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Ab Initio Calculations of Three and Four Body Dynamics M. Tomaselli a,b Th. Kühl a, D. Ursescu a a Gesellschaft für Schwerionenforschung, D-64291 Darmstadt,Germany b Technical University Darmstadt, D-64289 Darmstadt, Germany

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2006 Brasil: Few body 18 Marco Tomaselli Motivation

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2006 Brasil: Few body 18 Marco Tomaselli The Equation of Motion (EoM) in the zero order dynamic linearization (GLA)

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2006 Brasil: Few body 18 Marco Tomaselli The Equation of Motion (EoM)-II

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2006 Brasil: Few body 18 Marco Tomaselli The Equation of Motion (EoM) in the second order GLA-III

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2006 Brasil: Few body 18 Marco Tomaselli The Equation of Motion (EoM)-IV

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2006 Brasil: Few body 18 Marco Tomaselli The Hamilton's Operator

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2006 Brasil: Few body 18 Marco Tomaselli The Non-Linear Eigenvalue Equation

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2006 Brasil: Few body 18 Marco Tomaselli Analogy with cluster theory..... Perturbation approximation possible. We prefer to calculate the effective operators. Correlations can be introduced via e iS method The perturbative terms of the correlation operators S i correspond to the diagram of the dynamic theory. The particle–hole terms generated by the S 3 operator are put to zero in the ladder perturbation.

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2006 Brasil: Few body 18 Marco Tomaselli Configuration mixing wave functions (CMWF)

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2006 Brasil: Few body 18 Marco Tomaselli Effective Hamiltonian S 2

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2006 Brasil: Few body 18 Marco Tomaselli The wavefunction of the deuteron

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2006 Brasil: Few body 18 Marco Tomaselli Cluster model based on Dynamic Correlation Model (DCM)......

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2006 Brasil: Few body 18 Marco Tomaselli The Equation of Motion (EoM)-IV The Equation of Motion (EoM)-IV

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2006 Brasil: Few body 18 Marco Tomaselli Effect of linearisation on commutator chain for two body clusters Collect the resulting terms Within the GLA the higher order terms (4p-2h) are calculated with the Wick's theorem by neglecting the normal order. Dynamic eigenvalue equations for mixed mode amplitudes 2 particles => 3 particles – 1 hole

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2006 Brasil: Few body 18 Marco Tomaselli Dynamics eigenvalue equation for one dressed dressed nucleon clusters which is solvable self-consistently

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2006 Brasil: Few body 18 Marco Tomaselli Degree of spuriousity

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2006 Brasil: Few body 18 Marco Tomaselli Charge distributions of 6 He

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2006 Brasil: Few body 18 Marco Tomaselli Charge radii of 6 He

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2006 Brasil: Few body 18 Marco Tomaselli Charge distributions of 6 Li

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2006 Brasil: Few body 18 Marco Tomaselli Charge form factor for 6 Li

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2006 Brasil: Few body 18 Marco Tomaselli Elastic proton scattering on 6 Li

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2006 Brasil: Few body 18 Marco Tomaselli Medium effects on the two body matrix elements ( 18 O)

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2006 Brasil: Few body 18 Marco Tomaselli Positive and Negative parity states in 18 O

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2006 Brasil: Few body 18 Marco Tomaselli Comparison with V low-k potential: 18 O

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2006 Brasil: Few body 18 Marco Tomaselli The EoM of the Three Nucleon Clusters

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2006 Brasil: Few body 18 Marco Tomaselli Three particle Dynamic model

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2006 Brasil: Few body 18 Marco Tomaselli Nuclear results for Li isotopes

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2006 Brasil: Few body 18 Marco Tomaselli Elastic proton scattering on 11 Li

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Summary of Charge Radii References:Method: [1] I. Tanihata, Phys. Lett B 206,592 (1988)Interaction Cross Sections with Glauber model, HO distributions [2] P. Navratil, PRC 57,3119 (1998)Large-basis shell-model calculations [3] S. Pieper, Annu.Rev.Nucl.Part.Sci. 51, 53 (2001)Greens Function Monte Carlo AV18/IL2 [4] S. Pieper, PRC 66, 044310 (2002)Greens Function Monte Carlo AV18/IL2 [5] Suzuki, Progr.Theo.Phys.Suppl. 146, 413 (2002)Stochastic Variational Multicluster Method on a correlated gaussian basis [6] M. Tomaselli et al., Can. J. Phys. 80, 1347 (2002)Dynamic Correlation model [7] Penionzhkevich, Nucl.Phys. A 616, 247 (1997)coupled channel calculations, double-folding optical potential, M3Y effective interaction [8] C.W. de Jager, At.Dat.Nucl.Dat.Tab. 14, 479 (1974)Electron Scattering R c = charge radiusR p = point radius

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2006 Brasil: Few body 18 Marco Tomaselli Charge distributions for A=3 nuclei

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2006 Brasil: Few body 18 Marco Tomaselli Charge distribution: alpha particle RMS (Bonn)=1.50 fm RMS (Yale)=1.51 fm

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2006 Brasil: Few body 18 Marco Tomaselli Energy splitting and BE(E2;2 + 0 + ) transition for 16 C

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2006 Brasil: Few body 18 Marco Tomaselli Cluster Factorization Theory I

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2006 Brasil: Few body 18 Marco Tomaselli Cluster Factorization Theory II

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2006 Brasil: Few body 18 Marco Tomaselli Cluster Factorization Theory III

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2006 Brasil: Few body 18 Marco Tomaselli Cluster Factorization Theory IV

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2006 Brasil: Few body 18 Marco Tomaselli Factorisation of the model CMWFs in terms of cluster coefficients The factorisation method is presently applied to reduce complex Feynman diagrams to simple form Particle lineHole line Interaction between nucleons

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2006 Brasil: Few body 18 Marco Tomaselli

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