1. 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. 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 6,8 He, 11 Li and 11 Be  Calculations on 6,8 He nuclei  Results on charge radii of 6,8 He  Comparison with other models  Conclusion and Future Plans

3. I.Tanihata et al PRL 55, 2676 (1985) Proximity of the continuum 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. 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.,et al, PR 374, 271 (2003) J. Rotureau et al, PRL 95 042503 (2005) Gamow Shell Model (GSM) N. Michel et al, 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) Theories that incorporate the continuum

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

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

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

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

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

10. 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) center of mass of the nucleus 6 He 8 He EXPERIMENTAL RADII OF 6 He, 8 He, 11 Li R charge ( 6 He) > R charge ( 8 He) “Swelling” of the core is not negligible charge radii determines the correlations between valence particles AND reflects the radial extent of the halo nucleus 4 He 6 He 8 He L.B.Wang et al P.Mueller et al 1.43fm1.912fm 1.45fm1.925fm1.808fm 9 Li 11 Li R.Sanchez et al 2.217fm2.467fm Point proton charge radii Annu.Rev.Nucl.Part.Sci. 51, 53 (2001) 10 Be 11 Be W.Nortershauser et al 2.357fm 2.460fm

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

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

13. GSM calculations for 6,8 He nuclei 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,…N sh We limit ourselves to 2 particles occupying continuum orbits. Im[k] (fm -1 ) Re[k] (fm -1 ) 3.27 0p3/2 B (0.17,-0.15) A (2.0,0.0) Schematic two-body interactions employed 1.Modified Minnesota Interaction (MN) (NPA 286, 53) 2.Surface Delta Interaction (SDI) (PR 145, 830) 3.Surface Gaussian Interaction (SGI) (PRC 70, 064313) SGI MN SDI

14. GSM calculations for 6,8 He nuclei Forces with N max = 10 and b = 2fm 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. SGI/SDI parameters are adjusted to the g.s of 6 He The 2 + state of 6 He 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 6,8 He The matrix elements of the MN were calculated with the HO expansion method, like in the recoil case. In all the following we employed SGI+WS, SDI+WS and KKNN+MN for 6 He For 8 He we used the spherical HF potential obtained from each interaction

15. GSM calculations for 6,8 He nuclei Example: 6 He 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. n max = 5 and b = 2fm (We have for example 0d5/2, 1d5/2, 2d5/2, 3d5/2, 4d5/2 for n max = 5 ) Total dimension: dim(M) = 5303 g.s energies for 6 He Basis set 1 J π : 0 + = - 0.9801 MeV Basis set 2 J π : 0 + = - 0.9779 MeV Differences of the order of ~ 0.22 keV…

16. GSM calculations for 6,8 He nuclei 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)… Radial density of the 6 He g.s. red and green curves correspond to the two different basis sets.

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

18. Radial density of valence neutrons for the 6 He 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 cut

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

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

21. Results on charge radii of 6 He with SDI interaction r pp ( 6 He) = 1.92 fm θ nn = 82.13 o

22. Comparison with other models and experiment

23. Conclusion and Future Plans  The very precise measurements of 6,8 He halos 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 6 He 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 8 He, 11 Li, 11 Be assuming an 4 He 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 the thresholds and the position of the S-matrix poles

25. Different interactions lead to different configuration mixing. 6 He charge radius (R ch ) 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. Results and discussion

26. Complex Scaling

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

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

29. 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  Ψ J =ΣC ki {|k> A |i> B } J is picked by the overlap method  From the C ki we built the density matrix and the N_opt states are corresponding to the maximum eigenvalues of ρ.

30. ◊ 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 Experiment From radii...to stellar nucleosynthesis! Figures are taken from PRL 96, 033002 (2006) and PRL 93, 142501 (2004)

31. SHELL MODEL (as usually applied to closed quantum systems) Largest tractable M-scheme dimension ~ 10 9 single particle Harmonic Oscillator (HO) basis nice mathematical properties: Lawson method applicable…

32. HEAVIER SYSTEMS Explosion of dimension Hamiltonian Matrix is dense+non-hermitian Lanczos converges slowly Density Matrix Renormalization Group 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 bound-states resonances non- resonant continua A B  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

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

34. EXPERIMENTAL RADII OF Be ISOTOPES W.Norteshauser et all nucl-ex/0809.2607v1 interaction cross section measurements GFMC NCSM PRC 73 065801 (2006) and PRC 71 044312 (2005) PRC 66, 044310, (2002) and Annu.Rev.Nucl.Part.Sci. 51, 53 (2001) FMD 7 Be charge radius provides constraints for the S 17 determination Charge radius decreases from 7 Be to 10 Be and then increases for 11 Be 11 Be increase can be attributed to the c.m motion of the 10 Be core The message is that changes in charge distributions provides information about the interactions in the different subsystems of the strongly clustered nucleus! 11 Be 1-neutron halo

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