Transcript Towards a Universal Energy Density Functional Study of Odd-Mass Nuclei in EDF Theory N.

Towards a Universal Energy
Density Functional
Study of Odd-Mass
Nuclei in EDF Theory
N. Schunck
University of Tennessee, 401 Nielsen Physics , Knoxville,
TN-37996, USA
Oak Ridge National Laboratory, Bldg. 6025, MS6373, P.O. Box
2008, Oak Ridge, TN-37831, USA
Together with:
J. Dobaczewski, W. Nazarewicz, N. Nikolov and M.
Stoitsov
Nuclear EDF in a Nutshell
• Construct density fields in normal, spin, isospin space
• Use Local Density Approximation (LDA)
• Express the energy density as a scalar made of these
fields and their derivatives up to 2nd order
• Examples:
• Compute the total energy (variational principle):
EDF and Effective Interactions
• Skyrme effective interaction:
• Zero-range (Skyrme) or finite-range (Gogny)
• Many-body Hamiltonian reads:
• Express total energy as function of densities
Nuclear Energy Density Functional
• Energy density is real, scalar, time-even, iso-scalar but
constituting fields are not (necessarily) and therefore need
be computed…
• Time-even part (depends on time-even fields)
• Time-odd part (depends on time-odd fields)
• Scalar, vector and tensor terms depend on scalar, vector or
tensor fields
• Iso-scalar (t = 0), iso-vector (t = 1)
• Parameters C may be related to (t,x) set of Skyrme force but
this is not necessary
Pairing correlations
• Pairing functional a priori as rich as mean-field functional:
the choice of the pairing channel is not finalized yet
• Form considered here: surface-volume pairing
• Cut-off, regularization procedure
• (Some form of) projection on the number of particles is
required
– Breaking down of pairing correlations
– Fluctuations of particle number
• Current Difficulties:
– Constrained/unconstrained calculations with Lipkin-Nogami
prescription
– Blocking in odd nuclei with Lipkin-Nogami prescription
Determination of V0
Neutron Pairing Gap in 120Sn in
Skyrme HFB Calculations
Computational Aspects
• Variational principle
gives a set of equations: HF (no pairing) or HFB (pairing)
• Self-consistency requires iterative procedure
• Modern super-computers allow large-scale calculations
– Symmetry-restricted codes: one mass table in a couple of hours
(Mario)
– Symmetry-unrestricted codes: one mass table in a couple of days
• HFODD = HFB solver
– Basis made of 3D, deformed, cartesian, y-simplex conserving
harmonic oscillator eigenfunctions
– All symmetries broken (including time-reversal)
• MPI-HFODD: about 1.2 Gflops/core on Jaguar: HFODD core
and parallel interface
Treatment of Odd Nuclei
• Odd nuclei neglected in previous fit strategies
– Provide unique handle on time-odd fields (half of the functional !)
– Will help constrain high-spin physics better
• Practical difficulties:
– Break time-reversal symmetry (HF and HFB)
– Treatment of the odd particle
– Proton-neutron pairing for odd-odd nuclei (not addressed here)
• Blocking in HFB
• Several states around the
Fermi level need to be
blocked to find the g.s.
Normalized Q.P. Spectrum
SLy4
SLy4 + Tensor
Experiment
Rare Earth Region
Conclusions
• Progress:
– Computing core (HFODD) optimized
– Parallel architecture ready (can/will be improved)
– Information on deformation properties and odd nuclei spectra vs. functional
• Difficulties:
– Remaining problems with approximate particle number projection for odd
nuclei
– Stability problems of calculations with full time-odd terms
• Needs:
– Discussion of treatment of pairing and correlations
– Interface symmetry-restricted vs. symmetry-unrestricted codes
– Highly optimized minimization routines for fit of functional
• Small number of function evaluations
• Parallelized (or …-able)
– Automatic handling of divergences of self-consistency procedure