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Giessen-BUU: recent progressGiessen-BUU: recent progress

T. Gaitanos (JLU-Giessen)T. Gaitanos (JLU-Giessen)

Model outlineModel outlineRelativistic transport (GiBUU)(briefly) The transport Eq. in relativistic hadrodynamicsNot trivial: ground state in transport , momentum dependence in RMFground state in transport , momentum dependence in RMF

New ground state (GS) initialization for transport studiesResults of GS simulations

Application (Giant Monopol Resonances, GMR)Theoretical aspects and numerical results (Vlasov, Vlasov+coll.), comparison with expriments (excitation energy, widths of GMR)

Momentum dependence in relativistic hadrodynamicsNon-linear derivative terms in original Lagrangian of QHDarXiv: 0904.1130 [nucl-th]

Final remarksFinal remarks

Many thanks to GiBUU-group

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Outline of the model…Outline of the model…Outline of the model…Outline of the model…

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The MF approach of Quantum-Hadro-Dynamics (QHD)The MF approach of Quantum-Hadro-Dynamics (QHD)The MF approach of Quantum-Hadro-Dynamics (QHD)The MF approach of Quantum-Hadro-Dynamics (QHD)

Equations of motion for Dirac () and boson fields () in Mean-Field (MF) approach:

Equations of motion for Dirac () and boson fields () in Mean-Field (MF) approach:

The Energy-Momentum Tensor in MF:The Energy-Momentum Tensor in MF:

Infinite nuclear matter (B(x,t)=const.) no 4-derivatives, no Coulomb

Finite systems: Local Density Appr. (LDA)

Infinite nuclear matter (B(x,t)=const.) no 4-derivatives, no Coulomb

Finite systems: Local Density Appr. (LDA)

J.D. Walecka, Ann. Phys. (N.Y.) 8383 (1974) 497

Finite systems: beyond LDA, space-like derivarites included (surface effects)Finite systems: beyond LDA, space-like derivarites included (surface effects)

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The relativistic transport equation (BUU)The relativistic transport equation (BUU)The relativistic transport equation (BUU)The relativistic transport equation (BUU)

Starting basisStarting basis The MF-approach of QHD in terms of the effective Dirac equation:Starting basisStarting basis The MF-approach of QHD in terms of the effective Dirac equation:

Q.Li, J.Q. Wu, C.M. Ko, Phys. Rev. C39 (1989) 849

B. Blättel, V. Koch, U. Mosel, Rep. Prog. Phys. 56 (1993) 1

Wigner transform of the 1-b-density matrix („Wigner-matrix“):Wigner transform of the 1-b-density matrix („Wigner-matrix“):

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Numerical realization (Test-Particle Ansatz)…Numerical realization (Test-Particle Ansatz)…Numerical realization (Test-Particle Ansatz)…Numerical realization (Test-Particle Ansatz)…

Discretization of the phase-space distribution f(x,p*) in terms of N „test particles“ Discretization of the phase-space distribution f(x,p*) in terms of N „test particles“

http://www.physik.uni-giessen.de/http://www.physik.uni-giessen.de/GiBUU/GiBUU/http://www.physik.uni-giessen.de/http://www.physik.uni-giessen.de/GiBUU/GiBUU/

Energy momentum tensor energy density Energy momentum tensor energy density

++

Nuclear ground state (initialization)

„test particles“ initialized according to empirical density distributions

(Wood-Saxon type for heavy nuclei, harmonic-osz. Type for light systems)

Problem: density profiles not consistent with mean-field used in propagation

variational method of =[] in RMF different density distr. for ground state

nucleus not in its „real“ groundstate, but in an „excited“ state

affects stability

Nuclear ground state (initialization)

„test particles“ initialized according to empirical density distributions

(Wood-Saxon type for heavy nuclei, harmonic-osz. Type for light systems)

Problem: density profiles not consistent with mean-field used in propagation

variational method of =[] in RMF different density distr. for ground state

nucleus not in its „real“ groundstate, but in an „excited“ state

affects stability

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New initialization: method…New initialization: method…New initialization: method…New initialization: method…

Here: Relativistic Thomas-Fermi (RTF) model for spherical nuclei (Horst Lenske)

variational method for energy density functional [p,n]

Here: Relativistic Thomas-Fermi (RTF) model for spherical nuclei (Horst Lenske)

variational method for energy density functional [p,n]

Relativistic Thomas-Fermi (RTF) equationsRelativistic Thomas-Fermi (RTF) equationsRelativistic Thomas-Fermi (RTF) equationsRelativistic Thomas-Fermi (RTF) equations

+ meson field equations (for the different meson fields)+ meson field equations (for the different meson fields)+ meson field equations (for the different meson fields)+ meson field equations (for the different meson fields)

RTF densitiesRTF densities

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New initialization: relativistic fields (scalar, vector, etc) & New initialization: relativistic fields (scalar, vector, etc) & stability…stability…New initialization: relativistic fields (scalar, vector, etc) & New initialization: relativistic fields (scalar, vector, etc) & stability…stability…

old initialization

new initialization

Fluctuations in different Lorentz-components of nuclear self energy (scalar, vector)

V ~ (vector – scalar) fluctuates considerably!

Fluctuations in different Lorentz-components of nuclear self energy (scalar, vector)

V ~ (vector – scalar) fluctuates considerably!

Almost perfect stability + agreement with ground stateAlmost perfect stability + agreement with ground state

scalar partscalar partscalar partscalar part vector partvector partvector partvector part

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New initialization: relativistic potential (scalar-vector) & stability…New initialization: relativistic potential (scalar-vector) & stability…New initialization: relativistic potential (scalar-vector) & stability…New initialization: relativistic potential (scalar-vector) & stability…

old initialization

new initialization

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New initialization: relativistic fields (scalar, vector, isocvector, New initialization: relativistic fields (scalar, vector, isocvector, coulomb) & stability…coulomb) & stability…New initialization: relativistic fields (scalar, vector, isocvector, New initialization: relativistic fields (scalar, vector, isocvector, coulomb) & stability…coulomb) & stability…

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New initialization: density distributions & stability…New initialization: density distributions & stability…New initialization: density distributions & stability…New initialization: density distributions & stability…

old initialization

new initialization

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New initialization: Fermi energies in RTF…New initialization: Fermi energies in RTF…New initialization: Fermi energies in RTF…New initialization: Fermi energies in RTF…

EF (protons)EF (protons)

EF (neutrons)EF (neutrons)

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New initialization: Fermi energies in BUU…New initialization: Fermi energies in BUU…New initialization: Fermi energies in BUU…New initialization: Fermi energies in BUU…

old initialization

new initialization

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New initialization: Binding energy & rms-radius in BUU…New initialization: Binding energy & rms-radius in BUU…New initialization: Binding energy & rms-radius in BUU…New initialization: Binding energy & rms-radius in BUU…

Old initialization

Old initialization

new initialization

new initialization

RTF-binding energy

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Ground state in BUU-II: Ground state in BUU-II: Results (application to proton-induced reactions)Results (application to proton-induced reactions)Ground state in BUU-II: Ground state in BUU-II: Results (application to proton-induced reactions)Results (application to proton-induced reactions)

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Giant Resonances – preliminaries…Giant Resonances – preliminaries…Giant Resonances – preliminaries…Giant Resonances – preliminaries…

Giant resonances = highly collective modeshighly collective modes of nuclear excitation Giant resonances = highly collective modeshighly collective modes of nuclear excitation

CollectivityCollectivity

CoherentCoherent super-position of many single-particle transitions from one shell to another

Collective motionCollective motion of an appreciable fraction of nucleons of

nucleus

Monopol (L=0)Monopol (L=0)Monopol (L=0)Monopol (L=0) Dipol (L=1)Dipol (L=1)Dipol (L=1)Dipol (L=1) Oktupol (L=2)Oktupol (L=2)Oktupol (L=2)Oktupol (L=2) … … (L>3)(L>3)… … (L>3)(L>3)

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Giant Monopol Resonances (GMR) - Importance…Giant Monopol Resonances (GMR) - Importance…Giant Monopol Resonances (GMR) - Importance…Giant Monopol Resonances (GMR) - Importance…

Indirect determination of the nuclear compression modulus (important for EoS ~sat)

Microscopic approaches (RPA) : Determine E*E* with several nuclear models

and NM properties (compression moduluscompression modulus)

Indirect determination of the nuclear compression modulus (important for EoS ~sat)

Microscopic approaches (RPA) : Determine E*E* with several nuclear models

and NM properties (compression moduluscompression modulus)

Excitation energy of GMRExcitation energy of GMR

Compression modulus of NM

Compression modulus of NM

Effective interactionsEffective interactions

208Pb208Pb

90Zr90Zr

Exp. dataExp. data

EGMR~A-1/3EGMR~A-1/3

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Giant Monopol Resonances (GMR) in GiBUU (influence of init. Giant Monopol Resonances (GMR) in GiBUU (influence of init. method…)method…)Giant Monopol Resonances (GMR) in GiBUU (influence of init. Giant Monopol Resonances (GMR) in GiBUU (influence of init. method…)method…)

Vlasov calculations Vlasov calculations

0

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Giant Monopol Resonances (GMR) – excitation energy & width Giant Monopol Resonances (GMR) – excitation energy & width (Vlasov)…(Vlasov)…Giant Monopol Resonances (GMR) – excitation energy & width Giant Monopol Resonances (GMR) – excitation energy & width (Vlasov)…(Vlasov)…

rms

radiu

s (f

m)

rms

radiu

s (f

m)

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Giant Monopol Resonances (GMR) – excitation energy & width (Giant Monopol Resonances (GMR) – excitation energy & width (Vlasov+coll.Vlasov+coll.))……Giant Monopol Resonances (GMR) – excitation energy & width (Giant Monopol Resonances (GMR) – excitation energy & width (Vlasov+coll.Vlasov+coll.))……

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Giant Monopol Resonances (GMR) – excitation energy & width (Giant Monopol Resonances (GMR) – excitation energy & width (Vlasov+coll.Vlasov+coll.))……Giant Monopol Resonances (GMR) – excitation energy & width (Giant Monopol Resonances (GMR) – excitation energy & width (Vlasov+coll.Vlasov+coll.))……

VlasoVlasovv

VlasoVlasovv

VlasoVlasovv

VlasoVlasovv

Vlasov+coVlasov+coll.ll.

Vlasov+coVlasov+coll.ll.

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Non-Linear derivatives in relativistic hadrodynamics – MotivationNon-Linear derivatives in relativistic hadrodynamics – MotivationNon-Linear derivatives in relativistic hadrodynamics – MotivationNon-Linear derivatives in relativistic hadrodynamics – Motivation

Starting basisStarting basis QHD Lagrangian (original Walecka, 1974)Starting basisStarting basis QHD Lagrangian (original Walecka, 1974)

Dirac Equation: Dirac Equation: momenta and mass dressed by density dependent self energies

Dirac Equation: Dirac Equation: momenta and mass dressed by density dependent self energies

Schrödinger equivalent optical potential: Schrödinger equivalent optical potential: linear increase with energy!Schrödinger equivalent optical potential: Schrödinger equivalent optical potential: linear increase with energy!

Not consistent with Dirac phenomenology (Hama et al.)Not consistent with Dirac phenomenology (Hama et al.)

Microscopic Dirac-Brueckner: non-linear density AND density Microscopic Dirac-Brueckner: non-linear density AND density dependencedependence

Not consistent with Dirac phenomenology (Hama et al.)Not consistent with Dirac phenomenology (Hama et al.)

Microscopic Dirac-Brueckner: non-linear density AND density Microscopic Dirac-Brueckner: non-linear density AND density dependencedependence

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Non-Linear Derivatives (NLD) in relativistic hadrodynamics – The NLD Non-Linear Derivatives (NLD) in relativistic hadrodynamics – The NLD LagrangianLagrangianNon-Linear Derivatives (NLD) in relativistic hadrodynamics – The NLD Non-Linear Derivatives (NLD) in relativistic hadrodynamics – The NLD LagrangianLagrangian

Non-linear derivative operators:Non-linear derivative operators:Non-linear derivative operators:Non-linear derivative operators:

Auxiliary field: Auxiliary field: structure not relevant (no rearrangement in nuclear matter)

Mass term:Mass term: just to not re-normalize the standard QHD couplings

Parameter Parameter :: ~hadronic mass scale, e.g., =1 GeV

Auxiliary field: Auxiliary field: structure not relevant (no rearrangement in nuclear matter)

Mass term:Mass term: just to not re-normalize the standard QHD couplings

Parameter Parameter :: ~hadronic mass scale, e.g., =1 GeV

Starting basisStarting basis again QHD Lagrangian (original Walecka, 1974)Starting basisStarting basis again QHD Lagrangian (original Walecka, 1974)

Modified interaction: Modified interaction: non-linear operators in scalar and vector int. termsModified interaction: Modified interaction: non-linear operators in scalar and vector int. terms

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Non-Linear Derivatives (NLD) in relativistic hadrodynamics – Field equationsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Field equationsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Field equationsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Field equations

Dirac-equation in nuclear matter:Dirac-equation in nuclear matter:Dirac-equation in nuclear matter:Dirac-equation in nuclear matter:

NLD LagrangianNLD Lagrangian contains all higher order derivatives of the baryon fieldNLD LagrangianNLD Lagrangian contains all higher order derivatives of the baryon field

Generalized Euler-Lagrange (Noether-currents, etc…):Generalized Euler-Lagrange (Noether-currents, etc…):Generalized Euler-Lagrange (Noether-currents, etc…):Generalized Euler-Lagrange (Noether-currents, etc…):

Meson-field equations:Meson-field equations:Meson-field equations:Meson-field equations:

Density and energy Density and energy dependence of self dependence of self

energiesenergies

Density and energy Density and energy dependence of self dependence of self

energiesenergies

Non-linear density Non-linear density dependence of meson dependence of meson

fields, particularly, of the fields, particularly, of the vector fieldvector field

Non-linear density Non-linear density dependence of meson dependence of meson

fields, particularly, of the fields, particularly, of the vector fieldvector field

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Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(density dependence)(density dependence)

Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(density dependence)(density dependence)

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Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(density dependence)(density dependence)

Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(density dependence)(density dependence)

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Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(density & energy dependence)(density & energy dependence)

Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(density & energy dependence)(density & energy dependence)

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Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(energy dependence)(energy dependence)

Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(energy dependence)(energy dependence)

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Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(energy dependence, optical potential)(energy dependence, optical potential)

Non-Linear Derivatives (NLD) in relativistic hadrodynamics – ResultsNon-Linear Derivatives (NLD) in relativistic hadrodynamics – Results

(energy dependence, optical potential)(energy dependence, optical potential)