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Charge radii of 6,8 He and Halo nuclei in Gamow Shell Model

Charge radii of 6,8 He and Halo nuclei in Gamow Shell Model. G.Papadimitriou 1 N.Michel 6,7 , W.Nazarewicz 1,2,4 , 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 radii of 6,8 He and Halo nuclei in Gamow Shell Model

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  1. Charge radii of 6,8Heand Halo nucleiin Gamow Shell Model G.Papadimitriou1 N.Michel6,7,W.Nazarewicz1,2,4, 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 • Calculations on 6,8He nuclei • Results on charge radii of 6,8He • 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. 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 (2002) • R. Id Betan et.al PRL 89, 042501 (2002) • 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)

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

  6. 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

  7. GSM application for He chain PRC 70, 064313 (2004) N.Michel et al GHF+SGI p model space 0p3/2 resonance 0p1/2 resonance • Optimal basis for each nucleus via the GHF method • Borromean nature of 6,8He is manifested • Helium anomaly is well reproduced

  8. 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…

  9. Recoil term treatment ii) Expand in HO basis PRC 73 (2006) 064307 • Two methods which are equivalent from a numerical point of view i) Transformation to momentum space α,γ are oscillator shells a,c are Gamow states • No complex scaling is involved for the recoil matrix elements 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

  10. 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)

  11. Comparison of 6,8He radius data with nuclear theory models Charge radii provide a benchmark test for nuclear structure theory!

  12. GSM calculations for 6,8He nuclei Im[k] (fm-1) (2.0,0.0) Re[k] (fm-1) B 0p3/2 3.27 A (0.17,-0.15) • 4He + valence particles framework • WS or KKNN or HF basis • WS is parameterized to the 0p3/2 s.p of 5He • KKNNPTP 61, 1327 (1979) • Reproduces low-energy phase shifts of 4He-n system

  13. GSM calculations for 6,8He nuclei Im[k] (fm-1) (2.0,0.0) Re[k] (fm-1) B 0p3/2 3.27 A (0.17,-0.15) Model space p-sd waves 0p3/2 resonance only i{p3/2} complex non-resonant part i{s1/2}, i{p1/2}, i{d3/2}, i{d5/2} real continua (red line) with i=1,…Nsh We limit ourselves to 2 particles occupying continuum orbits. Schematic two-body interactions employed • Modified Minnesota Interaction (MN) (NPA 286, 53) • Surface Delta Interaction (SDI) (PR 145, 830) • Surface Gaussian Interaction (SGI) (PRC 70, 064313) MN SGI SDI

  14. GSM calculations for 6,8He nuclei Forces • SGI/SDI parameters are adjusted to the g.s of 6He • The 2+ state of 6He is used to adjust the V(J=2,T=1) strength of the SGI • The two strength parameters of the MN are adjusted to the g.s of 6,8He • The matrix elements of the MN were calculated with the HO expansion method, • like in the recoil case. with Nmax = 10 and b = 2fm • In all the following we employed SGI+WS, SDI+WS and KKNN+MN for 6He • For 8He we used the spherical HF potential obtained from each interaction States with high angular momentum (d5/2, d3/2) • The large centrifugal barrier results in an enhanced localization of the d-waves • ONLY for d5/2, d3/2 or higher orbits we may use HO basis states for our calculation • s-p waves are always generated by a complex WS or KKNN basis.

  15. GSM calculations for 6,8He nuclei Example: 6He g.s with MN interaction. Basis set 1: p-sd waves with 0p3/2 resonant and ALL the rest continua i{p3/2}, i{p1/2}, i{s1/2}, i{d3/2}, i{d5/2} 30 points along the complex p3/2 contour and 25 points for each real continuum Total dimension: dim(M) = 12552 Basis set 2: p-sd waves with 0p3/2 resonant and i{p3/2}, i{p1/2}, i{s1/2} non-resonant continua BUT d5/2 and d3/2 HO states. nmax = 5 and b = 2fm (We have for example 0d5/2, 1d5/2, 2d5/2, 3d5/2, 4d5/2 for nmax = 5) Total dimension: dim(M) = 5303 g.s energies for 6He Basis set 1 Basis set 2 Jπ : 0+ = - 0.9801 MeV Jπ : 0+ = - 0.9779 MeV Differences of the order of ~ 0.22 keV…

  16. GSM calculations for 6,8He nuclei Radial density of the 6He g.s. red and green curves correspond to the two different basis sets. • Energies and radial properties are equivalent in both representations. • The combination of Gamow states for low values of angular momentum and HO for higher, captures all the relevant physics while keeping the basis in a manageable size. • Applicable only with fully finite range forces (MN)…

  17. Charge Radii calculations for 6,8He nuclei Expression of charge radius in these coordinates Generalization to n-valence particles is straightforward complex scaling cannot be applied! Renormalization of the integral based on physical arguments (density) In our calculations we carried out the radial integration until 20fm

  18. 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

  19. We calculated also the correlation angle for 6He PRC 76, 051602 Our result is: Angles estimated from the available B(E1) data and the average distances between neutrons. To be compared with Results on charge radii of 6,8He with MN interaction rpp(6He)= 1.85 fm rpp (8He) = 1.801 fm 8He

  20. p3/2 d5/2 p1/2 d3/2 s1/2 Results on charge radii of 6,8He with SGI interaction rpp(8He) = 1.957 fm rpp(6He) = 1.924 fm GSM Jπ: 0+ = - 3.125 MeV Exp Jπ: 0+ = - 3.11 MeV θnn =81.80o

  21. Results on charge radii of 6He with SDI interaction rpp(6He) = 1.92 fm θnn =82.13o

  22. Comparison with other models and experiment

  23. Conclusion and Future Plans • The very precise measurements of 6,8Hehalos charge radii provide a valuable test of the configuration mixing and the effective interaction in nuclei close to the drip-lines. • The GSM description is appropriate for modelling weakly bound nuclei with large radial extension. • For 6He we observed an overall weak sensitivity for both correlation angle and charge radius. However, when adding 2 more neutrons in the system he details of each interaction are revealed. • 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 PRC 79 014304, 2009) • 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

  24. 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.

  25. Complex Scaling

  26. 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

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

  28. 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.

  29. 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)

  30. 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)

  31. 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

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

  33. 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

  34. 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…

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