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Charge radius of 6 He and Halo nuclei in Gamow Shell Model

Charge radius of 6 He and Halo nuclei in Gamow Shell Model. G.Papadimitriou 1 W.Nazarewicz 1,2,4 , N.Michel 6,7 , M.Ploszajczak 5 , J.Rotureau 8. 1 Department of Physics and Astronomy, University of Tennessee,Knoxville. 2 Physics Division, Oak Ridge National Laboratory, Oak Ridge.

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Charge radius of 6 He and Halo nuclei in Gamow Shell Model

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  1. Charge radius of 6Heand Halo nucleiin Gamow Shell Model G.Papadimitriou1 W.Nazarewicz1,2,4, N.Michel6,7, M.Ploszajczak5, J.Rotureau8 1 Department of Physics and Astronomy, University of Tennessee,Knoxville. 2 Physics Division, Oak Ridge National Laboratory, Oak Ridge. 3 Joint Institute for Heavy Ion Research, Oak Ridge National Laboratory, Oak Ridge 4 Institute of Theoretical Physics, University of Warsaw, Warsaw. 5 Grand Accélérateur National d'Ions Lourds (GANIL). 6 CEA/DSM, Caen, France 7 Department of Physics, Graduate School of Science, Kyoto University, Kyoto 8 Department of Physics, University of Arizona, Tucson, Arizona

  2. Outline • Drip line nuclei as Open Quantum Systems • Gamow Shell Model Formalism • Experimental Radii of 6,8He ,11Li and 11Be • Results on 6He charge radius calculation • Comparison with other models • Conclusion and Future Plans

  3. Proximity of the continuum I.Tanihata et al PRL 55, 2676 (1985) It is a major challenge of nuclear theory to develop theories and algorithms that would allows us to understand the properties of these exotic systems.

  4. scattering continuum resonance bound states Closed Quantum System Open quantum system (nuclei near the valley of stability) (nuclei far from stability) infinite well discrete states (HO) basis nice mathematical properties: Exact treatment of the c.m, analytical solution…

  5. Theories that incorporate the continuum Continuum Shell Model (CSM) • H.W.Bartz et al, NP A275 (1977) 111 • A.Volya and V.Zelevinsky PRC 74, 064314 (2006) Shell Model Embedded in Continuum (SMEC) • J. Okolowicz.,etal, PR 374, 271 (2003) • J. Rotureau etal, PRL 95 042503 (2005) Gamow Shell Model (GSM) • N. Michel etal, PRL 89 042502 • N. Michel et al., Phys. Rev. C67, 054311 (2003) • N. Michel et al., Phys. Rev. C70, 064311 (2004 • G. Hagen et al, Phys. Rev. C71, 044314 (2005) • N.Michel et al, J.Phys. G: Nucl.Part.Phys 36, 013101 (2009)

  6. N.Michel et.al 2002 PRL 89 042502 The Gamow Shell Model (Open Quantum System) Poles of the S-matrix

  7. Berggren’s Completeness relation T.Berggren (1968) NP A109, 265 Non-resonant Continuum along the contour resonant states (bound, resonances…) Many-body discrete basis Complex-Symmetric Hamiltonian matrix Matrix elements calculated via complex scaling

  8. GSM application for He chain PRC 70, 064313 (2004) GHF+SGI p model space 0p3/2 resonance • Optimal basis for each nucleus via the GHF method • Borromean nature of 6,8He is manifested • Helium anomaly is well reproduced

  9. GSM HAMILTONIAN “recoil” term coming from the expression of H in the COSM coordinates. No spurious states We want a Hamiltonian free from spurious CM motion Lawson method? Jacobi coordinates? Y.Suzuki and K.Ikeda PRC 38,1 (1988) • pipj matrix elements • complex scaling does not apply to this particular integral…

  10. Recoil term treatment • Two methods which are equivalent from a numerical point of view i) Transformation in momentum space ii) Expand in HO basis • disregard numerical derivatives α,γ are oscillator shells a,c are Gamow states Fourier transformation to return back to r-space • No complex scaling is involved • Gaussian fall-off of HO states provides convergence • Convergence is achieved with a truncation of about Nmax ~ 10 HO quanta PRC 73 (2006) 064307

  11. EXPERIMENTAL RADII OF 6He, 8He, 11Li 4He 6He 8He 1.43fm 1.912fm L.B.Wang et al 1.45fm 1.925fm 1.808fm P.Mueller et al 9Li 11Li 2.217fm 2.467fm R.Sanchez et al Rcharge(6He) > Rcharge (8He) Point proton charge radii 6He charge radii determines the correlations between valence particles AND reflects the radial extent of the halo nucleus center of mass of the nucleus 8He 10Be 11Be W.Nortershauser et al 2.357fm 2.460fm • “Swelling” of the core is not negligible L.B.Wang et al, PRL 93, 142501 (2004) P.Mueller et al, PRL 99, 252501 (2007) R.Sanchez et al PRL 96, 033002 (2006) W.Nortershauser et al nucl-ex/0809.2607v1 (2008) Annu.Rev.Nucl.Part.Sci. 51, 53 (2001)

  12. Comparison of 6He radius data with nuclear theory models Charge radii provide a benchmark test for nuclear structure theory!

  13. GSM calculations for 6He nucleus 5He basis Im[k] (fm-1) Re[k] (fm-1) 3.27 0p3/2 • Separable Gaussian Interaction (GI) (PRC 71 044314) • Surface Delta Interaction (SDI) (PR 145, 830) • Surface Gaussian Interaction (SGI) (PRC 70, 064313) • WS basis parameterized to 5He s.p energies p-sd Valence space 0p3/2 resonant state plus {ip3/2}, {ip1/2}, {is1/2}, {id5/2}, {id3/2} non-resonant continua With i=1,……Nsh . Nsh=60 with Gauss Legendre Schematic two-body interactions employed • The parameter(s) of each force • is(are) fitted on the g.s energy of 6He

  14. GSM calculations for 6He nucleus rnn Expression of charge radius in these coordinates r2 r1 Generalization to n-valence particles is straightforward rc-2n=(r1+r2)/2 complex scaling cannot be applied! Renormalization of the integral based on physical arguments (density) In our calculations we carried out the radial integration until 25fm

  15. Radial density of valence neutrons for the 6He cut • With an adequate number of points along the contour the fluctuations become minimal • We “cut” when for a given number of discretization points the fluctuations • are smeared out

  16. Results and discussion Angles estimated from the available B(E1) data and the average distances between neutrons. PRC 76, 051602 charge radii and angles for a p-sd model space employed Decomposition of the wavefunction ~91% • The p3/2 occupancy is a crucial quantity for the correct determination of the charge radius in 6He

  17. Results and discussion • Different interactions lead to different configuration mixing. • 6He charge radius (Rch) is primarily related to the p3/2 occupation of the 2-body wavefunction. • The recent measurements put a constraint in our GSM Hamiltonian which is related to the p3/2 occupation. • We observe an overall weak sensitivity for both radii and the correlation angle.

  18. Comparison with other structure Models

  19. Conclusion and Future Plans • The very precise measurements on 6He, 8He, 11Li and 11Be Halos charge radii give us the opportunity to constrain our GSM Hamiltonian. • The GSM description is appropriate for modelling weakly bound nuclei with large radial extension. • The next step: charge radii 8He, 11Li, 11Be assuming an 4He core. The rapid increase in the dimensionality of the space will be handled by the GSM+DMRG method. (J.Rotureau et al PRL 97 110603 (2006) and nucl-th/0810.0781.v1) • The 2+ state of 6He will be used to adjust the quadrupole strength V(J=2,T=1) of the interaction in 8He and 11Li. For 11Li the T=0 channel of the interaction will be fitted to the 6Li nucleus. • Develop effective interaction for GSM applications in the p and p-sd shells that will open a window for a detailed description of weakly bound systems. The effective GSM interaction depends on the valence space, but also in the position of thethresholds and the position of the S-matrix poles

  20. Complex Scaling

  21. Diagonalization of Hamiltonian matrix • Large Complex Symmetric Matrix • Two step procedure non-resonant continua “pole approximation” resonance Full space bound state resonance bound state • Identification of physical state by maximization of

  22. Integral regularization problem between scattering states For this integral it cannot be found an angle in the r-complex plane to regularize it

  23. Density Matrix Renormalization Group • From the main configuration space all the |k>A are built (in J-coupled scheme) • Succesivelly we add states from the non resonant continuum state and construct states |i>B • In the {|k>A|i>B}J the H is diagonalized • ΨJ=ΣCki {|k>A|i>B}J is picked by the overlap method • From the Cki we built the density matrix and the N_opt states are corresponding to the maximum eigenvalues of ρ.

  24. From radii...to stellar nucleosynthesis! Experiment • NCSM P.Navratil and W.E Ormand PRC 68 034305 • GFMC S.C.Pieper and R.B.Wiringa Annu.Rev.Nucl.Part.Sci. 51, 53 • Collective attempt to calculate the charge radius by all modern structure models • Very precise measurements on charge radii • Provide critical test of nuclear models Charge radii of Halo nuclei is a very important observable that needs theoretical justification Figures are taken from PRL 96, 033002 (2006) and PRL 93, 142501 (2004)

  25. single particle Harmonic Oscillator (HO) basis nice mathematical properties: Lawson method applicable… Largest tractable M-scheme dimension ~ 109 SHELL MODEL (as usually applied to closed quantum systems)

  26. HEAVIER SYSTEMS non-resonant continua bound-states resonances • Separation of configuration space in A and B • Truncation on B by choosing the most important configurations • Criterion is the largest eigenvalue of the density matrix • Explosion of dimension • Hamiltonian Matrix is dense+non-hermitian • Lanczos converges slowly B • Density Matrix Renormalization Group A S.R.White., 1992 PRL 69, 2863; PRB 48, 10345 T.Papenbrock.,D.Dean 2005., J.Phys.G31 S1377 J.Rotureau 2006., PRL 97, 110603 J.Rotureau et al. (2008), to be submitted

  27. Form of forces that are used SGI SDI Minnesota GI

  28. EXPERIMENTAL RADII OF Be ISOTOPES • 7Be charge radius provides constraints for the • S17 determination • Charge radius decreases from 7Be to 10Be and then increases for 11Be • 11Be increase can be attributed to the c.m motion of the 10Be core 11Be 1-neutron halo W.Norteshauser et all nucl-ex/0809.2607v1 interaction cross section measurements GFMC PRC 66, 044310, (2002) and Annu.Rev.Nucl.Part.Sci. 51, 53 (2001) NCSM PRC 73 065801 (2006) and PRC 71 044312 (2005) The message is that changes in charge distributions provides information about the interactions in the different subsystems of the strongly clustered nucleus! FMD

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