1 / 16

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

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

orli
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related