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Reaction mechanisms in transport theories:a test of the nuclear effective interaction
Maria Colonna INFN - Laboratori Nazionali del Sud (Catania)
NN201211TH INTERNATIONAL CONFERENCE ON
NUCLEUS-NUCLEUS COLLISIONSMay 27-June 1, 2012San Antonio, Texas
Collective excitations in neutron-rich systems
Dissipation and fragmentation mechanisms at Fermi energies
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)
Mean-field effects well described, but fluctuations underestimated…
collcoll IfIhft
tprf
dt
tprdf
,
,,,, Residual interaction:Correlations, Fluctuationsk δk
)(KC F
ff 1(1,2) (3,4)
ˆ H
E
EeffH
Effective interactions
EDF theories: The exact density functional is approximated with powers and gradients of one-body nucleon densities and currents.
Stochastic Mean-Field (SMF) model
Two-body Collision Integral
Fluctuations in collision integral
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:
<1 g Asysoft, >1 g Asystiff
)/( 0potsymE
Investigate the sensitivity of the reaction dynamics to this ingredient Put some constraints on the effective interactions
Symmetry energy
Asysoft
Asystiff
Neutron skinIsovector modesPigmy resonances
around ρ0
Effective interactions and symmetry energy
γ = 2
γ = 0.5
Isovector dipole response
PDR
GDR
The Isovector Dipole Response (DR) in neutron-rich nuclei
The DR in 132Sn: a study within semi-classical transport theories
The neutron skin is sensitive to asy-stiffness: larger in the stiff cases
X.Roca-Maza et al., PRC 85(2012)
Pygmy dipole strength
Giant DR
Pygmy DR
Klimkiewicz et al.
Isovector dipole moment (L = 1)
X neutrons – protons
Xc core neutrons-protons
Y excess neutrons - core
Neutron center of mass
Proton center of mass
The Isovector Dipole Response (DR) in neutron-rich nuclei
19.5%: too much !
Analysis of collective motion with transport theory (Vlasov)
• Pygmy-like initial conditions (Y)
X
Y
Fourier transform of D Strength of the Dipole Response
X c
Neutron skin and core are coupled
The low- energy response is not sensitive tothe asy-stiffness
-- soft-- stiff-- superstiff
Baran et al.
Interpretation in terms of isovector-like and isoscalar-like modes in asymmetric systems
D
Pygmy-like
GDR (isovector-like)
PDR (isoscalar-like)
θ
PDR is isoscalar-like 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 θ , but also Larger L larger neutron skin
2.7 % soft4.4 % stiff4.5 % superstiff
A.Carbone et al., PRC 81 (2010)
Diffusion: mass exchange, charge equilibration, energy dissipation
Dissipation and fragmentation in “MF” models
d-t/AB e (t)R ex-t/τ
kin ABkin e 0)(t E (t)E exd /ττ
kinkin ABAB 0)](tE/ [E R
R
SMF calculations, 124Sn +112Sn, 50 AMeV
Isovector modes faster than isoscalar modes: τd/τex < 1
Larger symmetry energy at low density:
Faster equilibration with soft
(PLF)
(neck)
Neck instabilities: important role of fluctuations….but still ‘mean-field’ dominated mechanism: isospin migration
soft
stiff
Relative weight of Is and Iv dissipation:
J. Rizzo et al., NPA(2008)
Isospin transport ratio
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
Look at the correlation betweencharge and velocity of PLF residues
and IMF’s (2<Z<9) multiplicity
N/Z of neck fragments can help to check the reaction dynamics Isospin as a tracer
• Mid-peripheral impact parameters: Results are model-dependentImportant to study the reaction dynamics (dissipation and nature of dissipation)
Summary
Mechanisms at Fermi energies
PLF
IMF’s V.Baran, B.Frecus (Bucharest),M. Di Toro (LNS)Y.Zhang (CIAE, Beijing)
In collaboration with
Collective excitation in n-rich nuclei:
transport theories predict a good sensitivity to asy-EoS (GDR energy) PDR is isoscalar-like, its relative strength depends on asy-EoS
Isospin transport and fragmentation mechanisms in semi-central collisions
Simple hydro picture
IDDj
IDDjIppp
Innn
I
EIEjj sym
sympn
)()(
Diffusion Driftdrift diffusion
-- Drift: Isospin migration
ρneck < ρPLF(TLF)
Asymmetry flux
-- Diffusion: charge equilibrationOverdamped dipole oscillation
d-t/e D(0) D(t) τd Esym
t
D(t)
neck instabilitiessoft
stiff
β=(ρn-ρp)/ρ
Sn112 Sn124
b = 6 fm, 50 AMeV
Neck fragments are neutron-richer than PLF-TLF
neck
PLF-TLF
B. Tsang et al. PRL 102 (2009)
Mass(A) ~ Mass(B) ; N/Z(A) = N/Z(B)
Isospin transport ratio R :
BBAA
BBAAAB
XX
XXXR
2
A dominance
mixing
B dominance
+1
0
-1
-- AA and BB refer to two symmetric reactions between n-rich and n-poor nucleiAB to the mixed reaction
R(t) = 2(xAB(t) – xm) / (xA – xB) RAB = e-t/τd τd Esym
-- X is an observable related to the N/Z of the projectile-like fragments (PLF)
A
B
Tools to study charge equilibration between A and B
xm = (xA + xB)/2
stiff
soft
More central collisions: larger contact time more dissipation, smaller R Good sensitivity to Asy-EoS
SMF calculations
124Sn +112Sn,50 AMeV
X = N/ZPLF
t = contact time
The strength in the PDR region depends on the asy-stiffness(increases with L)Larger L larger θ , but also Larger L larger neutron skin
1) GDR-like initial conditions (X)D
GDR (isovector-like)
PDR (isoscalar-like)
Pygmy-like
GDR (isovector-like)
PDR (isoscalar-like)
θ
2.7 % soft4.4 % stiff4.5 % superstiff
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' ),|(| 2 vKK F' )||,( 21 vF'
Average effect of the residual interaction
one-body
Fluctuations
TDHF
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