …)
E/A (ρ) = Es(ρ) + Esym(ρ) β²
β=(ρn-ρp)/ρThe density dependence of Esym is rathercontroversial, since there exist effective interactionsleading to a variety of shapes for Esym:
g1 Asysoft, g1 Asystiff
g )/( 0symE
Investigate the sensitivity of the reaction dynamics to this ingredient Put some constraints on the effective interactions EOS
Symmetry energy
Asysoft
Asystiff
Neutron skinIsovector modesPigmy resonances
around ρ0
Effective interactions and symmetry energy
γ = 1 γ = 0.5
γ = L/(3S0) ...3
)(0
00
LSEsym
Low-energy reaction mechanisms: Fusion, Deep-Inelastic, Charge equilibration
Fragmentation mechanisms at Fermi energies
Study of the PDR with transport theories
Low energy reaction mechanismsFusion vs Deep Inelastic
Competition between reaction mechanisms: fusion vs deep-inelastic
Fusion probabilities may depend on the N/Z of the reaction partners: - A mechanism to test the isovector part of the nuclear interaction Important role of fluctuations
36Ar + 96Zr , E/A = 9 MeV
Fusion or break-up ?
C.Rizzo et al., PRC83, 014604 (2011)
36Ar + 96Zr , E/A = 9 MeV,b = 6 fm
Starting from t = 200-300 fm/c, solve the Langevin Equation (LE) for selected degrees of freedom: Q (quadrupole), β3 (octupole), θ, and related velocities
Competition between reaction mechanisms
θ
Examples of trajectories
t = 0 t = 40 t = 100 t = 140 t = 200 fm/c
z (fm)
Break-up configurations
β2 , β3, E*~250 MeV, J ~70ћ
l (ћ)
softstiff
Shvedov, Colonna, Di Toro, PRC 81, 054605 (2010)
Break-up times of the order of 500-1000 fm/c !
Extract the fusion cross section
Charge equilibration in fusion and D.I. collisions
D(t) : bremss. dipole radiation CN: stat. GDRInitial Dipole
A1 A2
iDKD
pNZ
PPPtDK
xNZ
XtXtXA
NZtD
npinpnp
npinpnp
,,1)(
,1)()()(
,,
,,
Relative motion of neutron and proton centers of massIf N1/Z1 ≠ N2/Z2
132Sn + 58Ni , D0 = 45 fmE/A = 10 MeV
TDHF calculations
C.Rizzo et al., PRC 83, 014604 (2011)
40Ca + 100Mo
Simenel et al, PRC 76, 024609 (2007)
SMF simulations
2''2
3
2
)(3
2 gg
DA
NZEc
edEdP
Bremsstrahlung: Quantitative estimation
V.Baran, D.M.Brink, M.Colonna, M.Di Toro, PRL.87 (2001)
C.Rizzo et al., PRC83, 014604 (2011)
B.Martin et al., PLB 664 (2008) 47
36Ar + 96Zr
40Ar + 92Zr
Dynamical dipole emission
D.Pierroutsakou et al., PRC80, 024612 (2009)
Experimental evidence of the extra-yield (LNS data)
soft
stiff
stiff soft
Asysoftlarger symmetry energy, larger effect (30 %)
see also A.Corsi et al., PLB 679, 197 (2009), LNL experiments
-- Charge equilibration: from dipole oscillations to diffusion
Overdamped dipole oscillationd-t/e D(0) D(t) τd Esym
t
D(t)
Charge equilibration in D.I. collisions at ~ 30 MeV/Alow energy
Full N/Z equilibration not reached:Degree of eq. ruled by contact time and symmetry energy
Ediss / Ecm
Observable:(N/Z)CP = N/Z of light charge particlesemitted by PLF
as a function of
Dissipated energy: (impact parameter, contact time)
INDRA data, Galichet et al., PRC (2010)
TLF) (PLFE - E E kincmdiss
1(PLF)
2(TLF)
B. Tsang et al., PRL 102 (2009)
Mass(A) ~ Mass(B) ; N/Z(A) = N/Z(B)
Isospin transport ratio R :
BBAA
BBAAAB
XXXXXR
2
A dominance
mixing
B dominance
+10
-1-- AA and BB refer to two symmetric reactions between n-rich and n-poor nucleiAB to the mixed reaction -- X is an observable related to the N/Z of PLF (or TLF)
A
B
“Isospin diffusion” in D.I. collisions
stiffsoft
More central collisions: larger contact time more dissipation, smaller R Good sensitivity to Asy-EoS
X = N/ZPLF
SMF calculations, 124Sn +112Sn, 50 AMeV
J. Rizzo et al., NPA(2008)
(PLF)
(TLF)
X = Y(7Li)/ Y(7Be)
Comparison betweenImQMD calculations
and MSU data
B. Tsang et al., PRL 102 (2009)
“Exotic” fragments in reactionsat Fermi energies
TLF
PLF
)3,1(/ VIOLAREL VV
Y.Zhang et al., PRC(2011)
Fragmentation mechanisms at Fermi energies
IMF124Sn + 64Ni 35 AMeV:4π CHIMERA detector
Fragment emission at mid-rapidity (neck emission) neutron-enrichment of the neck region
dynamical emission
E. De Filippo et al., PRC(2012)
Charge Z
Isospin migration in neck fragmentation
Transfer of asymmetry from PLF and TLF to the low density neck region: neutron enrichment
of the neck region Effect related to the derivative of the symmetry
energy with respect to density
PLF, TLFneck
emittednucleons
ρIMF < ρPLF(TLF)
Asymmetry flux
Sn112 + Sn112
Sn124 + Sn124
b = 6 fm, 50 AMeV
Density gradients derivative of Esym
The asymmetry of the neck is larger than the asymmetry of PLF (TLF) in the stiff case
J.Rizzo et al. NPA806 (2008) 79
asy-soft
asy-stiff
Larger derivative with asy-stiff larger isospin migration effects
stiff - full
Comparison with Chimera data
124Sn + 64Ni 35 AMeVDESE
Properties of ‘dynamically emitted’ (DE) fragments
Charge distributionParallel velocity distribution
E. De Filippo et al., PRC(2012)
Good reproduction of overall dynamics The N/Z content of IMF’s in better reproducedby asystiff (L =75 MeV)
“Exotic” collective excitations in Nuclei
X neutrons – protons
Xc core neutrons-protons
Y excess neutrons - core
Neutron center of mass
Proton center of mass
Isovector dipole response
PDR
GDR
The Isovector Dipole Response (DR) in neutron-rich nuclei
X.Roca-Maza et al., PRC 85(2012)
Pygmy dipole strength
Giant DR
Pygmy DR
Klimkiewicz et al.
• Pygmy-like initial conditions (Y)
X
Y
X c
Neutron skin and core are coupled
The low- energy response is not sensitive tothe asy-stiffness
-- soft-- stiff-- superstiff
Baran et al.
25.0
see M.Urban, PRC85, 034322 (2012)
132Sn
• GDR-like initial conditions (X)
Fourier transform of D Strength of the Dipole Response
PDR is isoscalar-like frequency not dependent on Esym
GDR is isovector-like dependent on Esym
The strength in the PDR region depends on the asy-stiffness(increases with L)Larger L larger strength , but also
Larger L larger neutron skin
2.7 % soft4.4 % stiff4.5 % superstiff
see A.Carbone et al., PRC 81 (2010)
E (MeV)
The energy centroid of the PDR as a function of the mass for Sn isotopes, 48Ca, 68Ni, 86Kr, 208Pb ~ 41 A-1/3
PDR energy and strength (a systematic study)
Neutron skin extension with soft, stiff and superstiff
Correlation between the EWSR exhausted by the PDR
and the neutron skin extension
V.Baran et al., arXiv:1306.4969
soft
superstiffstiff
soft
stiffsuperstiff
A.Carbone et al., PRC 81 (2010)
“Summary” of Esym constraints
V.Baran (NIPNE HH,Bucharest) M.Di Toro, C.Rizzo, J.Rizzo, (LNS, Catania)
M.Zielinska-Pfabe (Smith College) H.H.Wolter (Munich)E.Galichet, P.Napolitani (IPN, Orsay)
Galichet 2010De Filippo 2012
Reduce experimental error bars
Perform more complete analyses
Improve theoretical description (overcome problems related to
model dependence)
Asysoft
Asystiff
Isospin diffusion
Heavy Ion Collisions (HIC) allow one to explore the behavior of nuclear matter under several conditions of density, temperature, spin, isospin, …
HIC from low to Fermi energies (~10-60 MeV/A) are a way to probe the density domain just around and below normal density.The reaction dynamics is largely affected by surface effects, at theborderline with nuclear structure. Increasing the beam energy, high density regions become accessible.
Varying the N/Z of the colliding nuclei (up to exotic systems) , it becomes possible to test the isovector part of the nuclear interaction(symmetry energy) around and below normal density.
A.Carbone et al., PRC 81 (2010)
M.B.Tsang et al., PRC (2012)
“Summary” of Esym constraints
V.Baran (NIPNE HH,Bucharest) M.Di Toro, C.Rizzo, J.Rizzo, (LNS, Catania)
M.Zielinska-Pfabe (Smith College) H.H.Wolter (Munich)E.Galichet, P.Napolitani (IPN, Orsay)
Calculations:- N/Z increases with the centrality of
collision for the two systems and energies(For Ni + Ni pre-equilibrium effects)- In Ni + Au systems more isospin
diffusion for asy-soft (as expected)- (N/Z)CP linearly correlated to (N/Z)QP
PLF
PLF
CP
CP
PLF
PLF
CP
CP
N/ZCP
-- stiff (γ=1) + SIMON-- soft(γ=0.6) + SIMON
SMF transport calculations:N/Z of the PLF (Quasi-Projectile)
Squares: soft Stars: stiff
After statistical decay :
N = Σi Ni , Z = Σi Zi
Charged particles: Z=1-4
forward n-n c.m.
forward PLF
Comparison with data
Data: open points higher than full points
(n-rich mid-rapidity particles) Isospin equilibration reached for
Ediss/Ecm = 0.7-0.8 ? (open and full dots converge)
Data fall between the two calculations
Interpretation in terms of isovector-like and isoscalar-like modes in asymmetric systems
22 2 nnnpnnpppp
)3
(8)2tan(
0
LEI
cba
c
sym
Isoscalar-likeIsovector-like
softstiff
Energy-density variation
Curvature matrix Normal modes
n
p
)tan(
α increases with L strength in the PDR region (isoscalar-like mode) increases with L
α =450 for symmetric matter
Zero temperature
stiff
soft
Fragment isotopic distribution and symmetry energy
Fragmentation can be associated with mean-field instabilities(growth of unstable collective modes)
Oscillations of the total (isoscalar-like density) fragment formation and average Z/A (see δρn /δρp )
Oscillations of the isovector density (ρn - ρp ) isotopic variance and distributions (Y2 / Y1 , isoscaling)
Study of isovector fluctuations, link with symmetry energy in fragmentation
Isovector density ρv = ρn –ρp
λ = 2π/k
nuclear matter in a box
At equilibrium, according to the fluctuation-dissipation theorem
What does really happen in fragmentation ?
Fv coincides with the free symmetry energy at the considered density
Full SMF simulationsT = 3 MeV, Density: ρ1 = 0.025 fm-3 , 2ρ1 , 3ρ1
I = 0.14
T
ρ
F’ follows the local equilibrium value !
The isospin distillation effect goes together with the isovector variance
Ihigh density low density
ρ
Nuclear matter in a box
freeze-out
F’ ~ T / σ
stiffsoft
stiff
soft
M.Colonna, PRL,110(2013) Average Z/A and isovector fluctuations as a function of local density ρ
Dissipation and fragmentation in “MD” models
ImQMD calculations, 112Sn +112Sn, 50 AMeV
More ‘explosive’ dynamics:more fragments and light clusters emittedmore ‘transparency’
What happens to charge equilibration ?
Rather flat behavior with impact parameter b:- Weak dependence on b of reaction dynamics ?- Other dissipation sources (not nucleon exchange) ? fluctuations, cluster emission weak nucleon exchange
z vy
Isospin transport ratio R
Y.Zhang et al., PRC(2011)
124Sn +112Sn, 50 AMeV
Comparison SMF-ImQMD
6 fm8 fm
γ = 0.5 SMF = dashed linesImQMD = full lines
For semi-central impact parameters:
Larger transparency in ImQMD (but not so a drastic effect)
Other sources of dissipation (in addition to nucleon exchange)More cluster emission
SMF
ImQMD
γ = 0.5
Different trends in ImQMD and SMF!
What about fragment N/Z ?
γ = 2 Isospin transport R around PLF rapidity:
Good agreement in peripheral reactions
Elsewhere the different dynamics(nucleon exchange less important in ImQMD) leads to less iso-equilibration
Liquid phase: ρ > 1/5 ρ0 Neighbouring cells are connected (coalescence procedure)
Extract random A nucleons among test particle distribution Coalescence procedureCheck energy and momentum conservationA.Bonasera et al, PLB244, 169 (1990)
Fragment excitation energy evaluated by subtracting Fermi motion (local density approx) from Kinetic energy
• Correlations are introduced in the time evolution of the one-body density: ρ ρ +δρ as corrections of the mean-field trajectory• Correlated density domains appear due to the occurrence of mean-field (spinodal) instabilities at low density
Fragmentation Mechanism: spinodal decomposition
Is it possible to reconstruct fragments and calculate their properties only from f ?
Several aspects of multifragmentation in central and semi-peripheral collisions well reproduced by the model
Statistical analysis of the fragmentation path
Comparison with AMD results
Chomaz,Colonna, Randrup Phys. Rep. 389 (2004)Baran,Colonna,Greco, Di Toro Phys. Rep. 410, 335 (2005)Tabacaru et al., NPA764, 371 (2006)
A.H. Raduta, Colonna, Baran, Di Toro, ., PRC 74,034604(2006) iPRC76, 024602 (2007) Rizzo, Colonna, Ono, PRC 76, 024611 (2007)
Details of SMF model
T
ρ
liquid gas
Fragment Recognition