1. Towards a Universal Energy Density Functional 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

2. 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 2 nd orderExpress the energy density as a scalar made of these fields and their derivatives up to 2 nd order Examples: Compute the total energy (variational principle): Nuclear EDF in a Nutshell

3. EDF and Effective Interactions effectiveSkyrme effective interaction: Zero-range (Skyrme) or finite-range (Gogny) Many-body Hamiltonian reads: Express total energy as function of densities

4. realscalartime-even,iso-scalarEnergy density is real, scalar, time-even, iso-scalar but constituting fields are not (necessarily) and therefore need be computed… Time-evenTime-even part (depends on time-even fields) Time-oddTime-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) not necessaryParameters C may be related to (t,x) set of Skyrme force but this is not necessary Nuclear Energy Density Functional

5. Pairing correlations a prioriPairing 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 V 0 Neutron Pairing Gap in 120 Sn in Skyrme HFB Calculations

7. Computational Aspects Variational principle gives a set of equations: HF (no pairing) or HFB (pairing) iterative procedureSelf-consistency requires iterative procedure Modern super-computers allow large-scale calculations couple of hours –Symmetry-restricted codes: one mass table in a couple of hours (Mario) couple of days –Symmetry-unrestricted codes: one mass table in a couple of days HFODD = HFB solver 3Ddeformedcartesiany-simplex conserving harmonic oscillator –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

8. Treatment of Odd Nuclei Odd nuclei neglected in previous fit strategies time-odd fields –Provide unique handle on time-odd fields (half of the functional !) high-spin –Will help constrain high-spin physics better Practical difficulties: time-reversal symmetry –Break time-reversal symmetry (HF and HFB) odd particle –Treatment of the odd particle –Proton-neutron pairing –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 SLy4ExperimentSLy4 + Tensor

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