1. 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, (2008)
Phys. Rev. C 77, (R) (2008)
Workshop on nuclei close to the dripline, CEA/SPhN Saclay 18-20th May 2009

2. Introduction and Motivations1
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. 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

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, (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. 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?

9. 8 Neutron Skins Neutron skin is defined by:
Results (2/3) 8
Neutron Skins
Phys. Rev. C 61, (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. 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, (2000)
D1S drip line
D1 drip line

11. Beyond Mean-field at the drip line: RVAP Method10
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. Illustration of the RVAP Method11
Illustration of the RVAP Method
Particle-number projected solution which approximates the VAP solution

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

14. Definition of the drip line: again…
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