Symmetry-Conserving Spherical Gogny HFB Calculations in a Woods-Saxon Basis

1. 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

2. Introduction (1/2) 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)

3. 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

4. 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

5. 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.

6. Method (3/4) 5 Convergences

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

8. 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 positiveS1n = 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 (effcombination 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?

9. 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?

10. Results (3/3) 9 Neutron Halos • 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

11. RVAP (1/4) 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

12. RVAP (2/4) 11 Illustration of the RVAP Method Particle-number projected solution which approximates the VAP solution

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

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

15. 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?

16. Appendix