A Study of Odd-Mass Nuclei Energy Density Functional in EDF Theory

1. Towards a Universal Energy Density FunctionalStudy 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

2. 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):

3. 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

4. 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-oddpart (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

5. 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

6. Determination of V0 Neutron Pairing Gap in 120Sn in Skyrme HFB Calculations

7. 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 conservingharmonic oscillator eigenfunctions • All symmetries broken (including time-reversal) • MPI-HFODD: about 1.2 Gflops/core on Jaguar: HFODD core and parallel interface

8. 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.

9. Normalized Q.P. Spectrum SLy4 SLy4 + Tensor Experiment

10. Rare Earth Region

11. 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