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, USA2) Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, USA3) Departamento de Fisica Teorica, Universidad Autonoma de Madrid, Cantoblanco 28049, Madrid, Spain

Workshop on nuclei close to the dripline, CEA/SPhN Saclay 18-20th May 2009

Phys. Rev. C 78, 064305 (2008) Phys. Rev. C 77, 011301(R) (2008)

Introduction and Motivations1Introduction (1/2)

• 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)

• Emphasis on heavy nuclei near, or at, the driplineMicroscopy

• 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

General Framework2Introduction (2/2)

The Basis3Method (1/4)

• 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

The Energy Functional4• 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 (2/4)

Convergences5Method (3/4)

Neutron densities6Method (4/4)

Phys. Rev. C 53, 2809 (1996)

A comment: definition of the drip line7

• 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?

Results (1/3)

Neutron Skins8Results (2/3)

• 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?

Phys. Rev. C 61, 044326 (2000)

Neutron Halos9Results (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…

D1S drip line

D1 drip line

Phys. Rev. Lett. 79, 3841 (1997)Phys. Rev. Lett. 80, 460 (1998)

Phys. Rev. C 61, 044326 (2000)

SLy4

Beyond Mean-field at the drip line: RVAP Method

10• 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

RVAP (1/4)

Illustration of the RVAP Method11

Particle-number projected solution which approximates the VAP solution

RVAP (2/4)

Application: 11Li…12

Vanishing pairing regime

Non-zero pairing regime

Increase of radius induced by correlations

RVAP (3/4)

Definition of the drip line: again…13RVAP (3/4)

Halos: a light nuclei phenomenon only ?

Conclusions14• 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