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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 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, USA

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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.Michel7,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, USA 3 Joint Institute for Heavy Ion Research, Oak Ridge National Laboratory, Oak Ridge, USA 4 Institute of Theoretical Physics, University of Warsaw, Warsaw. 5 Grand Accélérateur National d'Ions Lourds (GANIL) CEA/DSM Caen Cedex, France 7 Department of Physics, Post Office Box 35 (YFL), FI-40014 University of Jyväskylä, Finland 8 Department of Physics, University of Arizona, Tucson, Arizona, USA

  2. Outline • Drip line nuclei as Open Quantum Systems • Gamow Shell Model Formalism • Experimental Radii of 6,8He (11Li and 11Be) • Spin-orbit density effect on charge radii • Calculations on charge radii of 6,8He • DMRG application to 8He • Conclusions 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. Shell Model Theories that incorporate the continuum, selected references 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. 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 (G.Hagen et al PRL 103 062503 (2009)) Lawson method? Jacobi coordinates? Y.Suzuki and K.Ikeda PRC 38,1 (1988) • Appropriate treatment for proper description of the recoil of the core • and the removal of the spurious CoM motion.

  6. EXPERIMENTAL RADII OF 6He, 8He, 11Li 4He 6He 8He 1.67fm 2.054(18)fm L.B.Wang et al 1.67fm 2.068(11)fm 1.929(26)fm P.Mueller et al 9Li 11Li 2.217(35)fm 2.467(37)fm R.Sanchez et al Rcharge(6He) > Rcharge (8He) RMS charge radii charge radii determines the correlations between valence particles AND reflects the radial extent of the halo nucleus • “Swelling” of the core is not negligible ~8% ~4% 10Be 11Be Annu.Rev.Nucl.Part.Sci. 51, 53 (2001) W.Nortershauser et al 2.357(16)fm 2.460(16)fm 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 PRL 102, 062503 (2009)

  7. Spin-orbit contribution to the charge radius Usually point radii are converted to charge radii through: Darwin-Foldy term finite size effects • It was proposed that the D.F term should be treated as a part of the charge radius because it • appears in the charge density of the proton (J.L. Friar et al PRA 56, 6 (1997)) Additionally the spin-orbit density could have a non-negligible effect on the charge radius.  Contributes on a noticeable change to the charge radius between 40Ca and 48Ca (W. Bertozzi et al , PLB 41, 408 (1972)) Both terms appear explicitly in the expression of the single nucleon charge operator and they enter a non-relativistic calculation to an order 1/m2 . BUT … • Finite size effects and relativistic D.F term are consistently considered in theoretical calculations H.Esbensen et al PRC 76, 024302 (2007) • The s.o effect is almost never considered… (except maybe) A.Ong et al PRC 82, 014320 (2010)

  8. Spin-orbit contribution to the charge radius  The formulas to calculate the s.o correction are the following: (J.Friar et al Adv.Nucl.Phys. 8, 219, (1975)) • As we shall see , the s.o can have a comparable contribution with the finite size effects!! • The charge distribution in Helium halos is consistently described by: • The orbital motion of the core around the center of mass of the nucleus • The polarization of the core by the valence neutrons  The s.o contribution caused by the anomalous magnetic moment of the neutron

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

  10. 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 Nsh = 30 for p3/2 contour and Nsh= 20 for each real cont. Total 111 single particle states. We limit ourselves to 2 particles occupying continuum orbits... Modified Minnesota Interaction (MN) (NPA 286, 53) Parameterizations • The two strength parameters of the MN are adjusted to the g.s of 6,8He.

  11. Charge Radii calculations for 6,8He nuclei Expression of charge radius in these coordinates Generalization to n-valence particles is straightforward • The ingredients of the calculation are the OBME and TBME • Same formulas for heavier systems 11Li, 11Be

  12. Results on charge radii of 6,8He with MMN interaction • The s.o corrections are comparable to the D.W term 6He and comparable to • the finite size corrections in 8He. •  The s.o as compared to a maximal estimate they are not very different. •  Good overall agreement of the radii with experiment. Experimental trend is satisfied.

  13. 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 MMN interaction (configuration mixing and correlation angle (for 6He) ) 8He

  14. The charge radius of 6He as a function of the S2n Black line: Core polarization was not included. Red line: Core polarization is taken into account. We use a 4% increase of the α-core pp radius as it was estimated by GFMC calculations. Blue line:Core polarization + s.o effect • The narrow experimental error bars • suggest that the S2n should be • calculated with a high precision if one • aims in a detailed description of the 6He • radial extent. • When this condition is met the p3/2 state • had a dominant occupation of about 90% • in the 6He g.s. • For this p3/2 percentage and the correct S2n, the geometry of the neutrons (correlations) • and the radial extent is such, so as the calculated radius is in a satisfactory agreement • with the experiment.

  15. Comparison with other models and experiment

  16. Density Matrix Renormalization Group (DMRG) S.R White PRL 69 (1992) 2863 T.Papenbrock and D.Dean J.Phys.G 31 (2005) S1377 S.Pittel et al PRC 73 (2006) 014301 J.Rotureau et al PRC 79 (2009) 014304 • Truncation Method applied to lattice models, spin chains, atomic nuclei…. • Basic idea: where A and B are partitions of the system. in terms of m < M basis states (truncation)  Approximate or  These m states are eigenstates of the density matrix • The partition of the system has to be decided by the practitioner. In GSM+DMRG we optimize the number of non-resonant states along the scattering contours.  The difference between the exact and the approximated , has the minimal norm.

  17. Density Matrix Renormalization Group application to 8He radius • Key point: In DMRG the wave function is not stored. But the second quantized • operators that define the Hamiltonian are calculated and stored in each step… • The radius operator has the same form (in second quantization) with H • We calculate OBMEs and TBMEs of rpp • In each DMRG step we calculate the expectation value the radius

  18. Density Matrix Renormalization Group application to 8He radius • In the following we slightly renormalized the strengths of the MN interaction so as to reproduce • the g.s 0+ energy of 8He.

  19. 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 modeling weakly bound nuclei with large radial extension. • Using a finite range force (MN) and adjusting the strengths to the g.s energies • of 6,8He we reproduced the experimental trend of Helium halo charge radii. • Charge radii are primarily sensitive on the p3/2 occupation and the S2n. • The core polarization by the valence neutrons is a small but NOT negligible • effect. • Our calculations showed that the s.o contribution in the conversion of the • point-proton radii can be comparable to the D.F term and the finite size effects • The next step: charge radii and properties of 11Li, 11Be assuming an 4He core in • a GSM+DMRG framework. • Develop the 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.

  20. 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

  21. Density Matrix Renormalization Group application to 8He proton radius  Convergence properties of the DMRG are met for both radius and energy.  DMRG converges on the right value. We compare a 2p-2h calculation with a full (4p-4h).  The differences depict the model space effect on the observables (energy/radius). The energy in DMRG is more attractive and the radius is smaller compared to the 2p-2h. ε = 10-8

  22. 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…

  23. 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)…

  24. 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

  25. 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.

  26. Density Matrix Renormalization Group application to 8He p-sd shells (5 partial-waves) , 47 shells total. Edmrg = -3.284 MeV, EGSM = -3.112 MeV  the system gained energy in DMRG as a result of the larger model space (4p-4h). The difference is ~150keV. Remember that EGSM (2p-2h) is the experimental value (MN was fitted in this way.) Truncation in the DMRG sector is governed by the trace of the density matrix  dim GSM = 9384683 dim DMRG = 3859 ε = 10-8

  27. Complex Scaling

  28. 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

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

  30. 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.

  31. 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)

  32. 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)

  33. 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

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

  35. 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

  36. 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…

  37. 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.

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