Ab Initio Calculations of Three and Four Body Dynamics M. Tomasellia,b Th. Kühla, D. Ursescua a Gesellschaft für Schwerionenforschung, D-64291 Darmstadt,Germany b Technical University Darmstadt, D-64289 Darmstadt, Germany Motivation Brasil: Few body 18 Marco Tomaselli 2006 The Equation of Motion (EoM) in the zero order dynamic linearization (GLA) Brasil: Few body 18 Marco Tomaselli 2006 The Equation of Motion (EoM) in the second order GLA-III Brasil: Few body 18 Marco Tomaselli 2006 The Equation of Motion (EoM)-IV Brasil: Few body 18 Marco Tomaselli 2006 The Hamilton's Operator Brasil: Few body 18 Marco Tomaselli 2006 The Non-Linear Eigenvalue Equation Brasil: Few body 18 Marco Tomaselli 2006 Analogy with cluster theory Correlations can be introduced via eiS method N N i 1 1 2 i 1 eiS ai 0 (1 S1 S 2 S3 ...) ai 0 S2 a1 a2a3 a4 S1 a a S3 a1 a2a3 a4a5 a6 ..... Perturbation approximation possible. We prefer to calculate the effective operators. H eff e iS He iS Oeff e iS Oe iS The perturbative terms of the correlation operators Si correspond to the diagram of the dynamic theory. The particle–hole terms generated by the S3 operator are put to zero in the ladder perturbation. Brasil: Few body 18 Marco Tomaselli 2006 Configuration mixing wave functions (CMWF) Brasil: Few body 18 Marco Tomaselli 2006 Effective Hamiltonian S2 Brasil: Few body 18 Marco Tomaselli 2006 The wavefunction of the deuteron Brasil: Few body 18 Marco Tomaselli 2006 Cluster model based on Dynamic Correlation Model (DCM) N N ai 0 i 1 N N N 1 [ H , a ] 0 a 0 ai a1 0 i i i 1 N 1 i 1 N 1 i 1 N 2 2 i 1 i 1 i 1 i '1 [ H , ai a1 ] 0 ai a1 0 ai ai ' 0 ...... N N 1 N 2 2 i 1 i 1 i 1 i '1 tot N N 1,1 N 2,2 ... ai 0 ai a1 0 ai ai ' 0 ... Brasil: Few body 18 Marco Tomaselli 2006 Effect of linearisation on commutator chain for two body clusters Within the GLA the higher order terms (4p-2h) are calculated with the Wick's theorem by neglecting the normal order. Collect the resulting terms E E2 p 2 p V 2 p 3 p1h V 2 p 2 p 0 E E3 p1h 3 p1h V 3 p1h 3 p1h 2 p V 3 p1h Dynamic eigenvalue equations for mixed mode amplitudes 2 particles => 3 particles – 1 hole Brasil: Few body 18 Marco Tomaselli 2006 Dynamics eigenvalue equation for one dressed dressed nucleon clusters which is solvable self-consistently Brasil: Few body 18 Marco Tomaselli 2006 Degree of spuriousity Brasil: Few body 18 Marco Tomaselli 2006 Charge distributions of 6He Brasil: Few body 18 Marco Tomaselli 2006 Charge radii of 6He Brasil: Few body 18 Marco Tomaselli 2006 Charge distributions of 6Li Brasil: Few body 18 Marco Tomaselli 2006 Charge form factor for 6Li Brasil: Few body 18 Marco Tomaselli 2006 Elastic proton scattering on 6Li Brasil: Few body 18 Marco Tomaselli 2006 Medium effects on the two body matrix elements (18O) Brasil: Few body 18 Marco Tomaselli 2006 Positive and Negative parity states in 18O Brasil: Few body 18 Marco Tomaselli 2006 Comparison with Vlow-k potential: 18O Brasil: Few body 18 Marco Tomaselli 2006 The EoM of the Three Nucleon Clusters Brasil: Few body 18 Marco Tomaselli 2006 Three particle Dynamic model Brasil: Few body 18 Marco Tomaselli 2006 Nuclear results for Li isotopes Brasil: Few body 18 Marco Tomaselli 2006 Elastic proton scattering on 11Li Brasil: Few body 18 Marco Tomaselli 2006 Summary of Charge Radii Li - Exp.[GSI] rms R c Exp.[8] rms R c 6 7 8 9 11 2,55 2,46 2,37 2,30 2,47 2.55 (4) 2.37 (3) Rc = charge radius Exp.+Th.[1] Exp+Th[1] rms R p rms R c 2.32 (3) 2.27 (2) 2.26 (2) 2.18 (2) 2.88 (2) 2.46 (2) 2.40 (2) Th.[2] rms R p 2,045 1,941 1,946 1,986 Th.[3] rms R p Th.[4] rms Rp 2,39 2,25 2,09 2,04 Th.[5] rms R p 2,27 2,18 2,10 Th.[6] rms R c 2,55 2,41 2,40 2,42 2,67 Exp+Th[7] rms R c 2,235 Rp = point radius References: [1] I. Tanihata, Phys. Lett B 206,592 (1988) Method: 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 Charge distributions for A=3 nuclei Brasil: Few body 18 Marco Tomaselli 2006 Charge distribution: alpha particle RMS (Bonn)=1.50 fm RMS (Yale)=1.51 fm Brasil: Few body 18 Marco Tomaselli 2006 Energy splitting and BE(E2;2+ transition for 16C Brasil: Few body 18 Marco Tomaselli 0+) 2006 Cluster Factorization Theory I Brasil: Few body 18 Marco Tomaselli 2006 Cluster Factorization Theory II Brasil: Few body 18 Marco Tomaselli 2006 Cluster Factorization Theory III Brasil: Few body 18 Marco Tomaselli 2006 Cluster Factorization Theory IV Brasil: Few body 18 Marco Tomaselli 2006 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 line Brasil: Few body 18 Interaction between nucleons Hole line Marco Tomaselli 2006 Brasil: Few body 18 Marco Tomaselli 2006