1. 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 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 6,8 He ( 11 Li and 11 Be)  Spin-orbit density effect on charge radii  Calculations on charge radii of 6,8 He  DMRG application to 8 He  Conclusions 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) Shell Model Theories that incorporate the continuum, selected references

6. 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) We want a Hamiltonian free from spurious CM motion Lawson method? Jacobi coordinates? Appropriate treatment for proper description of the recoil of the core and the removal of the spurious CoM motion. (G.Hagen et al PRL 103 062503 (2009))

7. 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) EXPERIMENTAL RADII OF 6 He, 8 He, 11 Li R charge ( 6 He) > R charge ( 8 He) “Swelling” of the core is not negligible 4 He 6 He 8 He L.B.Wang et al P.Mueller et al 1.67fm2.054(18)fm 1.67fm2.068(11)fm1.929(26)fm 9 Li 11 Li R.Sanchez et al 2.217(35)fm2.467(37)fm RMS charge radii charge radii determines the correlations between valence particles AND reflects the radial extent of the halo nucleus 10 Be 11 Be W.Nortershauser et al 2.357(16)fm 2.460(16)fm ~4% ~8% Annu.Rev.Nucl.Part.Sci. 51, 53 (2001)

8. Spin-orbit contribution to the charge radius  Usually point radii are converted to charge radii through: finite size effects Darwin-Foldy term  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 40 Ca and 48 Ca (W. Bertozzi et al, PLB 41, 408 (1972)) Finite size effects and relativistic D.F term are consistently considered in theoretical calculations The s.o effect is almost never considered… (except maybe ) H.Esbensen et al PRC 76, 024302 (2007) A.Ong et al PRC 82, 014320 (2010) Both terms appear explicitly in the expression of the single nucleon charge operator and they enter a non-relativistic calculation to an order 1/m 2. BUT …

9. Spin-orbit contribution to the charge radius As we shall see, the s.o can have a comparable contribution with the finite size effects!!  The formulas to calculate the s.o correction are the following: (J.Friar et al Adv.Nucl.Phys. 8, 219, (1975)) 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

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

11. 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 N sh = 30 for p3/2 contour and N sh = 20 for each real cont. Total 111 single particle states. 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) Modified Minnesota Interaction (MN) (NPA 286, 53) Parameterizations The two strength parameters of the MN are adjusted to the g.s of 6,8 He.

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

13. Results on charge radii of 6,8 He with MMN interaction  The s.o corrections are comparable to the D.W term 6 He and comparable to the finite size corrections in 8 He.  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.

14. Results on charge radii of 6,8 He with MMN interaction (configuration mixing and correlation angle (for 6 He) ) 8 He 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

15. The charge radius of 6 He as a function of the S 2n 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. The narrow experimental error bars suggest that the S 2n should be calculated with a high precision if one aims in a detailed description of the 6 He radial extent. When this condition is met the p 3/2 state had a dominant occupation of about 90% in the 6 He g.s. For this p 3/2 percentage and the correct S 2n, 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. Blue line: Core polarization + s.o effect

16. Comparison with other models and experiment

17. 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.  Approximate in terms of m < M basis states (truncation)  These m states are eigenstates of the density matrix or  The difference between the exact and the approximated, has the minimal norm. 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.

18.  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 r pp  In each DMRG step we calculate the expectation value the radius Density Matrix Renormalization Group application to 8 He radius

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

20. 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 modeling weakly bound nuclei with large radial extension.  Using a finite range force (MN) and adjusting the strengths to the g.s energies of 6,8 He we reproduced the experimental trend of Helium halo charge radii.  Charge radii are primarily sensitive on the p 3/2 occupation and the S 2n.  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 11 Li, 11 Be assuming an 4 He 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.

22. 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

23.  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. Density Matrix Renormalization Group application to 8 He proton radius ε = 10 -8

24. 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…

25. 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.

26. 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

27. 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

28. p-sd shells (5 partial-waves), 47 shells total. E dmrg = -3.284 MeV, E GSM = -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 E GSM (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 Density Matrix Renormalization Group application to 8 He 

29. Complex Scaling

30. 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

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

32. 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.

33. ◊ 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)

34. 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…

35. 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

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

37. 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

39. 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…

40. 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