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Understanding nuclear structure and reactions microscopically, including the continuum. March 17-21, 2014, GANIL, France Many-body methods for the description of bound weakly bound and unbound nuclear states George Papadimitriou georgios@iastate.edu B. Barrett N. Michel, W. Nazarewicz, M.Ploszajczak, J. Rotureau, J. Vary, P. Maris. Outline • Nuclear Physics on the edge of stability Experimental and Theoretical endeavors • The Gamow Shell Model (GSM) Applications on charge radii of Helium Halos and neutron correlations • Alternative method for extracting resonance parameters: The Complex Scaling Method in a Slater basis. • Outlook, conclusions and Future Plans Life on the edge of nuclear stability: Experimental highlights • New decay modes: 2n radioactivity? • Shell structure revisited: Magic numbers disappear, other arise. From: A.Gade A.Spyrou et al Marques et al (conflicting experiment) • • • • New exotic resonant states: 7H, 13Li, Nuclear Physics News 2013 10He,26O… PRC 87, 011304, PRL 110 152501, PRL 108 142503, PRL 109, 232501 recently) Metastable states embedded in the continuum are measured. Very dilute matter distribution Extreme clusterization close to particle thresholds. Provide stringent constraints to theory But also: Theory is in need for predictions and supporting certain experimental aspects Dimension of the problem increases One size does not fit all! Fig: Bertsch, Dean, Nazarewicz, SciDAC review 2007 Life on the edge of nuclear stability: Theory Dobaczewski et al Prog.Part.Nucl.Phys. 59, 432 (2007) Dobaczewski, Nazarewicz Phil. Trans. R.Soc. A 356 (1998) • The very notion of the mean-field and shell structure is under question • Nuclei are open quantum system and the openness is governed by the Sn • Weak binding and the proximity of the continuum affects bulk properties and spectra of nuclei. Especially the clusterization of matter is a generic property of the coupling to the continuum (or the impact of the open reaction channels). Clusterization does not depend on the specific characteristics of the NN interaction Okolowicz, Ploszajczak, Nazarewicz Fortschr. Phys. 61, 69 (2013) Life on the edge of nuclear stability: Theory Physics of nuclei close to the drip-line Input Forces Many-body Methods techniques Open Channels Coupling to continuum Additionally, complementary to the above: a new aspect is quality control 1) Cross check of codes/benchmarking 2) Statistical tools to estimate errors of calculations… Recent Paradigms: DFT functionals, new chiral forces, new extrapolation techniques Resonant and non-resonant states (how do they appear?) d2 l l 1 2 2 vr 2 k ul k , r 0 r dr k Solutions with outgoing boundary conditions Solution of the one-body Schrödinger equation with outgoing boundary conditions and a finite depth potential ul (k , r ) ~ C H l (k , r ) , r bound states, resonances ul (k , r ) ~ C H l (k , r ) C H l (k , r ) , r scattering states 2mE 2 The Berggren basis (cont’d) T.Berggren (1968) NP A109, 265 The eigenstates of the 1b Shrödinger equtaion form a complete basis, IF: we also consider the L+ scattering states are complex continuum states along the L+ contour (they satisfy scattering b.c) The shape of the contour is arbitrary, and any state between the contour and the real axis can be expanded in such as basis (proof by T. Berggren) In practice the continuum is discretized via a quadrature rule (e.g Gauss-Legendre): with Berggren’s Completeness relation and Gamow Shell Model resonant states (bound, resonances…) N.Michel et.al 2002 PRL 89 042502 Non-resonant Continuum along the contour The GSM in 4 steps Hermitian Hamiltonian Many-body SDi basis Hamiltonian matrix is built (complex symmetric): SDi ui1uiA Hamiltonian diagonalized Many body correlations and coupling to continuum are taken into account simultaneously GSM HAMILTONIAN We assume an alpha core in our calculations.. Hamiltonian free from spurious CM motion “recoil” term coming from the expression of H in relative coordinates. No spurious states Appropriate treatment for proper description of the recoil of the core and the removal of the spurious CoM motion. Y.Suzuki and K.Ikeda PRC 38,1 (1988) Vij is a phenomelogical NN interaction, fitted to spectra of nuclei: Minnesota force is used, unless otherwise indicated. Applications of the Berggren basis –SpectraHelium isotopic chain (4He core plus valence neutrons in the p-shell) Schematic NN force Applications 6,8He charge radii L.B.Wang et al, PRL 93, 142501 (2004) P.Mueller et al, PRL 99, 252501 (2007) M. Brodeur et al, PRL 108, 052504 (2012) RMS charge radii 4He 6He 8He L.B.Wang et al 1.67fm 2.054(18)fm M.Brodeur et al 1.67fm 2.059(7)fm 1.959(16)fm • 2n Very precisecorrelated data based as a strong pairon Isotopic Shifts measurements 6He: 8He: are distributed more symmetrically around thecalculations with • 4n Extraction of radii via Quantum Chemistry charged core a precision of up to 20 figures! (Hyllerraas basis calculations) Z.-T.Lu, P.Mueller, G.Drake,W.Nörtershäuser, S.C. Pieper, Z.-C.Yan Rev.Mod.Phys. 2013, 85, (2013). “Laser probing of neutron rich nuclei in light atoms” Other • effects Modelalso… independence of results testquantify for the nuclear Hamiltonian Can• weStringent calculate and these correlations? G. Papadimitriou et al PRC 84, 051304 6He 8He s.o density and radii also calculated by S. Bacca et al PRC 86, 064316 rpp2 ( AC +n ) X = rpp2 ( ) AC X + n 1 ( AC +n) 2 å i=1 ri2 + n 2 ( AC +n) 2 å r ×r i i< j j Radii (and other operators different than Hamiltonian) are challenging Example: Courtesy of P. Maris 1) How to reliably extrapolate radial operators to the infinite basis? Sid Coon et al, Furnstahl et al methods? 2) Renormalized operators? 3) Different basis? Neutron correlations in 6He ground state Halo tail Probability of finding the particles at distance r from the core with an angle θnn See also I. Brida and F. Nunes NPA 847,1 (2010) and P. Navratil talk Coupling to the continuum crucial for clusterization G. Papadimitriou et al PRC 84, 051304 Only p3/2 Full continuum • In the absence of continuum p1/2-sd states the neutrons show no preference • S=0 component (spin-antiparallel) dominant Manifestation of the Pauli effect • Average opening angle calculated from the density: θnn = 68o Neutron correlations in 6He 2+ excited state and spectroscopy 2+1 : [1.82, 0.1] MeV 2+2 : [4.13, 3.17] MeV 0+2 : [4.75, 8.6] MeV 1+1 : [4.4, 5.5 ] MeV G.P et al PRC(R) 84, 051304, 2011 02+ 1+ 22+ GSM MN force fitted just to the g.s. energy of 6,8He. 21+ 2+ neutrons almost uncorrelated… Fig. from http://www.tunl.duke.edu/nucldata/ Constructing an effective interaction in GSM in the p and sd shell. Effective interactions depend on the position of thresholds… Additional tools in our arsenal The complex scaling Belongs to the category of: Bound state technique to calculate resonant parameters and/or states in the continuum (see also talks by Lazauskas, Bacca, Orlandini) • Prog. Part. Nucl. Phys. 74, 55 (2014) and 68, 158 (2013) (reviews of bound state methods) Nuclear Physics Nuttal and Cohen PR 188, 1542 (1969) Lazauskas and Carbonell PRC 72 034003 (2005) Witala and Glöeckle PRC 60 024002 (1999) Aoyama et al PTP 116, 1 (2006) Horiuchi, Suzuki, Arai PRC 85, 054002 (2012) • • • • • Chemistry • • Moiseyev Phys. Rep 302 212 (1998) Y. K. Ho Phys. Rep. 99 1, (1983) Additional tools in our arsenal Complex Scaling Method in a Slater basis A.T.Kruppa, G.Papadimitriou, W.Nazarewicz, N. Michel PRC 89 014330 (2014) 1) Basic idea is to rotate coordinates and momenta i.e. r reiθ Hamiltonian is transformed to H(θ) = U(θ)HoriginalU(θ)-1 H(θ)Ψ(θ) = ΕΨ(θ) complex eigenvalue problem • The spectrum of H(θ) contains bound, resonances and continuum states. 2) Slater basis or Slater Type Orbitals (STOs): Basically, exponential decaying functions Powerful method to obtain resonance parameters in Quantum Chemistry Involves L2 square integrable functions. Can (in general) be applied to available bound state methods techniques (i.e. NCSM, Faddeev, CC etc) Some results • Comparison between CS Slater and CSM 0+ g.s, 2+ 1st excited Force Minnesota, α-n interaction KKNN 0+ • Test the HO expansion of the NN force in GSM for the unbound 2+ state. In GSM the force is expanded in a HO basis: • • • 2+ Talmi-Moshinsky transformation Numerical effort: Overlaps between HO and Gamow states. Very weak dependence of results on b nnmax. Some results 6He 0+ g.s. Valence neutrons radial density Phenomenological NN Minnesota interaction Correct asymptotic behavior Some results 2+ first excited state in 6He The 2+ state is a many-body resonance (outgoing wave) GSM exhibits naturally this behavior but CS is decaying for large distances, even for a resonance state This is OK. The solution Ψ(θ) is known to “die” off (L2 function) Solution Perform a direct back-rotation. What is that? Back-rotation In the case of the density this becomes: The CS density has the correct asymptotic behavior (outgoing wave) • Back rotation is very unstable numerically. An Ill posed inverse problem. Long standing problem in the CS community (in Quantum Chemistry as well) • The problem lies in the analytical continuation of a square integrable function in the complex plane. • We are using the theory of Fourier transformations and a regularization process (Tikhonov) to minimize the ultraviolet numerical noise of the inversion process. 2+ densities in 6He (real and imaginary part) Conclusions/Future plans Berggren basis appropriate for calculations of weakly bound/unbound nuclei. • GSM calculations provided insight behind the charge differences of Helium halo nuclei. Construct effective interaction in the p and sd shell. Use realistic effective interactions for GSM calculations that stem from NCSM with a core, or Coupled Cluster or IM-SRG… GSM is the Shell Model technique to: i) study 3N forces effects and continuum coupling for the detailed spectroscopy of heavy drip line nuclei. ii) exact treatment of many body correlations and coupling to continuum Complementary method to describe resonant states: Complex Scaling in a Slater basis L2 integrable basis formulation. Slater basis correct asymptotic behavior Back rotation inverse problem solved. Apart from complex arithmetics the computational expense is as “tough” or as “easy” as for the solution of the bound state. Explore complex scaling in more depth Back up Solution Back rotation is very unstable numerically. Unsolved problem in the CS community (in QC as well) The problem lies in the analytical continuation of a square integrable function in the complex plane. We are using the theory of Fourier transformations and Tikhonov regularization process to obtain the original (GSM) density To apply theory of F.T to the density, it should be defined in (-∞,+∞) Now defined from (-∞,+∞) F.T Value of (1) for x+iy (analytical continuation) Tikhonov regularization x = -lnr , y = θ Last slide before conclusions/future plans NN force: JISP16 (A. Shirokov et al PRC79, 014610) and NNLOopt (A. Ekstrom et al PRL 110, 192502) Quality control: Verification/Validation, cross check of codes MFDn: Vary, Maris NCGSM: G.P, Rotureau, Michel… MFDn/NC-GSM + computer scientists at LBNL (Ng, Yang, Aktulga), collaboration Goal: Scalable diagonalizations of complex symmetric matrices Dimension comparison Lanczos: “brute” force diagon of H. DMRG: Diagon of H in the space where only the most important degrees of “freedom” are considered Similar treatment by Caprio, Vary, Maris in Sturmian basis Complex Scaling construction of a block in : • Construct all many-body states associated with the pole space P • Construct all many-body states associated with the space of the discrete continua C. • Create many-body basis by coupling states in P and C. block construction of a superblock : superblock truncation with the density matrix : Nopt states that correspond to the largest eigenvalues of the density matrix are kept • • • • The process is reversed… In each step (shell added) the Hamiltonian is diagonalized and Nopt states are kept. Iterative method to take into account all the degrees of freedom in an effective manner. In the end of the process the result is the same (within keV) with the one obtained by “brute” force diagonalization of H. Sweep-down Sweep-up truncation “down” “up” truncation Results: 4He against Fadeev-Yakubovsky G.P., J.Rotureau, N. Michel, M.Ploszajczak, B. Barrett arXiv:1301.7140 2 neutrons 2 protons Pole space A:0s1/2 (p/n) Continuum space B: p3/2,p1/2,s1/2 real energy continua d5/2-d3/2 f5/2-f7/2 g7/2-g9/2 H.O states 156 s.p. states total Dim for direct diagon: 119,864,088 Eab-initio = -29.15 MeV EFY = -29.19 MeV Neutron correlations in 8He ground state G.Papadimitriou PhD thesis Neutron correlations in 6He 2+ excited state G.P et al PRC(R) 84, 051304, 2011 2+ neutrons almost uncorrelated…