Symmetry-Conserving Spherical Gogny HFB Calculations in a Woods-Saxon Basis
description
Transcript of 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, 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