N. Schunck(1,2,3) and J. L. Egido(3) Symmetry-Conserving Spherical Gogny HFB Calculations in a Woods-Saxon Basis N. Schunck(1,2,3) and J. L. Egido(3) 1) Department of Physics  Astronomy, University of Tennessee, Knoxville, TN-37996, USA 2) Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, USA 3) Departamento de Fisica Teorica, Universidad Autonoma de Madrid, Cantoblanco 28049, Madrid, Spain Phys. Rev. C 78, 064305 (2008) Phys. Rev. C 77, 011301(R) (2008) Workshop on nuclei close to the dripline, CEA/SPhN Saclay 18-20th May 2009

Introduction and Motivations 1 Introduction and Motivations Challenges of nuclear structure near the driplines This workshop: Importance of including continuum effects within a given theoretical framework : HFB, RMF, Shell Model, Cluster Models, etc. Robustness of the effective interaction or Lagrangian: iso-vector dependence, all relevant terms (tensor), etc. Case of EDF approaches: Crucial role of super-fluidity in weakly-bound nuclei (ground-state) Strategies for EDF theories with continuum: HFB calculations in coordinate space: Box-boundary conditions (Skyrme and RMF/RHB) Outgoing-wave boundary conditions (Skyrme) HFB calculations in configuration space: Transformed Harmonic Oscillator (Skyrme) Gamow basis (Skyrme)

2 General Framework Emphasis on heavy nuclei near, or at, the dripline Introduction (2/2) 2 General Framework Emphasis on heavy nuclei near, or at, the dripline Microscopy Finite-range Gogny interaction Hamiltonian picture: interaction defines intrinsic Hamiltonian Particle-hole and particle-particle channel treated on the same footing No divergence problem in the p.p. channel Beyond mean-field correlations: PNP (after variation) Continuum Basis embedding discretized continuum states Better adapted to finite-range forces Easy inclusion of symmetry-breaking terms and beyond mean-field effects Flexibility: study the influence of the basis Box-boundary conditions and spherical symmetry

Method (1/4) 3 The Basis Realistic one-body potential in a box: eigenstates of the Woods-Saxon potential Early application in RMF - Phys. Rev. C 68, 034323 (2003) Basis states obtained numerically on a mesh Set of discrete bound-state and discretized positive energy states Essentially equivalent to Localized Atomic Orbital Bases used in condensed matter

Method (2/4) 4 The Energy Functional Changing the basis in spherical HFB calculations: Only the radial part of the matrix elements need be re-calculated Gogny Interaction (finite-range) Remarks: Only central term differs from Skyrme family: SO and density-dependent terms are formally identical Same interaction in the p.h. and p.p. channels All exchange terms taken into account (this includes Coulomb), and all terms of the p.h. and p.p. functional included: Coulomb, center-of-mass, etc.

Method (3/4) 5 Convergences

Method (4/4) 6 Neutron densities Phys. Rev. C 53, 2809 (1996)

A comment: definition of the drip line Results (1/3) 7 A comment: definition of the drip line Several possible definitions of the dripline: 2-particle separation energy becomes positive S2n = B(N+2) – B(N) 1-particle separation energy becomes positive S1n = B(N+1) – B(N) Chemical potential becomes positive ≈ dB/dN Several problems: Concept of chemical potential does not apply: At HF level because of pairing collapse When approximate particle number projection (Lipkin-Nogami) is used (eff combination of  and 2) When exact projection is used (N is well-defined) 1-particle separation energy requires breaking time-reversal symmetry and blocking calculations: not done yet near the dripline Only the 2-particle separation energy is somewhat model-independent and robust enough - Is it enough?

8 Neutron Skins Neutron skin is defined by: Results (2/3) 8 Neutron Skins Phys. Rev. C 61, 044326 (2000) Neutron skin is defined by: Similar results with calculations based on Skyrme and Gogny interaction Values of the neutron skin directly related to neutron-proton asymetry Can neutron skin help differentiate functionals?

9 Neutron Halos Results (3/3) Different definitions of the halo size (see Karim’s talk). Here: Very large fluctuations from one interaction/functional to another (much larger than for neutron skins) No giant halo… SLy4 Phys. Rev. Lett. 79, 3841 (1997) Phys. Rev. Lett. 80, 460 (1998) Phys. Rev. C 61, 044326 (2000) D1S drip line D1 drip line

Beyond Mean-field at the drip line: RVAP Method 10 Beyond Mean-field at the drip line: RVAP Method Observation: in the (static) EDF theory, the coupling to the continuum is mediated by the pairing correlations Avoiding pairing collapse of the HFB theory with particle number projection (PNP) Projection after variation (PAV) does not always help Projection before variation (VAP) is very costly Good approximation: Restricted Variation After Projection (RVAP) method Introduce a scaling factor  and generate pairing-enhanced wave-functions by scaling, at each iteration, the pairing field At convergence calculate expectation value of the projected, original Gogny Hamiltonian: Repeat for different scaling factors: RVAP solution is the minimum of the curve

Illustration of the RVAP Method 11 Illustration of the RVAP Method Particle-number projected solution which approximates the VAP solution

12 Application: 11Li… RVAP (3/4) Increase of radius induced by correlations Vanishing pairing regime Non-zero pairing regime

Definition of the drip line: again… RVAP (3/4) 13 Definition of the drip line: again… Halos: a light nuclei phenomenon only ?

14 Conclusions First example of spherical Gogny HFB calculations at the dripline by using an expansion on WS eigenstates: Give the correct asymptotic behavior of nuclear wave-functions Robust and precise, amenable to beyond mean-field extensions and large-scale calculations Limitation: box-boundary conditions Neutron skins are directly correlated to neutron-proton asymmetry Neutron halos are small No giant halo: do halos really exist in heavy nuclei at all? Large model-dependence (interaction and type of mean-field) RVAP method is a simple, inexpensive but effective method to simulate VAP since it ensures a non-zero pairing regime Possible extensions: Replace vanishing box-boundary conditions with outgoing-wave? Parallelization?

Appendix