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
1
Introduction (1/2)
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)
2
General Framework
Introduction (2/2)
• 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
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
•
•
•
•
4
The Energy Functional
Method (2/4)
• 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.
5
Convergences
Method (3/4)
6
Neutron densities
Method (4/4)
Phys. Rev. C 53, 2809 (1996)
7
Results (1/3)
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 modelindependent and robust enough - Is it enough?
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?
Results (2/3)
9
Neutron Halos
• Different definitions of the halo size (see
Karim’s talk). Here:
Results (3/3)
SLy4
• Very large fluctuations from one
interaction/functional to another (much
larger than for neutron skins)
• No giant halo…
Phys. Rev. Lett. 79, 3841 (1997)
Phys. Rev. Lett. 80, 460 (1998)
D1S drip
line
D1 drip
line
Phys. Rev. C 61, 044326 (2000)
10
RVAP (1/4)
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 wavefunctions
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
11
Illustration of the RVAP Method
Particle-number projected
solution which approximates
the VAP solution
RVAP (2/4)
12
Increase of
radius
induced by
correlations
Application: 11Li…
Vanishing
pairing regime
Non-zero
pairing regime
RVAP (3/4)
13
RVAP (3/4)
Definition of the drip line: again…
Halos: a light nuclei
phenomenon only ?
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 largescale 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